Subgroups — C₆₇×#60907 #30740158

Order 17152.

1910 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3444	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3256	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3255	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3254	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3253	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#3252	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3251	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3076	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3054	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3035	C₂	1	4	yes	maximal, complemented
11	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	#533×C₆₇	C₂²	1	2	yes	—
15	#533×C₆₇	C₂²	1	2	yes	—
16	#533×C₆₇	C₂²	1	2	yes	—
17	#532×C₆₇	C₂²	1	2	yes	—
18	#532×C₆₇	C₂²	1	2	yes	—
19	#532×C₆₇	C₂²	1	2	yes	—
20	#529×C₆₇	—	2	2	no	—
21	#529×C₆₇	—	2	2	no	—
22	#529×C₆₇	—	2	2	no	—
23	#529×C₆₇	—	2	2	no	—
24	#529×C₆₇	C₂²	1	4	yes	—
25	#529×C₆₇	C₂²	1	4	yes	—
26	#525×C₆₇	—	2	2	no	—
27	#525×C₆₇	—	2	2	no	—
28	#525×C₆₇	C₂²	1	4	yes	—
29	C₄×#41×C₆₇	—	2	2	no	—
30	C₄×#41×C₆₇	—	2	2	no	—
31	C₄×#41×C₆₇	C₂²	1	4	yes	—
32	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	#479×C₆₇	C₂²	1	8	yes	complemented
34	#478×C₆₇	C₂²	1	8	yes	complemented
35	#477×C₆₇	C₂²	1	4	yes	complemented
36	#476×C₆₇	C₂²	1	4	yes	complemented
37	#475×C₆₇	C₂²	1	4	yes	complemented
38	#474×C₆₇	C₂²	1	4	yes	complemented
39	#470×C₆₇	C₂²	1	4	yes	—
40	#470×C₆₇	C₂²	1	4	yes	—
41	#469×C₆₇	C₂²	1	4	yes	—
42	#469×C₆₇	C₂²	1	4	yes	—
43	#468×C₆₇	C₂²	1	8	yes	—
44	#468×C₆₇	C₂²	1	8	yes	—
45	#464×C₆₇	C₂²	1	8	yes	complemented
46	#463×C₆₇	C₂²	1	8	yes	complemented
47	#462×C₆₇	C₂²	1	4	yes	complemented
48	#461×C₆₇	C₂²	1	4	yes	complemented
49	#460×C₆₇	C₂²	1	4	yes	complemented
50	#459×C₆₇	C₂²	1	4	yes	complemented
51	#427×C₆₇	C₂²	1	4	yes	—
52	#423×C₆₇	C₂²	1	4	yes	—
53	#423×C₆₇	C₂²	1	1	yes	characteristic
54	#423×C₆₇	C₂²	1	1	yes	characteristic
55	#423×C₆₇	C₂²	1	2	yes	—
56	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
57	#416×C₆₇	C₂²	1	8	yes	—
58	C₂×#41×C₆₇	—	2	2	no	—
59	C₂×#41×C₆₇	—	2	2	no	—
60	C₂×#41×C₆₇	C₂³	1	4	yes	—
61	C₂³×C₄×C₆₇	D₄	1	2	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	#135×C₆₇	—	4	8	no	—
64	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
65	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
66	C₂×D₈×C₆₇	C₂³	1	4	yes	—
67	C₂×M₁₆×C₆₇	—	4	4	no	—
68	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
70	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
72	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
75	#127×C₆₇	C₂³	1	2	yes	—
76	#126×C₆₇	—	2	4	no	—
77	#126×C₆₇	—	2	4	no	—
78	#125×C₆₇	—	2	2	no	—
79	#125×C₆₇	—	2	2	no	—
80	#125×C₆₇	C₂³	1	4	yes	—
81	#124×C₆₇	—	2	2	no	—
82	#124×C₆₇	—	2	2	no	—
83	#124×C₆₇	C₂³	1	4	yes	—
84	#123×C₆₇	—	2	4	no	—
85	#123×C₆₇	—	2	4	no	—
86	#123×C₆₇	—	2	4	no	—
87	C₂²⋊Q₈×C₆₇	—	2	2	no	—
88	C₂²⋊Q₈×C₆₇	—	2	2	no	—
89	C₂²⋊Q₈×C₆₇	—	2	2	no	—
90	C₂²⋊Q₈×C₆₇	—	2	2	no	—
91	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
92	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
93	#121×C₆₇	—	2	2	no	—
94	#121×C₆₇	—	2	2	no	—
95	#121×C₆₇	—	2	2	no	—
96	#121×C₆₇	—	2	2	no	—
97	#121×C₆₇	C₂³	1	4	yes	—
98	#121×C₆₇	C₂³	1	4	yes	—
99	#120×C₆₇	—	2	4	no	—
100	C₄×Q₈×C₆₇	—	2	2	no	—
