Subgroups — C₈×#14523966 #30746604

Order 17152.

518 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523966	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863126	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#22558050	C₂	1	1	yes	maximal, characteristic
6	#22558043	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₃₄	—	2	2	no	—
10	C₈×Dic₁₃₄	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₈	—	2	1	no	—
13	C₂×#14523966	C₄	1	2	yes	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₇	—	2	1	no	—
17	#16863158	C₄	1	2	yes	—
18	#16863126	C₄	1	4	yes	complemented
19	C₈×C₆₇⋊C₈	—	2	2	no	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	#336×C₆₇	—	2	1	no	—
25	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	—	2	1	no	—
27	C₂²×C₆₇⋊C₈	—	2	1	no	—
28	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₃₄	—	2	1	no	—
30	C₄×Dic₁₃₄	—	2	1	no	—
31	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×Dic₆₇⋊C₄	—	2	1	no	—
35	C₂×C₄×Dic₆₇	—	2	1	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
37	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
38	Dic₆₇⋊C₈	—	2	4	no	—
39	#14523966	C₈	1	8	yes	direct product factor
40	C₄×C₆₇⋊C₈	—	2	1	no	—
41	C₄×C₆₇⋊C₈	—	2	1	no	—
42	C₄×C₆₇⋊C₈	—	2	1	no	—
43	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
44	C₈×Dic₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	—	2	1	no	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₆₇	—	2	1	no	—
52	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₆₇	—	4	2	no	—
54	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₆₇	D₄	1	2	yes	—
56	#95×C₆₇	—	2	1	no	—
57	C₂×Dic₁₃₄	—	2	1	no	—
58	C₂×Dic₁₃₄	—	2	2	no	—
59	C₂×Dic₁₃₄	C₄²	1	2	yes	—
60	C₂×Dic₁₃₄	C₂×C₈	1	4	yes	—
61	C₂²×Dic₆₇	—	2	1	no	—
62	C₂²×Dic₆₇	—	2	1	no	—
63	C₂×C₆₇⋊C₈	—	2	4	no	—
64	C₂×C₆₇⋊C₈	—	2	2	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	—	4	1	no	—
69	C₂×C₆₇⋊C₈	—	4	1	no	—
70	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
71	C₄⋊Dic₆₇	—	2	1	no	—
72	C₄⋊Dic₆₇	C₄²	1	2	yes	—
73	Dic₆₇⋊C₄	—	2	2	no	—
74	Dic₆₇⋊C₄	—	2	2	no	—
75	C₄×Dic₆₇	—	2	2	no	—
76	C₄×Dic₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₆₇	—	2	1	no	—
82	C₂×C₈×C₆₇	—	2	1	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	4	1	no	—
85	C₂×C₈×C₆₇	—	4	1	no	—
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
89	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
92	C₄²×C₆₇	—	4	1	no	—
93	C₄²×C₆₇	—	4	1	no	—
94	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₇	#41	1	1	yes	characteristic
98	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₆₇	#131	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
115	C₂×Dic₆₇	—	2	2	no	—
116	C₂×Dic₆₇	—	2	4	no	—
117	C₂×Dic₆₇	—	2	2	no	—
118	C₂×Dic₆₇	—	4	2	no	—
119	Dic₁₃₄	—	2	2	no	—
120	Dic₁₃₄	—	2	4	no	—
121	Dic₁₃₄	C₄×C₈	1	4	yes	complemented
122	C₈×C₆₇	—	2	2	no	—
123	C₈×C₆₇	—	4	2	no	—
124	C₈×C₆₇	#103	1	4	yes	complemented
125	C₆₇⋊C₈	—	2	4	no	—
126	C₆₇⋊C₈	—	2	4	no	—
127	C₆₇⋊C₈	—	4	2	no	—
128	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₆₇	C₈×D₄	1	2	yes	—
130	C₂²×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
131	C₂²×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
132	C₂²×C₆₇	#443	1	2	yes	—
133	C₄×C₆₇	—	2	1	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	4	4	no	—
136	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₆₇	#405	1	1	yes	characteristic
138	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
139	C₄×C₆₇	#416	1	1	yes	characteristic
140	C₄×C₆₇	C₈×D₄	1	2	yes	—
141	C₄×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
142	C₄×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
143	Dic₆₇	—	2	4	no	—
144	Dic₆₇	—	4	2	no	—
145	C₈×#103	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	C₈×SD₁₆	1	2	yes	—
147	C₂×C₆₇	C₈×Q₁₆	1	2	yes	—
148	C₂×C₆₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₆₇	C₄×#103	1	1	yes	characteristic
150	C₂×C₆₇	#1764	1	1	yes	characteristic
151	C₂×C₈×Q₈	—	67	1	no	—
152	C₈×C₄⋊C₄	D₆₇	1	1	yes	characteristic
153	C₄×#103	—	67	1	no	—
154	C₂×Q₈⋊C₈	—	67	1	no	—
155	C₂×C₈²	—	67	1	no	—
156	#1279	—	67	1	no	—
157	#1277	—	67	1	no	—
158	C₆₇	C₈×#103	1	1	yes	characteristic, complemented, Sylow
159	C₂×C₄×Q₈	—	67	1	no	—
160	C₈×Q₈	—	67	4	no	—
161	C₈×Q₈	—	134	2	no	—
162	C₂×C₄⋊C₈	—	67	1	no	—
163	C₂×C₄⋊C₈	—	134	1	no	—
164	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
165	C₂×#103	—	67	2	no	—
166	C₂×C₄×C₈	—	2	1	no	—
167	C₂×C₄×C₈	—	67	1	no	—
168	C₂×C₄×C₈	—	67	1	no	—
169	C₂×C₄×C₈	—	134	1	no	—
