Subgroups — C₆₇×#17877 #30697128

Order 17152.

726 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2975	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1586	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1581	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1478	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1478	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×D₈×C₆₇	C₄	1	2	yes	—
14	C₄×D₈×C₆₇	C₄	1	2	yes	—
15	C₄×D₈×C₆₇	C₂²	1	2	yes	—
16	C₄×D₈×C₆₇	C₂²	1	2	yes	—
17	#436×C₆₇	—	2	2	no	—
18	#436×C₆₇	C₂²	1	4	yes	complemented
19	C₈×D₄×C₆₇	—	2	2	no	—
20	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
21	#431×C₆₇	C₂²	1	1	yes	characteristic
22	#431×C₆₇	C₂²	1	1	yes	characteristic
23	#431×C₆₇	C₂²	1	1	yes	characteristic
24	#431×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
26	C₂×#102×C₆₇	C₄	1	1	yes	characteristic
27	C₂×#102×C₆₇	C₄	1	1	yes	characteristic
28	#407×C₆₇	—	2	2	no	—
29	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#335×C₆₇	C₂²	1	2	yes	complemented
32	#334×C₆₇	C₂²	1	2	yes	complemented
33	D₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
34	D₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
35	#322×C₆₇	C₂²	1	2	yes	complemented
36	#322×C₆₇	C₂²	1	2	yes	complemented
37	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	—	2	1	no	—
40	C₂³×C₄×C₆₇	—	2	1	no	—
41	C₂×D₈×C₆₇	—	2	4	no	—
42	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	—	2	4	no	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₆₇	—	2	1	no	—
49	C₄×D₄×C₆₇	—	2	1	no	—
50	C₄×D₄×C₆₇	—	2	1	no	—
51	C₄×D₄×C₆₇	—	2	1	no	—
52	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₆₇	C₂³	1	2	yes	—
55	C₄×D₄×C₆₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#107×C₆₇	C₂×C₄	1	2	yes	—
62	#107×C₆₇	C₂×C₄	1	2	yes	—
63	C₄⋊C₈×C₆₇	—	2	1	no	—
64	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
65	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
66	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
67	#102×C₆₇	C₂×C₄	1	2	yes	—
68	#102×C₆₇	C₂×C₄	1	2	yes	—
69	#102×C₆₇	C₂×C₄	1	2	yes	—
70	#102×C₆₇	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈×C₆₇	—	2	1	no	—
72	C₂²⋊C₈×C₆₇	—	2	1	no	—
73	C₂²⋊C₈×C₆₇	—	2	1	no	—
74	C₂²⋊C₈×C₆₇	—	2	4	no	—
75	C₂²⋊C₈×C₆₇	—	2	4	no	—
76	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
77	C₄×C₈×C₆₇	—	2	1	no	—
78	C₄×C₈×C₆₇	—	2	1	no	—
79	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈×C₆₇	C₂³	1	2	yes	—
82	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₆₇	C₂³	1	2	yes	—
85	C₂⁴×C₆₇	—	2	1	no	—
86	C₂⁴×C₆₇	—	2	1	no	—
87	C₂×D₄×C₆₇	—	2	1	no	—
88	C₂×D₄×C₆₇	—	2	1	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	M₁₆	1	4	yes	complemented
93	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	8	2	no	—
102	C₂²×C₄×C₆₇	—	8	2	no	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
106	D₈×C₆₇	—	2	4	no	—
107	D₈×C₆₇	—	2	4	no	—
108	M₁₆×C₆₇	—	8	4	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	8	4	no	—
115	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²⋊C₄×C₆₇	—	2	2	no	—
125	C₂²⋊C₄×C₆₇	—	2	2	no	—
126	C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₂²⋊C₄×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	#41	1	1	yes	characteristic
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	4	no	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	4	2	no	—
136	C₂³×C₆₇	—	8	1	no	—
137	C₂³×C₆₇	—	8	1	no	—
138	C₂³×C₆₇	—	8	1	no	—
139	C₂³×C₆₇	—	8	1	no	—
140	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
141	D₄×C₆₇	—	2	2	no	—
142	D₄×C₆₇	—	2	2	no	—
143	D₄×C₆₇	—	2	2	no	—
144	D₄×C₆₇	—	2	4	no	—
145	D₄×C₆₇	—	4	2	no	—
146	D₄×C₆₇	C₂×M₁₆	1	4	yes	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	4	1	no	—
151	C₂×C₄×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	—	4	4	no	—
153	C₂×C₄×C₆₇	—	8	2	no	—
154	C₂×C₄×C₆₇	—	8	2	no	—
155	C₂×C₄×C₆₇	—	8	1	no	—
156	C₂×C₄×C₆₇	—	8	1	no	—
157	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
161	C₂×C₄×C₆₇	#131	1	2	yes	—
162	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
165	C₈×C₆₇	—	2	1	no	—
166	C₈×C₆₇	—	2	1	no	—
167	C₈×C₆₇	—	2	1	no	—
