Subgroups — C₆₇×#61565 #30740816

Order 17152.

2142 conjugacy classes of subgroups.

The subgroup lattice has 678 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×#547×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3303	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3302	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3284	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3283	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3206	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3190	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3101	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3100	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	C₂²	1	2	yes	complemented
21	#547×C₆₇	—	2	2	no	—
22	#547×C₆₇	—	2	2	no	—
23	#547×C₆₇	C₂²	1	4	yes	—
24	#547×C₆₇	C₂²	1	4	yes	—
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×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	#525×C₆₇	C₂²	1	4	yes	—
32	C₂×#123×C₆₇	—	2	2	no	—
33	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
34	C₂×#121×C₆₇	—	2	1	no	—
35	#520×C₆₇	—	2	1	no	—
36	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
39	#494×C₆₇	C₂²	1	2	yes	complemented
40	#492×C₆₇	C₂²	1	2	yes	complemented
41	#487×C₆₇	C₂²	1	2	yes	complemented
42	#487×C₆₇	C₂²	1	2	yes	complemented
43	#485×C₆₇	C₂²	1	8	yes	complemented
44	#483×C₆₇	C₂²	1	2	yes	—
45	#480×C₆₇	C₂²	1	2	yes	—
46	#476×C₆₇	C₂²	1	2	yes	complemented
47	#474×C₆₇	C₂²	1	2	yes	complemented
48	#467×C₆₇	C₂²	1	8	yes	complemented
49	#465×C₆₇	C₂²	1	8	yes	complemented
50	#462×C₆₇	C₂²	1	4	yes	complemented
51	#461×C₆₇	C₂²	1	2	yes	complemented
52	#461×C₆₇	C₂²	1	4	yes	complemented
53	#459×C₆₇	C₂²	1	2	yes	complemented
54	#452×C₆₇	C₂²	1	8	yes	complemented
55	#452×C₆₇	C₂²	1	8	yes	complemented
56	#451×C₆₇	C₂²	1	4	yes	—
57	#450×C₆₇	C₂²	1	2	yes	—
58	#449×C₆₇	C₂²	1	8	yes	complemented
59	#448×C₆₇	C₂²	1	4	yes	—
60	#447×C₆₇	C₂²	1	2	yes	—
61	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
62	C₄×D₈×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	2	yes	—
75	C₂²×D₄×C₆₇	—	2	2	no	—
76	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
77	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	—	2	1	no	—
79	C₂³×C₄×C₆₇	—	2	2	no	—
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	#135×C₆₇	—	4	16	no	—
82	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
83	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
84	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
85	C₂×D₈×C₆₇	C₂³	1	4	yes	—
86	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
87	C₄⋊Q₈×C₆₇	—	2	1	no	—
88	C₄⋊Q₈×C₆₇	—	2	1	no	—
89	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
90	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
92	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
94	#127×C₆₇	—	2	2	no	—
95	#127×C₆₇	C₂³	1	1	yes	characteristic
