Subgroups — #467×D₁₅₇ #34109706

Order 20096.

512 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Q₈×D₁₅₇	C₂	1	2	yes	maximal
4	#23757515	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757514	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757513	C₂	1	2	yes	maximal, complemented
7	#23757512	C₂	1	2	yes	maximal, complemented
8	#23757511	C₂	1	1	yes	maximal, characteristic, complemented
9	#107×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#103×D₁₅₇	C₂	1	2	yes	maximal, complemented
11	#467×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₅₇	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₆₂₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×#14578944	C₂²	1	2	yes	complemented
16	Q₁₆×D₁₅₇	—	2	4	no	—
17	C₂²×#11929434	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₅₇	—	2	4	no	—
19	C₂×C₈×D₁₅₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₅₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₅₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₅₇	C₂²	1	1	yes	characteristic
23	#19097751	C₂²	1	2	yes	—
24	#19097750	C₂²	1	2	yes	—
25	#19097749	—	2	2	no	—
26	#19097748	—	2	2	no	—
27	#19097747	C₂²	1	2	yes	—
28	C₄⋊C₄×D₁₅₇	—	2	2	no	—
29	C₄⋊C₄×D₁₅₇	C₂²	1	2	yes	—
30	C₂²⋊C₄×D₁₅₇	—	2	2	no	—
31	#19097705	C₂²	1	1	yes	characteristic, complemented
32	#19097704	C₂²	1	2	yes	complemented
33	#19097696	C₂²	1	2	yes	complemented
34	#19097695	C₂²	1	2	yes	complemented
35	#19097694	C₂²	1	2	yes	complemented
36	C₂×Q₁₆×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
38	C₂²⋊Q₈×C₁₅₇	C₂²	1	2	yes	—
39	#107×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
40	#103×C₁₅₇	C₂²	1	2	yes	complemented
41	C₂³×D₁₅₇	D₄	1	1	yes	characteristic
42	C₂×Dic₃₁₄	—	2	2	no	—
43	C₂×Dic₃₁₄	C₂³	1	2	yes	—
44	Q₈×D₁₅₇	—	2	4	no	—
45	Q₈×D₁₅₇	—	2	4	no	—
46	C₂²×Dic₁₅₇	D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₅₇	—	2	2	no	—
48	C₂×C₄×D₁₅₇	—	2	2	no	—
49	C₂×C₄×D₁₅₇	—	2	2	no	—
50	C₂×C₄×D₁₅₇	—	2	2	no	—
51	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
53	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
55	Dic₆₂₈	—	2	2	no	—
56	#14578944	—	2	4	no	—
57	C₂×#11929434	—	2	2	no	—
58	C₂×#11929434	C₂³	1	1	yes	characteristic
59	C₂×#11929434	C₂³	1	1	yes	characteristic
60	C₈×D₁₅₇	—	2	2	no	—
61	C₈×D₁₅₇	—	2	2	no	—
62	C₈×D₁₅₇	—	2	2	no	—
63	C₈×D₁₅₇	D₄	1	4	yes	—
64	C₄⋊Dic₁₅₇	—	2	2	no	—
