Subgroups — C₆₇×#61694 #30740945

Order 17152.

2298 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3334	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3294	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3226	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3186	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3100	C₂	1	2	yes	maximal, complemented
8	C₂×#480×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₆₇	—	2	1	no	—
11	C₂²×#41×C₆₇	—	2	1	no	—
12	#547×C₆₇	—	2	4	no	—
13	#547×C₆₇	C₂²	1	8	yes	complemented
14	#547×C₆₇	C₂²	1	4	yes	complemented
15	#546×C₆₇	—	2	4	no	—
16	D₄²×C₆₇	C₂²	1	4	yes	complemented
17	#537×C₆₇	—	2	1	no	—
18	#534×C₆₇	—	2	2	no	—
19	#532×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
21	C₂×#123×C₆₇	—	2	2	no	—
22	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
23	C₂×#121×C₆₇	C₂²	1	4	yes	—
24	C₂×#121×C₆₇	C₂²	1	2	yes	complemented
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
28	#500×C₆₇	C₂²	1	4	yes	complemented
29	#486×C₆₇	C₂²	1	4	yes	complemented
30	#480×C₆₇	C₂²	1	16	yes	complemented
31	#480×C₆₇	C₂²	1	8	yes	complemented
32	#480×C₆₇	C₂²	1	4	yes	complemented
33	#480×C₆₇	C₂²	1	8	yes	complemented
34	#474×C₆₇	C₂²	1	8	yes	complemented
35	#474×C₆₇	C₂²	1	4	yes	complemented
36	#466×C₆₇	C₂²	1	16	yes	complemented
37	#459×C₆₇	C₂²	1	4	yes	complemented
38	#447×C₆₇	C₂²	1	8	yes	complemented
39	#447×C₆₇	C₂²	1	4	yes	complemented
40	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
41	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
42	#431×C₆₇	C₂²	1	2	yes	complemented
43	C₂×#107×C₆₇	C₂²	1	8	yes	complemented
44	C₂×#102×C₆₇	C₂²	1	8	yes	complemented
45	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
46	H(5,𝔽₂)×C₆₇	—	4	8	no	—
47	C₂×#41×C₆₇	—	2	4	no	—
48	C₂×#41×C₆₇	—	2	8	no	—
49	C₂×#41×C₆₇	—	2	2	no	—
50	C₂×#41×C₆₇	—	4	8	no	—
51	C₂²×Q₈×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	—	2	4	no	—
53	C₂²×D₄×C₆₇	—	2	2	no	—
54	C₂²×D₄×C₆₇	—	2	4	no	—
55	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
56	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
58	C₂³×C₄×C₆₇	—	2	2	no	—
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×D₈×C₆₇	C₂³	1	8	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
63	C₄⋊Q₈×C₆₇	—	2	2	no	—
64	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
65	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
66	#127×C₆₇	C₂³	1	2	yes	—
67	#124×C₆₇	—	2	2	no	—
68	#123×C₆₇	—	2	16	no	—
69	#123×C₆₇	—	2	8	no	—
70	C₂²⋊Q₈×C₆₇	—	2	8	no	—
71	C₂²⋊Q₈×C₆₇	—	2	4	no	—
72	#121×C₆₇	—	2	4	no	—
73	#121×C₆₇	—	2	4	no	—
74	#121×C₆₇	—	2	4	no	—
75	#121×C₆₇	—	2	2	no	—
76	#121×C₆₇	—	2	4	no	—
77	#121×C₆₇	C₂³	1	8	yes	—
78	#121×C₆₇	C₂³	1	8	yes	—
79	#121×C₆₇	C₂³	1	4	yes	—
80	#120×C₆₇	—	2	8	no	—
81	C₄×D₄×C₆₇	—	2	2	no	—
82	C₄×D₄×C₆₇	—	2	2	no	—
83	C₄×D₄×C₆₇	D₄	1	4	yes	—
84	C₄×D₄×C₆₇	C₂³	1	4	yes	—
85	C₄×D₄×C₆₇	C₂³	1	4	yes	—
86	C₄×D₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
99	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
100	#107×C₆₇	C₂³	1	16	yes	—
101	#107×C₆₇	C₂³	1	8	yes	—
102	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
103	#102×C₆₇	C₂³	1	8	yes	—
104	#102×C₆₇	C₂³	1	8	yes	—
105	#102×C₆₇	C₂³	1	16	yes	—