96	#126×C₆₇	—	2	4	no	—
97	#125×C₆₇	—	2	1	no	—
98	#124×C₆₇	—	2	2	no	—
99	#124×C₆₇	—	2	4	no	—
100	#124×C₆₇	—	2	1	no	—
101	#123×C₆₇	—	2	4	no	—
102	#123×C₆₇	—	2	8	no	—
103	#123×C₆₇	—	2	8	no	—
104	#123×C₆₇	—	2	4	no	—
105	C₂²⋊Q₈×C₆₇	—	2	4	no	—
106	C₂²⋊Q₈×C₆₇	—	2	1	no	—
107	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
108	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
109	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
110	#121×C₆₇	—	2	4	no	—
111	#121×C₆₇	—	2	4	no	—
112	#121×C₆₇	—	2	2	no	—
113	#121×C₆₇	—	2	2	no	—
114	#121×C₆₇	—	2	1	no	—
115	#121×C₆₇	—	2	4	no	—
116	#121×C₆₇	C₂³	1	2	yes	—
117	#121×C₆₇	C₂³	1	4	yes	—
118	#120×C₆₇	—	2	4	no	—
119	C₄×Q₈×C₆₇	—	2	2	no	—
120	C₄×Q₈×C₆₇	—	2	1	no	—
121	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
122	C₄×D₄×C₆₇	—	2	4	no	—
123	C₄×D₄×C₆₇	—	2	4	no	—
124	C₄×D₄×C₆₇	—	2	2	no	—
125	C₄×D₄×C₆₇	—	2	1	no	—
126	C₄×D₄×C₆₇	—	2	1	no	—
127	C₄×D₄×C₆₇	—	2	1	no	—
128	C₄×D₄×C₆₇	D₄	1	2	yes	—
129	C₄×D₄×C₆₇	D₄	1	2	yes	—
130	C₄×D₄×C₆₇	C₂³	1	2	yes	—
131	C₄×D₄×C₆₇	C₂³	1	2	yes	—
132	C₄×D₄×C₆₇	C₂³	1	2	yes	—
133	C₄×D₄×C₆₇	C₂³	1	2	yes	—
134	C₄×D₄×C₆₇	C₂³	1	2	yes	—
135	#117×C₆₇	—	2	1	no	—
136	#117×C₆₇	—	2	2	no	—
137	#117×C₆₇	C₂³	1	4	yes	—
138	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
142	C₂×C₄⋊C₄×C₆₇	—	2	4	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	—	2	1	no	—
148	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
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	8	yes	—
155	#103×C₆₇	C₂³	1	8	yes	—
156	#103×C₆₇	C₂³	1	4	yes	—
157	#102×C₆₇	C₂³	1	4	yes	—
158	#102×C₆₇	C₂³	1	4	yes	—
159	#102×C₆₇	C₂³	1	4	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	1	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	1	no	—
173	C₂×Q₈×C₆₇	—	2	2	no	—
174	C₂×Q₈×C₆₇	—	2	1	no	—
175	C₂×Q₈×C₆₇	—	4	4	no	—
176	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
177	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
178	C₂×D₄×C₆₇	—	2	4	no	—
179	C₂×D₄×C₆₇	—	2	2	no	—
180	C₂×D₄×C₆₇	—	2	8	no	—
181	C₂×D₄×C₆₇	—	2	4	no	—
182	C₂×D₄×C₆₇	—	2	4	no	—
183	C₂×D₄×C₆₇	—	2	4	no	—
184	C₂×D₄×C₆₇	—	2	1	no	—
185	C₂×D₄×C₆₇	—	2	1	no	—
186	C₂×D₄×C₆₇	—	2	1	no	—
187	C₂×D₄×C₆₇	—	4	8	no	—
188	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
189	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
190	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
191	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₆₇	—	2	1	no	—
193	C₂²×C₄×C₆₇	—	2	2	no	—
194	C₂²×C₄×C₆₇	—	2	2	no	—
195	C₂²×C₄×C₆₇	—	2	2	no	—
196	C₂²×C₄×C₆₇	—	2	8	no	—
197	C₂²×C₄×C₆₇	—	2	2	no	—
198	C₂²×C₄×C₆₇	—	2	4	no	—
199	C₂²×C₄×C₆₇	—	4	4	no	—
200	C₂²×C₄×C₆₇	—	4	8	no	—
201	C₂²×C₄×C₆₇	—	4	4	no	—
202	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
205	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
207	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
208	Q₁₆×C₆₇	—	4	8	no	—
