Subgroups — C₆₇×#18087 #30697338

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3168	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2994	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1588	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
7	C₈×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1483	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#529×C₆₇	C₂²	1	2	yes	—
13	#499×C₆₇	C₄	1	1	yes	characteristic
14	#495×C₆₇	C₄	1	1	yes	characteristic
15	#488×C₆₇	C₄	1	2	yes	—
16	#467×C₆₇	C₄	1	1	yes	characteristic
17	#466×C₆₇	C₄	1	1	yes	characteristic
18	#465×C₆₇	C₄	1	2	yes	—
19	#446×C₆₇	C₂²	1	2	yes	—
20	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
23	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
24	#436×C₆₇	C₂²	1	2	yes	—
25	#435×C₆₇	—	2	2	no	—
26	C₈×D₄×C₆₇	—	2	2	no	—
27	C₈×D₄×C₆₇	C₂²	1	2	yes	—
28	#433×C₆₇	—	2	2	no	—
29	#431×C₆₇	C₂²	1	1	yes	characteristic
30	#424×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
32	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	#335×C₆₇	C₂²	1	1	yes	characteristic
35	#335×C₆₇	C₂²	1	2	yes	—
36	#335×C₆₇	C₂²	1	1	yes	characteristic
37	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
38	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
39	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
40	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
41	#322×C₆₇	C₂²	1	1	yes	characteristic
42	#322×C₆₇	C₂²	1	1	yes	characteristic
43	C₈²×C₆₇	C₂²	1	1	yes	characteristic
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆×C₆₇	—	2	1	no	—
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	#126×C₆₇	—	2	2	no	—
55	#125×C₆₇	C₂×C₄	1	2	yes	—
56	#124×C₆₇	C₂×C₄	1	2	yes	—
57	#123×C₆₇	—	2	2	no	—
58	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
59	#121×C₆₇	C₂×C₄	1	2	yes	—
60	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	C₂³	1	2	yes	—
63	#117×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
66	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
67	#106×C₆₇	C₂×C₄	1	2	yes	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	—	2	2	no	—
71	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
75	#103×C₆₇	C₂×C₄	1	2	yes	—
76	#103×C₆₇	C₂×C₄	1	2	yes	—
77	#102×C₆₇	C₂×C₄	1	2	yes	—
78	#102×C₆₇	C₂×C₄	1	2	yes	—
79	C₂²⋊C₈×C₆₇	—	2	2	no	—
80	C₂²⋊C₈×C₆₇	—	2	2	no	—
81	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
82	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
83	#97×C₆₇	—	2	2	no	—
84	#97×C₆₇	—	2	1	no	—
85	C₄×C₈×C₆₇	—	2	2	no	—
86	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
101	Q₁₆×C₆₇	—	2	2	no	—
102	SD₁₆×C₆₇	—	4	2	no	—
103	D₈×C₆₇	—	2	2	no	—
104	M₁₆×C₆₇	—	2	4	no	—
105	M₁₆×C₆₇	—	8	2	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	4	1	no	—
112	C₂×C₈×C₆₇	—	8	2	no	—
113	C₂×C₈×C₆₇	—	8	2	no	—
114	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₆₇	—	2	2	no	—
129	C₂²⋊C₄×C₆₇	—	2	2	no	—
130	C₂²⋊C₄×C₆₇	—	2	2	no	—
131	C₂²⋊C₄×C₆₇	—	2	2	no	—
132	C₂²⋊C₄×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₆₇	#41	1	1	yes	characteristic
137	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
140	Q₈×C₆₇	—	2	2	no	—
141	Q₈×C₆₇	—	2	2	no	—
142	D₄×C₆₇	—	2	2	no	—
143	D₄×C₆₇	—	2	2	no	—
144	D₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
161	C₈×C₆₇	—	2	2	no	—
162	C₈×C₆₇	—	2	2	no	—
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	2	1	no	—
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	1	no	—
170	C₈×C₆₇	—	4	2	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	#131	1	2	yes	—
173	C₂²×C₆₇	—	2	1	no	—
174	C₂²×C₆₇	—	2	1	no	—
175	C₂²×C₆₇	—	4	2	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	#443	1	2	yes	—
179	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	—	2	1	no	—
