Subgroups — C₆₇×#61566 #30740817

Order 17152.

2194 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3506	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3305	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3304	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3287	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3284	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3208	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3205	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3188	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3104	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3002	C₂	1	4	yes	maximal, complemented
13	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₆₇	C₂²	1	2	yes	—
15	C₂×#135×C₆₇	C₂²	1	8	yes	complemented
16	#559×C₆₇	—	2	2	no	—
17	#556×C₆₇	C₂²	1	2	yes	complemented
18	#555×C₆₇	C₂²	1	2	yes	complemented
19	#553×C₆₇	—	2	2	no	—
20	#550×C₆₇	—	2	2	no	—
21	#550×C₆₇	—	2	2	no	—
22	#550×C₆₇	C₂²	1	4	yes	—
23	#550×C₆₇	C₂²	1	4	yes	—
24	#550×C₆₇	C₂²	1	2	yes	complemented
25	#547×C₆₇	C₂²	1	2	yes	complemented
26	#538×C₆₇	—	2	2	no	—
27	#535×C₆₇	—	2	1	no	—
28	#533×C₆₇	C₂²	1	1	yes	characteristic
29	#532×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#127×C₆₇	—	2	1	no	—
31	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
32	#525×C₆₇	C₂²	1	4	yes	—
33	C₂×#121×C₆₇	—	2	2	no	—
34	C₂×#121×C₆₇	C₂²	1	2	yes	—
35	#520×C₆₇	—	2	1	no	—
36	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₆₇	—	2	1	no	—
38	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
39	#493×C₆₇	C₂²	1	2	yes	complemented
40	#492×C₆₇	C₂²	1	2	yes	complemented
41	#486×C₆₇	C₂²	1	2	yes	complemented
42	#486×C₆₇	C₂²	1	2	yes	complemented
43	#484×C₆₇	C₂²	1	2	yes	—
44	#482×C₆₇	C₂²	1	8	yes	complemented
45	#481×C₆₇	C₂²	1	2	yes	—
46	#477×C₆₇	C₂²	1	2	yes	complemented
47	#475×C₆₇	C₂²	1	2	yes	complemented
48	#466×C₆₇	C₂²	1	8	yes	complemented
49	#465×C₆₇	C₂²	1	8	yes	complemented
50	#462×C₆₇	C₂²	1	2	yes	complemented
51	#460×C₆₇	C₂²	1	2	yes	complemented
52	#460×C₆₇	C₂²	1	4	yes	complemented
53	#459×C₆₇	C₂²	1	4	yes	complemented
54	#452×C₆₇	C₂²	1	8	yes	complemented
55	#451×C₆₇	C₂²	1	2	yes	—
56	#450×C₆₇	C₂²	1	4	yes	—
57	#449×C₆₇	C₂²	1	8	yes	complemented
58	#449×C₆₇	C₂²	1	8	yes	complemented
59	#448×C₆₇	C₂²	1	2	yes	—
60	#447×C₆₇	C₂²	1	4	yes	—
61	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
62	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
63	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
64	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
65	#431×C₆₇	C₂²	1	1	yes	characteristic
66	#431×C₆₇	C₂²	1	1	yes	characteristic
67	#416×C₆₇	C₂²	1	8	yes	complemented
68	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
69	C₂×#41×C₆₇	—	2	1	no	—
70	C₂×#41×C₆₇	—	2	4	no	—
71	C₂×#41×C₆₇	—	4	8	no	—
72	C₂×#41×C₆₇	C₂³	1	8	yes	—
73	C₂×#41×C₆₇	C₂³	1	4	yes	—
74	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×D₄×C₆₇	—	2	2	no	—
76	C₂²×D₄×C₆₇	—	2	2	no	—
77	C₂²×D₄×C₆₇	—	2	1	no	—
78	C₂²×D₄×C₆₇	—	2	1	no	—
79	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
80	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	—	2	2	no	—