209	SD₁₆×C₆₇	—	2	8	no	—
210	SD₁₆×C₆₇	—	4	8	no	—
211	D₈×C₆₇	—	2	8	no	—
212	C₂×C₈×C₆₇	—	4	8	no	—
213	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
214	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
215	C₄⋊C₄×C₆₇	—	2	2	no	—
216	C₄⋊C₄×C₆₇	—	2	2	no	—
217	C₄⋊C₄×C₆₇	—	2	8	no	—
218	C₄⋊C₄×C₆₇	—	2	1	no	—
219	C₄⋊C₄×C₆₇	—	2	4	no	—
220	C₄⋊C₄×C₆₇	—	2	2	no	—
221	C₄⋊C₄×C₆₇	—	2	2	no	—
222	C₄⋊C₄×C₆₇	—	2	1	no	—
223	C₄⋊C₄×C₆₇	—	2	1	no	—
224	C₄⋊C₄×C₆₇	—	2	2	no	—
225	C₄⋊C₄×C₆₇	—	2	1	no	—
226	C₄⋊C₄×C₆₇	—	2	1	no	—
227	C₄⋊C₄×C₆₇	—	2	2	no	—
228	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
229	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
230	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
231	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
232	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
233	C₂²⋊C₄×C₆₇	—	2	8	no	—
234	C₂²⋊C₄×C₆₇	—	2	2	no	—
235	C₂²⋊C₄×C₆₇	—	2	4	no	—
236	C₂²⋊C₄×C₆₇	—	2	4	no	—
237	C₂²⋊C₄×C₆₇	—	2	2	no	—
238	C₂²⋊C₄×C₆₇	—	2	2	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	4	no	—
245	C₄²×C₆₇	—	2	2	no	—
246	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
247	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
248	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
249	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
250	C₂³×C₆₇	—	2	4	no	—
251	C₂³×C₆₇	—	2	2	no	—
252	C₂³×C₆₇	—	2	4	no	—
253	C₂³×C₆₇	—	2	2	no	—
254	C₂³×C₆₇	—	4	2	no	—
255	C₂³×C₆₇	—	4	2	no	—
256	C₂³×C₆₇	—	4	2	no	—
257	C₂³×C₆₇	—	4	2	no	—
258	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
259	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
260	Q₈×C₆₇	—	2	4	no	—
261	Q₈×C₆₇	—	2	8	no	—
262	Q₈×C₆₇	—	4	4	no	—
263	Q₈×C₆₇	—	4	2	no	—
264	Q₈×C₆₇	—	8	1	no	—
265	D₄×C₆₇	—	2	4	no	—
266	D₄×C₆₇	—	2	4	no	—
267	D₄×C₆₇	—	2	4	no	—
268	D₄×C₆₇	—	2	2	no	—
269	D₄×C₆₇	—	2	4	no	—
270	D₄×C₆₇	—	4	16	no	—
271	D₄×C₆₇	—	4	4	no	—
272	D₄×C₆₇	—	4	2	no	—
273	D₄×C₆₇	—	8	1	no	—
274	D₄×C₆₇	—	8	4	no	—
275	D₄×C₆₇	—	8	4	no	—
276	D₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
277	C₂×C₄×C₆₇	—	2	8	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	2	no	—
282	C₂×C₄×C₆₇	—	2	2	no	—
283	C₂×C₄×C₆₇	—	2	4	no	—
284	C₂×C₄×C₆₇	—	4	2	no	—
285	C₂×C₄×C₆₇	—	4	4	no	—
286	C₂×C₄×C₆₇	—	4	1	no	—
287	C₂×C₄×C₆₇	—	4	16	no	—
288	C₂×C₄×C₆₇	—	4	4	no	—
289	C₂×C₄×C₆₇	—	4	2	no	—
290	C₂×C₄×C₆₇	—	4	4	no	—
291	C₂×C₄×C₆₇	—	4	4	no	—
292	C₂×C₄×C₆₇	—	4	4	no	—
293	C₂×C₄×C₆₇	—	8	4	no	—
294	C₂×C₄×C₆₇	#135	1	4	yes	complemented
295	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
296	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
302	C₈×C₆₇	—	2	4	no	—
303	C₈×C₆₇	—	2	4	no	—
304	C₂²×C₆₇	—	2	1	no	—
305	C₂²×C₆₇	—	2	1	no	—
306	C₂²×C₆₇	—	2	4	no	—
307	C₂²×C₆₇	—	2	4	no	—
308	C₂²×C₆₇	—	2	4	no	—
309	C₂²×C₆₇	—	2	4	no	—
310	C₂²×C₆₇	—	4	8	no	—
311	C₂²×C₆₇	—	4	2	no	—
312	C₂²×C₆₇	—	4	2	no	—
313	C₂²×C₆₇	—	4	2	no	—