183	C₄×C₆₇	—	2	1	no	—
184	C₄×C₆₇	—	4	2	no	—
185	C₄×C₆₇	—	4	2	no	—
186	C₄×C₆₇	—	4	2	no	—
187	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
188	C₄×C₆₇	#568	1	1	yes	characteristic
189	C₄×C₆₇	#576	1	1	yes	characteristic
190	C₄×C₆₇	#578	1	1	yes	characteristic
191	#18087	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂×C₆₇	—	2	2	no	—
193	C₂×C₆₇	—	8	2	no	—
194	C₂×C₆₇	#2935	1	1	yes	characteristic
195	C₂×C₆₇	#2951	1	1	yes	characteristic
196	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
197	#3168	C₂×C₆₇	1	1	yes	characteristic
198	#2994	C₂×C₆₇	1	2	yes	complemented
199	#1591	C₂×C₆₇	1	1	yes	characteristic, complemented
200	#1589	C₂×C₆₇	1	1	yes	characteristic, complemented
201	#1588	C₂×C₆₇	1	2	yes	complemented
202	#1585	C₂×C₆₇	1	2	yes	complemented
203	C₈×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
204	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
205	C₂×#335	C₂×C₆₇	1	1	yes	characteristic, complemented
206	#1483	C₂×C₆₇	1	2	yes	complemented
207	C₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
208	C₆₇	#18087	1	1	yes	characteristic, direct product factor, Sylow
209	#529	C₂²×C₆₇	1	2	yes	—
210	#499	C₄×C₆₇	1	1	yes	characteristic
211	#495	C₄×C₆₇	1	1	yes	characteristic
212	#488	C₄×C₆₇	1	2	yes	—
213	#467	C₄×C₆₇	1	1	yes	characteristic
214	#466	C₄×C₆₇	1	1	yes	characteristic
215	#465	C₄×C₆₇	1	2	yes	—
216	#446	C₂²×C₆₇	1	2	yes	—
217	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
218	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
219	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
220	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
221	#436	C₂²×C₆₇	1	2	yes	—
222	#435	—	2	2	no	—
223	C₈×D₄	—	2	2	no	—
224	C₈×D₄	C₂²×C₆₇	1	2	yes	—
225	#433	—	2	2	no	—
226	#431	C₂²×C₆₇	1	1	yes	characteristic
227	#424	—	2	2	no	—
228	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
229	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
230	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
231	#335	C₂²×C₆₇	1	1	yes	characteristic
232	#335	C₂²×C₆₇	1	2	yes	—
233	#335	C₂²×C₆₇	1	1	yes	characteristic
234	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
235	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
236	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
237	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
238	#322	C₂²×C₆₇	1	1	yes	characteristic
239	#322	C₂²×C₆₇	1	1	yes	characteristic
240	C₈²	C₂²×C₆₇	1	1	yes	characteristic
241	C₈²	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
243	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
244	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
245	C₂×M₁₆	—	2	1	no	—
246	C₂×M₁₆	—	2	2	no	—
247	C₂²×C₈	—	2	2	no	—
248	C₂²×C₈	—	2	2	no	—
249	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
250	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
251	#126	—	2	2	no	—
252	#125	C₂×C₄×C₆₇	1	2	yes	—
253	#124	C₂×C₄×C₆₇	1	2	yes	—
254	#123	—	2	2	no	—
255	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
256	#121	C₂×C₄×C₆₇	1	2	yes	—
257	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
258	C₄×D₄	—	2	2	no	—
259	C₄×D₄	C₂³×C₆₇	1	2	yes	—
260	#117	—	2	2	no	—
261	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
262	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
263	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
264	#106	C₂×C₄×C₆₇	1	2	yes	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	2	2	no	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
269	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
270	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
271	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
272	#103	C₂×C₄×C₆₇	1	2	yes	—
273	#103	C₂×C₄×C₆₇	1	2	yes	—