101	C₄×Q₈×C₆₇	—	2	2	no	—
102	C₄×Q₈×C₆₇	—	2	2	no	—
103	C₄×Q₈×C₆₇	—	2	2	no	—
104	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
105	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
106	C₄×D₄×C₆₇	—	2	4	no	—
107	C₄×D₄×C₆₇	—	2	4	no	—
108	C₄×D₄×C₆₇	—	2	4	no	—
109	C₄×D₄×C₆₇	—	2	4	no	—
110	C₄×D₄×C₆₇	—	2	4	no	—
111	C₄×D₄×C₆₇	—	2	4	no	—
112	C₄×D₄×C₆₇	—	2	2	no	—
113	C₄×D₄×C₆₇	—	2	2	no	—
114	C₄×D₄×C₆₇	—	2	2	no	—
115	C₄×D₄×C₆₇	—	2	2	no	—
116	C₄×D₄×C₆₇	C₂³	1	4	yes	—
117	C₄×D₄×C₆₇	C₂³	1	4	yes	—
118	#117×C₆₇	—	2	2	no	—
119	#117×C₆₇	—	2	2	no	—
120	#117×C₆₇	—	2	4	no	—
121	#117×C₆₇	—	2	4	no	—
122	#117×C₆₇	—	2	4	no	—
123	#117×C₆₇	C₂³	1	4	yes	—
124	C₂×C₄²×C₆₇	—	2	4	no	—
125	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₆₇	D₄	1	2	yes	—
127	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	#107×C₆₇	C₂³	1	8	yes	—
135	#106×C₆₇	C₂³	1	4	yes	—
136	#106×C₆₇	C₂³	1	4	yes	—
137	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
138	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
139	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
140	#103×C₆₇	C₂³	1	8	yes	—
141	#103×C₆₇	C₂³	1	8	yes	—
142	#102×C₆₇	C₂³	1	8	yes	—
143	#102×C₆₇	C₂³	1	8	yes	—
144	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
145	#41×C₆₇	—	2	8	no	—
146	#41×C₆₇	—	2	8	no	—
147	#41×C₆₇	—	4	8	no	—
148	#41×C₆₇	—	4	8	no	—
149	C₂×Q₈×C₆₇	—	2	2	no	—
150	C₂×Q₈×C₆₇	—	2	2	no	—
151	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	—	2	2	no	—
157	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	4	4	no	—
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
176	Q₁₆×C₆₇	—	4	4	no	—
177	SD₁₆×C₆₇	—	4	8	no	—
178	D₈×C₆₇	—	4	4	no	—
179	M₁₆×C₆₇	—	4	2	no	—
180	M₁₆×C₆₇	—	4	4	no	—
181	M₁₆×C₆₇	—	4	2	no	—
182	C₂×C₈×C₆₇	—	2	8	no	—
183	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
184	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	4	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	—	2	4	no	—
195	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	4	no	—
199	C₂²⋊C₄×C₆₇	—	2	4	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	4	no	—
208	C₄²×C₆₇	—	2	4	no	—
209	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₆₇	C₂×D₄	1	2	yes	—
211	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₆₇	C₂×D₄	1	2	yes	—
213	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₆₇	C₂×D₄	1	2	yes	—
216	C₄²×C₆₇	C₂×D₄	1	2	yes	—
217	C₄²×C₆₇	C₂⁴	1	2	yes	—
218	C₄²×C₆₇	C₂⁴	1	2	yes	—
219	C₂³×C₆₇	—	2	2	no	—
220	C₂³×C₆₇	—	2	4	no	—
221	C₂³×C₆₇	—	4	2	no	—
222	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
223	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₆₇	—	2	4	no	—
226	Q₈×C₆₇	—	2	4	no	—
227	Q₈×C₆₇	—	4	4	no	—
228	Q₈×C₆₇	—	4	4	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	2	4	no	—
231	D₄×C₆₇	—	4	4	no	—
232	D₄×C₆₇	—	4	8	no	—
233	D₄×C₆₇	—	4	8	no	—
234	D₄×C₆₇	—	4	8	no	—
235	D₄×C₆₇	—	4	4	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	1	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	4	2	no	—
251	C₂×C₄×C₆₇	—	4	2	no	—
252	C₂×C₄×C₆₇	—	4	8	no	—
253	C₂×C₄×C₆₇	—	4	8	no	—
254	C₂×C₄×C₆₇	—	4	8	no	—
255	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
263	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
264	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
265	C₈×C₆₇	—	2	4	no	—
266	C₈×C₆₇	—	2	4	no	—