314	C₂²×C₆₇	—	4	2	no	—
315	C₂²×C₆₇	—	4	2	no	—
316	C₂²×C₆₇	—	8	4	no	—
317	C₂²×C₆₇	C₂×#135	1	2	yes	—
318	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
319	C₄×C₆₇	—	2	1	no	—
320	C₄×C₆₇	—	2	1	no	—
321	C₄×C₆₇	—	2	4	no	—
322	C₄×C₆₇	—	2	1	no	—
323	C₄×C₆₇	—	4	2	no	—
324	C₄×C₆₇	—	4	4	no	—
325	C₄×C₆₇	—	4	2	no	—
326	C₄×C₆₇	—	4	2	no	—
327	C₄×C₆₇	—	4	2	no	—
328	C₄×C₆₇	C₂×#135	1	2	yes	—
329	C₄×C₆₇	C₂×#135	1	2	yes	—
330	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
331	#61565	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
332	C₂×C₆₇	—	2	2	no	—
333	C₂×C₆₇	—	4	4	no	—
334	C₂×C₆₇	—	4	4	no	—
335	C₂×C₆₇	—	8	2	no	—
336	C₂×C₆₇	#3484	1	1	yes	characteristic
337	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
338	C₂×C₆₇	#3586	1	1	yes	characteristic
339	#3506	C₂×C₆₇	1	1	yes	characteristic, complemented
340	C₂×#547	C₂×C₆₇	1	1	yes	characteristic, complemented
341	#3303	C₂×C₆₇	1	4	yes	complemented
342	#3302	C₂×C₆₇	1	4	yes	complemented
343	#3284	C₂×C₆₇	1	4	yes	complemented
344	#3283	C₂×C₆₇	1	4	yes	complemented
345	#3209	C₂×C₆₇	1	1	yes	characteristic, complemented
346	#3206	C₂×C₆₇	1	1	yes	characteristic, complemented
347	#3190	C₂×C₆₇	1	4	yes	complemented
348	#3101	C₂×C₆₇	1	1	yes	characteristic, complemented
349	#3100	C₂×C₆₇	1	1	yes	characteristic, complemented
350	#3002	C₂×C₆₇	1	4	yes	complemented
351	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
352	C₆₇	#61565	1	1	yes	characteristic, direct product factor, Sylow
353	C₂²×#41	C₂²×C₆₇	1	2	yes	—
354	C₂×#135	C₂²×C₆₇	1	8	yes	complemented
355	#559	—	2	2	no	—
356	#556	C₂²×C₆₇	1	2	yes	complemented
357	#555	C₂²×C₆₇	1	2	yes	complemented
358	#553	—	2	2	no	—
359	#550	C₂²×C₆₇	1	2	yes	complemented
360	#547	—	2	2	no	—
361	#547	—	2	2	no	—
362	#547	C₂²×C₆₇	1	4	yes	—
363	#547	C₂²×C₆₇	1	4	yes	—
364	#547	C₂²×C₆₇	1	2	yes	complemented
365	#538	—	2	2	no	—
366	#535	—	2	1	no	—
367	#533	C₂²×C₆₇	1	1	yes	characteristic
368	#532	C₂²×C₆₇	1	1	yes	characteristic
369	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
370	#525	C₂²×C₆₇	1	4	yes	—
371	C₂×#123	—	2	2	no	—
372	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
373	C₂×#121	—	2	1	no	—
374	#520	—	2	1	no	—
375	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
376	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
377	C₂²×C₄⋊C₄	—	2	1	no	—
378	#494	C₂²×C₆₇	1	2	yes	complemented
379	#492	C₂²×C₆₇	1	2	yes	complemented
380	#487	C₂²×C₆₇	1	2	yes	complemented
381	#487	C₂²×C₆₇	1	2	yes	complemented
382	#485	C₂²×C₆₇	1	8	yes	complemented
383	#483	C₂²×C₆₇	1	2	yes	—
384	#480	C₂²×C₆₇	1	2	yes	—
385	#476	C₂²×C₆₇	1	2	yes	complemented
386	#474	C₂²×C₆₇	1	2	yes	complemented
387	#467	C₂²×C₆₇	1	8	yes	complemented
388	#465	C₂²×C₆₇	1	8	yes	complemented
389	#462	C₂²×C₆₇	1	4	yes	complemented
390	#461	C₂²×C₆₇	1	2	yes	complemented
391	#461	C₂²×C₆₇	1	4	yes	complemented
392	#459	C₂²×C₆₇	1	2	yes	complemented
