Subgroups — C₆₇×#61584 #30740835

Order 17152.

2086 conjugacy classes of subgroups.

The subgroup lattice has 660 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
2	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3329	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3328	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3278	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3276	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×#3196	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×#3004	C₂	1	4	yes	maximal, complemented
13	C₆₇×#2975	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#137×C₆₇	C₂²	1	8	yes	complemented
16	#559×C₆₇	—	2	2	no	—
17	#556×C₆₇	C₂²	1	2	yes	—
18	#555×C₆₇	C₂²	1	2	yes	—
19	#553×C₆₇	—	2	2	no	—
20	#550×C₆₇	C₂²	1	2	yes	—
21	D₄×Q₈×C₆₇	—	2	2	no	—
22	D₄×Q₈×C₆₇	—	2	2	no	—
23	D₄×Q₈×C₆₇	C₂²	1	4	yes	—
24	D₄×Q₈×C₆₇	C₂²	1	4	yes	—
25	#547×C₆₇	C₂²	1	2	yes	—
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	—	2	1	no	—
31	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	#525×C₆₇	C₂²	1	4	yes	—
33	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
34	C₂×C₂²⋊Q₈×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	#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	#484×C₆₇	C₂²	1	2	yes	—
45	#481×C₆₇	C₂²	1	2	yes	—
46	#477×C₆₇	C₂²	1	2	yes	complemented
47	#475×C₆₇	C₂²	1	2	yes	complemented
48	#470×C₆₇	C₂²	1	8	yes	complemented
49	#468×C₆₇	C₂²	1	8	yes	complemented
50	#462×C₆₇	C₂²	1	2	yes	complemented
51	#462×C₆₇	C₂²	1	4	yes	complemented
52	#461×C₆₇	C₂²	1	4	yes	complemented
53	#460×C₆₇	C₂²	1	2	yes	complemented
54	#452×C₆₇	C₂²	1	8	yes	complemented
55	#451×C₆₇	C₂²	1	2	yes	—
56	#451×C₆₇	C₂²	1	4	yes	—
57	#451×C₆₇	C₂²	1	8	yes	complemented
58	#448×C₆₇	C₂²	1	2	yes	—
59	#448×C₆₇	C₂²	1	4	yes	—
60	#448×C₆₇	C₂²	1	8	yes	complemented
61	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
62	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
63	#436×C₆₇	C₂²	1	4	yes	—
64	#436×C₆₇	C₂²	1	4	yes	—
65	#431×C₆₇	C₂²	1	1	yes	characteristic
66	#431×C₆₇	C₂²	1	1	yes	characteristic
67	#419×C₆₇	C₂²	1	8	yes	complemented
68	#407×C₆₇	C₂²	1	4	yes	—
69	C₂×#41×C₆₇	—	2	1	no	—
70	C₂×#41×C₆₇	C₂³	1	4	yes	—
71	C₂²×Q₈×C₆₇	—	2	2	no	—
72	C₂²×Q₈×C₆₇	—	2	4	no	—
73	C₂²×Q₈×C₆₇	—	4	8	no	—
74	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
76	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
77	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	—	2	2	no	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	#137×C₆₇	—	4	16	no	—
81	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
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×M₁₆×C₆₇	C₂³	1	8	yes	—
86	C₄⋊Q₈×C₆₇	—	2	4	no	—
87	C₄⋊Q₈×C₆₇	—	2	4	no	—
88	C₄⋊Q₈×C₆₇	—	2	1	no	—
89	C₄⋊Q₈×C₆₇	—	2	1	no	—
90	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
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	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
95	#127×C₆₇	—	2	2	no	—
96	#127×C₆₇	C₂³	1	1	yes	characteristic
97	#126×C₆₇	—	2	4	no	—
98	#125×C₆₇	—	2	1	no	—
99	#124×C₆₇	—	2	2	no	—
100	#124×C₆₇	—	2	4	no	—
101	#124×C₆₇	—	2	1	no	—
102	#123×C₆₇	—	2	4	no	—
103	#123×C₆₇	—	2	4	no	—
