Subgroups — C₆₇×#17855 #30697106

Order 17152.

730 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2982	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×SD₁₆⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₈×SD₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1478	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₆₇	—	2	2	no	—
14	C₈×Q₈×C₆₇	C₂²	1	4	yes	complemented
15	C₄×SD₁₆×C₆₇	C₄	1	2	yes	—
16	C₄×SD₁₆×C₆₇	C₄	1	2	yes	—
17	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
18	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
19	C₈×D₄×C₆₇	—	2	2	no	—
20	C₈×D₄×C₆₇	C₂²	1	4	yes	—
21	#431×C₆₇	—	2	2	no	—
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	#431×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#106×C₆₇	C₄	1	1	yes	characteristic
27	C₂×#103×C₆₇	C₄	1	1	yes	characteristic
28	C₂×#102×C₆₇	C₄	1	1	yes	characteristic
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	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	complemented
34	D₄⋊C₈×C₆₇	C₂²	1	4	yes	—
35	#322×C₆₇	C₂²	1	2	yes	complemented
36	C₈²×C₆₇	C₂²	1	2	yes	complemented
37	C₂²×Q₈×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	C₂×SD₁₆×C₆₇	C₈	1	8	yes	complemented
41	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₆₇	—	2	1	no	—
47	C₄×Q₈×C₆₇	—	2	1	no	—
48	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
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	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄²×C₆₇	—	2	1	no	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	#106×C₆₇	C₂×C₄	1	2	yes	—
61	#106×C₆₇	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₆₇	—	2	4	no	—
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	#103×C₆₇	C₂×C₄	1	2	yes	—
68	#103×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	D₄	1	2	yes	—
76	C₄×C₈×C₆₇	—	2	4	no	—
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	C₂×Q₈×C₆₇	—	2	1	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	complemented
89	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
93	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	—	2	1	no	—
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	—	8	2	no	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	SD₁₆×C₆₇	C₂×C₈	1	8	yes	complemented
104	SD₁₆×C₆₇	C₂×C₈	1	8	yes	complemented
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	8	4	no	—
110	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	complemented
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
121	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
122	C₂²⋊C₄×C₆₇	—	2	2	no	—
123	C₂²⋊C₄×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	#41	1	1	yes	characteristic
129	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
130	C₂³×C₆₇	—	2	2	no	—
131	C₂³×C₆₇	—	2	4	no	—
132	C₂³×C₆₇	—	8	1	no	—
133	C₂³×C₆₇	—	8	1	no	—
134	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₆₇	—	2	2	no	—
136	Q₈×C₆₇	—	2	4	no	—
137	Q₈×C₆₇	C₂²×C₈	1	4	yes	—
138	D₄×C₆₇	—	2	2	no	—
139	D₄×C₆₇	—	2	4	no	—
140	D₄×C₆₇	C₂²×C₈	1	4	yes	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	4	4	no	—
148	C₂×C₄×C₆₇	—	8	2	no	—
149	C₂×C₄×C₆₇	—	8	1	no	—
150	C₂×C₄×C₆₇	—	8	1	no	—
151	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
155	C₂×C₄×C₆₇	#131	1	2	yes	—
156	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
159	C₈×C₆₇	—	2	1	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	4	no	—
162	C₈×C₆₇	—	4	2	no	—
163	C₈×C₆₇	—	4	2	no	—
164	C₈×C₆₇	—	4	2	no	—
165	C₈×C₆₇	C₂²×C₈	1	2	yes	—
166	C₈×C₆₇	C₂²×C₈	1	2	yes	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	—	2	4	no	—
170	C₂²×C₆₇	—	4	2	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	—	8	4	no	—
173	C₂²×C₆₇	C₈×D₄	1	2	yes	—
174	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₆₇	—	2	4	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	4	2	no	—