65	C₄⋊Dic₁₅₇	C₂³	1	2	yes	—
66	#14578933	—	2	2	no	—
67	#14578933	—	2	2	no	—
68	#14578933	—	2	2	no	—
69	C₂²⋊Dic₁₅₇	—	2	2	no	—
70	#14578931	—	2	2	no	—
71	#14578931	—	2	2	no	—
72	C₂×Q₈×C₁₅₇	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
74	Q₁₆×C₁₅₇	—	2	2	no	—
75	C₂×C₈×C₁₅₇	—	2	2	no	—
76	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₅₇	—	2	2	no	—
79	C₄⋊C₄×C₁₅₇	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
81	C₂²×D₁₅₇	—	2	2	no	—
82	C₂²×D₁₅₇	—	2	2	no	—
83	C₂²×D₁₅₇	Q₁₆	1	4	yes	complemented
84	C₂²×D₁₅₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₁₅₇	C₂×D₄	1	1	yes	characteristic
86	Dic₃₁₄	—	2	2	no	—
87	Dic₃₁₄	—	2	2	no	—
88	Dic₃₁₄	—	4	4	no	—
89	C₂×Dic₁₅₇	—	2	2	no	—
90	C₂×Dic₁₅₇	—	2	2	no	—
91	C₂×Dic₁₅₇	—	2	2	no	—
92	C₂×Dic₁₅₇	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₁₅₇	C₂×D₄	1	1	yes	characteristic
94	C₄×D₁₅₇	—	2	2	no	—
95	C₄×D₁₅₇	—	2	1	no	—
96	C₄×D₁₅₇	—	2	1	no	—
97	C₄×D₁₅₇	—	2	2	no	—
98	C₄×D₁₅₇	—	4	4	no	—
99	C₄×D₁₅₇	—	4	4	no	—
100	C₄×D₁₅₇	C₂×D₄	1	2	yes	—
101	C₄×D₁₅₇	#41	1	2	yes	—
102	#11929434	—	2	1	no	—
103	#11929434	C₂×D₄	1	2	yes	—
104	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₁₅₇	—	2	2	no	—
106	Q₈×C₁₅₇	—	2	2	no	—
107	C₂×C₄×C₁₅₇	—	2	1	no	—
108	C₂×C₄×C₁₅₇	—	2	1	no	—
109	C₂×C₄×C₁₅₇	—	2	2	no	—
110	C₂×C₄×C₁₅₇	—	2	2	no	—
111	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
113	C₈×C₁₅₇	—	2	1	no	—
114	C₈×C₁₅₇	C₂×D₄	1	2	yes	—
115	C₂²×C₁₅₇	—	2	1	no	—
116	C₂²×C₁₅₇	—	2	1	no	—
117	C₂²×C₁₅₇	C₂×Q₁₆	1	2	yes	complemented
118	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₁₅₇	—	2	1	no	—
120	C₂×D₁₅₇	—	2	1	no	—
121	C₂×D₁₅₇	—	2	8	no	—
122	C₂×D₁₅₇	#121	1	2	yes	—
123	C₂×D₁₅₇	C₂×Q₁₆	1	2	yes	—
124	C₂×D₁₅₇	C₂×Q₁₆	1	2	yes	complemented
125	C₂×D₁₅₇	#135	1	2	yes	—
126	C₄×C₁₅₇	—	2	1	no	—
127	C₄×C₁₅₇	—	4	2	no	—
128	C₄×C₁₅₇	—	4	2	no	—
129	C₄×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
131	Dic₁₅₇	—	2	1	no	—
132	Dic₁₅₇	—	4	2	no	—
133	Dic₁₅₇	—	4	2	no	—
134	Dic₁₅₇	#121	1	2	yes	—
135	C₂×C₁₅₇	—	2	2	no	—
136	C₂×C₁₅₇	C₂×#121	1	1	yes	characteristic
137	C₂×C₁₅₇	C₂²×Q₁₆	1	1	yes	characteristic
138	C₂×C₁₅₇	C₂×#135	1	1	yes	characteristic
139	D₁₅₇	—	2	2	no	—
140	D₁₅₇	#467	1	4	yes	direct product factor
141	C₁₅₇	C₂×#467	1	1	yes	characteristic, complemented, Sylow
142	C₂×#467	—	157	1	no	maximal, Sylow
143	C₂²×Q₁₆	—	157	1	no	—
144	C₂³×C₈	—	157	1	no	—
145	C₂×C₂²⋊Q₈	—	157	2	no	—
