Subgroups — #34109199

Order 20096.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757820	C₂	1	1	yes	maximal, characteristic
2	#23757352	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757351	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757348	C₂	1	2	yes	maximal, complemented
5	#23757338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₁₅₇	C₂	1	2	yes	maximal, complemented
7	#23757325	C₂	1	1	yes	maximal, characteristic, complemented
8	#23757324	C₂	1	1	yes	maximal, characteristic, complemented
9	#23757239	C₂	1	2	yes	maximal, complemented
10	#23757238	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14578954	C₂²	1	2	yes	—
13	C₂×#14578938	C₂²	1	2	yes	—
14	C₂×#14578937	C₂²	1	2	yes	—
15	C₂×C₈×D₁₅₇	C₂²	1	2	yes	—
16	#19097770	C₄	1	1	yes	characteristic
17	#19097767	C₄	1	1	yes	characteristic
18	#19097757	C₄	1	2	yes	—
19	#19097746	C₄	1	1	yes	characteristic
20	#19097744	C₄	1	2	yes	—
21	#19097742	C₄	1	1	yes	characteristic
22	C₄×D₆₂₈	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₅₇	C₂²	1	2	yes	—
24	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
25	C₄⋊#11929434	C₂²	1	1	yes	characteristic
26	#19097700	C₂²	1	1	yes	characteristic
27	#19097700	C₂²	1	1	yes	characteristic
28	C₂²⋊#11929434	C₂²	1	2	yes	—
29	#19097683	C₂²	1	1	yes	characteristic
30	#19097683	C₂²	1	1	yes	characteristic
31	#19097683	C₂²	1	2	yes	—
32	#19097683	C₂²	1	2	yes	—
33	#19097683	C₂²	1	2	yes	—
34	C₄×#11929434	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₅₇	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
37	C₄×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₅₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
41	#14578954	—	4	2	no	—
42	#14578954	—	4	2	no	—
43	C₂×Dic₃₁₄	C₂×C₄	1	2	yes	—
44	C₂×D₆₂₈	C₂×C₄	1	1	yes	characteristic
45	C₂×D₆₂₈	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₁₅₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₁₅₇	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₅₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₁₅₇	C₂³	1	2	yes	—
53	#14578938	—	4	2	no	—
54	#14578937	—	4	2	no	—
55	C₂×#11929434	C₂³	1	2	yes	—
56	C₂×#11929434	C₂³	1	1	yes	characteristic
57	C₂×#11929434	C₂³	1	1	yes	characteristic
58	C₈×D₁₅₇	—	4	2	no	—
59	C₄⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
60	#14578933	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
62	#14578931	C₂×C₄	1	2	yes	—
63	#14578931	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
69	C₂×C₈×C₁₅₇	—	4	2	no	—
70	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
73	C₄⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
75	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