82	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
83	#135×C₆₇	—	4	16	no	—
84	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
85	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
86	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
87	C₂×D₈×C₆₇	C₂³	1	8	yes	—
88	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
89	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
92	#127×C₆₇	—	2	4	no	—
93	#127×C₆₇	—	2	4	no	—
94	#127×C₆₇	—	2	2	no	—
95	#127×C₆₇	—	2	1	no	—
96	#127×C₆₇	—	2	1	no	—
97	#127×C₆₇	C₂³	1	2	yes	—
98	#127×C₆₇	C₂³	1	2	yes	—
99	#127×C₆₇	C₂³	1	1	yes	characteristic
100	#126×C₆₇	—	2	4	no	—
101	#125×C₆₇	—	2	1	no	—
102	#124×C₆₇	—	2	2	no	—
103	#124×C₆₇	—	2	4	no	—
104	#124×C₆₇	—	2	1	no	—
105	#123×C₆₇	—	2	4	no	—
106	#123×C₆₇	—	2	4	no	—
107	C₂²⋊Q₈×C₆₇	—	2	1	no	—
108	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
109	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
110	#121×C₆₇	—	2	2	no	—
111	#121×C₆₇	—	2	2	no	—
112	#121×C₆₇	—	2	4	no	—
113	#121×C₆₇	—	2	1	no	—
114	#121×C₆₇	—	2	8	no	—
115	#121×C₆₇	—	2	8	no	—
116	#121×C₆₇	—	2	4	no	—
117	#121×C₆₇	C₂³	1	2	yes	—
118	#121×C₆₇	C₂³	1	4	yes	—
119	#121×C₆₇	C₂³	1	8	yes	—
120	#120×C₆₇	—	2	4	no	—
121	C₄×Q₈×C₆₇	—	2	2	no	—
122	C₄×Q₈×C₆₇	—	2	1	no	—
123	C₄×Q₈×C₆₇	—	2	1	no	—
124	C₄×Q₈×C₆₇	—	2	1	no	—
125	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
126	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
127	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
128	C₄×D₄×C₆₇	—	2	4	no	—
129	C₄×D₄×C₆₇	—	2	4	no	—
130	C₄×D₄×C₆₇	—	2	4	no	—
131	C₄×D₄×C₆₇	—	2	4	no	—
132	C₄×D₄×C₆₇	—	2	2	no	—
133	C₄×D₄×C₆₇	—	2	1	no	—
134	C₄×D₄×C₆₇	D₄	1	2	yes	—
135	C₄×D₄×C₆₇	D₄	1	2	yes	—
136	C₄×D₄×C₆₇	C₂³	1	2	yes	—
137	C₄×D₄×C₆₇	C₂³	1	2	yes	—
138	C₄×D₄×C₆₇	C₂³	1	2	yes	—
139	#117×C₆₇	—	2	1	no	—
140	#117×C₆₇	—	2	2	no	—
141	#117×C₆₇	C₂³	1	4	yes	—
142	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
144	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
145	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
146	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
147	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
148	C₂×C₄²×C₆₇	—	2	1	no	—
149	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
150	#107×C₆₇	C₂³	1	4	yes	—
151	#106×C₆₇	C₂³	1	4	yes	—
152	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
153	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
154	#103×C₆₇	C₂³	1	4	yes	—
155	#103×C₆₇	C₂³	1	4	yes	—
156	#103×C₆₇	C₂³	1	4	yes	—
157	#102×C₆₇	C₂³	1	4	yes	—
158	#102×C₆₇	C₂³	1	8	yes	—
159	#102×C₆₇	C₂³	1	8	yes	—
160	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
161	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
162	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
163	C₄×C₈×C₆₇	C₂³	1	2	yes	—
164	C₄×C₈×C₆₇	C₂³	1	2	yes	—
165	C₂⁴×C₆₇	—	2	2	no	—
166	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
167	#41×C₆₇	—	2	16	no	—
168	#41×C₆₇	—	4	8	no	—
169	#41×C₆₇	—	4	8	no	—
170	#41×C₆₇	—	4	16	no	—
171	#41×C₆₇	—	4	4	no	—
172	C₂×Q₈×C₆₇	—	2	4	no	—
173	C₂×Q₈×C₆₇	—	2	1	no	—
174	C₂×Q₈×C₆₇	—	2	1	no	—