274	#102	C₂×C₄×C₆₇	1	2	yes	—
275	#102	C₂×C₄×C₆₇	1	2	yes	—
276	C₂²⋊C₈	—	2	2	no	—
277	C₂²⋊C₈	—	2	2	no	—
278	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
279	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
280	#97	—	2	2	no	—
281	#97	—	2	1	no	—
282	C₄×C₈	—	2	2	no	—
283	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
284	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
285	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
286	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
287	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
288	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
289	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
290	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
291	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
298	Q₁₆	—	2	2	no	—
299	SD₁₆	—	4	2	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	2	4	no	—
302	M₁₆	—	8	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	—	8	2	no	—
310	C₂×C₈	—	8	2	no	—
311	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
312	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
313	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
314	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
315	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
316	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
318	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	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	C₂²⋊C₄	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
332	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
333	C₄²	#41×C₆₇	1	1	yes	characteristic
334	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
335	C₂³	—	2	2	no	—
336	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
337	Q₈	—	2	2	no	—
338	Q₈	—	2	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	4	4	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
354	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
358	C₈	—	2	2	no	—
359	C₈	—	2	2	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	2	1	no	—
365	C₈	—	2	1	no	—
366	C₈	—	2	1	no	—
367	C₈	—	4	2	no	—
368	C₈	—	4	2	no	—
369	C₈	#131×C₆₇	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	#443×C₆₇	1	2	yes	—
376	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	4	2	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	2	no	—
384	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
385	C₄	#568×C₆₇	1	1	yes	characteristic
386	C₄	#576×C₆₇	1	1	yes	characteristic
387	C₄	#578×C₆₇	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	8	2	no	—
390	C₂	C₆₇×#2935	1	1	yes	characteristic
391	C₂	C₆₇×#2951	1	1	yes	characteristic
392	C₂	C₂×#443×C₆₇	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	#466×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
2. C₆₇×#2994	C₂	semidirect
3. C₆₇×#1591	C₂	semidirect
4. C₆₇×#1589	C₂	semidirect
5. C₆₇×#1588	C₂	semidirect
6. C₆₇×#1585	C₂	semidirect
7. C₈×Q₁₆×C₆₇	C₂	semidirect
8. C₈×D₈×C₆₇	C₂	semidirect
9. C₂×#335×C₆₇	C₂	semidirect
10. C₆₇×#1483	C₂	semidirect
11. C₈×M₁₆×C₆₇	C₂	semidirect
191. #18087	C₆₇	direct (N × H)
198. #2994	C₂×C₆₇	semidirect
199. #1591	C₂×C₆₇	semidirect
200. #1589	C₂×C₆₇	semidirect
201. #1588	C₂×C₆₇	semidirect
202. #1585	C₂×C₆₇	semidirect
203. C₈×Q₁₆	C₂×C₆₇	semidirect
204. C₈×D₈	C₂×C₆₇	semidirect
205. C₂×#335	C₂×C₆₇	semidirect
206. #1483	C₂×C₆₇	semidirect
207. C₈×M₁₆	C₂×C₆₇	semidirect
208. C₆₇	#18087	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3168	2
C₆₇×#2994	2
C₆₇×#1591	2
C₆₇×#1589	2
C₆₇×#1588	2
C₆₇×#1585	2
C₈×Q₁₆×C₆₇	2
C₈×D₈×C₆₇	2
C₂×#335×C₆₇	2
C₆₇×#1483	2
C₈×M₁₆×C₆₇	2
#18087	67

Subgroups — C67×#18087 #30697338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18087 #30697338