267	C₂²×C₆₇	—	2	2	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	—	2	1	no	—
270	C₂²×C₆₇	—	2	2	no	—
271	C₂²×C₆₇	—	2	2	no	—
272	C₂²×C₆₇	—	4	4	no	—
273	C₂²×C₆₇	—	4	4	no	—
274	C₂²×C₆₇	—	4	4	no	—
275	C₂²×C₆₇	—	4	4	no	—
276	C₂²×C₆₇	#537	1	2	yes	—
277	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
278	C₄×C₆₇	—	2	1	no	—
279	C₄×C₆₇	—	2	1	no	—
280	C₄×C₆₇	—	2	2	no	—
281	C₄×C₆₇	—	2	2	no	—
282	C₄×C₆₇	—	2	2	no	—
283	C₄×C₆₇	—	4	4	no	—
284	C₄×C₆₇	—	4	4	no	—
285	C₄×C₆₇	—	4	4	no	—
286	C₄×C₆₇	#537	1	2	yes	—
287	C₄×C₆₇	#575	1	4	yes	—
288	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
289	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
290	#60907	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂×C₆₇	—	2	2	no	—
292	C₂×C₆₇	—	4	1	no	—
293	C₂×C₆₇	—	4	1	no	—
294	C₂×C₆₇	—	8	4	no	—
295	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
296	C₂×C₆₇	C₂×#575	1	2	yes	—
297	#3444	C₂×C₆₇	1	2	yes	complemented
298	#3256	C₂×C₆₇	1	4	yes	complemented
299	#3255	C₂×C₆₇	1	1	yes	characteristic, complemented
300	#3254	C₂×C₆₇	1	2	yes	complemented
301	#3253	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#3252	C₂×C₆₇	1	4	yes	complemented
303	#3251	C₂×C₆₇	1	4	yes	complemented
304	#3076	C₂×C₆₇	1	4	yes	complemented
305	#3054	C₂×C₆₇	1	4	yes	complemented
306	#3035	C₂×C₆₇	1	4	yes	complemented
307	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
308	C₆₇	#60907	1	1	yes	characteristic, direct product factor, Sylow
309	C₂×#135	C₂²×C₆₇	1	4	yes	—
310	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
311	#533	C₂²×C₆₇	1	2	yes	—
312	#533	C₂²×C₆₇	1	2	yes	—
313	#533	C₂²×C₆₇	1	2	yes	—
314	#532	C₂²×C₆₇	1	2	yes	—
315	#532	C₂²×C₆₇	1	2	yes	—
316	#532	C₂²×C₆₇	1	2	yes	—
317	#529	—	2	2	no	—
318	#529	—	2	2	no	—
319	#529	—	2	2	no	—
320	#529	—	2	2	no	—
321	#529	C₂²×C₆₇	1	4	yes	—
322	#529	C₂²×C₆₇	1	4	yes	—
323	#525	—	2	2	no	—
324	#525	—	2	2	no	—
325	#525	C₂²×C₆₇	1	4	yes	—
326	C₄×#41	—	2	2	no	—
327	C₄×#41	—	2	2	no	—
328	C₄×#41	C₂²×C₆₇	1	4	yes	—
329	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
330	#479	C₂²×C₆₇	1	8	yes	complemented
331	#478	C₂²×C₆₇	1	8	yes	complemented
332	#477	C₂²×C₆₇	1	4	yes	complemented
333	#476	C₂²×C₆₇	1	4	yes	complemented
334	#475	C₂²×C₆₇	1	4	yes	complemented
335	#474	C₂²×C₆₇	1	4	yes	complemented
336	#470	C₂²×C₆₇	1	4	yes	—
337	#470	C₂²×C₆₇	1	4	yes	—
338	#469	C₂²×C₆₇	1	4	yes	—
339	#469	C₂²×C₆₇	1	4	yes	—
340	#468	C₂²×C₆₇	1	8	yes	—
341	#468	C₂²×C₆₇	1	8	yes	—
342	#464	C₂²×C₆₇	1	8	yes	complemented
343	#463	C₂²×C₆₇	1	8	yes	complemented
344	#462	C₂²×C₆₇	1	4	yes	complemented
345	#461	C₂²×C₆₇	1	4	yes	complemented
346	#460	C₂²×C₆₇	1	4	yes	complemented
347	#459	C₂²×C₆₇	1	4	yes	complemented
348	#427	C₂²×C₆₇	1	4	yes	—
349	#423	C₂²×C₆₇	1	4	yes	—
350	#423	C₂²×C₆₇	1	1	yes	characteristic
351	#423	C₂²×C₆₇	1	1	yes	characteristic
352	#423	C₂²×C₆₇	1	2	yes	—
353	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
354	#416	C₂²×C₆₇	1	8	yes	—
355	C₂×#41	—	2	2	no	—
356	C₂×#41	—	2	2	no	—
357	C₂×#41	C₂³×C₆₇	1	4	yes	—
358	C₂³×C₄	D₄×C₆₇	1	2	yes	—
359	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
360	#135	—	4	8	no	—
361	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
362	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
363	C₂×D₈	C₂³×C₆₇	1	4	yes	—
364	C₂×M₁₆	—	4	4	no	—
365	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