175	C₂×Q₈×C₆₇	—	2	1	no	—
176	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
177	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
178	C₂×D₄×C₆₇	—	2	2	no	—
179	C₂×D₄×C₆₇	—	2	8	no	—
180	C₂×D₄×C₆₇	—	2	4	no	—
181	C₂×D₄×C₆₇	—	2	4	no	—
182	C₂×D₄×C₆₇	—	2	8	no	—
183	C₂×D₄×C₆₇	—	2	4	no	—
184	C₂×D₄×C₆₇	—	2	1	no	—
185	C₂×D₄×C₆₇	—	2	2	no	—
186	C₂×D₄×C₆₇	—	2	2	no	—
187	C₂×D₄×C₆₇	—	2	2	no	—
188	C₂×D₄×C₆₇	—	2	1	no	—
189	C₂×D₄×C₆₇	—	2	2	no	—
190	C₂×D₄×C₆₇	—	2	2	no	—
191	C₂×D₄×C₆₇	—	4	4	no	—
192	C₂×D₄×C₆₇	—	4	8	no	—
193	C₂×D₄×C₆₇	—	4	4	no	—
194	C₂×D₄×C₆₇	—	8	2	no	—
195	C₂×D₄×C₆₇	—	8	4	no	—
196	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
197	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
198	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
199	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₆₇	—	2	1	no	—
201	C₂²×C₄×C₆₇	—	2	1	no	—
202	C₂²×C₄×C₆₇	—	2	2	no	—
203	C₂²×C₄×C₆₇	—	2	8	no	—
204	C₂²×C₄×C₆₇	—	2	4	no	—
205	C₂²×C₄×C₆₇	—	4	4	no	—
206	C₂²×C₄×C₆₇	—	4	8	no	—
207	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
209	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
210	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
212	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
213	Q₁₆×C₆₇	—	2	8	no	—
214	SD₁₆×C₆₇	—	2	8	no	—
215	SD₁₆×C₆₇	—	4	8	no	—
216	D₈×C₆₇	—	4	8	no	—
217	C₂×C₈×C₆₇	—	4	8	no	—
218	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
219	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
220	C₄⋊C₄×C₆₇	—	2	1	no	—
221	C₄⋊C₄×C₆₇	—	2	4	no	—
222	C₄⋊C₄×C₆₇	—	2	2	no	—
223	C₄⋊C₄×C₆₇	—	2	2	no	—
224	C₄⋊C₄×C₆₇	—	2	1	no	—
225	C₄⋊C₄×C₆₇	—	2	1	no	—
226	C₄⋊C₄×C₆₇	—	2	2	no	—
227	C₄⋊C₄×C₆₇	—	2	1	no	—
228	C₄⋊C₄×C₆₇	—	2	1	no	—
229	C₄⋊C₄×C₆₇	—	2	2	no	—
230	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
231	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
232	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
233	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
234	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
235	C₂²⋊C₄×C₆₇	—	2	2	no	—
236	C₂²⋊C₄×C₆₇	—	2	4	no	—
237	C₂²⋊C₄×C₆₇	—	2	4	no	—
238	C₂²⋊C₄×C₆₇	—	2	8	no	—
239	C₂²⋊C₄×C₆₇	—	2	4	no	—
240	C₂²⋊C₄×C₆₇	—	2	2	no	—
241	C₂²⋊C₄×C₆₇	—	2	2	no	—
242	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
243	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
244	C₄²×C₆₇	—	2	2	no	—
245	C₄²×C₆₇	—	2	2	no	—
246	C₄²×C₆₇	—	2	4	no	—
247	C₄²×C₆₇	—	2	2	no	—
248	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
249	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
250	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
251	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
252	C₂³×C₆₇	—	2	4	no	—
253	C₂³×C₆₇	—	2	2	no	—
254	C₂³×C₆₇	—	2	4	no	—
255	C₂³×C₆₇	—	4	2	no	—
256	C₂³×C₆₇	—	4	2	no	—
257	C₂³×C₆₇	—	4	4	no	—
258	C₂³×C₆₇	—	4	4	no	—
259	C₂³×C₆₇	—	4	4	no	—
260	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
261	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	Q₈×C₆₇	—	2	4	no	—
263	Q₈×C₆₇	—	2	2	no	—
264	Q₈×C₆₇	—	2	4	no	—
265	Q₈×C₆₇	—	4	2	no	—
266	Q₈×C₆₇	—	8	1	no	—
267	Q₈×C₆₇	H(5,𝔽₂)	1	4	yes	—