106	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
107	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
108	C₄×C₈×C₆₇	C₂³	1	4	yes	—
109	C₂⁴×C₆₇	—	2	1	no	—
110	C₂⁴×C₆₇	—	2	4	no	—
111	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
112	#41×C₆₇	—	4	32	no	—
113	#41×C₆₇	—	4	16	no	—
114	#41×C₆₇	—	4	8	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	4	no	—
117	C₂×Q₈×C₆₇	—	4	4	no	—
118	C₂×D₄×C₆₇	—	2	8	no	—
119	C₂×D₄×C₆₇	—	2	4	no	—
120	C₂×D₄×C₆₇	—	2	8	no	—
121	C₂×D₄×C₆₇	—	2	1	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	4	2	no	—
126	C₂×D₄×C₆₇	—	4	8	no	—
127	C₂×D₄×C₆₇	—	4	8	no	—
128	C₂×D₄×C₆₇	—	4	16	no	—
129	C₂×D₄×C₆₇	—	4	16	no	—
130	C₂×D₄×C₆₇	D₈	1	8	yes	complemented
131	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
132	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
133	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	8	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	4	4	no	—
141	C₂²×C₄×C₆₇	—	4	8	no	—
142	C₂²×C₄×C₆₇	—	4	4	no	—
143	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
149	D₈×C₆₇	—	4	8	no	—
150	C₂×C₈×C₆₇	—	4	8	no	—
151	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
152	C₄⋊C₄×C₆₇	—	2	1	no	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	8	no	—
155	C₄⋊C₄×C₆₇	—	2	4	no	—
156	C₄⋊C₄×C₆₇	—	2	4	no	—
157	C₄⋊C₄×C₆₇	—	2	2	no	—
158	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
159	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
160	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
161	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
162	C₂²⋊C₄×C₆₇	—	2	4	no	—
163	C₂²⋊C₄×C₆₇	—	2	4	no	—
164	C₂²⋊C₄×C₆₇	—	2	8	no	—
165	C₂²⋊C₄×C₆₇	—	2	8	no	—
166	C₂²⋊C₄×C₆₇	—	2	4	no	—
167	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
168	C₄²×C₆₇	C₂×D₄	1	2	yes	—
169	C₄²×C₆₇	C₂⁴	1	2	yes	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	—	2	4	no	—
172	C₂³×C₆₇	—	4	2	no	—
173	C₂³×C₆₇	—	4	2	no	—
174	C₂³×C₆₇	—	4	2	no	—
175	C₂³×C₆₇	—	4	2	no	—
176	C₂³×C₆₇	—	4	2	no	—
177	C₂³×C₆₇	—	4	8	no	—
178	C₂³×C₆₇	—	8	4	no	—
179	C₂³×C₆₇	—	8	4	no	—
180	C₂³×C₆₇	C₂×D₈	1	8	yes	—
181	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
182	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₆₇	—	4	8	no	—
184	Q₈×C₆₇	—	4	4	no	—
185	D₄×C₆₇	—	2	4	no	—
186	D₄×C₆₇	—	2	8	no	—
187	D₄×C₆₇	—	4	2	no	—
188	D₄×C₆₇	—	4	8	no	—
189	D₄×C₆₇	—	4	16	no	—
190	D₄×C₆₇	—	4	16	no	—
191	D₄×C₆₇	—	4	4	no	—
192	D₄×C₆₇	—	8	2	no	—
193	D₄×C₆₇	—	8	4	no	—
194	D₄×C₆₇	C₂×D₈	1	16	yes	—
195	C₂×C₄×C₆₇	—	2	8	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	4	4	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	4	4	no	—
205	C₂×C₄×C₆₇	—	4	16	no	—
206	C₂×C₄×C₆₇	—	4	8	no	—
207	C₂×C₄×C₆₇	—	4	8	no	—
208	C₂×C₄×C₆₇	—	4	8	no	—
209	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
213	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
215	C₈×C₆₇	—	2	8	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	2	1	no	—
219	C₂²×C₆₇	—	2	4	no	—
220	C₂²×C₆₇	—	4	8	no	—
221	C₂²×C₆₇	—	4	4	no	—
222	C₂²×C₆₇	—	4	2	no	—
223	C₂²×C₆₇	—	4	2	no	—
224	C₂²×C₆₇	—	4	4	no	—
225	C₂²×C₆₇	—	4	2	no	—
226	C₂²×C₆₇	—	4	4	no	—
227	C₂²×C₆₇	—	8	4	no	—
228	C₂²×C₆₇	—	16	8	no	—
229	C₂²×C₆₇	C₂²×D₈	1	2	yes	—
230	C₂²×C₆₇	C₂²×D₈	1	8	yes	—