104	C₂²⋊Q₈×C₆₇	—	2	4	no	—
105	C₂²⋊Q₈×C₆₇	—	2	1	no	—
106	C₂²⋊Q₈×C₆₇	—	2	8	no	—
107	C₂²⋊Q₈×C₆₇	—	2	8	no	—
108	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
109	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
110	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
111	#121×C₆₇	—	2	2	no	—
112	#121×C₆₇	—	2	2	no	—
113	#121×C₆₇	—	2	1	no	—
114	#121×C₆₇	—	2	4	no	—
115	#121×C₆₇	C₂³	1	2	yes	—
116	#121×C₆₇	C₂³	1	4	yes	—
117	#120×C₆₇	—	2	4	no	—
118	C₄×Q₈×C₆₇	—	2	2	no	—
119	C₄×Q₈×C₆₇	—	2	1	no	—
120	C₄×Q₈×C₆₇	—	2	1	no	—
121	C₄×Q₈×C₆₇	—	2	1	no	—
122	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
123	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
124	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
125	C₄×D₄×C₆₇	—	2	4	no	—
126	C₄×D₄×C₆₇	—	2	4	no	—
127	C₄×D₄×C₆₇	—	2	4	no	—
128	C₄×D₄×C₆₇	—	2	4	no	—
129	C₄×D₄×C₆₇	—	2	2	no	—
130	C₄×D₄×C₆₇	—	2	1	no	—
131	C₄×D₄×C₆₇	D₄	1	2	yes	—
132	C₄×D₄×C₆₇	D₄	1	2	yes	—
133	C₄×D₄×C₆₇	C₂³	1	2	yes	—
134	C₄×D₄×C₆₇	C₂³	1	2	yes	—
135	C₄×D₄×C₆₇	C₂³	1	2	yes	—
136	#117×C₆₇	—	2	1	no	—
137	#117×C₆₇	—	2	2	no	—
138	#117×C₆₇	C₂³	1	4	yes	—
139	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
140	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
142	C₂×C₄⋊C₄×C₆₇	—	2	2	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	8	yes	—
156	#103×C₆₇	C₂³	1	4	yes	—
157	#103×C₆₇	C₂³	1	8	yes	—
158	#103×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	2	yes	—
162	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
163	C₄×C₈×C₆₇	C₂³	1	2	yes	—
164	C₄×C₈×C₆₇	C₂³	1	2	yes	—
165	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
166	#41×C₆₇	—	4	8	no	—
167	#41×C₆₇	—	4	4	no	—
168	C₂×Q₈×C₆₇	—	2	4	no	—
169	C₂×Q₈×C₆₇	—	2	1	no	—
170	C₂×Q₈×C₆₇	—	2	1	no	—
171	C₂×Q₈×C₆₇	—	2	8	no	—
172	C₂×Q₈×C₆₇	—	2	2	no	—
173	C₂×Q₈×C₆₇	—	2	1	no	—
174	C₂×Q₈×C₆₇	—	2	16	no	—
175	C₂×Q₈×C₆₇	—	4	8	no	—
176	C₂×Q₈×C₆₇	—	4	4	no	—
177	C₂×Q₈×C₆₇	—	4	8	no	—
178	C₂×Q₈×C₆₇	—	4	16	no	—
179	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
180	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
181	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
182	C₂×D₄×C₆₇	—	2	2	no	—
183	C₂×D₄×C₆₇	—	2	4	no	—
184	C₂×D₄×C₆₇	—	2	4	no	—
185	C₂×D₄×C₆₇	—	2	4	no	—
186	C₂×D₄×C₆₇	—	2	1	no	—
187	C₂×D₄×C₆₇	—	2	1	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	C₂²×C₄×C₆₇	—	2	1	no	—
192	C₂²×C₄×C₆₇	—	2	1	no	—
193	C₂²×C₄×C₆₇	—	2	2	no	—
194	C₂²×C₄×C₆₇	—	2	8	no	—
195	C₂²×C₄×C₆₇	—	2	2	no	—
196	C₂²×C₄×C₆₇	—	2	4	no	—
197	C₂²×C₄×C₆₇	—	4	4	no	—
198	C₂²×C₄×C₆₇	—	4	8	no	—
199	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
201	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
202	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
204	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
205	Q₁₆×C₆₇	—	2	8	no	—
206	Q₁₆×C₆₇	—	4	8	no	—
207	SD₁₆×C₆₇	—	2	8	no	—
208	SD₁₆×C₆₇	—	4	8	no	—
209	M₁₆×C₆₇	—	4	8	no	—
210	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
211	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
212	C₄⋊C₄×C₆₇	—	2	1	no	—
213	C₄⋊C₄×C₆₇	—	2	2	no	—
214	C₄⋊C₄×C₆₇	—	2	2	no	—
215	C₄⋊C₄×C₆₇	—	2	4	no	—
216	C₄⋊C₄×C₆₇	—	2	2	no	—
217	C₄⋊C₄×C₆₇	—	2	2	no	—