146	#467	—	157	1	no	—
147	#467	—	157	1	no	—
148	#467	—	157	2	no	—
149	#467	—	157	1	no	—
150	#467	—	157	2	no	—
151	#467	D₁₅₇	1	1	yes	characteristic, direct product factor
152	C₂×#107	—	157	1	no	—
153	C₂×#103	—	157	2	no	—
154	C₂²×Q₈	—	157	2	no	—
155	C₂³×C₄	—	157	1	no	—
156	C₂×Q₁₆	—	314	4	no	—
157	C₂×Q₁₆	—	157	2	no	—
158	C₂×Q₁₆	—	157	1	no	—
159	C₂×Q₁₆	C₂×D₁₅₇	1	1	yes	characteristic, complemented
160	C₂²×C₈	—	314	4	no	—
161	C₂²×C₈	—	157	1	no	—
162	C₂²×C₈	—	157	1	no	—
163	C₂²×C₈	—	157	1	no	—
164	C₂²×C₈	—	157	1	no	—
165	C₂²×C₈	—	157	1	no	—
166	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
167	C₂²⋊Q₈	—	314	2	no	—
168	C₂²⋊Q₈	—	314	2	no	—
169	C₂²⋊Q₈	—	157	2	no	—
170	C₂²⋊Q₈	—	157	2	no	—
171	C₂²⋊Q₈	—	157	2	no	—
172	C₂²⋊Q₈	C₂×D₁₅₇	1	2	yes	—
173	C₂×C₄⋊C₄	—	314	2	no	—
174	C₂×C₄⋊C₄	—	157	2	no	—
175	C₂×C₂²⋊C₄	—	314	2	no	—
176	#107	—	157	1	no	—
177	#107	—	157	2	no	—
178	#107	C₂×D₁₅₇	1	1	yes	characteristic, complemented
179	#103	—	157	2	no	—
180	#103	—	157	2	no	—
181	#103	—	157	2	no	—
182	#103	C₂×D₁₅₇	1	2	yes	complemented
183	C₂⁴	—	157	1	no	—
184	C₂×Q₈	—	314	2	no	—
185	C₂×Q₈	—	314	4	no	—
186	C₂×Q₈	—	314	4	no	—
187	C₂×Q₈	—	157	2	no	—
188	C₂×Q₈	C₂²×D₁₅₇	1	2	yes	—
189	C₂²×C₄	—	314	2	no	—
190	C₂²×C₄	—	314	2	no	—
191	C₂²×C₄	—	314	2	no	—
192	C₂²×C₄	—	314	2	no	—
193	C₂²×C₄	—	157	1	no	—
194	C₂²×C₄	—	157	1	no	—
195	C₂²×C₄	—	157	1	no	—
196	C₂²×C₄	—	157	1	no	—
197	C₂²×C₄	—	157	1	no	—
198	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	314	4	no	—
201	Q₁₆	—	314	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	314	2	no	—
204	C₂×C₈	—	314	2	no	—
205	C₂×C₈	—	314	2	no	—
206	C₂×C₈	—	314	2	no	—
207	C₂×C₈	—	157	1	no	—
208	C₂×C₈	—	157	1	no	—
209	C₂×C₈	—	157	4	no	—
210	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	314	2	no	—
214	C₄⋊C₄	—	314	2	no	—
215	C₄⋊C₄	—	314	2	no	—
216	C₄⋊C₄	—	314	2	no	—
217	C₄⋊C₄	—	157	2	no	—
218	C₄⋊C₄	C₂²×D₁₅₇	1	2	yes	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	314	2	no	—
221	C₂²⋊C₄	—	314	2	no	—
222	C₂²⋊C₄	—	314	2	no	—
223	C₂³	—	314	2	no	—
224	C₂³	—	314	2	no	—
225	C₂³	—	157	4	no	—
226	C₂³	—	157	1	no	—
227	C₂³	—	157	1	no	—
228	C₂³	D₄×D₁₅₇	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	2	2	no	—
231	Q₈	—	314	2	no	—
232	Q₈	—	314	2	no	—
233	Q₈	—	628	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	314	2	no	—
239	C₂×C₄	—	314	2	no	—