393	#452	C₂²×C₆₇	1	8	yes	complemented
394	#452	C₂²×C₆₇	1	8	yes	complemented
395	#451	C₂²×C₆₇	1	4	yes	—
396	#450	C₂²×C₆₇	1	2	yes	—
397	#449	C₂²×C₆₇	1	8	yes	complemented
398	#448	C₂²×C₆₇	1	4	yes	—
399	#447	C₂²×C₆₇	1	2	yes	—
400	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
401	C₄×D₈	C₂²×C₆₇	1	4	yes	complemented
402	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
403	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
404	#431	C₂²×C₆₇	1	1	yes	characteristic
405	#431	C₂²×C₆₇	1	1	yes	characteristic
406	#416	C₂²×C₆₇	1	8	yes	complemented
407	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
408	C₂×#41	—	2	1	no	—
409	C₂×#41	—	2	4	no	—
410	C₂×#41	—	4	8	no	—
411	C₂×#41	C₂³×C₆₇	1	8	yes	—
412	C₂×#41	C₂³×C₆₇	1	4	yes	—
413	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
414	C₂²×D₄	—	2	2	no	—
415	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
416	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
417	C₂³×C₄	—	2	1	no	—
418	C₂³×C₄	—	2	2	no	—
419	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
420	#135	—	4	16	no	—
421	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
422	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
423	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
424	C₂×D₈	C₂³×C₆₇	1	4	yes	—
425	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
426	C₄⋊Q₈	—	2	1	no	—
427	C₄⋊Q₈	—	2	1	no	—
428	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
429	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
430	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
431	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
432	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
433	#127	—	2	2	no	—
434	#127	C₂³×C₆₇	1	1	yes	characteristic
435	#126	—	2	4	no	—
436	#125	—	2	1	no	—
437	#124	—	2	2	no	—
438	#124	—	2	4	no	—
439	#124	—	2	1	no	—
440	#123	—	2	4	no	—
441	#123	—	2	8	no	—
442	#123	—	2	8	no	—
443	#123	—	2	4	no	—
444	C₂²⋊Q₈	—	2	4	no	—
445	C₂²⋊Q₈	—	2	1	no	—
446	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
447	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
448	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
449	#121	—	2	4	no	—
450	#121	—	2	4	no	—
451	#121	—	2	2	no	—
452	#121	—	2	2	no	—
453	#121	—	2	1	no	—
454	#121	—	2	4	no	—
455	#121	C₂³×C₆₇	1	2	yes	—
456	#121	C₂³×C₆₇	1	4	yes	—
457	#120	—	2	4	no	—
458	C₄×Q₈	—	2	2	no	—
459	C₄×Q₈	—	2	1	no	—
460	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
461	C₄×D₄	—	2	4	no	—
462	C₄×D₄	—	2	4	no	—
463	C₄×D₄	—	2	2	no	—
464	C₄×D₄	—	2	1	no	—
465	C₄×D₄	—	2	1	no	—
466	C₄×D₄	—	2	1	no	—
467	C₄×D₄	D₄×C₆₇	1	2	yes	—
468	C₄×D₄	D₄×C₆₇	1	2	yes	—
469	C₄×D₄	C₂³×C₆₇	1	2	yes	—
470	C₄×D₄	C₂³×C₆₇	1	2	yes	—
471	C₄×D₄	C₂³×C₆₇	1	2	yes	—
472	C₄×D₄	C₂³×C₆₇	1	2	yes	—
473	C₄×D₄	C₂³×C₆₇	1	2	yes	—
474	#117	—	2	1	no	—
475	#117	—	2	2	no	—
476	#117	C₂³×C₆₇	1	4	yes	—