231	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	2	1	no	—
234	C₄×C₆₇	—	2	4	no	—
235	C₄×C₆₇	—	4	2	no	—
236	C₄×C₆₇	—	4	4	no	—
237	C₄×C₆₇	—	4	4	no	—
238	C₄×C₆₇	C₂²×D₈	1	4	yes	—
239	C₄×C₆₇	#585	1	2	yes	—
240	#61694	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂×C₆₇	—	2	2	no	—
242	C₂×C₆₇	—	2	8	no	—
243	C₂×C₆₇	—	8	2	no	—
244	C₂×C₆₇	—	8	4	no	—
245	C₂×C₆₇	#3485	1	1	yes	characteristic
246	C₂×C₆₇	C₂³×D₈	1	1	yes	characteristic
247	C₂×C₆₇	#3586	1	1	yes	characteristic
248	#3496	C₂×C₆₇	1	1	yes	characteristic, complemented
249	C₂×#547	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#3334	C₂×C₆₇	1	8	yes	complemented
251	#3294	C₂×C₆₇	1	8	yes	complemented
252	#3226	C₂×C₆₇	1	2	yes	complemented
253	#3186	C₂×C₆₇	1	4	yes	complemented
254	#3100	C₂×C₆₇	1	2	yes	complemented
255	C₂×#480	C₂×C₆₇	1	4	yes	complemented
256	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
257	C₆₇	#61694	1	1	yes	characteristic, direct product factor, Sylow
258	C₂×H(5,𝔽₂)	—	2	1	no	—
259	C₂²×#41	—	2	1	no	—
260	#547	—	2	4	no	—
261	#547	C₂²×C₆₇	1	4	yes	complemented
262	#547	C₂²×C₆₇	1	8	yes	complemented
263	#546	—	2	4	no	—
264	D₄²	C₂²×C₆₇	1	4	yes	complemented
265	#537	—	2	1	no	—
266	#534	—	2	2	no	—
267	#532	C₂²×C₆₇	1	2	yes	—
268	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
269	C₂×#123	—	2	2	no	—
270	C₂×C₂²⋊Q₈	—	2	1	no	—
271	C₂×#121	C₂²×C₆₇	1	4	yes	—
272	C₂×#121	C₂²×C₆₇	1	2	yes	complemented
273	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
275	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
276	#500	C₂²×C₆₇	1	4	yes	complemented
277	#486	C₂²×C₆₇	1	4	yes	complemented
278	#480	C₂²×C₆₇	1	8	yes	complemented
279	#480	C₂²×C₆₇	1	16	yes	complemented
280	#480	C₂²×C₆₇	1	4	yes	complemented
281	#480	C₂²×C₆₇	1	8	yes	complemented
282	#474	C₂²×C₆₇	1	4	yes	complemented
283	#474	C₂²×C₆₇	1	8	yes	complemented
284	#466	C₂²×C₆₇	1	16	yes	complemented
285	#459	C₂²×C₆₇	1	4	yes	complemented
286	#447	C₂²×C₆₇	1	4	yes	complemented
287	#447	C₂²×C₆₇	1	8	yes	complemented
288	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
289	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
290	#431	C₂²×C₆₇	1	2	yes	complemented
291	C₂×#107	C₂²×C₆₇	1	8	yes	complemented
292	C₂×#102	C₂²×C₆₇	1	8	yes	complemented
293	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
294	H(5,𝔽₂)	—	4	8	no	—
295	C₂×#41	—	2	4	no	—
296	C₂×#41	—	2	8	no	—
297	C₂×#41	—	2	2	no	—
298	C₂×#41	—	4	8	no	—
299	C₂²×Q₈	—	2	1	no	—
300	C₂²×D₄	—	2	4	no	—
301	C₂²×D₄	—	2	2	no	—
302	C₂²×D₄	—	2	4	no	—
303	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
304	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
305	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
306	C₂³×C₄	—	2	2	no	—
307	C₂³×C₄	—	2	1	no	—
308	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
309	C₂×D₈	C₂³×C₆₇	1	8	yes	—
310	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
313	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
314	#127	C₂³×C₆₇	1	2	yes	—
315	#124	—	2	2	no	—
316	#123	—	2	16	no	—
317	#123	—	2	8	no	—
318	C₂²⋊Q₈	—	2	8	no	—
319	C₂²⋊Q₈	—	2	4	no	—
320	#121	—	2	4	no	—
321	#121	—	2	4	no	—
322	#121	—	2	4	no	—
323	#121	—	2	2	no	—
324	#121	—	2	4	no	—
325	#121	C₂³×C₆₇	1	4	yes	—