218	C₄⋊C₄×C₆₇	—	2	2	no	—
219	C₄⋊C₄×C₆₇	—	2	2	no	—
220	C₄⋊C₄×C₆₇	—	2	8	no	—
221	C₄⋊C₄×C₆₇	—	2	1	no	—
222	C₄⋊C₄×C₆₇	—	2	1	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	C₂×D₄	1	2	yes	—
228	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
229	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
230	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
231	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
232	C₂²⋊C₄×C₆₇	—	2	8	no	—
233	C₂²⋊C₄×C₆₇	—	2	2	no	—
234	C₂²⋊C₄×C₆₇	—	2	4	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	C₂×D₄	1	4	yes	—
240	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
241	C₄²×C₆₇	—	2	2	no	—
242	C₄²×C₆₇	—	2	2	no	—
243	C₄²×C₆₇	—	2	4	no	—
244	C₄²×C₆₇	—	2	2	no	—
245	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
246	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
247	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
248	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
249	C₂³×C₆₇	—	2	4	no	—
250	C₂³×C₆₇	—	2	2	no	—
251	C₂³×C₆₇	—	4	2	no	—
252	C₂³×C₆₇	—	4	2	no	—
253	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
254	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
255	Q₈×C₆₇	—	2	4	no	—
256	Q₈×C₆₇	—	2	2	no	—
257	Q₈×C₆₇	—	2	8	no	—
258	Q₈×C₆₇	—	2	4	no	—
259	Q₈×C₆₇	—	4	16	no	—
260	Q₈×C₆₇	—	4	4	no	—
261	Q₈×C₆₇	—	4	2	no	—
262	Q₈×C₆₇	—	8	1	no	—
263	Q₈×C₆₇	H(5,𝔽₂)	1	4	yes	—
264	D₄×C₆₇	—	2	4	no	—
265	D₄×C₆₇	—	2	4	no	—
266	D₄×C₆₇	—	2	4	no	—
267	D₄×C₆₇	—	4	4	no	—
268	D₄×C₆₇	—	4	2	no	—
269	D₄×C₆₇	—	8	1	no	—
270	D₄×C₆₇	—	8	4	no	—
271	D₄×C₆₇	—	8	4	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	8	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	4	no	—
281	C₂×C₄×C₆₇	—	4	4	no	—
282	C₂×C₄×C₆₇	—	4	2	no	—
283	C₂×C₄×C₆₇	—	4	1	no	—
284	C₂×C₄×C₆₇	—	4	16	no	—
285	C₂×C₄×C₆₇	—	4	2	no	—
286	C₂×C₄×C₆₇	—	4	4	no	—
287	C₂×C₄×C₆₇	—	4	4	no	—
288	C₂×C₄×C₆₇	—	8	4	no	—
289	C₂×C₄×C₆₇	#137	1	4	yes	complemented
290	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
291	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
297	C₈×C₆₇	—	2	4	no	—
298	C₈×C₆₇	—	2	4	no	—
299	C₂²×C₆₇	—	2	1	no	—
300	C₂²×C₆₇	—	2	1	no	—
301	C₂²×C₆₇	—	2	4	no	—
302	C₂²×C₆₇	—	2	4	no	—
303	C₂²×C₆₇	—	2	4	no	—
304	C₂²×C₆₇	—	4	8	no	—
305	C₂²×C₆₇	—	4	2	no	—
306	C₂²×C₆₇	—	4	2	no	—
307	C₂²×C₆₇	—	4	2	no	—
308	C₂²×C₆₇	C₂×#137	1	2	yes	—
309	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
310	C₄×C₆₇	—	2	1	no	—
311	C₄×C₆₇	—	2	1	no	—
312	C₄×C₆₇	—	2	4	no	—
313	C₄×C₆₇	—	2	1	no	—
314	C₄×C₆₇	—	2	4	no	—
315	C₄×C₆₇	—	4	2	no	—
316	C₄×C₆₇	—	4	2	no	—
317	C₄×C₆₇	—	4	4	no	—
318	C₄×C₆₇	—	4	2	no	—
319	C₄×C₆₇	—	4	2	no	—
320	C₄×C₆₇	C₂×#137	1	2	yes	—
321	C₄×C₆₇	C₂×#137	1	2	yes	—
322	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
323	#61584	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
324	C₂×C₆₇	—	2	2	no	—
325	C₂×C₆₇	—	4	4	no	—
326	C₂×C₆₇	—	8	2	no	—
327	C₂×C₆₇	#3484	1	1	yes	characteristic
328	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
329	C₂×C₆₇	#3586	1	1	yes	characteristic
330	#3506	C₂×C₆₇	1	1	yes	characteristic
331	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
332	#3329	C₂×C₆₇	1	4	yes	complemented
333	#3328	C₂×C₆₇	1	4	yes	complemented