477	C₂×C₄⋊C₄	—	2	1	no	—
478	C₂×C₄⋊C₄	—	2	2	no	—
479	C₂×C₄⋊C₄	—	2	1	no	—
480	C₂×C₄⋊C₄	—	2	4	no	—
481	C₂×C₄⋊C₄	—	2	4	no	—
482	C₂×C₄⋊C₄	—	2	2	no	—
483	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
484	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
485	C₂×C₂²⋊C₄	—	2	2	no	—
486	C₂×C₂²⋊C₄	—	2	1	no	—
487	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
488	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
489	#107	C₂³×C₆₇	1	4	yes	—
490	#106	C₂³×C₆₇	1	4	yes	—
491	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
492	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
493	#103	C₂³×C₆₇	1	8	yes	—
494	#103	C₂³×C₆₇	1	8	yes	—
495	#103	C₂³×C₆₇	1	4	yes	—
496	#102	C₂³×C₆₇	1	4	yes	—
497	#102	C₂³×C₆₇	1	4	yes	—
498	#102	C₂³×C₆₇	1	4	yes	—
499	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
500	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
501	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
502	C₄×C₈	C₂³×C₆₇	1	2	yes	—
503	C₄×C₈	C₂³×C₆₇	1	2	yes	—
504	C₂⁴	—	2	1	no	—
505	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
506	#41	—	2	16	no	—
507	#41	—	4	8	no	—
508	#41	—	4	8	no	—
509	#41	—	4	16	no	—
510	#41	—	4	4	no	—
511	C₂×Q₈	—	2	1	no	—
512	C₂×Q₈	—	2	2	no	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	—	4	4	no	—
515	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
516	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
517	C₂×D₄	—	2	4	no	—
518	C₂×D₄	—	2	2	no	—
519	C₂×D₄	—	2	8	no	—
520	C₂×D₄	—	2	4	no	—
521	C₂×D₄	—	2	4	no	—
522	C₂×D₄	—	2	4	no	—
523	C₂×D₄	—	2	1	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	1	no	—
526	C₂×D₄	—	4	8	no	—
527	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
528	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
529	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
530	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	8	no	—
536	C₂²×C₄	—	2	2	no	—
537	C₂²×C₄	—	2	4	no	—
538	C₂²×C₄	—	4	4	no	—
539	C₂²×C₄	—	4	8	no	—
540	C₂²×C₄	—	4	4	no	—
541	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
542	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
543	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
544	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
545	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
546	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
547	Q₁₆	—	4	8	no	—
548	SD₁₆	—	2	8	no	—
549	SD₁₆	—	4	8	no	—
550	D₈	—	2	8	no	—
551	C₂×C₈	—	4	8	no	—
552	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
553	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
554	C₄⋊C₄	—	2	2	no	—
555	C₄⋊C₄	—	2	2	no	—
556	C₄⋊C₄	—	2	8	no	—
557	C₄⋊C₄	—	2	1	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	2	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	—	2	1	no	—
562	C₄⋊C₄	—	2	1	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
