Subgroups — C₈×C₆₇⋊#95 #30760987

Order 17152.

1174 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₆₇	C₂	1	2	yes	maximal
2	C₂×#16863151	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863150	C₂	1	2	yes	maximal
4	C₄×C₆₇⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
8	C₂×C₈×Dic₆₇	—	2	4	no	—
9	C₂×C₈×Dic₆₇	—	2	4	no	—
10	C₂×C₈×Dic₆₇	C₄	1	8	yes	complemented
11	C₂×C₆₇⋊#95	C₄	1	4	yes	—
12	#16863151	C₄	1	4	yes	complemented
13	#16863150	C₄	1	8	yes	complemented
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	—	2	4	no	—
18	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	complemented
19	C₂³×Dic₆₇	C₂×C₄	1	4	yes	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₆₇	—	2	2	no	—
28	C₂×C₄×Dic₆₇	—	2	2	no	—
29	C₂×C₄×Dic₆₇	—	2	2	no	—
30	C₂×C₄×Dic₆₇	—	2	2	no	—
31	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
33	C₈×Dic₆₇	—	2	16	no	—
34	C₆₇⋊#95	C₈	1	16	yes	direct product factor
35	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	—	2	1	no	—
39	C₂²×C₈×C₆₇	—	2	1	no	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	C₂²×C₈×C₆₇	D₄	1	4	yes	—
46	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
47	C₂×C₄²×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₆₇	—	2	8	no	—
52	C₂²×Dic₆₇	—	2	4	no	—
53	C₂²×Dic₆₇	—	2	4	no	—
54	C₂²×Dic₆₇	—	2	8	no	—
55	C₂²×Dic₆₇	C₄²	1	8	yes	—
56	C₂²×Dic₆₇	C₂×C₈	1	16	yes	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	2	4	no	—
62	C₂×C₆₇⋊C₈	—	2	4	no	—
63	C₂×C₆₇⋊C₈	C₄²	1	8	yes	—
64	C₄×Dic₆₇	—	2	4	no	—
65	C₄×Dic₆₇	—	2	4	no	—
66	C₄×Dic₆₇	—	2	4	no	—
67	C₄×Dic₆₇	—	2	4	no	—
68	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	C₄²	1	4	yes	—
88	C₂×C₈×C₆₇	C₄²	1	4	yes	—
89	C₂×C₈×C₆₇	C₂²⋊C₄	1	8	yes	—
90	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
91	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
92	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	4	no	—
96	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
97	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
98	C₂³×C₆₇	C₄×D₄	1	2	yes	—
99	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
100	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
101	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
102	C₂³×C₆₇	#131	1	4	yes	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	16	no	—
106	C₂×C₄×C₆₇	C₄×C₈	1	8	yes	complemented
107	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#117	1	2	yes	—
115	C₂×C₄×C₆₇	#117	1	2	yes	—
116	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
125	C₂×Dic₆₇	—	2	8	no	—
126	C₂×Dic₆₇	—	2	8	no	—
127	C₂×Dic₆₇	—	2	16	no	—
128	C₂×Dic₆₇	—	2	16	no	—
129	C₂×Dic₆₇	C₄×C₈	1	16	yes	complemented
130	C₈×C₆₇	—	2	4	no	—
131	C₈×C₆₇	#95	1	8	yes	complemented
132	C₆₇⋊C₈	—	2	8	no	—
133	C₂²×C₆₇	C₄³	1	1	yes	characteristic
134	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
136	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
138	C₂²×C₆₇	C₂×C₄×C₈	1	4	yes	—
139	C₂²×C₆₇	C₂×C₄×C₈	1	2	yes	—
140	C₂²×C₆₇	#405	1	4	yes	—
141	C₂²×C₆₇	#405	1	2	yes	—
142	C₂²×C₆₇	C₈×D₄	1	8	yes	—
143	C₂²×C₆₇	C₈×D₄	1	4	yes	—
144	C₂²×C₆₇	C₈×Q₈	1	4	yes	—
145	C₄×C₆₇	—	2	8	no	—