268	D₄×C₆₇	—	2	4	no	—
269	D₄×C₆₇	—	2	4	no	—
270	D₄×C₆₇	—	2	4	no	—
271	D₄×C₆₇	—	2	8	no	—
272	D₄×C₆₇	—	4	16	no	—
273	D₄×C₆₇	—	4	4	no	—
274	D₄×C₆₇	—	4	4	no	—
275	D₄×C₆₇	—	4	4	no	—
276	D₄×C₆₇	—	4	4	no	—
277	D₄×C₆₇	—	4	2	no	—
278	D₄×C₆₇	—	8	1	no	—
279	D₄×C₆₇	—	8	2	no	—
280	D₄×C₆₇	—	8	2	no	—
281	D₄×C₆₇	—	8	4	no	—
282	D₄×C₆₇	—	8	8	no	—
283	D₄×C₆₇	—	8	4	no	—
284	C₂×C₄×C₆₇	—	2	4	no	—
285	C₂×C₄×C₆₇	—	2	2	no	—
286	C₂×C₄×C₆₇	—	2	8	no	—
287	C₂×C₄×C₆₇	—	2	2	no	—
288	C₂×C₄×C₆₇	—	2	2	no	—
289	C₂×C₄×C₆₇	—	2	2	no	—
290	C₂×C₄×C₆₇	—	2	2	no	—
291	C₂×C₄×C₆₇	—	2	2	no	—
292	C₂×C₄×C₆₇	—	4	2	no	—
293	C₂×C₄×C₆₇	—	4	4	no	—
294	C₂×C₄×C₆₇	—	4	1	no	—
295	C₂×C₄×C₆₇	—	4	16	no	—
296	C₂×C₄×C₆₇	—	4	2	no	—
297	C₂×C₄×C₆₇	—	4	4	no	—
298	C₂×C₄×C₆₇	—	8	4	no	—
299	C₂×C₄×C₆₇	#135	1	4	yes	complemented
300	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
301	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
303	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
305	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
307	C₈×C₆₇	—	2	4	no	—
308	C₈×C₆₇	—	2	4	no	—
309	C₂²×C₆₇	—	2	1	no	—
310	C₂²×C₆₇	—	2	1	no	—
311	C₂²×C₆₇	—	2	4	no	—
312	C₂²×C₆₇	—	2	4	no	—
313	C₂²×C₆₇	—	2	4	no	—
314	C₂²×C₆₇	—	4	4	no	—
315	C₂²×C₆₇	—	4	2	no	—
316	C₂²×C₆₇	—	4	4	no	—
317	C₂²×C₆₇	—	4	2	no	—
318	C₂²×C₆₇	—	4	2	no	—
319	C₂²×C₆₇	—	4	4	no	—
320	C₂²×C₆₇	—	8	4	no	—
321	C₂²×C₆₇	—	8	8	no	—
322	C₂²×C₆₇	C₂×#135	1	2	yes	—
323	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
324	C₄×C₆₇	—	2	1	no	—
325	C₄×C₆₇	—	2	1	no	—
326	C₄×C₆₇	—	2	4	no	—
327	C₄×C₆₇	—	2	4	no	—
328	C₄×C₆₇	—	2	1	no	—
329	C₄×C₆₇	—	4	2	no	—
330	C₄×C₆₇	—	4	2	no	—
331	C₄×C₆₇	—	4	2	no	—
332	C₄×C₆₇	—	4	2	no	—
333	C₄×C₆₇	C₂×#135	1	2	yes	—
334	C₄×C₆₇	C₂×#135	1	2	yes	—
335	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
336	#61566	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
337	C₂×C₆₇	—	2	2	no	—
338	C₂×C₆₇	—	4	4	no	—
339	C₂×C₆₇	—	8	2	no	—
340	C₂×C₆₇	—	8	4	no	—
341	C₂×C₆₇	#3484	1	1	yes	characteristic
342	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
343	C₂×C₆₇	#3585	1	1	yes	characteristic
344	#3506	C₂×C₆₇	1	1	yes	characteristic, complemented
345	C₂×#550	C₂×C₆₇	1	1	yes	characteristic, complemented
346	#3305	C₂×C₆₇	1	4	yes	complemented
347	#3304	C₂×C₆₇	1	4	yes	complemented
348	#3287	C₂×C₆₇	1	4	yes	complemented
349	#3284	C₂×C₆₇	1	4	yes	complemented
350	#3208	C₂×C₆₇	1	1	yes	characteristic, complemented
351	#3205	C₂×C₆₇	1	1	yes	characteristic, complemented
352	#3188	C₂×C₆₇	1	4	yes	complemented
353	#3104	C₂×C₆₇	1	1	yes	characteristic, complemented
354	#3103	C₂×C₆₇	1	1	yes	characteristic, complemented
355	#3002	C₂×C₆₇	1	4	yes	complemented
356	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
357	C₆₇	#61566	1	1	yes	characteristic, direct product factor, Sylow
358	C₂²×#41	C₂²×C₆₇	1	2	yes	—
359	C₂×#135	C₂²×C₆₇	1	8	yes	complemented
360	#559	—	2	2	no	—
361	#556	C₂²×C₆₇	1	2	yes	complemented
362	#555	C₂²×C₆₇	1	2	yes	complemented
363	#553	—	2	2	no	—
364	#550	—	2	2	no	—
365	#550	—	2	2	no	—
366	#550	C₂²×C₆₇	1	4	yes	—
367	#550	C₂²×C₆₇	1	4	yes	—
368	#550	C₂²×C₆₇	1	2	yes	complemented
