Subgroups — #428×D₁₅₇ #34108717

Order 20096.

536 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₅₇	C₂	1	1	yes	maximal, characteristic
2	#117×D₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#23757316	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757315	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757314	C₂	1	2	yes	maximal, complemented
7	#23757313	C₂	1	1	yes	maximal, characteristic, complemented
8	#23757312	C₂	1	2	yes	maximal, complemented
9	#107×D₁₅₇	C₂	1	2	yes	maximal, complemented
10	#106×D₁₅₇	C₂	1	2	yes	maximal, complemented
11	#428×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₅₇	—	2	1	no	—
13	C₂²×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
14	C₂×#14578938	C₂²	1	1	yes	characteristic
15	C₂×#14578937	C₂²	1	2	yes	—
16	M₁₆×D₁₅₇	C₄	1	8	yes	complemented
17	C₂×C₈×D₁₅₇	C₂²	1	2	yes	—
18	#19097730	C₂²	1	1	yes	characteristic
19	#19097729	C₂²	1	1	yes	characteristic
20	#19097729	C₂²	1	1	yes	characteristic
21	#19097728	—	2	1	no	—
22	#19097727	—	2	1	no	—
23	C₂×C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
24	C₂×#14578933	—	2	1	no	—
25	C₄⋊C₄×D₁₅₇	—	2	1	no	—
26	C₄⋊C₄×D₁₅₇	—	2	1	no	—
27	C₄⋊C₄×D₁₅₇	C₂²	1	2	yes	—
28	C₄⋊C₄×D₁₅₇	C₂²	1	2	yes	—
29	C₄⋊C₄×D₁₅₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₁₅₇	C₂²	1	1	yes	characteristic
31	C₂²⋊C₄×D₁₅₇	—	2	1	no	—
32	C₄²×D₁₅₇	—	2	1	no	—
33	#19097705	C₂²	1	2	yes	complemented
34	#19097704	C₂²	1	2	yes	complemented
35	#19097704	C₂²	1	2	yes	complemented
36	#19097703	C₂²	1	2	yes	complemented
37	#19097702	C₂²	1	2	yes	complemented
38	#19097702	C₂²	1	2	yes	complemented
39	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
40	#117×C₁₅₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
42	#107×C₁₅₇	C₂²	1	2	yes	complemented
43	#106×C₁₅₇	C₂²	1	2	yes	complemented
44	C₂³×D₁₅₇	D₄	1	1	yes	characteristic
45	C₂²×Dic₁₅₇	—	2	1	no	—
46	C₂²×Dic₁₅₇	D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₅₇	—	2	1	no	—
48	C₂×C₄×D₁₅₇	—	2	1	no	—
49	C₂×C₄×D₁₅₇	—	2	2	no	—
50	C₂×C₄×D₁₅₇	—	2	2	no	—
51	C₂×C₄×D₁₅₇	—	4	4	no	—
52	C₂×C₄×D₁₅₇	C₂×C₄	1	4	yes	—
53	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
55	C₂×C₄×D₁₅₇	Q₈	1	4	yes	—
56	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
58	#14578938	C₂×C₄	1	4	yes	complemented
59	#14578937	C₂×C₄	1	8	yes	complemented
60	C₂×#11929434	C₂³	1	2	yes	—
61	C₈×D₁₅₇	C₂×C₄	1	8	yes	complemented
62	C₄⋊Dic₁₅₇	—	2	1	no	—
63	C₄⋊Dic₁₅₇	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₁₅₇	C₂³	1	2	yes	—
65	C₄⋊Dic₁₅₇	C₂³	1	1	yes	characteristic
66	#14578933	—	2	2	no	—
67	#14578933	—	2	2	no	—
68	#14578933	—	2	1	no	—
69	#14578933	—	2	1	no	—
70	C₂²⋊Dic₁₅₇	—	2	1	no	—
71	#14578931	—	2	1	no	—