326	#121	C₂³×C₆₇	1	8	yes	—
327	#121	C₂³×C₆₇	1	8	yes	—
328	#120	—	2	8	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	—	2	2	no	—
331	C₄×D₄	D₄×C₆₇	1	4	yes	—
332	C₄×D₄	C₂³×C₆₇	1	4	yes	—
333	C₄×D₄	C₂³×C₆₇	1	4	yes	—
334	C₄×D₄	C₂³×C₆₇	1	4	yes	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	4	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
342	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
343	C₂×C₂²⋊C₄	—	2	1	no	—
344	C₂×C₂²⋊C₄	—	2	4	no	—
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
347	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
348	#107	C₂³×C₆₇	1	8	yes	—
349	#107	C₂³×C₆₇	1	16	yes	—
350	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
351	#102	C₂³×C₆₇	1	8	yes	—
352	#102	C₂³×C₆₇	1	16	yes	—
353	#102	C₂³×C₆₇	1	8	yes	—
354	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
355	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
356	C₄×C₈	C₂³×C₆₇	1	4	yes	—
357	C₂⁴	—	2	1	no	—
358	C₂⁴	—	2	4	no	—
359	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
360	#41	—	4	32	no	—
361	#41	—	4	16	no	—
362	#41	—	4	8	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	4	4	no	—
366	C₂×D₄	—	2	8	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	4	8	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	—	4	16	no	—
377	C₂×D₄	—	4	16	no	—
378	C₂×D₄	D₈×C₆₇	1	8	yes	complemented
379	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
380	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
381	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	8	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	4	4	no	—
389	C₂²×C₄	—	4	8	no	—
390	C₂²×C₄	—	4	4	no	—
391	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
393	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
394	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
397	D₈	—	4	8	no	—
398	C₂×C₈	—	4	8	no	—
399	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
407	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
408	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
409	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₂²⋊C₄	—	2	8	no	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₂²⋊C₄	—	2	4	no	—
415	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
416	C₄²	C₂×D₄×C₆₇	1	2	yes	—
417	C₄²	C₂⁴×C₆₇	1	2	yes	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	8	no	—
426	C₂³	—	8	4	no	—
427	C₂³	—	8	4	no	—
428	C₂³	C₂×D₈×C₆₇	1	8	yes	—
429	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
430	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
431	Q₈	—	4	8	no	—
432	Q₈	—	4	4	no	—
433	D₄	—	2	4	no	—
434	D₄	—	2	8	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	8	no	—
437	D₄	—	4	16	no	—
438	D₄	—	4	16	no	—
439	D₄	—	4	4	no	—
440	D₄	—	8	4	no	—
441	D₄	—	8	2	no	—
442	D₄	C₂×D₈×C₆₇	1	16	yes	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	4	4	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	16	no	—
454	C₂×C₄	—	4	8	no	—
455	C₂×C₄	—	4	8	no	—
456	C₂×C₄	—	4	8	no	—
457	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
458	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
461	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
462	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