369	#547	C₂²×C₆₇	1	2	yes	complemented
370	#538	—	2	2	no	—
371	#535	—	2	1	no	—
372	#533	C₂²×C₆₇	1	1	yes	characteristic
373	#532	C₂²×C₆₇	1	1	yes	characteristic
374	C₂×#127	—	2	1	no	—
375	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
376	#525	C₂²×C₆₇	1	4	yes	—
377	C₂×#121	—	2	2	no	—
378	C₂×#121	C₂²×C₆₇	1	2	yes	—
379	#520	—	2	1	no	—
380	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
381	C₂×C₄×D₄	—	2	1	no	—
382	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
383	#493	C₂²×C₆₇	1	2	yes	complemented
384	#492	C₂²×C₆₇	1	2	yes	complemented
385	#486	C₂²×C₆₇	1	2	yes	complemented
386	#486	C₂²×C₆₇	1	2	yes	complemented
387	#484	C₂²×C₆₇	1	2	yes	—
388	#482	C₂²×C₆₇	1	8	yes	complemented
389	#481	C₂²×C₆₇	1	2	yes	—
390	#477	C₂²×C₆₇	1	2	yes	complemented
391	#475	C₂²×C₆₇	1	2	yes	complemented
392	#466	C₂²×C₆₇	1	8	yes	complemented
393	#465	C₂²×C₆₇	1	8	yes	complemented
394	#462	C₂²×C₆₇	1	2	yes	complemented
395	#460	C₂²×C₆₇	1	2	yes	complemented
396	#460	C₂²×C₆₇	1	4	yes	complemented
397	#459	C₂²×C₆₇	1	4	yes	complemented
398	#452	C₂²×C₆₇	1	8	yes	complemented
399	#451	C₂²×C₆₇	1	2	yes	—
400	#450	C₂²×C₆₇	1	4	yes	—
401	#449	C₂²×C₆₇	1	8	yes	complemented
402	#449	C₂²×C₆₇	1	8	yes	complemented
403	#448	C₂²×C₆₇	1	2	yes	—
404	#447	C₂²×C₆₇	1	4	yes	—
405	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
406	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
407	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
408	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
409	#431	C₂²×C₆₇	1	1	yes	characteristic
410	#431	C₂²×C₆₇	1	1	yes	characteristic
411	#416	C₂²×C₆₇	1	8	yes	complemented
412	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
413	C₂×#41	—	2	1	no	—
414	C₂×#41	—	2	4	no	—
415	C₂×#41	—	4	8	no	—
416	C₂×#41	C₂³×C₆₇	1	8	yes	—
417	C₂×#41	C₂³×C₆₇	1	4	yes	—
418	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
419	C₂²×D₄	—	2	2	no	—
420	C₂²×D₄	—	2	2	no	—
421	C₂²×D₄	—	2	1	no	—
422	C₂²×D₄	—	2	1	no	—
423	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
424	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
425	C₂³×C₄	—	2	2	no	—
426	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
427	#135	—	4	16	no	—
428	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
429	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
430	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
431	C₂×D₈	C₂³×C₆₇	1	8	yes	—
432	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
433	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
434	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
435	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
436	#127	—	2	4	no	—
437	#127	—	2	4	no	—
438	#127	—	2	2	no	—
439	#127	—	2	1	no	—
440	#127	—	2	1	no	—
441	#127	C₂³×C₆₇	1	2	yes	—
442	#127	C₂³×C₆₇	1	2	yes	—
443	#127	C₂³×C₆₇	1	1	yes	characteristic
444	#126	—	2	4	no	—
445	#125	—	2	1	no	—
446	#124	—	2	2	no	—
447	#124	—	2	4	no	—
448	#124	—	2	1	no	—
449	#123	—	2	4	no	—
450	#123	—	2	4	no	—
451	C₂²⋊Q₈	—	2	1	no	—
452	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
453	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