568	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
569	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
570	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
571	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
572	C₂²⋊C₄	—	2	8	no	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	4	no	—
575	C₂²⋊C₄	—	2	4	no	—
576	C₂²⋊C₄	—	2	2	no	—
577	C₂²⋊C₄	—	2	2	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	2	no	—
580	C₂²⋊C₄	—	2	2	no	—
581	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
582	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
583	C₄²	—	2	4	no	—
584	C₄²	—	2	2	no	—
585	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
586	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
587	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
588	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
589	C₂³	—	2	4	no	—
590	C₂³	—	2	2	no	—
591	C₂³	—	2	4	no	—
592	C₂³	—	2	2	no	—
593	C₂³	—	4	2	no	—
594	C₂³	—	4	2	no	—
595	C₂³	—	4	2	no	—
596	C₂³	—	4	2	no	—
597	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
598	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
599	Q₈	—	2	4	no	—
600	Q₈	—	2	8	no	—
601	Q₈	—	4	4	no	—
602	Q₈	—	4	2	no	—
603	Q₈	—	8	1	no	—
604	D₄	—	2	4	no	—
605	D₄	—	2	4	no	—
606	D₄	—	2	4	no	—
607	D₄	—	2	2	no	—
608	D₄	—	2	4	no	—
609	D₄	—	4	16	no	—
610	D₄	—	4	4	no	—
611	D₄	—	4	2	no	—
612	D₄	—	8	1	no	—
613	D₄	—	8	4	no	—
614	D₄	—	8	4	no	—
615	D₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
616	C₂×C₄	—	2	8	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	4	no	—
623	C₂×C₄	—	4	2	no	—
624	C₂×C₄	—	4	4	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	16	no	—
627	C₂×C₄	—	4	4	no	—
628	C₂×C₄	—	4	2	no	—
629	C₂×C₄	—	4	4	no	—
630	C₂×C₄	—	4	4	no	—
631	C₂×C₄	—	4	4	no	—
632	C₂×C₄	—	8	4	no	—
633	C₂×C₄	#135×C₆₇	1	4	yes	complemented
634	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
635	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
636	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
637	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
638	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
639	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
640	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
641	C₈	—	2	4	no	—
642	C₈	—	2	4	no	—
643	C₂²	—	2	1	no	—
644	C₂²	—	2	1	no	—
645	C₂²	—	2	4	no	—
646	C₂²	—	2	4	no	—
647	C₂²	—	2	4	no	—
648	C₂²	—	2	4	no	—
649	C₂²	—	4	8	no	—
650	C₂²	—	4	2	no	—
651	C₂²	—	4	2	no	—
652	C₂²	—	4	2	no	—
653	C₂²	—	4	2	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	8	4	no	—
656	C₂²	C₂×#135×C₆₇	1	2	yes	—
657	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
658	C₄	—	2	1	no	—
659	C₄	—	2	1	no	—
660	C₄	—	2	4	no	—
661	C₄	—	2	1	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	2	no	—
664	C₄	—	4	2	no	—
665	C₄	—	4	4	no	—
666	C₄	—	4	2	no	—
667	C₄	C₂×#135×C₆₇	1	2	yes	—
668	C₄	C₂×#135×C₆₇	1	2	yes	—