168	C₈×C₆₇	—	2	1	no	—
169	C₈×C₆₇	—	2	4	no	—
170	C₈×C₆₇	—	4	2	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	—	2	1	no	—
174	C₂²×C₆₇	—	2	1	no	—
175	C₂²×C₆₇	—	2	4	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	—	4	2	no	—
179	C₂²×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	4	2	no	—
181	C₂²×C₆₇	—	8	4	no	—
182	C₂²×C₆₇	—	8	4	no	—
183	C₂²×C₆₇	#435	1	2	yes	—
184	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	4	2	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	#435	1	2	yes	—
191	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
192	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
193	C₄×C₆₇	#576	1	2	yes	—
194	#17877	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂×C₆₇	—	2	2	no	—
196	C₂×C₆₇	—	4	4	no	—
197	C₂×C₆₇	—	8	2	no	—
198	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
199	C₂×C₆₇	#2948	1	1	yes	characteristic
200	C₂×C₆₇	#2957	1	1	yes	characteristic
201	#2975	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
203	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
204	#1586	C₂×C₆₇	1	4	yes	complemented
205	#1581	C₂×C₆₇	1	4	yes	complemented
206	#1564	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#1478	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#1478	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#1452	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₆₇	#17877	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic
212	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
214	C₄×D₈	C₄×C₆₇	1	2	yes	—
215	C₄×D₈	C₄×C₆₇	1	2	yes	—
216	C₄×D₈	C₂²×C₆₇	1	2	yes	—
217	C₄×D₈	C₂²×C₆₇	1	2	yes	—
218	#436	—	2	2	no	—
219	#436	C₂²×C₆₇	1	4	yes	complemented
220	C₈×D₄	—	2	2	no	—
221	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
222	#431	C₂²×C₆₇	1	1	yes	characteristic
223	#431	C₂²×C₆₇	1	1	yes	characteristic
224	#431	C₂²×C₆₇	1	1	yes	characteristic
225	#431	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×#107	C₄×C₆₇	1	1	yes	characteristic
227	C₂×#102	C₄×C₆₇	1	1	yes	characteristic
228	C₂×#102	C₄×C₆₇	1	1	yes	characteristic
229	#407	—	2	2	no	—
230	C₂×C₂²⋊C₈	—	2	2	no	—
231	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
232	#335	C₂²×C₆₇	1	2	yes	complemented
233	#334	C₂²×C₆₇	1	2	yes	complemented
234	D₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
235	D₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
236	#322	C₂²×C₆₇	1	2	yes	complemented
237	#322	C₂²×C₆₇	1	2	yes	complemented
238	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
239	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
240	C₂³×C₄	—	2	1	no	—
241	C₂³×C₄	—	2	1	no	—
242	C₂×D₈	—	2	4	no	—
243	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
244	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
245	C₂×M₁₆	—	2	4	no	—
246	C₂²×C₈	—	2	4	no	—
247	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
248	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
249	C₄×D₄	—	2	1	no	—
250	C₄×D₄	—	2	1	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
254	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
255	C₄×D₄	C₂³×C₆₇	1	2	yes	—
256	C₄×D₄	C₂³×C₆₇	1	2	yes	—
257	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
259	C₂×C₂²⋊C₄	—	2	1	no	—
260	C₂×C₂²⋊C₄	—	2	1	no	—
261	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
262	#107	C₂×C₄×C₆₇	1	2	yes	—
263	#107	C₂×C₄×C₆₇	1	2	yes	—
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
266	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
267	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
268	#102	C₂×C₄×C₆₇	1	2	yes	—
269	#102	C₂×C₄×C₆₇	1	2	yes	—
270	#102	C₂×C₄×C₆₇	1	2	yes	—
271	#102	C₂×C₄×C₆₇	1	2	yes	—
272	C₂²⋊C₈	—	2	1	no	—
273	C₂²⋊C₈	—	2	1	no	—
274	C₂²⋊C₈	—	2	1	no	—
275	C₂²⋊C₈	—	2	4	no	—
276	C₂²⋊C₈	—	2	4	no	—
277	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
278	C₄×C₈	—	2	1	no	—
279	C₄×C₈	—	2	1	no	—
280	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
281	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
282	C₄×C₈	C₂³×C₆₇	1	2	yes	—
283	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