72	#14578931	—	2	1	no	—
73	C₄×Dic₁₅₇	—	2	1	no	—
74	C₄×Dic₁₅₇	—	2	1	no	—
75	C₄×Dic₁₅₇	—	2	1	no	—
76	C₂²×C₄×C₁₅₇	—	2	1	no	—
77	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
78	M₁₆×C₁₅₇	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
80	C₄⋊C₄×C₁₅₇	—	2	1	no	—
81	C₄⋊C₄×C₁₅₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₅₇	C₂³	1	2	yes	—
83	C₄⋊C₄×C₁₅₇	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₁₅₇	—	2	1	no	—
85	C₄²×C₁₅₇	—	2	1	no	—
86	C₂²×D₁₅₇	—	2	2	no	—
87	C₂²×D₁₅₇	—	2	2	no	—
88	C₂²×D₁₅₇	C₄⋊C₄	1	4	yes	—
89	C₂²×D₁₅₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₁₅₇	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₁₅₇	—	2	2	no	—
92	C₂×Dic₁₅₇	—	2	1	no	—
93	C₂×Dic₁₅₇	—	2	1	no	—
94	C₂×Dic₁₅₇	—	4	2	no	—
95	C₂×Dic₁₅₇	C₄⋊C₄	1	4	yes	—
96	C₂×Dic₁₅₇	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₁₅₇	C₂×D₄	1	1	yes	characteristic
98	C₄×D₁₅₇	—	4	4	no	—
99	C₄×D₁₅₇	—	4	2	no	—
100	C₄×D₁₅₇	—	4	4	no	—
101	C₄×D₁₅₇	—	4	2	no	—
102	C₄×D₁₅₇	C₄⋊C₄	1	4	yes	—
103	C₄×D₁₅₇	C₄⋊C₄	1	4	yes	—
104	C₄×D₁₅₇	C₂²×C₄	1	2	yes	—
105	C₄×D₁₅₇	C₂²×C₄	1	2	yes	—
106	C₄×D₁₅₇	C₂×Q₈	1	2	yes	—
107	C₄×D₁₅₇	C₂×Q₈	1	2	yes	—
108	#11929434	C₂²×C₄	1	4	yes	complemented
109	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₅₇	—	2	2	no	—
111	C₂×C₄×C₁₅₇	—	2	1	no	—
112	C₂×C₄×C₁₅₇	—	2	1	no	—
113	C₂×C₄×C₁₅₇	—	4	2	no	—
114	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
117	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
118	C₈×C₁₅₇	C₂²×C₄	1	4	yes	complemented
119	C₂²×C₁₅₇	—	2	1	no	—
120	C₂²×C₁₅₇	—	2	1	no	—
121	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₅₇	—	2	1	no	—
124	C₂×D₁₅₇	—	2	1	no	—
125	C₂×D₁₅₇	—	2	8	no	—
126	C₂×D₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×D₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×D₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×D₁₅₇	#137	1	2	yes	—
130	C₄×C₁₅₇	—	4	2	no	—
131	C₄×C₁₅₇	—	4	1	no	—
132	C₄×C₁₅₇	—	4	1	no	—
133	C₄×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₄×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
136	Dic₁₅₇	—	4	2	no	—
137	Dic₁₅₇	—	4	1	no	—
138	Dic₁₅₇	—	4	1	no	—
139	Dic₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
140	Dic₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₁₅₇	—	2	2	no	—
142	C₂×C₁₅₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₁₅₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₁₅₇	C₂×#137	1	1	yes	characteristic
145	D₁₅₇	—	2	2	no	—
146	D₁₅₇	#428	1	4	yes	direct product factor
147	C₁₅₇	C₂×#428	1	1	yes	characteristic, complemented, Sylow
148	C₂×#428	—	157	1	no	maximal, Sylow
149	C₂²×M₁₆	—	157	1	no	—
150	C₂×#117	—	157	1	no	—
151	C₂²×C₄⋊C₄	—	157	1	no	—
152	#428	—	157	1	no	—
