Subgroups — C₈×#14523971 #30746539

Order 17152.

554 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11761628	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523971	C₂	1	1	yes	maximal, characteristic
3	#22558592	C₂	1	1	yes	maximal, characteristic
4	C₈×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558046	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558042	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11761628	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₃₄	C₄	1	2	yes	complemented
10	C₈×D₂₆₈	C₄	1	2	yes	complemented
11	C₈×C₂²⋊D₆₇	—	2	2	no	—
12	#16863449	C₂²	1	1	yes	characteristic
13	#16863449	C₂²	1	1	yes	characteristic
14	#16863447	—	2	1	no	—
15	C₂×#14523971	C₄	1	2	yes	—
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₆₇	—	2	1	no	—
18	#16863156	C₄	1	2	yes	—
19	#16863128	C₄	1	2	yes	complemented
20	#16863125	C₄	1	2	yes	complemented
21	#16863118	—	2	1	no	—
22	C₈×C₆₇⋊C₈	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	—	2	2	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₄³×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
28	C₂×#11761612	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₆₇	—	2	2	no	—
30	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
33	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₆₇	—	2	1	no	—
35	C₄×C₂²⋊D₆₇	—	2	1	no	—
36	#14524002	C₂×C₄	1	1	yes	characteristic
37	#14524000	—	2	1	no	—
38	C₄²×D₆₇	—	2	1	no	—
39	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
40	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
41	Dic₆₇⋊C₈	—	2	2	no	—
42	#14523971	C₈	1	8	yes	direct product factor
43	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
44	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
45	(C₂×D₆₇)⋊C₈	—	2	2	no	—
46	#14523954	—	2	1	no	—
47	C₄×C₆₇⋊C₈	—	2	1	no	—
48	C₄×C₆₇⋊C₈	—	2	1	no	—
49	C₄×C₆₇⋊C₈	—	2	1	no	—
50	C₈×Dic₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	—	2	1	no	—
52	C₂²×C₈×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	—	2	1	no	—
54	C₂×C₄²×C₆₇	—	2	1	no	—
55	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₆₇	—	2	8	no	—
58	C₄×C₈×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
63	#11761628	—	2	2	no	—
64	#11761628	C₂×C₈	1	4	yes	—
65	C₂×Dic₁₃₄	C₄²	1	1	yes	characteristic
66	C₂×D₂₆₈	C₄²	1	1	yes	characteristic
67	C₂×C₂²⋊D₆₇	—	2	1	no	—
68	C₂×C₄×D₆₇	—	2	1	no	—
69	C₂×C₄×D₆₇	—	2	1	no	—
70	C₂×C₄×D₆₇	—	2	1	no	—
71	#11761612	—	2	2	no	—
72	#11761612	C₂×C₈	1	4	yes	complemented
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₈×D₆₇	—	4	4	no	—
77	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
78	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
79	Dic₆₇⋊C₄	—	2	1	no	—
80	Dic₆₇⋊C₄	—	2	1	no	—
81	C₂²⋊Dic₆₇	—	2	1	no	—
82	(C₂×D₆₇)⋊C₄	—	2	1	no	—
83	(C₂×D₆₇)⋊C₄	—	2	1	no	—
84	C₄×Dic₆₇	—	2	1	no	—
85	C₄×Dic₆₇	—	2	1	no	—
86	C₄×Dic₆₇	—	2	1	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
96	C₄²×C₆₇	—	2	4	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	C₂×C₈	1	4	yes	complemented
100	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₆₇	#41	1	1	yes	characteristic
104	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₆₇	#131	1	2	yes	—