334	#3278	C₂×C₆₇	1	4	yes	complemented
335	#3276	C₂×C₆₇	1	4	yes	complemented
336	#3209	C₂×C₆₇	1	1	yes	characteristic, complemented
337	#3206	C₂×C₆₇	1	1	yes	characteristic, complemented
338	#3196	C₂×C₆₇	1	4	yes	complemented
339	#3104	C₂×C₆₇	1	1	yes	characteristic, complemented
340	#3103	C₂×C₆₇	1	1	yes	characteristic, complemented
341	#3004	C₂×C₆₇	1	4	yes	complemented
342	#2975	C₂×C₆₇	1	1	yes	characteristic, complemented
343	C₆₇	#61584	1	1	yes	characteristic, direct product factor, Sylow
344	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
345	C₂×#137	C₂²×C₆₇	1	8	yes	complemented
346	#559	—	2	2	no	—
347	#556	C₂²×C₆₇	1	2	yes	—
348	#555	C₂²×C₆₇	1	2	yes	—
349	#553	—	2	2	no	—
350	#550	C₂²×C₆₇	1	2	yes	—
351	D₄×Q₈	—	2	2	no	—
352	D₄×Q₈	—	2	2	no	—
353	D₄×Q₈	C₂²×C₆₇	1	4	yes	—
354	D₄×Q₈	C₂²×C₆₇	1	4	yes	—
355	#547	C₂²×C₆₇	1	2	yes	—
356	#538	—	2	2	no	—
357	#535	—	2	1	no	—
358	#533	C₂²×C₆₇	1	1	yes	characteristic
359	#532	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×C₄⋊Q₈	—	2	1	no	—
361	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
362	#525	C₂²×C₆₇	1	4	yes	—
363	C₂×C₂²⋊Q₈	—	2	2	no	—
364	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
365	#520	—	2	1	no	—
366	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
367	C₂×C₄×D₄	—	2	1	no	—
368	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
369	#494	C₂²×C₆₇	1	2	yes	complemented
370	#492	C₂²×C₆₇	1	2	yes	complemented
371	#487	C₂²×C₆₇	1	2	yes	complemented
372	#487	C₂²×C₆₇	1	2	yes	complemented
373	#485	C₂²×C₆₇	1	8	yes	complemented
374	#484	C₂²×C₆₇	1	2	yes	—
375	#481	C₂²×C₆₇	1	2	yes	—
376	#477	C₂²×C₆₇	1	2	yes	complemented
377	#475	C₂²×C₆₇	1	2	yes	complemented
378	#470	C₂²×C₆₇	1	8	yes	complemented
379	#468	C₂²×C₆₇	1	8	yes	complemented
380	#462	C₂²×C₆₇	1	4	yes	complemented
381	#462	C₂²×C₆₇	1	2	yes	complemented
382	#461	C₂²×C₆₇	1	4	yes	complemented
383	#460	C₂²×C₆₇	1	2	yes	complemented
384	#452	C₂²×C₆₇	1	8	yes	complemented
385	#451	C₂²×C₆₇	1	2	yes	—
386	#451	C₂²×C₆₇	1	4	yes	—
387	#451	C₂²×C₆₇	1	8	yes	complemented
388	#448	C₂²×C₆₇	1	2	yes	—
389	#448	C₂²×C₆₇	1	4	yes	—
390	#448	C₂²×C₆₇	1	8	yes	complemented
391	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
392	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
393	#436	C₂²×C₆₇	1	4	yes	—
394	#436	C₂²×C₆₇	1	4	yes	—
395	#431	C₂²×C₆₇	1	1	yes	characteristic
396	#431	C₂²×C₆₇	1	1	yes	characteristic
397	#419	C₂²×C₆₇	1	8	yes	complemented
398	#407	C₂²×C₆₇	1	4	yes	—
399	C₂×#41	—	2	1	no	—
400	C₂×#41	C₂³×C₆₇	1	4	yes	—
401	C₂²×Q₈	—	2	2	no	—
402	C₂²×Q₈	—	2	4	no	—
403	C₂²×Q₈	—	4	8	no	—
404	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
405	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
406	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
407	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
408	C₂³×C₄	—	2	2	no	—
409	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
410	#137	—	4	16	no	—
411	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
412	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
413	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
414	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