180	C₄×C₆₇	—	4	2	no	—
181	C₄×C₆₇	C₈×D₄	1	2	yes	—
182	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
183	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
184	C₄×C₆₇	#577	1	2	yes	—
185	#17855	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂×C₆₇	—	2	2	no	—
187	C₂×C₆₇	—	4	4	no	—
188	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
189	C₂×C₆₇	#2949	1	1	yes	characteristic
190	C₂×C₆₇	#2957	1	1	yes	characteristic
191	#2982	C₂×C₆₇	1	1	yes	characteristic, complemented
192	#2972	C₂×C₆₇	1	1	yes	characteristic
193	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
194	SD₁₆⋊C₈	C₂×C₆₇	1	4	yes	complemented
195	C₈×SD₁₆	C₂×C₆₇	1	4	yes	complemented
196	#1564	C₂×C₆₇	1	1	yes	characteristic, complemented
197	#1479	C₂×C₆₇	1	1	yes	characteristic, complemented
198	#1478	C₂×C₆₇	1	1	yes	characteristic
199	#1450	C₂×C₆₇	1	1	yes	characteristic, complemented
200	C₆₇	#17855	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²×SD₁₆	C₄×C₆₇	1	1	yes	characteristic
202	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
203	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
204	C₈×Q₈	—	2	2	no	—
205	C₈×Q₈	C₂²×C₆₇	1	4	yes	complemented
206	C₄×SD₁₆	C₄×C₆₇	1	2	yes	—
207	C₄×SD₁₆	C₄×C₆₇	1	2	yes	—
208	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
209	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
210	C₈×D₄	—	2	2	no	—
211	C₈×D₄	C₂²×C₆₇	1	4	yes	—
212	#431	—	2	2	no	—
213	#431	C₂²×C₆₇	1	1	yes	characteristic
214	#431	C₂²×C₆₇	1	1	yes	characteristic
215	#431	C₂²×C₆₇	1	1	yes	characteristic
216	#431	C₂²×C₆₇	1	1	yes	characteristic
217	C₂×#106	C₄×C₆₇	1	1	yes	characteristic
218	C₂×#103	C₄×C₆₇	1	1	yes	characteristic
219	C₂×#102	C₄×C₆₇	1	1	yes	characteristic
220	C₂×C₂²⋊C₈	—	2	2	no	—
221	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
222	#335	C₂²×C₆₇	1	2	yes	complemented
223	#334	C₂²×C₆₇	1	2	yes	complemented
224	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	complemented
225	D₄⋊C₈	C₂²×C₆₇	1	4	yes	—
226	#322	C₂²×C₆₇	1	2	yes	complemented
227	C₈²	C₂²×C₆₇	1	2	yes	complemented
228	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
229	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
230	C₂³×C₄	—	2	1	no	—
231	C₂×SD₁₆	C₈×C₆₇	1	8	yes	complemented
232	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
233	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
234	C₂²×C₈	—	2	4	no	—
235	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
236	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
240	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	2	1	no	—
243	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
244	C₄×D₄	C₂³×C₆₇	1	2	yes	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
247	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
248	C₂×C₂²⋊C₄	—	2	1	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
251	#106	C₂×C₄×C₆₇	1	2	yes	—
252	#106	C₂×C₄×C₆₇	1	2	yes	—
253	C₄⋊C₈	—	2	4	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
256	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
257	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
258	#103	C₂×C₄×C₆₇	1	2	yes	—
259	#103	C₂×C₄×C₆₇	1	2	yes	—
260	#102	C₂×C₄×C₆₇	1	2	yes	—
261	#102	C₂×C₄×C₆₇	1	2	yes	—
262	C₂²⋊C₈	—	2	1	no	—
263	C₂²⋊C₈	—	2	1	no	—
264	C₂²⋊C₈	—	2	1	no	—
265	C₂²⋊C₈	—	2	4	no	—
266	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
267	C₄×C₈	—	2	4	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	1	no	—
270	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
271	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
272	C₄×C₈	C₂³×C₆₇	1	2	yes	—
273	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
274	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