119	C₂²⋊D₆₇	—	2	4	no	—
120	C₂×Dic₆₇	—	2	1	no	—
121	C₂×Dic₆₇	—	2	1	no	—
122	C₂×Dic₆₇	—	2	1	no	—
123	D₂₆₈	—	2	1	no	—
124	D₂₆₈	C₄×C₈	1	2	yes	complemented
125	C₄×D₆₇	—	2	4	no	—
126	C₄×D₆₇	—	4	2	no	—
127	C₄×D₆₇	—	4	2	no	—
128	Dic₁₃₄	—	2	1	no	—
129	Dic₁₃₄	C₄×C₈	1	2	yes	complemented
130	C₈×C₆₇	—	2	4	no	—
131	C₈×C₆₇	—	2	2	no	—
132	C₈×C₆₇	#104	1	4	yes	complemented
133	C₆₇⋊C₈	—	2	4	no	—
134	C₆₇⋊C₈	—	4	2	no	—
135	C₆₇⋊C₈	—	4	2	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₆₇	C₈×D₄	1	2	yes	—
140	C₂×D₆₇	—	2	2	no	—
141	C₂×D₆₇	—	4	1	no	—
142	C₂×D₆₇	—	4	1	no	—
143	C₄×C₆₇	—	2	8	no	—
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	—	2	1	no	—
146	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
147	C₄×C₆₇	#405	1	1	yes	characteristic
148	C₄×C₆₇	C₂×#104	1	2	yes	—
149	C₄×C₆₇	C₈×D₄	1	2	yes	—
150	C₄×C₆₇	#443	1	2	yes	—
151	Dic₆₇	—	2	2	no	—
152	Dic₆₇	—	4	1	no	—
153	Dic₆₇	—	4	1	no	—
154	C₈×#104	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₆₇	C₄×#104	1	1	yes	characteristic
158	C₂×C₆₇	#1764	1	1	yes	characteristic
159	D₆₇	—	4	2	no	—
160	C₈×#41	—	67	1	no	—
161	C₄×#104	—	67	1	no	—
162	C₄²×C₈	D₆₇	1	1	yes	characteristic, complemented
163	#1476	—	67	1	no	—
164	C₈×M₁₆	—	67	1	no	—
165	#1278	—	67	1	no	—
166	#1276	—	67	1	no	—
167	C₆₇	C₈×#104	1	1	yes	characteristic, complemented, Sylow
168	C₄×#41	—	67	1	no	—
169	C₈×Q₈	—	67	2	no	—
170	C₈×D₄	—	67	2	no	—
171	C₈×D₄	—	134	2	no	—
172	#431	—	67	1	no	—
173	#431	—	67	1	no	—
174	#431	—	134	1	no	—
175	C₂×#104	—	67	2	no	—
176	C₄×M₁₆	—	67	1	no	—
177	C₂×C₄×C₈	—	2	2	no	—
178	C₂×C₄×C₈	—	134	1	no	—
179	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
181	C₄³	C₂×D₆₇	1	1	yes	characteristic
182	#339	—	67	2	no	—
183	Q₈⋊C₈	—	67	2	no	—
184	D₄⋊C₈	—	67	2	no	—
185	#322	—	134	1	no	—
186	C₈²	—	134	1	no	—
187	C₂×#41	—	67	1	no	—
188	C₂×M₁₆	—	67	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	134	2	no	—
191	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
192	C₄×Q₈	—	67	1	no	—
193	C₄×Q₈	—	67	1	no	—
194	C₄×D₄	—	67	1	no	—
195	C₄×D₄	—	67	1	no	—
196	C₄×D₄	—	134	1	no	—
197	C₄×D₄	—	134	1	no	—
198	#117	—	67	1	no	—
199	#117	—	134	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	134	1	no	—
203	C₂×C₄²	C₄×D₆₇	1	2	yes	—
204	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
205	C₄⋊C₈	—	67	2	no	—
206	C₄⋊C₈	—	67	2	no	—
207	C₄⋊C₈	—	134	2	no	—
208	#104	—	67	8	no	—
209	C₂²⋊C₈	—	134	1	no	—
210	C₂²⋊C₈	—	134	1	no	—
211	C₂²⋊C₈	—	134	2	no	—
212	#97	—	134	1	no	—
213	C₄×C₈	—	2	8	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	134	2	no	—
216	C₄×C₈	—	134	1	no	—
217	C₄×C₈	—	134	1	no	—
218	C₄×C₈	—	134	1	no	—
219	C₄×C₈	C₄×D₆₇	1	2	yes	—
220	C₄×C₈	C₄×D₆₇	1	2	yes	—
221	C₄×C₈	D₂₆₈	1	1	yes	characteristic
222	C₄×C₈	D₂₆₈	1	1	yes	characteristic
223	#41	—	67	4	no	—
224	#41	—	134	2	no	—
225	C₂×Q₈	—	67	1	no	—
226	C₂×D₄	—	67	1	no	—
227	C₂×D₄	—	134	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	134	1	no	—
230	C₂²×C₄	—	134	1	no	—
231	C₂²×C₄	—	134	1	no	—
232	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
234	C₂²×C₄	#11761628	1	1	yes	characteristic
235	M₁₆	—	67	4	no	—
236	M₁₆	—	134	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	268	4	no	—
241	C₂×C₈	—	134	2	no	—