240	C₂×C₄	—	314	2	no	—
241	C₂×C₄	—	314	1	no	—
242	C₂×C₄	—	314	1	no	—
243	C₂×C₄	—	314	2	no	—
244	C₂×C₄	—	314	2	no	—
245	C₂×C₄	—	628	4	no	—
246	C₂×C₄	—	628	4	no	—
247	C₂×C₄	—	157	1	no	—
248	C₂×C₄	—	157	1	no	—
249	C₂×C₄	—	157	2	no	—
250	C₂×C₄	—	157	2	no	—
251	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	314	1	no	—
255	C₈	—	157	2	no	—
256	C₈	D₄×D₁₅₇	1	2	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	314	8	no	—
260	C₂²	—	314	1	no	—
261	C₂²	—	314	1	no	—
262	C₂²	—	157	2	no	—
263	C₂²	—	157	2	no	—
264	C₂²	—	157	2	no	—
265	C₂²	—	157	2	no	—
266	C₂²	Q₁₆×D₁₅₇	1	2	yes	complemented
267	C₂²	C₂×D₄×D₁₅₇	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	314	1	no	—
272	C₄	—	628	2	no	—
273	C₄	—	628	2	no	—
274	C₄	—	157	2	no	—
275	C₄	C₂×D₄×D₁₅₇	1	1	yes	characteristic
276	C₄	#41×D₁₅₇	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	314	2	no	—
279	C₂	—	157	4	no	—
280	C₂	#121×D₁₅₇	1	1	yes	characteristic
281	C₂	C₂×Q₁₆×D₁₅₇	1	1	yes	characteristic
282	C₂	#135×D₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#467×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
prime radical	C₂³×D₁₅₇	D₄
hypercenter	#467	D₁₅₇
hypocenter	C₁₅₇	C₂×#467
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×D₁₅₇	C₂	semidirect
4. #23757515	C₂	semidirect
5. #23757514	C₂	semidirect
6. #23757513	C₂	semidirect
7. #23757512	C₂	semidirect
8. #23757511	C₂	semidirect
9. #107×D₁₅₇	C₂	semidirect
10. #103×D₁₅₇	C₂	semidirect
11. #467×C₁₅₇	C₂	semidirect
14. C₂×Dic₆₂₈	C₂²	semidirect
15. C₂×#14578944	C₂²	semidirect
31. #19097705	C₂²	semidirect
32. #19097704	C₂²	semidirect
33. #19097696	C₂²	semidirect
34. #19097695	C₂²	semidirect
35. #19097694	C₂²	semidirect
36. C₂×Q₁₆×C₁₅₇	C₂²	semidirect
39. #107×C₁₅₇	C₂²	semidirect
40. #103×C₁₅₇	C₂²	semidirect
83. C₂²×D₁₅₇	Q₁₆	semidirect
117. C₂²×C₁₅₇	C₂×Q₁₆	semidirect
124. C₂×D₁₅₇	C₂×Q₁₆	semidirect
140. D₁₅₇	#467	direct (N × H)
141. C₁₅₇	C₂×#467	semidirect
151. #467	D₁₅₇	direct (N × H)
159. C₂×Q₁₆	C₂×D₁₅₇	semidirect
178. #107	C₂×D₁₅₇	semidirect
182. #103	C₂×D₁₅₇	semidirect
266. C₂²	Q₁₆×D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×D₁₅₇	2
C₂²×C₈×D₁₅₇	2
C₂²⋊Q₈×D₁₅₇	2
#23757515	2
#23757514	2
#23757513	2
#23757512	2
#23757511	2
#107×D₁₅₇	2
#103×D₁₅₇	2
#467×C₁₅₇	2
C₂×#467	157

Subgroups — #467×D157 #34109706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #467×D₁₅₇ #34109706