454	#121	—	2	2	no	—
455	#121	—	2	2	no	—
456	#121	—	2	4	no	—
457	#121	—	2	1	no	—
458	#121	—	2	8	no	—
459	#121	—	2	8	no	—
460	#121	—	2	4	no	—
461	#121	C₂³×C₆₇	1	8	yes	—
462	#121	C₂³×C₆₇	1	2	yes	—
463	#121	C₂³×C₆₇	1	4	yes	—
464	#120	—	2	4	no	—
465	C₄×Q₈	—	2	2	no	—
466	C₄×Q₈	—	2	1	no	—
467	C₄×Q₈	—	2	1	no	—
468	C₄×Q₈	—	2	1	no	—
469	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
470	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
471	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
472	C₄×D₄	—	2	4	no	—
473	C₄×D₄	—	2	4	no	—
474	C₄×D₄	—	2	4	no	—
475	C₄×D₄	—	2	4	no	—
476	C₄×D₄	—	2	2	no	—
477	C₄×D₄	—	2	1	no	—
478	C₄×D₄	D₄×C₆₇	1	2	yes	—
479	C₄×D₄	D₄×C₆₇	1	2	yes	—
480	C₄×D₄	C₂³×C₆₇	1	2	yes	—
481	C₄×D₄	C₂³×C₆₇	1	2	yes	—
482	C₄×D₄	C₂³×C₆₇	1	2	yes	—
483	#117	—	2	1	no	—
484	#117	—	2	2	no	—
485	#117	C₂³×C₆₇	1	4	yes	—
486	C₂×C₄⋊C₄	—	2	1	no	—
487	C₂×C₄⋊C₄	—	2	2	no	—
488	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
489	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
490	C₂×C₂²⋊C₄	—	2	2	no	—
491	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
492	C₂×C₄²	—	2	1	no	—
493	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
494	#107	C₂³×C₆₇	1	4	yes	—
495	#106	C₂³×C₆₇	1	4	yes	—
496	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
497	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
498	#103	C₂³×C₆₇	1	4	yes	—
499	#103	C₂³×C₆₇	1	4	yes	—
500	#103	C₂³×C₆₇	1	4	yes	—
501	#102	C₂³×C₆₇	1	8	yes	—
502	#102	C₂³×C₆₇	1	8	yes	—
503	#102	C₂³×C₆₇	1	4	yes	—
504	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
505	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
506	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
507	C₄×C₈	C₂³×C₆₇	1	2	yes	—
508	C₄×C₈	C₂³×C₆₇	1	2	yes	—
509	C₂⁴	—	2	2	no	—
510	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
511	#41	—	2	16	no	—
512	#41	—	4	8	no	—
513	#41	—	4	8	no	—
514	#41	—	4	16	no	—
515	#41	—	4	4	no	—
516	C₂×Q₈	—	2	4	no	—
517	C₂×Q₈	—	2	1	no	—
518	C₂×Q₈	—	2	1	no	—
519	C₂×Q₈	—	2	1	no	—
520	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
521	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
522	C₂×D₄	—	2	2	no	—
523	C₂×D₄	—	2	8	no	—
524	C₂×D₄	—	2	4	no	—
525	C₂×D₄	—	2	4	no	—
526	C₂×D₄	—	2	8	no	—
527	C₂×D₄	—	2	4	no	—
528	C₂×D₄	—	2	1	no	—
529	C₂×D₄	—	2	2	no	—
530	C₂×D₄	—	2	2	no	—
531	C₂×D₄	—	2	2	no	—
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	2	2	no	—
534	C₂×D₄	—	2	2	no	—
535	C₂×D₄	—	4	4	no	—
536	C₂×D₄	—	4	8	no	—
537	C₂×D₄	—	4	4	no	—
538	C₂×D₄	—	8	2	no	—
539	C₂×D₄	—	8	4	no	—
540	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
541	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
542	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
543	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	2	no	—
547	C₂²×C₄	—	2	8	no	—
548	C₂²×C₄	—	2	4	no	—
549	C₂²×C₄	—	4	4	no	—
550	C₂²×C₄	—	4	8	no	—
551	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
552	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
553	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
554	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