242	C₂×C₈	—	134	2	no	—
243	C₂×C₈	—	134	2	no	—
244	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
245	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
246	C₄⋊C₄	—	67	1	no	—
247	C₄⋊C₄	—	67	1	no	—
248	C₄⋊C₄	—	134	1	no	—
249	C₄⋊C₄	—	134	1	no	—
250	C₂²⋊C₄	—	134	1	no	—
251	C₂²⋊C₄	—	134	1	no	—
252	C₂²⋊C₄	—	134	1	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	134	1	no	—
257	C₄²	—	134	1	no	—
258	C₄²	—	134	1	no	—
259	C₄²	C₈×D₆₇	1	4	yes	complemented
260	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
262	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
263	C₄²	#11761628	1	1	yes	characteristic
264	C₂³	—	134	1	no	—
265	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
266	Q₈	—	67	2	no	—
267	Q₈	—	134	1	no	—
268	D₄	—	67	2	no	—
269	D₄	—	134	4	no	—
270	D₄	—	134	1	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	268	2	no	—
277	C₂×C₄	—	268	2	no	—
278	C₂×C₄	—	134	1	no	—
279	C₂×C₄	—	134	1	no	—
280	C₂×C₄	—	134	1	no	—
281	C₂×C₄	—	134	4	no	—
282	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
283	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#14524000	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
286	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
287	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
289	C₂×C₄	#14524049	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	2	2	no	—
292	C₈	—	268	2	no	—
293	C₈	—	268	2	no	—
294	C₈	—	134	4	no	—
295	C₈	#14523971	1	4	yes	direct product factor
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	268	1	no	—
299	C₂²	—	268	1	no	—
300	C₂²	—	134	2	no	—
301	C₂²	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
302	C₂²	C₈×C₂²⋊D₆₇	1	2	yes	—
303	C₄	—	2	8	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	268	1	no	—
307	C₄	—	268	1	no	—
308	C₄	—	134	2	no	—
309	C₄	C₄×C₈×D₆₇	1	1	yes	characteristic
310	C₄	#16863350	1	1	yes	characteristic
311	C₄	C₂×#14523971	1	2	yes	—
312	C₄	C₈×D₂₆₈	1	2	yes	—
313	C₄	#16863514	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	268	2	no	—
316	C₂	C₈×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
317	C₂	C₄×#14523971	1	1	yes	characteristic
318	C₂	#22560046	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	D₂₆₈
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₄×C₂²⋊C₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	C₈×#104
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
4. C₈×#11761612	C₂	semidirect
5. #22558046	C₂	semidirect
6. #22558042	C₂	semidirect
7. C₄²×C₈×C₆₇	C₂	semidirect
9. C₈×Dic₁₃₄	C₄	semidirect
10. C₈×D₂₆₈	C₄	semidirect
19. #16863128	C₄	semidirect
20. #16863125	C₄	semidirect
42. #14523971	C₈	direct (N × H)
72. #11761612	C₂×C₈	semidirect
99. C₄²×C₆₇	C₂×C₈	semidirect
124. D₂₆₈	C₄×C₈	semidirect
129. Dic₁₃₄	C₄×C₈	semidirect
132. C₈×C₆₇	#104	semidirect
162. C₄²×C₈	D₆₇	semidirect
167. C₆₇	C₈×#104	semidirect
259. C₄²	C₈×D₆₇	semidirect
295. C₈	#14523971	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#11761628	2
C₄×#14523971	2
#22558592	2
C₈×#11761612	2
#22558046	2
#22558042	2
C₄²×C₈×C₆₇	2
C₈×#104	67

Subgroups — C8×#14523971 #30746539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14523971 #30746539