669	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
670	C₂	—	2	2	no	—
671	C₂	—	4	4	no	—
672	C₂	—	4	4	no	—
673	C₂	—	8	2	no	—
674	C₂	C₆₇×#3484	1	1	yes	characteristic
675	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
676	C₂	C₆₇×#3586	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×#547×C₆₇	C₂	semidirect
3. C₆₇×#3303	C₂	semidirect
4. C₆₇×#3302	C₂	semidirect
5. C₆₇×#3284	C₂	semidirect
6. C₆₇×#3283	C₂	semidirect
7. C₆₇×#3209	C₂	semidirect
8. C₆₇×#3206	C₂	semidirect
9. C₆₇×#3190	C₂	semidirect
10. C₆₇×#3101	C₂	semidirect
11. C₆₇×#3100	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
20. #550×C₆₇	C₂²	semidirect
25. #547×C₆₇	C₂²	semidirect
39. #494×C₆₇	C₂²	semidirect
40. #492×C₆₇	C₂²	semidirect
41. #487×C₆₇	C₂²	semidirect
42. #487×C₆₇	C₂²	semidirect
43. #485×C₆₇	C₂²	semidirect
46. #476×C₆₇	C₂²	semidirect
47. #474×C₆₇	C₂²	semidirect
48. #467×C₆₇	C₂²	semidirect
49. #465×C₆₇	C₂²	semidirect
50. #462×C₆₇	C₂²	semidirect
51. #461×C₆₇	C₂²	semidirect
52. #461×C₆₇	C₂²	semidirect
53. #459×C₆₇	C₂²	semidirect
54. #452×C₆₇	C₂²	semidirect
55. #452×C₆₇	C₂²	semidirect
58. #449×C₆₇	C₂²	semidirect
61. C₄×SD₁₆×C₆₇	C₂²	semidirect
62. C₄×D₈×C₆₇	C₂²	semidirect
63. C₈×D₄×C₆₇	C₂²	semidirect
64. C₈×D₄×C₆₇	C₂²	semidirect
67. #416×C₆₇	C₂²	semidirect
294. C₂×C₄×C₆₇	#135	semidirect
331. #61565	C₆₇	direct (N × H)
339. #3506	C₂×C₆₇	semidirect
340. C₂×#547	C₂×C₆₇	semidirect
341. #3303	C₂×C₆₇	semidirect
342. #3302	C₂×C₆₇	semidirect
343. #3284	C₂×C₆₇	semidirect
344. #3283	C₂×C₆₇	semidirect
345. #3209	C₂×C₆₇	semidirect
346. #3206	C₂×C₆₇	semidirect
347. #3190	C₂×C₆₇	semidirect
348. #3101	C₂×C₆₇	semidirect
349. #3100	C₂×C₆₇	semidirect
350. #3002	C₂×C₆₇	semidirect
351. #2972	C₂×C₆₇	semidirect
352. C₆₇	#61565	direct (N × H)
354. C₂×#135	C₂²×C₆₇	semidirect
356. #556	C₂²×C₆₇	semidirect
357. #555	C₂²×C₆₇	semidirect
359. #550	C₂²×C₆₇	semidirect
364. #547	C₂²×C₆₇	semidirect
378. #494	C₂²×C₆₇	semidirect
379. #492	C₂²×C₆₇	semidirect
380. #487	C₂²×C₆₇	semidirect
381. #487	C₂²×C₆₇	semidirect
382. #485	C₂²×C₆₇	semidirect
385. #476	C₂²×C₆₇	semidirect
386. #474	C₂²×C₆₇	semidirect
387. #467	C₂²×C₆₇	semidirect
388. #465	C₂²×C₆₇	semidirect
389. #462	C₂²×C₆₇	semidirect
390. #461	C₂²×C₆₇	semidirect
391. #461	C₂²×C₆₇	semidirect
392. #459	C₂²×C₆₇	semidirect
393. #452	C₂²×C₆₇	semidirect
394. #452	C₂²×C₆₇	semidirect
397. #449	C₂²×C₆₇	semidirect
400. C₄×SD₁₆	C₂²×C₆₇	semidirect
401. C₄×D₈	C₂²×C₆₇	semidirect
402. C₈×D₄	C₂²×C₆₇	semidirect
403. C₈×D₄	C₂²×C₆₇	semidirect
406. #416	C₂²×C₆₇	semidirect
633. 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×#547×C₆₇	2
C₆₇×#3303	2
C₆₇×#3302	2
C₆₇×#3284	2
C₆₇×#3283	2
C₆₇×#3209	2
C₆₇×#3206	2
C₆₇×#3190	2
C₆₇×#3101	2
C₆₇×#3100	2
C₆₇×#3002	2
C₆₇×#2972	2
#61565	67

Subgroups — C67×#61565 #30740816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61565 #30740816