463	C₈	—	2	8	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	2	4	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	4	8	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	8	4	no	—
476	C₂²	—	16	8	no	—
477	C₂²	C₂²×D₈×C₆₇	1	2	yes	—
478	C₂²	C₂²×D₈×C₆₇	1	8	yes	—
479	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	4	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	C₂²×D₈×C₆₇	1	4	yes	—
487	C₄	#585×C₆₇	1	2	yes	—
488	C₂	—	2	2	no	—
489	C₂	—	2	8	no	—
490	C₂	—	8	2	no	—
491	C₂	—	8	4	no	—
492	C₂	C₆₇×#3485	1	1	yes	characteristic
493	C₂	C₂³×D₈×C₆₇	1	1	yes	characteristic
494	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×#3496	C₂	semidirect
2. C₂×#547×C₆₇	C₂	semidirect
3. C₆₇×#3334	C₂	semidirect
4. C₆₇×#3294	C₂	semidirect
5. C₆₇×#3226	C₂	semidirect
6. C₆₇×#3186	C₂	semidirect
7. C₆₇×#3100	C₂	semidirect
8. C₂×#480×C₆₇	C₂	semidirect
9. C₆₇×#2972	C₂	semidirect
13. #547×C₆₇	C₂²	semidirect
14. #547×C₆₇	C₂²	semidirect
16. D₄²×C₆₇	C₂²	semidirect
20. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
24. C₂×#121×C₆₇	C₂²	semidirect
26. C₂×C₄×D₄×C₆₇	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
28. #500×C₆₇	C₂²	semidirect
29. #486×C₆₇	C₂²	semidirect
30. #480×C₆₇	C₂²	semidirect
31. #480×C₆₇	C₂²	semidirect
32. #480×C₆₇	C₂²	semidirect
33. #480×C₆₇	C₂²	semidirect
34. #474×C₆₇	C₂²	semidirect
35. #474×C₆₇	C₂²	semidirect
36. #466×C₆₇	C₂²	semidirect
37. #459×C₆₇	C₂²	semidirect
38. #447×C₆₇	C₂²	semidirect
39. #447×C₆₇	C₂²	semidirect
40. C₄×D₈×C₆₇	C₂²	semidirect
41. C₈×D₄×C₆₇	C₂²	semidirect
42. #431×C₆₇	C₂²	semidirect
43. C₂×#107×C₆₇	C₂²	semidirect
44. C₂×#102×C₆₇	C₂²	semidirect
45. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
130. C₂×D₄×C₆₇	D₈	semidirect
240. #61694	C₆₇	direct (N × H)
248. #3496	C₂×C₆₇	semidirect
249. C₂×#547	C₂×C₆₇	semidirect
250. #3334	C₂×C₆₇	semidirect
251. #3294	C₂×C₆₇	semidirect
252. #3226	C₂×C₆₇	semidirect
253. #3186	C₂×C₆₇	semidirect
254. #3100	C₂×C₆₇	semidirect
255. C₂×#480	C₂×C₆₇	semidirect
256. #2972	C₂×C₆₇	semidirect
257. C₆₇	#61694	direct (N × H)
261. #547	C₂²×C₆₇	semidirect
262. #547	C₂²×C₆₇	semidirect
264. D₄²	C₂²×C₆₇	semidirect
268. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
272. C₂×#121	C₂²×C₆₇	semidirect
274. C₂×C₄×D₄	C₂²×C₆₇	semidirect
275. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
276. #500	C₂²×C₆₇	semidirect
277. #486	C₂²×C₆₇	semidirect
278. #480	C₂²×C₆₇	semidirect
279. #480	C₂²×C₆₇	semidirect
280. #480	C₂²×C₆₇	semidirect
281. #480	C₂²×C₆₇	semidirect
282. #474	C₂²×C₆₇	semidirect
283. #474	C₂²×C₆₇	semidirect
284. #466	C₂²×C₆₇	semidirect
285. #459	C₂²×C₆₇	semidirect
286. #447	C₂²×C₆₇	semidirect
287. #447	C₂²×C₆₇	semidirect
288. C₄×D₈	C₂²×C₆₇	semidirect
289. C₈×D₄	C₂²×C₆₇	semidirect
290. #431	C₂²×C₆₇	semidirect
291. C₂×#107	C₂²×C₆₇	semidirect
292. C₂×#102	C₂²×C₆₇	semidirect
293. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
378. C₂×D₄	D₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3496	2
C₂×#547×C₆₇	2
C₆₇×#3334	2
C₆₇×#3294	2
C₆₇×#3226	2
C₆₇×#3186	2
C₆₇×#3100	2
C₂×#480×C₆₇	2
C₆₇×#2972	2
#61694	67

Subgroups — C67×#61694 #30740945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61694 #30740945