153	#428	—	157	1	no	—
154	#428	—	157	1	no	—
155	#428	—	157	2	no	—
156	#428	—	157	2	no	—
157	#428	D₁₅₇	1	1	yes	characteristic, direct product factor
158	C₂×#107	—	157	2	no	—
159	C₂×#106	—	157	2	no	—
160	C₂³×C₄	—	314	1	no	—
161	C₂³×C₄	—	157	1	no	—
162	C₂×M₁₆	—	157	1	no	—
163	C₂×M₁₆	—	157	8	no	—
164	C₂×M₁₆	—	157	2	no	—
165	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
166	C₂²×C₈	—	157	2	no	—
167	#117	—	314	1	no	—
168	#117	—	314	1	no	—
169	#117	—	157	1	no	—
170	#117	—	157	1	no	—
171	#117	—	157	1	no	—
172	#117	C₂×D₁₅₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	—	314	1	no	—
174	C₂×C₄⋊C₄	—	314	1	no	—
175	C₂×C₄⋊C₄	—	314	1	no	—
176	C₂×C₄⋊C₄	—	157	1	no	—
177	C₂×C₄⋊C₄	—	157	2	no	—
178	C₂×C₄⋊C₄	—	157	2	no	—
179	C₂×C₄⋊C₄	—	157	1	no	—
180	C₂×C₄⋊C₄	—	157	1	no	—
181	C₂×C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	—	314	1	no	—
183	C₂×C₄²	—	314	1	no	—
184	#107	—	157	2	no	—
185	#107	—	157	2	no	—
186	#107	—	157	2	no	—
187	#107	C₂×D₁₅₇	1	2	yes	complemented
188	#106	—	157	2	no	—
189	#106	—	157	2	no	—
190	#106	—	157	2	no	—
191	#106	C₂×D₁₅₇	1	2	yes	complemented
192	C₂⁴	—	157	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	314	1	no	—
195	C₂²×C₄	—	314	1	no	—
196	C₂²×C₄	—	314	2	no	—
197	C₂²×C₄	—	314	2	no	—
198	C₂²×C₄	—	314	1	no	—
199	C₂²×C₄	—	628	4	no	—
200	C₂²×C₄	—	157	1	no	—
201	C₂²×C₄	—	157	4	no	—
202	C₂²×C₄	—	157	1	no	—
203	C₂²×C₄	—	157	1	no	—
204	C₂²×C₄	—	157	4	no	—
205	C₂²×C₄	—	157	1	no	—
206	C₂²×C₄	—	157	1	no	—
207	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
208	M₁₆	—	157	4	no	—
209	M₁₆	—	157	8	no	—
210	M₁₆	C₄×D₁₅₇	1	4	yes	complemented
211	C₂×C₈	—	157	2	no	—
212	C₂×C₈	—	157	8	no	—
213	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	314	2	no	—
216	C₄⋊C₄	—	314	2	no	—
217	C₄⋊C₄	—	314	1	no	—
218	C₄⋊C₄	—	314	1	no	—
219	C₄⋊C₄	—	314	1	no	—
220	C₄⋊C₄	—	157	1	no	—
221	C₄⋊C₄	—	157	2	no	—
222	C₄⋊C₄	—	157	1	no	—
223	C₄⋊C₄	C₂²×D₁₅₇	1	1	yes	characteristic
224	C₄⋊C₄	C₂²×D₁₅₇	1	2	yes	—
225	C₄⋊C₄	C₂²×D₁₅₇	1	1	yes	characteristic
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	314	1	no	—
228	C₂²⋊C₄	—	314	1	no	—
229	C₂²⋊C₄	—	314	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	314	1	no	—
232	C₄²	—	314	1	no	—
233	C₄²	—	314	1	no	—
234	C₂³	—	314	2	no	—
235	C₂³	—	314	2	no	—
236	C₂³	—	157	4	no	—
237	C₂³	—	157	1	no	—
238	C₂³	—	157	1	no	—
239	C₂³	D₄×D₁₅₇	1	1	yes	characteristic
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	314	1	no	—
245	C₂×C₄	—	314	1	no	—
246	C₂×C₄	—	314	2	no	—
247	C₂×C₄	—	628	4	no	—
248	C₂×C₄	—	628	4	no	—
249	C₂×C₄	—	628	2	no	—
250	C₂×C₄	—	628	2	no	—
251	C₂×C₄	—	628	2	no	—
252	C₂×C₄	—	157	4	no	—
253	C₂×C₄	—	157	1	no	—
254	C₂×C₄	—	157	1	no	—