555	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
556	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
557	Q₁₆	—	2	8	no	—
558	SD₁₆	—	2	8	no	—
559	SD₁₆	—	4	8	no	—
560	D₈	—	4	8	no	—
561	C₂×C₈	—	4	8	no	—
562	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
563	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
575	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
576	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
577	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
578	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
579	C₂²⋊C₄	—	2	2	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	4	no	—
582	C₂²⋊C₄	—	2	8	no	—
583	C₂²⋊C₄	—	2	4	no	—
584	C₂²⋊C₄	—	2	2	no	—
585	C₂²⋊C₄	—	2	2	no	—
586	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
587	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
588	C₄²	—	2	2	no	—
589	C₄²	—	2	2	no	—
590	C₄²	—	2	4	no	—
591	C₄²	—	2	2	no	—
592	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
593	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
594	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
595	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
596	C₂³	—	2	4	no	—
597	C₂³	—	2	2	no	—
598	C₂³	—	2	4	no	—
599	C₂³	—	4	2	no	—
600	C₂³	—	4	2	no	—
601	C₂³	—	4	4	no	—
602	C₂³	—	4	4	no	—
603	C₂³	—	4	4	no	—
604	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
605	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
606	Q₈	—	2	4	no	—
607	Q₈	—	2	2	no	—
608	Q₈	—	2	4	no	—
609	Q₈	—	4	2	no	—
610	Q₈	—	8	1	no	—
611	Q₈	H(5,𝔽₂)×C₆₇	1	4	yes	—
612	D₄	—	2	4	no	—
613	D₄	—	2	4	no	—
614	D₄	—	2	4	no	—
615	D₄	—	2	8	no	—
616	D₄	—	4	16	no	—
617	D₄	—	4	4	no	—
618	D₄	—	4	4	no	—
619	D₄	—	4	4	no	—
620	D₄	—	4	4	no	—
621	D₄	—	4	2	no	—
622	D₄	—	8	2	no	—
623	D₄	—	8	2	no	—
624	D₄	—	8	1	no	—
625	D₄	—	8	8	no	—
626	D₄	—	8	4	no	—
627	D₄	—	8	4	no	—
628	C₂×C₄	—	2	4	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	8	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	4	2	no	—
637	C₂×C₄	—	4	4	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	16	no	—
640	C₂×C₄	—	4	2	no	—
641	C₂×C₄	—	4	4	no	—
642	C₂×C₄	—	8	4	no	—
643	C₂×C₄	#135×C₆₇	1	4	yes	complemented
644	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
645	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
646	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
647	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
648	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
649	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
650	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
651	C₈	—	2	4	no	—
652	C₈	—	2	4	no	—
653	C₂²	—	2	1	no	—
654	C₂²	—	2	1	no	—
655	C₂²	—	2	4	no	—
656	C₂²	—	2	4	no	—
657	C₂²	—	2	4	no	—
658	C₂²	—	4	2	no	—
659	C₂²	—	4	4	no	—
660	C₂²	—	4	4	no	—
661	C₂²	—	4	2	no	—
662	C₂²	—	4	4	no	—
663	C₂²	—	4	2	no	—
664	C₂²	—	8	4	no	—
665	C₂²	—	8	8	no	—
666	C₂²	C₂×#135×C₆₇	1	2	yes	—
667	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
668	C₄	—	2	1	no	—
669	C₄	—	2	1	no	—
670	C₄	—	2	4	no	—
671	C₄	—	2	4	no	—
672	C₄	—	2	1	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	2	no	—