146	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
147	C₄×C₆₇	#376	1	1	yes	characteristic
148	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
150	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
152	Dic₆₇	—	2	16	no	—
153	C₈×#95	—	67	1	no	maximal, Sylow
154	C₂×C₆₇	C₄×#95	1	1	yes	characteristic
155	C₂×C₆₇	C₄²×C₈	1	1	yes	characteristic
156	C₂×C₆₇	#1726	1	1	yes	characteristic
157	C₂×C₆₇	C₈×C₂²⋊C₄	1	2	yes	—
158	C₂×C₆₇	C₈×C₂²⋊C₄	1	4	yes	—
159	C₂×C₆₇	C₈×C₄⋊C₄	1	2	yes	—
160	C₂×C₆₇	C₈×C₄⋊C₄	1	4	yes	—
161	C₂²×C₄×C₈	—	67	2	no	—
162	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
163	C₂×#336	—	67	1	no	—
164	C₂×#336	—	67	2	no	—
165	C₄×#95	—	67	1	no	—
166	C₆₇	C₈×#95	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₈	—	67	2	no	—
168	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
169	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
170	C₂²×C₄²	—	67	2	no	—
171	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
172	C₂×C₄×C₈	—	2	4	no	—
173	C₂×C₄×C₈	—	67	8	no	—
174	C₂×C₄×C₈	—	134	4	no	—
175	C₂×C₄×C₈	—	134	4	no	—
176	C₂×C₄×C₈	Dic₆₇	1	4	yes	complemented
177	C₂×#95	—	67	4	no	—
178	#336	—	67	8	no	—
179	#336	—	67	4	no	—
180	C₂³×C₄	—	67	4	no	—
181	C₂³×C₄	C₄×D₆₇	1	2	yes	—
182	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	67	4	no	—
187	C₂²×C₈	—	67	4	no	—
188	C₂²×C₈	—	67	4	no	—
189	C₂²×C₈	—	134	2	no	—
190	C₂²×C₈	—	134	2	no	—
191	C₂²×C₈	—	134	2	no	—
192	C₂²×C₈	—	134	2	no	—
193	C₂²×C₈	Dic₁₃₄	1	2	yes	—
194	C₂²×C₈	C₄×D₆₇	1	4	yes	—
195	C₂²×C₈	C₄×D₆₇	1	4	yes	—
196	C₂²×C₈	D₂₆₈	1	2	yes	—
197	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
198	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
199	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	67	4	no	—
203	C₂×C₄²	—	67	4	no	—
204	C₂×C₄²	—	134	2	no	—
205	C₂×C₄²	—	134	2	no	—
206	C₂×C₄²	—	134	2	no	—
207	C₂×C₄²	—	134	2	no	—
208	C₂×C₄²	C₄×D₆₇	1	2	yes	—
209	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
210	C₄×C₈	—	2	8	no	—
211	C₄×C₈	—	134	16	no	—
212	#95	—	67	16	no	—
213	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	67	8	no	—
217	C₂²×C₄	—	67	16	no	—
218	C₂²×C₄	—	134	4	no	—
219	C₂²×C₄	—	134	4	no	—
220	C₂²×C₄	—	134	8	no	—
221	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
222	C₂²×C₄	C₈×D₆₇	1	8	yes	—
223	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
228	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
230	C₂²×C₄	#11761628	1	1	yes	characteristic
231	C₂²×C₄	#11761628	1	1	yes	characteristic
232	C₂²×C₄	#11761628	1	2	yes	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	67	8	no	—
238	C₂×C₈	—	134	4	no	—
239	C₂×C₈	—	134	4	no	—
240	C₂×C₈	—	134	4	no	—
241	C₂×C₈	—	134	4	no	—
242	C₂×C₈	—	134	4	no	—
243	C₂×C₈	—	134	4	no	—
244	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
245	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
246	C₂×C₈	(C₂×D₆₇)⋊C₄	1	8	yes	—
247	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
248	C₂×C₈	Dic₆₇⋊C₄	1	8	yes	—
249	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	134	4	no	—
254	C₄²	—	134	4	no	—
255	C₄²	—	134	4	no	—
256	C₄²	—	134	4	no	—
257	C₂³	C₄²×D₆₇	1	2	yes	—