415	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
416	C₄⋊Q₈	—	2	4	no	—
417	C₄⋊Q₈	—	2	4	no	—
418	C₄⋊Q₈	—	2	1	no	—
419	C₄⋊Q₈	—	2	1	no	—
420	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
421	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
422	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
423	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
424	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
425	#127	—	2	2	no	—
426	#127	C₂³×C₆₇	1	1	yes	characteristic
427	#126	—	2	4	no	—
428	#125	—	2	1	no	—
429	#124	—	2	2	no	—
430	#124	—	2	4	no	—
431	#124	—	2	1	no	—
432	#123	—	2	4	no	—
433	#123	—	2	4	no	—
434	C₂²⋊Q₈	—	2	4	no	—
435	C₂²⋊Q₈	—	2	1	no	—
436	C₂²⋊Q₈	—	2	8	no	—
437	C₂²⋊Q₈	—	2	8	no	—
438	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
439	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
440	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
441	#121	—	2	2	no	—
442	#121	—	2	2	no	—
443	#121	—	2	1	no	—
444	#121	—	2	4	no	—
445	#121	C₂³×C₆₇	1	2	yes	—
446	#121	C₂³×C₆₇	1	4	yes	—
447	#120	—	2	4	no	—
448	C₄×Q₈	—	2	2	no	—
449	C₄×Q₈	—	2	1	no	—
450	C₄×Q₈	—	2	1	no	—
451	C₄×Q₈	—	2	1	no	—
452	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
453	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
454	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
455	C₄×D₄	—	2	4	no	—
456	C₄×D₄	—	2	4	no	—
457	C₄×D₄	—	2	4	no	—
458	C₄×D₄	—	2	4	no	—
459	C₄×D₄	—	2	2	no	—
460	C₄×D₄	—	2	1	no	—
461	C₄×D₄	D₄×C₆₇	1	2	yes	—
462	C₄×D₄	D₄×C₆₇	1	2	yes	—
463	C₄×D₄	C₂³×C₆₇	1	2	yes	—
464	C₄×D₄	C₂³×C₆₇	1	2	yes	—
465	C₄×D₄	C₂³×C₆₇	1	2	yes	—
466	#117	—	2	1	no	—
467	#117	—	2	2	no	—
468	#117	C₂³×C₆₇	1	4	yes	—
469	C₂×C₄⋊C₄	—	2	1	no	—
470	C₂×C₄⋊C₄	—	2	1	no	—
471	C₂×C₄⋊C₄	—	2	1	no	—
472	C₂×C₄⋊C₄	—	2	2	no	—
473	C₂×C₄⋊C₄	—	2	2	no	—
474	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
475	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
476	C₂×C₂²⋊C₄	—	2	2	no	—
477	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
478	C₂×C₄²	—	2	1	no	—
479	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
480	#107	C₂³×C₆₇	1	4	yes	—
481	#106	C₂³×C₆₇	1	4	yes	—
482	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
483	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
484	#103	C₂³×C₆₇	1	4	yes	—
485	#103	C₂³×C₆₇	1	4	yes	—
486	#103	C₂³×C₆₇	1	8	yes	—
487	#103	C₂³×C₆₇	1	8	yes	—
488	#103	C₂³×C₆₇	1	4	yes	—
489	#102	C₂³×C₆₇	1	4	yes	—
490	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
491	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
492	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
493	C₄×C₈	C₂³×C₆₇	1	2	yes	—
494	C₄×C₈	C₂³×C₆₇	1	2	yes	—
495	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
496	#41	—	4	8	no	—
497	#41	—	4	4	no	—
498	C₂×Q₈	—	2	4	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	—	2	8	no	—
502	C₂×Q₈	—	2	2	no	—
503	C₂×Q₈	—	2	1	no	—
504	C₂×Q₈	—	2	16	no	—
505	C₂×Q₈	—	4	8	no	—
506	C₂×Q₈	—	4	4	no	—
507	C₂×Q₈	—	4	8	no	—
508	C₂×Q₈	—	4	16	no	—
509	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
510	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
511	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	4	no	—