366	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
367	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
368	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
369	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
370	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
371	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
372	#127	C₂³×C₆₇	1	2	yes	—
373	#126	—	2	4	no	—
374	#126	—	2	4	no	—
375	#125	—	2	2	no	—
376	#125	—	2	2	no	—
377	#125	C₂³×C₆₇	1	4	yes	—
378	#124	—	2	2	no	—
379	#124	—	2	2	no	—
380	#124	C₂³×C₆₇	1	4	yes	—
381	#123	—	2	4	no	—
382	#123	—	2	4	no	—
383	#123	—	2	4	no	—
384	C₂²⋊Q₈	—	2	2	no	—
385	C₂²⋊Q₈	—	2	2	no	—
386	C₂²⋊Q₈	—	2	2	no	—
387	C₂²⋊Q₈	—	2	2	no	—
388	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
389	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
390	#121	—	2	2	no	—
391	#121	—	2	2	no	—
392	#121	—	2	2	no	—
393	#121	—	2	2	no	—
394	#121	C₂³×C₆₇	1	4	yes	—
395	#121	C₂³×C₆₇	1	4	yes	—
396	#120	—	2	4	no	—
397	C₄×Q₈	—	2	2	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	—	2	2	no	—
401	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
402	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
403	C₄×D₄	—	2	4	no	—
404	C₄×D₄	—	2	4	no	—
405	C₄×D₄	—	2	4	no	—
406	C₄×D₄	—	2	4	no	—
407	C₄×D₄	—	2	4	no	—
408	C₄×D₄	—	2	4	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	C₂³×C₆₇	1	4	yes	—
414	C₄×D₄	C₂³×C₆₇	1	4	yes	—
415	#117	—	2	2	no	—
416	#117	—	2	2	no	—
417	#117	—	2	4	no	—
418	#117	—	2	4	no	—
419	#117	—	2	4	no	—
420	#117	C₂³×C₆₇	1	4	yes	—
421	C₂×C₄²	—	2	4	no	—
422	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
423	C₂×C₄²	D₄×C₆₇	1	2	yes	—
424	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
425	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
426	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
428	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
430	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
431	#107	C₂³×C₆₇	1	8	yes	—
432	#106	C₂³×C₆₇	1	4	yes	—
433	#106	C₂³×C₆₇	1	4	yes	—
434	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
435	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
436	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
437	#103	C₂³×C₆₇	1	8	yes	—
438	#103	C₂³×C₆₇	1	8	yes	—
439	#102	C₂³×C₆₇	1	8	yes	—
440	#102	C₂³×C₆₇	1	8	yes	—
441	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
442	#41	—	2	8	no	—
443	#41	—	2	8	no	—
444	#41	—	4	8	no	—
445	#41	—	4	8	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
465	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
466	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
468	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
471	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
472	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
473	Q₁₆	—	4	4	no	—
474	SD₁₆	—	4	8	no	—
475	D₈	—	4	4	no	—
476	M₁₆	—	4	2	no	—
477	M₁₆	—	4	4	no	—
478	M₁₆	—	4	2	no	—
479	C₂×C₈	—	2	8	no	—
480	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
481	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
493	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₄²	—	2	4	no	—
504	C₄²	—	2	4	no	—
505	C₄²	—	2	4	no	—
506	C₄²	C₂×D₄×C₆₇	1	2	yes	—
507	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
508	C₄²	C₂×D₄×C₆₇	1	2	yes	—
509	C₄²	C₂×D₄×C₆₇	1	2	yes	—
510	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