255	C₂×C₄	—	157	4	no	—
256	C₂×C₄	—	157	4	no	—
257	C₂×C₄	—	157	2	no	—
258	C₂×C₄	—	157	2	no	—
259	C₂×C₄	—	157	2	no	—
260	C₂×C₄	—	157	2	no	—
261	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
262	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
263	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
264	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
265	C₈	—	157	4	no	—
266	C₈	C₂×C₄×D₁₅₇	1	4	yes	complemented
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	314	8	no	—
270	C₂²	—	314	1	no	—
271	C₂²	—	314	1	no	—
272	C₂²	—	157	2	no	—
273	C₂²	—	157	2	no	—
274	C₂²	—	157	2	no	—
275	C₂²	—	157	2	no	—
276	C₂²	C₄⋊C₄×D₁₅₇	1	2	yes	—
277	C₂²	C₂×D₄×D₁₅₇	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	628	2	no	—
282	C₄	—	628	1	no	—
283	C₄	—	628	1	no	—
284	C₄	—	157	2	no	—
285	C₄	—	157	2	no	—
286	C₄	C₄⋊C₄×D₁₅₇	1	2	yes	—
287	C₄	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
288	C₄	C₂×Q₈×D₁₅₇	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	314	2	no	—
291	C₂	—	157	4	no	—
292	C₂	C₂×C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
293	C₂	AGL(1,ℤ/8)×D₁₅₇	1	1	yes	characteristic
294	C₂	#137×D₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₇
commutator	C₄×C₁₅₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#428×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
prime radical	C₂³×D₁₅₇	D₄
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	#428	D₁₅₇
hypocenter	C₁₅₇	C₂×#428
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #23757316	C₂	semidirect
5. #23757315	C₂	semidirect
6. #23757314	C₂	semidirect
7. #23757313	C₂	semidirect
8. #23757312	C₂	semidirect
9. #107×D₁₅₇	C₂	semidirect
10. #106×D₁₅₇	C₂	semidirect
11. #428×C₁₅₇	C₂	semidirect
16. M₁₆×D₁₅₇	C₄	semidirect
33. #19097705	C₂²	semidirect
34. #19097704	C₂²	semidirect
35. #19097704	C₂²	semidirect
36. #19097703	C₂²	semidirect
37. #19097702	C₂²	semidirect
38. #19097702	C₂²	semidirect
42. #107×C₁₅₇	C₂²	semidirect
43. #106×C₁₅₇	C₂²	semidirect
58. #14578938	C₂×C₄	semidirect
59. #14578937	C₂×C₄	semidirect
61. C₈×D₁₅₇	C₂×C₄	semidirect
78. M₁₆×C₁₅₇	C₂×C₄	semidirect
108. #11929434	C₂²×C₄	semidirect
118. C₈×C₁₅₇	C₂²×C₄	semidirect
146. D₁₅₇	#428	direct (N × H)
147. C₁₅₇	C₂×#428	semidirect
157. #428	D₁₅₇	direct (N × H)
187. #107	C₂×D₁₅₇	semidirect
191. #106	C₂×D₁₅₇	semidirect
210. M₁₆	C₄×D₁₅₇	semidirect
266. C₈	C₂×C₄×D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₅₇	2
#117×D₁₅₇	2
C₂×C₄⋊C₄×D₁₅₇	2
#23757316	2
#23757315	2
#23757314	2
#23757313	2
#23757312	2
#107×D₁₅₇	2
#106×D₁₅₇	2
#428×C₁₅₇	2
C₂×#428	157

Subgroups — #428×D157 #34108717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #428×D₁₅₇ #34108717