170	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
171	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
172	#340	—	67	2	no	—
173	#336	—	2	1	no	—
174	Q₈⋊C₈	—	67	4	no	—
175	C₈²	—	134	2	no	—
176	C₂²×Q₈	—	67	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	67	2	no	—
179	C₂²×C₈	—	134	1	no	—
180	C₂²×C₈	—	134	1	no	—
181	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
183	C₄×Q₈	—	67	2	no	—
184	C₄×Q₈	—	67	2	no	—
185	C₄×Q₈	—	134	1	no	—
186	C₄×Q₈	—	134	1	no	—
187	C₂×C₄⋊C₄	—	67	1	no	—
188	C₂×C₄⋊C₄	—	134	1	no	—
189	C₂×C₄⋊C₄	C₄×D₆₇	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	134	1	no	—
192	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	67	2	no	—
195	C₄⋊C₈	—	134	4	no	—
196	C₄⋊C₈	—	134	1	no	—
197	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
198	#103	—	67	8	no	—
199	C₄×C₈	—	4	2	no	—
200	C₄×C₈	—	67	2	no	—
201	C₄×C₈	—	134	4	no	—
202	C₄×C₈	—	134	1	no	—
203	C₄×C₈	—	134	1	no	—
204	C₄×C₈	—	134	1	no	—
205	C₄×C₈	C₄×D₆₇	1	2	yes	—
206	C₄×C₈	D₂₆₈	1	2	yes	—
207	#95	—	2	1	no	—
208	C₂×Q₈	—	67	2	no	—
209	C₂×Q₈	—	67	4	no	—
210	C₂×Q₈	—	134	1	no	—
211	C₂×Q₈	—	134	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	134	1	no	—
214	C₂²×C₄	—	134	1	no	—
215	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
217	C₂²×C₄	#11761628	1	1	yes	characteristic
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	268	1	no	—
224	C₂×C₈	—	268	1	no	—
225	C₂×C₈	—	67	4	no	—
226	C₂×C₈	—	134	2	no	—
227	C₂×C₈	—	134	4	no	—
228	C₂×C₈	—	134	2	no	—
229	C₂×C₈	—	134	2	no	—
230	C₂×C₈	—	134	2	no	—
231	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
232	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
233	C₂×C₈	#11761616	1	2	yes	—
234	C₂×C₈	#11761618	1	2	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	67	2	no	—
237	C₄⋊C₄	—	134	2	no	—
238	C₄⋊C₄	—	134	1	no	—
239	C₄⋊C₄	—	134	2	no	—
240	C₄⋊C₄	C₈×D₆₇	1	4	yes	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	134	2	no	—
244	C₄²	—	134	2	no	—
245	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
247	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
248	C₄²	#11761628	1	1	yes	characteristic
249	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
250	Q₈	—	67	4	no	—
251	Q₈	—	134	2	no	—
252	Q₈	—	134	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	268	2	no	—
258	C₂×C₄	—	134	2	no	—
259	C₂×C₄	—	134	2	no	—
260	C₂×C₄	—	134	4	no	—
261	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
262	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	#14524000	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
265	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
266	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
268	C₂×C₄	#14524049	1	2	yes	—
269	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
270	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
271	C₂×C₄	#14524063	1	1	yes	characteristic
272	C₂×C₄	#14524063	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	4	2	no	—
275	C₈	—	268	2	no	—
276	C₈	—	134	4	no	—
277	C₈	—	134	4	no	—
278	C₈	#14523966	1	4	yes	direct product factor
279	C₂²	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
280	C₂²	C₈×C₂²⋊D₆₇	1	2	yes	—
281	C₂²	C₄×#11761616	1	1	yes	characteristic
282	C₂²	C₄×#11761618	1	1	yes	characteristic
283	C₂²	#16863511	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	268	2	no	—
288	C₄	—	134	4	no	—
289	C₄	C₄×C₈×D₆₇	1	1	yes	characteristic
290	C₄	#16863350	1	1	yes	characteristic
291	C₄	C₂×#14523966	1	1	yes	characteristic
292	C₄	#16863390	1	1	yes	characteristic
293	C₄	C₈×D₂₆₈	1	2	yes	—
294	C₄	C₄×#11761616	1	1	yes	characteristic
295	C₄	C₄×#11761618	1	1	yes	characteristic
296	C₂	C₈×#11761616	1	2	yes	—
297	C₂	C₈×#11761618	1	2	yes	—
298	C₂	C₈×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
299	C₂	C₄×#14523966	1	1	yes	characteristic
300	C₂	#22560048	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₈×C₄⋊C₄×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₆₇
hypocenter	C₆₇	C₈×#103
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₁₃₄	C₄	semidirect
18. #16863126	C₄	semidirect
39. #14523966	C₈	direct (N × H)
121. Dic₁₃₄	C₄×C₈	semidirect
124. C₈×C₆₇	#103	semidirect
158. C₆₇	C₈×#103	semidirect
278. C₈	#14523966	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
C₄×#14523966	2
C₂×#16863126	2
C₂×C₈×C₆₇⋊C₈	2
#22558050	2
#22558043	2
C₈×C₄⋊C₄×C₆₇	2
C₈×#103	67

Subgroups — C8×#14523966 #30746604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14523966 #30746604