275	C₄×C₈	C₂³×C₆₇	1	2	yes	—
276	C₂⁴	—	2	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	complemented
280	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	C₂×C₈×C₆₇	1	4	yes	—
284	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	8	2	no	—
291	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
293	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
294	SD₁₆	C₂×C₈×C₆₇	1	8	yes	complemented
295	SD₁₆	C₂×C₈×C₆₇	1	8	yes	complemented
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	8	4	no	—
301	C₂×C₈	C₂×C₈×C₆₇	1	4	yes	complemented
302	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
303	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
304	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
306	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
312	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
319	C₄²	#41×C₆₇	1	1	yes	characteristic
320	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	—	2	4	no	—
323	C₂³	—	8	1	no	—
324	C₂³	—	8	1	no	—
325	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
326	Q₈	—	2	2	no	—
327	Q₈	—	2	4	no	—
328	Q₈	C₂²×C₈×C₆₇	1	4	yes	—
329	D₄	—	2	2	no	—
330	D₄	—	2	4	no	—
331	D₄	C₂²×C₈×C₆₇	1	4	yes	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	4	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	8	1	no	—
342	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
343	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
346	C₂×C₄	#131×C₆₇	1	2	yes	—
347	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	4	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₈	—	4	2	no	—
356	C₈	C₂²×C₈×C₆₇	1	2	yes	—
357	C₈	C₂²×C₈×C₆₇	1	2	yes	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	4	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	8	4	no	—
364	C₂²	C₈×D₄×C₆₇	1	2	yes	—
365	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
366	C₄	—	2	4	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	C₈×D₄×C₆₇	1	2	yes	—
373	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
374	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
375	C₄	#577×C₆₇	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	4	4	no	—
378	C₂	C₂×C₈×D₄×C₆₇	1	1	yes	characteristic
379	C₂	C₆₇×#2949	1	1	yes	characteristic
380	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₄	C₂³×C₈×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₄×C₆₇	C₂×C₄
LCS	G ▷ 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×#2982	C₂	semidirect
4. C₆₇×SD₁₆⋊C₈	C₂	semidirect
5. C₈×SD₁₆×C₆₇	C₂	semidirect
6. C₆₇×#1564	C₂	semidirect
7. C₆₇×#1479	C₂	semidirect
9. C₆₇×#1450	C₂	semidirect
14. C₈×Q₈×C₆₇	C₂²	semidirect
31. #335×C₆₇	C₂²	semidirect
32. #334×C₆₇	C₂²	semidirect
33. Q₈⋊C₈×C₆₇	C₂²	semidirect
35. #322×C₆₇	C₂²	semidirect
36. C₈²×C₆₇	C₂²	semidirect
40. C₂×SD₁₆×C₆₇	C₈	semidirect
88. C₂×Q₈×C₆₇	C₂×C₈	semidirect
103. SD₁₆×C₆₇	C₂×C₈	semidirect
104. SD₁₆×C₆₇	C₂×C₈	semidirect
110. C₂×C₈×C₆₇	C₂×C₈	semidirect
185. #17855	C₆₇	direct (N × H)
191. #2982	C₂×C₆₇	semidirect
194. SD₁₆⋊C₈	C₂×C₆₇	semidirect
195. C₈×SD₁₆	C₂×C₆₇	semidirect
196. #1564	C₂×C₆₇	semidirect
197. #1479	C₂×C₆₇	semidirect
199. #1450	C₂×C₆₇	semidirect
200. C₆₇	#17855	direct (N × H)
205. C₈×Q₈	C₂²×C₆₇	semidirect
222. #335	C₂²×C₆₇	semidirect
223. #334	C₂²×C₆₇	semidirect
224. Q₈⋊C₈	C₂²×C₆₇	semidirect
226. #322	C₂²×C₆₇	semidirect
227. C₈²	C₂²×C₆₇	semidirect
231. C₂×SD₁₆	C₈×C₆₇	semidirect
279. C₂×Q₈	C₂×C₈×C₆₇	semidirect
294. SD₁₆	C₂×C₈×C₆₇	semidirect
295. SD₁₆	C₂×C₈×C₆₇	semidirect
301. C₂×C₈	C₂×C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2982	2
C₆₇×#2972	2
C₂×C₄×SD₁₆×C₆₇	2
C₆₇×SD₁₆⋊C₈	2
C₈×SD₁₆×C₆₇	2
C₆₇×#1564	2
C₆₇×#1479	2
C₆₇×#1478	2
C₆₇×#1450	2
#17855	67

Subgroups — C67×#17855 #30697106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17855 #30697106