675	C₄	—	4	2	no	—
676	C₄	—	4	2	no	—
677	C₄	C₂×#135×C₆₇	1	2	yes	—
678	C₄	C₂×#135×C₆₇	1	2	yes	—
679	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
680	C₂	—	2	2	no	—
681	C₂	—	4	4	no	—
682	C₂	—	8	4	no	—
683	C₂	—	8	2	no	—
684	C₂	C₆₇×#3484	1	1	yes	characteristic
685	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
686	C₂	C₆₇×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#3506	C₂	semidirect
2. C₂×#550×C₆₇	C₂	semidirect
3. C₆₇×#3305	C₂	semidirect
4. C₆₇×#3304	C₂	semidirect
5. C₆₇×#3287	C₂	semidirect
6. C₆₇×#3284	C₂	semidirect
7. C₆₇×#3208	C₂	semidirect
8. C₆₇×#3205	C₂	semidirect
9. C₆₇×#3188	C₂	semidirect
10. C₆₇×#3104	C₂	semidirect
11. C₆₇×#3103	C₂	semidirect
12. C₆₇×#3002	C₂	semidirect
13. C₆₇×#2972	C₂	semidirect
15. C₂×#135×C₆₇	C₂²	semidirect
17. #556×C₆₇	C₂²	semidirect
18. #555×C₆₇	C₂²	semidirect
24. #550×C₆₇	C₂²	semidirect
25. #547×C₆₇	C₂²	semidirect
39. #493×C₆₇	C₂²	semidirect
40. #492×C₆₇	C₂²	semidirect
41. #486×C₆₇	C₂²	semidirect
42. #486×C₆₇	C₂²	semidirect
44. #482×C₆₇	C₂²	semidirect
46. #477×C₆₇	C₂²	semidirect
47. #475×C₆₇	C₂²	semidirect
48. #466×C₆₇	C₂²	semidirect
49. #465×C₆₇	C₂²	semidirect
50. #462×C₆₇	C₂²	semidirect
51. #460×C₆₇	C₂²	semidirect
52. #460×C₆₇	C₂²	semidirect
53. #459×C₆₇	C₂²	semidirect
54. #452×C₆₇	C₂²	semidirect
57. #449×C₆₇	C₂²	semidirect
58. #449×C₆₇	C₂²	semidirect
61. C₄×Q₁₆×C₆₇	C₂²	semidirect
62. C₄×SD₁₆×C₆₇	C₂²	semidirect
63. C₈×D₄×C₆₇	C₂²	semidirect
64. C₈×D₄×C₆₇	C₂²	semidirect
67. #416×C₆₇	C₂²	semidirect
299. C₂×C₄×C₆₇	#135	semidirect
336. #61566	C₆₇	direct (N × H)
344. #3506	C₂×C₆₇	semidirect
345. C₂×#550	C₂×C₆₇	semidirect
346. #3305	C₂×C₆₇	semidirect
347. #3304	C₂×C₆₇	semidirect
348. #3287	C₂×C₆₇	semidirect
349. #3284	C₂×C₆₇	semidirect
350. #3208	C₂×C₆₇	semidirect
351. #3205	C₂×C₆₇	semidirect
352. #3188	C₂×C₆₇	semidirect
353. #3104	C₂×C₆₇	semidirect
354. #3103	C₂×C₆₇	semidirect
355. #3002	C₂×C₆₇	semidirect
356. #2972	C₂×C₆₇	semidirect
357. C₆₇	#61566	direct (N × H)
359. C₂×#135	C₂²×C₆₇	semidirect
361. #556	C₂²×C₆₇	semidirect
362. #555	C₂²×C₆₇	semidirect
368. #550	C₂²×C₆₇	semidirect
369. #547	C₂²×C₆₇	semidirect
383. #493	C₂²×C₆₇	semidirect
384. #492	C₂²×C₆₇	semidirect
385. #486	C₂²×C₆₇	semidirect
386. #486	C₂²×C₆₇	semidirect
388. #482	C₂²×C₆₇	semidirect
390. #477	C₂²×C₆₇	semidirect
391. #475	C₂²×C₆₇	semidirect
392. #466	C₂²×C₆₇	semidirect
393. #465	C₂²×C₆₇	semidirect
394. #462	C₂²×C₆₇	semidirect
395. #460	C₂²×C₆₇	semidirect
396. #460	C₂²×C₆₇	semidirect
397. #459	C₂²×C₆₇	semidirect
398. #452	C₂²×C₆₇	semidirect
401. #449	C₂²×C₆₇	semidirect
402. #449	C₂²×C₆₇	semidirect
405. C₄×Q₁₆	C₂²×C₆₇	semidirect
406. C₄×SD₁₆	C₂²×C₆₇	semidirect
407. C₈×D₄	C₂²×C₆₇	semidirect
408. C₈×D₄	C₂²×C₆₇	semidirect
411. #416	C₂²×C₆₇	semidirect
643. C₂×C₄	#135×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3506	2
C₂×#550×C₆₇	2
C₆₇×#3305	2
C₆₇×#3304	2
C₆₇×#3287	2
C₆₇×#3284	2
C₆₇×#3208	2
C₆₇×#3205	2
C₆₇×#3188	2
C₆₇×#3104	2
C₆₇×#3103	2
C₆₇×#3002	2
C₆₇×#2972	2
#61566	67

Subgroups — C67×#61566 #30740817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61566 #30740817