511	C₄²	C₂×D₄×C₆₇	1	2	yes	—
512	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
514	C₄²	C₂⁴×C₆₇	1	2	yes	—
515	C₄²	C₂⁴×C₆₇	1	2	yes	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	4	2	no	—
519	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
520	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
521	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
522	Q₈	—	2	4	no	—
523	Q₈	—	2	4	no	—
524	Q₈	—	4	4	no	—
525	Q₈	—	4	4	no	—
526	D₄	—	2	4	no	—
527	D₄	—	2	4	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	8	no	—
530	D₄	—	4	8	no	—
531	D₄	—	4	8	no	—
532	D₄	—	4	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	8	no	—
550	C₂×C₄	—	4	8	no	—
551	C₂×C₄	—	4	8	no	—
552	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
555	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
559	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
560	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
561	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
562	C₈	—	2	4	no	—
563	C₈	—	2	4	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	4	4	no	—
573	C₂²	#537×C₆₇	1	2	yes	—
574	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	1	no	—
579	C₄	—	2	1	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	4	no	—
583	C₄	#537×C₆₇	1	2	yes	—
584	C₄	#575×C₆₇	1	4	yes	—
585	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
586	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
587	C₂	—	2	2	no	—
588	C₂	—	4	1	no	—
589	C₂	—	4	1	no	—
590	C₂	—	8	4	no	—
591	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
592	C₂	C₂×#575×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3444	C₂	semidirect
2. C₆₇×#3256	C₂	semidirect
3. C₆₇×#3255	C₂	semidirect
4. C₆₇×#3254	C₂	semidirect
5. C₆₇×#3253	C₂	semidirect
6. C₆₇×#3252	C₂	semidirect
7. C₆₇×#3251	C₂	semidirect
8. C₆₇×#3076	C₂	semidirect
9. C₆₇×#3054	C₂	semidirect
10. C₆₇×#3035	C₂	semidirect
11. C₂×#423×C₆₇	C₂	semidirect
33. #479×C₆₇	C₂²	semidirect
34. #478×C₆₇	C₂²	semidirect
35. #477×C₆₇	C₂²	semidirect
36. #476×C₆₇	C₂²	semidirect
37. #475×C₆₇	C₂²	semidirect
38. #474×C₆₇	C₂²	semidirect
45. #464×C₆₇	C₂²	semidirect
46. #463×C₆₇	C₂²	semidirect
47. #462×C₆₇	C₂²	semidirect
48. #461×C₆₇	C₂²	semidirect
49. #460×C₆₇	C₂²	semidirect
50. #459×C₆₇	C₂²	semidirect
290. #60907	C₆₇	direct (N × H)
297. #3444	C₂×C₆₇	semidirect
298. #3256	C₂×C₆₇	semidirect
299. #3255	C₂×C₆₇	semidirect
300. #3254	C₂×C₆₇	semidirect
301. #3253	C₂×C₆₇	semidirect
302. #3252	C₂×C₆₇	semidirect
303. #3251	C₂×C₆₇	semidirect
304. #3076	C₂×C₆₇	semidirect
305. #3054	C₂×C₆₇	semidirect
306. #3035	C₂×C₆₇	semidirect
307. C₂×#423	C₂×C₆₇	semidirect
308. C₆₇	#60907	direct (N × H)
330. #479	C₂²×C₆₇	semidirect
331. #478	C₂²×C₆₇	semidirect
332. #477	C₂²×C₆₇	semidirect
333. #476	C₂²×C₆₇	semidirect
334. #475	C₂²×C₆₇	semidirect
335. #474	C₂²×C₆₇	semidirect
342. #464	C₂²×C₆₇	semidirect
343. #463	C₂²×C₆₇	semidirect
344. #462	C₂²×C₆₇	semidirect
345. #461	C₂²×C₆₇	semidirect
346. #460	C₂²×C₆₇	semidirect
347. #459	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3444	2
C₆₇×#3256	2
C₆₇×#3255	2
C₆₇×#3254	2
C₆₇×#3253	2
C₆₇×#3252	2
C₆₇×#3251	2
C₆₇×#3076	2
C₆₇×#3054	2
C₆₇×#3035	2
C₂×#423×C₆₇	2
#60907	67

Subgroups — C67×#60907 #30740158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60907 #30740158