514	C₂×D₄	—	2	4	no	—
515	C₂×D₄	—	2	4	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	1	no	—
518	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
519	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
520	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	8	no	—
525	C₂²×C₄	—	2	2	no	—
526	C₂²×C₄	—	2	4	no	—
527	C₂²×C₄	—	4	4	no	—
528	C₂²×C₄	—	4	8	no	—
529	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
530	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
531	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
532	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
533	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
534	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
535	Q₁₆	—	2	8	no	—
536	Q₁₆	—	4	8	no	—
537	SD₁₆	—	2	8	no	—
538	SD₁₆	—	4	8	no	—
539	M₁₆	—	4	8	no	—
540	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
541	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	2	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	2	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	8	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	—	2	2	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
558	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
559	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
560	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
561	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
562	C₂²⋊C₄	—	2	8	no	—
563	C₂²⋊C₄	—	2	2	no	—
564	C₂²⋊C₄	—	2	4	no	—
565	C₂²⋊C₄	—	2	4	no	—
566	C₂²⋊C₄	—	2	4	no	—
567	C₂²⋊C₄	—	2	2	no	—
568	C₂²⋊C₄	—	2	2	no	—
569	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
570	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
571	C₄²	—	2	2	no	—
572	C₄²	—	2	2	no	—
573	C₄²	—	2	4	no	—
574	C₄²	—	2	2	no	—
575	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
576	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
577	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
578	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
579	C₂³	—	2	4	no	—
580	C₂³	—	2	2	no	—
581	C₂³	—	4	2	no	—
582	C₂³	—	4	2	no	—
583	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
584	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
585	Q₈	—	2	4	no	—
586	Q₈	—	2	2	no	—
587	Q₈	—	2	8	no	—
588	Q₈	—	2	4	no	—
589	Q₈	—	4	16	no	—
590	Q₈	—	4	4	no	—
591	Q₈	—	4	2	no	—
592	Q₈	—	8	1	no	—
593	Q₈	H(5,𝔽₂)×C₆₇	1	4	yes	—
594	D₄	—	2	4	no	—
595	D₄	—	2	4	no	—
596	D₄	—	2	4	no	—
597	D₄	—	4	4	no	—
598	D₄	—	4	2	no	—
599	D₄	—	8	1	no	—
600	D₄	—	8	4	no	—
601	D₄	—	8	4	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	8	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	4	4	no	—
612	C₂×C₄	—	4	2	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	16	no	—
615	C₂×C₄	—	4	4	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	4	no	—
618	C₂×C₄	—	8	4	no	—
619	C₂×C₄	#137×C₆₇	1	4	yes	complemented
620	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
621	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
622	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
623	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
624	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