258	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
259	C₂³	C₄×C₂²⋊D₆₇	1	2	yes	—
260	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
261	C₂³	C₄×Dic₁₃₄	1	1	yes	characteristic
262	C₂³	C₂×C₈×D₆₇	1	4	yes	—
263	C₂³	#14524049	1	4	yes	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	16	no	—
267	C₂×C₄	—	67	16	no	—
268	C₂×C₄	—	134	8	no	—
269	C₂×C₄	—	134	8	no	—
270	C₂×C₄	—	134	16	no	—
271	C₂×C₄	—	134	16	no	—
272	C₂×C₄	C₈×Dic₆₇	1	8	yes	complemented
273	C₂×C₄	C₄²×D₆₇	1	2	yes	—
274	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
277	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
278	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
279	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
280	C₂×C₄	#14524000	1	2	yes	—
281	C₂×C₄	#14524000	1	2	yes	—
282	C₂×C₄	#14524002	1	1	yes	characteristic
283	C₂×C₄	#14524002	1	1	yes	characteristic
284	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
285	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
286	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
287	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
288	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
289	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
290	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	134	8	no	—
293	C₈	C₆₇⋊#95	1	8	yes	direct product factor
294	C₂²	C₄²×Dic₆₇	1	1	yes	characteristic
295	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
296	C₂²	C₄×(C₂×D₆₇)⋊C₄	1	2	yes	—
297	C₂²	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
298	C₂²	C₄×Dic₆₇⋊C₄	1	2	yes	—
299	C₂²	C₄×C₈×D₆₇	1	4	yes	—
300	C₂²	C₂×C₈×Dic₆₇	1	2	yes	—
301	C₂²	#16863350	1	4	yes	—
302	C₂²	#16863351	1	2	yes	—
303	C₂²	C₈×C₂²⋊D₆₇	1	8	yes	—
304	C₂²	C₈×D₂₆₈	1	4	yes	—
305	C₂²	C₈×Dic₁₃₄	1	4	yes	—
306	C₄	—	2	8	no	—
307	C₄	—	134	16	no	—
308	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
309	C₄	#16863242	1	1	yes	characteristic
310	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
311	C₄	C₄×(C₂×D₆₇)⋊C₄	1	2	yes	—
312	C₄	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
313	C₄	C₄×Dic₆₇⋊C₄	1	2	yes	—
314	C₂	C₄×C₆₇⋊#95	1	1	yes	characteristic
315	C₂	C₄×C₈×Dic₆₇	1	1	yes	characteristic
316	C₂	#22559895	1	1	yes	characteristic
317	C₂	C₈×C₂²⋊Dic₆₇	1	2	yes	—
318	C₂	C₈×(C₂×D₆₇)⋊C₄	1	4	yes	—
319	C₂	C₈×C₄⋊Dic₆₇	1	2	yes	—
320	C₂	C₈×Dic₆₇⋊C₄	1	4	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₆₇
commutator	C₂×C₆₇	C₄²×C₈
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	C₈×#95
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. C₂×C₈×Dic₆₇	C₄	semidirect
12. #16863151	C₄	semidirect
13. #16863150	C₄	semidirect
18. C₂×C₄×C₈×C₆₇	C₄	semidirect
34. C₆₇⋊#95	C₈	direct (N × H)
106. C₂×C₄×C₆₇	C₄×C₈	semidirect
129. C₂×Dic₆₇	C₄×C₈	semidirect
131. C₈×C₆₇	#95	semidirect
166. C₆₇	C₈×#95	semidirect
176. C₂×C₄×C₈	Dic₆₇	semidirect
272. C₂×C₄	C₈×Dic₆₇	semidirect
293. C₈	C₆₇⋊#95	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₆₇	2
C₂×#16863151	2
C₂×#16863150	2
C₄×C₆₇⋊#95	2
C₂²×C₄×C₈×C₆₇	2
C₈×#95	67

Subgroups — C8×C67⋊#95 #30760987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₆₇⋊#95 #30760987