284	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
285	C₄×C₈	C₂³×C₆₇	1	2	yes	—
286	C₂⁴	—	2	1	no	—
287	C₂⁴	—	2	1	no	—
288	C₂×D₄	—	2	1	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	M₁₆×C₆₇	1	4	yes	complemented
294	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
295	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	8	2	no	—
303	C₂²×C₄	—	8	2	no	—
304	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
307	D₈	—	2	4	no	—
308	D₈	—	2	4	no	—
309	M₁₆	—	8	4	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	8	4	no	—
316	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
317	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
318	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
319	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
320	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
324	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
330	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₄²	#41×C₆₇	1	1	yes	characteristic
332	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
333	C₂³	—	2	2	no	—
334	C₂³	—	2	4	no	—
335	C₂³	—	2	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	8	1	no	—
338	C₂³	—	8	1	no	—
339	C₂³	—	8	1	no	—
340	C₂³	—	8	1	no	—
341	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
342	D₄	—	2	2	no	—
343	D₄	—	2	2	no	—
344	D₄	—	2	2	no	—
345	D₄	—	2	4	no	—
346	D₄	—	4	2	no	—
347	D₄	C₂×M₁₆×C₆₇	1	4	yes	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	4	no	—
353	C₂×C₄	—	4	4	no	—
354	C₂×C₄	—	8	2	no	—
355	C₂×C₄	—	8	2	no	—
356	C₂×C₄	—	8	1	no	—
357	C₂×C₄	—	8	1	no	—
358	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
359	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
360	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
361	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
362	C₂×C₄	#131×C₆₇	1	2	yes	—
363	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
364	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
365	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	4	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	4	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	8	4	no	—
383	C₂²	—	8	4	no	—
384	C₂²	#435×C₆₇	1	2	yes	—
385	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	4	2	no	—
390	C₄	—	4	2	no	—
391	C₄	#435×C₆₇	1	2	yes	—
392	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
393	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
394	C₄	#576×C₆₇	1	2	yes	—
395	C₂	—	2	2	no	—
396	C₂	—	4	4	no	—
397	C₂	—	8	2	no	—
398	C₂	C₂×#435×C₆₇	1	1	yes	characteristic
399	C₂	C₆₇×#2948	1	1	yes	characteristic
400	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₈×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ #431×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2975	C₂	semidirect
2. C₆₇×#2972	C₂	semidirect
4. C₆₇×#1586	C₂	semidirect
5. C₆₇×#1581	C₂	semidirect
6. C₆₇×#1564	C₂	semidirect
7. C₆₇×#1478	C₂	semidirect
8. C₆₇×#1478	C₂	semidirect
9. C₆₇×#1452	C₂	semidirect
18. #436×C₆₇	C₂²	semidirect
20. C₈×D₄×C₆₇	C₂²	semidirect
31. #335×C₆₇	C₂²	semidirect
32. #334×C₆₇	C₂²	semidirect
33. D₄⋊C₈×C₆₇	C₂²	semidirect
34. D₄⋊C₈×C₆₇	C₂²	semidirect
35. #322×C₆₇	C₂²	semidirect
36. #322×C₆₇	C₂²	semidirect
92. C₂×D₄×C₆₇	M₁₆	semidirect
194. #17877	C₆₇	direct (N × H)
201. #2975	C₂×C₆₇	semidirect
202. #2972	C₂×C₆₇	semidirect
204. #1586	C₂×C₆₇	semidirect
205. #1581	C₂×C₆₇	semidirect
206. #1564	C₂×C₆₇	semidirect
207. #1478	C₂×C₆₇	semidirect
208. #1478	C₂×C₆₇	semidirect
209. #1452	C₂×C₆₇	semidirect
210. C₆₇	#17877	direct (N × H)
219. #436	C₂²×C₆₇	semidirect
221. C₈×D₄	C₂²×C₆₇	semidirect
232. #335	C₂²×C₆₇	semidirect
233. #334	C₂²×C₆₇	semidirect
234. D₄⋊C₈	C₂²×C₆₇	semidirect
235. D₄⋊C₈	C₂²×C₆₇	semidirect
236. #322	C₂²×C₆₇	semidirect
237. #322	C₂²×C₆₇	semidirect
293. C₂×D₄	M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2975	2
C₆₇×#2972	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#1586	2
C₆₇×#1581	2
C₆₇×#1564	2
C₆₇×#1478	2
C₆₇×#1452	2
#17877	67

Subgroups — C67×#17877 #30697128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17877 #30697128