625	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
626	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
627	C₈	—	2	4	no	—
628	C₈	—	2	4	no	—
629	C₂²	—	2	1	no	—
630	C₂²	—	2	1	no	—
631	C₂²	—	2	4	no	—
632	C₂²	—	2	4	no	—
633	C₂²	—	2	4	no	—
634	C₂²	—	4	8	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	C₂×#137×C₆₇	1	2	yes	—
639	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
640	C₄	—	2	1	no	—
641	C₄	—	2	1	no	—
642	C₄	—	2	4	no	—
643	C₄	—	2	1	no	—
644	C₄	—	2	4	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	4	no	—
649	C₄	—	4	2	no	—
650	C₄	C₂×#137×C₆₇	1	2	yes	—
651	C₄	C₂×#137×C₆₇	1	2	yes	—
652	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
653	C₂	—	2	2	no	—
654	C₂	—	4	4	no	—
655	C₂	—	8	2	no	—
656	C₂	C₆₇×#3484	1	1	yes	characteristic
657	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
658	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3506	C₂
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
2. C₂×D₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#3329	C₂	semidirect
4. C₆₇×#3328	C₂	semidirect
5. C₆₇×#3278	C₂	semidirect
6. C₆₇×#3276	C₂	semidirect
7. C₆₇×#3209	C₂	semidirect
8. C₆₇×#3206	C₂	semidirect
9. C₆₇×#3196	C₂	semidirect
10. C₆₇×#3104	C₂	semidirect
11. C₆₇×#3103	C₂	semidirect
12. C₆₇×#3004	C₂	semidirect
13. C₆₇×#2975	C₂	semidirect
15. C₂×#137×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. #477×C₆₇	C₂²	semidirect
47. #475×C₆₇	C₂²	semidirect
48. #470×C₆₇	C₂²	semidirect
49. #468×C₆₇	C₂²	semidirect
50. #462×C₆₇	C₂²	semidirect
51. #462×C₆₇	C₂²	semidirect
52. #461×C₆₇	C₂²	semidirect
53. #460×C₆₇	C₂²	semidirect
54. #452×C₆₇	C₂²	semidirect
57. #451×C₆₇	C₂²	semidirect
60. #448×C₆₇	C₂²	semidirect
61. C₄×Q₁₆×C₆₇	C₂²	semidirect
62. C₄×SD₁₆×C₆₇	C₂²	semidirect
67. #419×C₆₇	C₂²	semidirect
289. C₂×C₄×C₆₇	#137	semidirect
323. #61584	C₆₇	direct (N × H)
331. C₂×D₄×Q₈	C₂×C₆₇	semidirect
332. #3329	C₂×C₆₇	semidirect
333. #3328	C₂×C₆₇	semidirect
334. #3278	C₂×C₆₇	semidirect
335. #3276	C₂×C₆₇	semidirect
336. #3209	C₂×C₆₇	semidirect
337. #3206	C₂×C₆₇	semidirect
338. #3196	C₂×C₆₇	semidirect
339. #3104	C₂×C₆₇	semidirect
340. #3103	C₂×C₆₇	semidirect
341. #3004	C₂×C₆₇	semidirect
342. #2975	C₂×C₆₇	semidirect
343. C₆₇	#61584	direct (N × H)
345. C₂×#137	C₂²×C₆₇	semidirect
369. #494	C₂²×C₆₇	semidirect
370. #492	C₂²×C₆₇	semidirect
371. #487	C₂²×C₆₇	semidirect
372. #487	C₂²×C₆₇	semidirect
373. #485	C₂²×C₆₇	semidirect
376. #477	C₂²×C₆₇	semidirect
377. #475	C₂²×C₆₇	semidirect
378. #470	C₂²×C₆₇	semidirect
379. #468	C₂²×C₆₇	semidirect
380. #462	C₂²×C₆₇	semidirect
381. #462	C₂²×C₆₇	semidirect
382. #461	C₂²×C₆₇	semidirect
383. #460	C₂²×C₆₇	semidirect
384. #452	C₂²×C₆₇	semidirect
387. #451	C₂²×C₆₇	semidirect
390. #448	C₂²×C₆₇	semidirect
391. C₄×Q₁₆	C₂²×C₆₇	semidirect
392. C₄×SD₁₆	C₂²×C₆₇	semidirect
397. #419	C₂²×C₆₇	semidirect
619. C₂×C₄	#137×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×D₄×Q₈×C₆₇	2
C₆₇×#3329	2
C₆₇×#3328	2
C₆₇×#3278	2
C₆₇×#3276	2
C₆₇×#3209	2
C₆₇×#3206	2
C₆₇×#3196	2
C₆₇×#3104	2
C₆₇×#3103	2
C₆₇×#3004	2
C₆₇×#2975	2
#61584	67

Subgroups — C67×#61584 #30740835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61584 #30740835