76	C₂²×D₁₅₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₁₅₇	C₂²×C₄	1	2	yes	—
78	Dic₃₁₄	—	4	2	no	—
79	D₆₂₈	—	2	2	no	—
80	D₆₂₈	—	4	2	no	—
81	C₂²⋊D₁₅₇	—	2	4	no	—
82	C₂²⋊D₁₅₇	—	4	2	no	—
83	C₂²⋊D₁₅₇	—	4	2	no	—
84	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₅₇	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
88	C₄×D₁₅₇	—	2	2	no	—
89	C₄×D₁₅₇	—	2	1	no	—
90	C₄×D₁₅₇	—	2	1	no	—
91	C₄×D₁₅₇	—	2	2	no	—
92	C₄×D₁₅₇	—	2	2	no	—
93	C₄×D₁₅₇	—	2	2	no	—
94	#11929434	—	2	1	no	—
95	#11929434	—	2	2	no	—
96	#11929434	—	2	1	no	—
97	C₂³×C₁₅₇	C₂²×C₄	1	2	yes	—
98	D₄×C₁₅₇	—	2	2	no	—
99	C₂×C₄×C₁₅₇	—	2	2	no	—
100	C₂×C₄×C₁₅₇	—	2	2	no	—
101	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
105	C₈×C₁₅₇	—	2	1	no	—
106	C₈×C₁₅₇	—	2	2	no	—
107	C₈×C₁₅₇	—	2	1	no	—
108	C₂²×C₁₅₇	—	2	2	no	—
109	C₂²×C₁₅₇	—	2	2	no	—
110	C₂²×C₁₅₇	—	2	2	no	—
111	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₁₅₇	—	2	1	no	—
113	C₂×D₁₅₇	—	2	2	no	—
114	C₂×D₁₅₇	—	2	1	no	—
115	C₂×D₁₅₇	—	2	1	no	—
116	C₂×D₁₅₇	—	2	2	no	—
117	C₂×D₁₅₇	—	2	2	no	—
118	C₄×C₁₅₇	—	2	2	no	—
119	C₄×C₁₅₇	—	2	1	no	—
120	C₄×C₁₅₇	#131	1	2	yes	—
121	C₄×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₁₅₇	—	2	1	no	—
124	Dic₁₅₇	—	2	2	no	—
125	Dic₁₅₇	—	2	1	no	—
126	Dic₁₅₇	#131	1	2	yes	—
127	C₂×C₁₅₇	—	4	2	no	—
128	C₂×C₁₅₇	#518	1	1	yes	characteristic
129	C₂×C₁₅₇	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₅₇	C₂×#131	1	1	yes	characteristic
131	D₁₅₇	—	4	1	no	—
132	D₁₅₇	—	4	2	no	—
133	C₁₅₇	#2979	1	1	yes	characteristic, complemented, Sylow
134	#2979	—	157	1	no	maximal, Sylow
135	#532	—	157	1	no	—
136	#436	—	157	1	no	—
137	#436	—	157	2	no	—
138	#436	—	157	1	no	—
139	C₈×D₄	—	157	2	no	—
140	C₈×D₄	—	157	1	no	—
141	C₈×D₄	D₁₅₇	1	1	yes	characteristic, complemented
142	#431	—	157	1	no	—
143	#431	—	157	1	no	—
144	#408	—	157	2	no	—
145	#408	—	157	2	no	—
146	C₂×#41	—	157	2	no	—
147	C₂×M₁₆	—	157	2	no	—
148	C₂×M₁₆	—	157	2	no	—
149	C₂²×C₈	—	157	2	no	—
150	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
151	C₄⋊Q₈	—	157	1	no	—
152	#127	—	157	1	no	—
153	#124	—	157	2	no	—
154	#121	—	157	1	no	—
155	#121	—	157	1	no	—
156	#121	—	157	2	no	—
157	C₄×D₄	—	157	2	no	—
158	C₄×D₄	—	157	1	no	—
159	C₄×D₄	C₂×D₁₅₇	1	1	yes	characteristic
160	C₂×C₄²	—	157	1	no	—
161	C₄⋊C₈	—	157	1	no	—
162	C₄⋊C₈	—	157	1	no	—
163	C₄⋊C₈	—	157	1	no	—
164	C₄⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
165	C₂²⋊C₈	—	157	2	no	—
166	C₂²⋊C₈	—	157	1	no	—
167	C₂²⋊C₈	—	157	2	no	—
168	C₂²⋊C₈	—	157	1	no	—
169	C₂²⋊C₈	—	157	2	no	—
170	C₂²⋊C₈	—	157	2	no	—
171	C₂²⋊C₈	C₂×D₁₅₇	1	2	yes	—
172	C₄×C₈	—	157	2	no	—
173	C₄×C₈	—	157	1	no	—
174	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
175	#41	—	628	2	no	—
176	#41	—	628	2	no	—
177	C₂×Q₈	—	157	2	no	—
178	C₂×D₄	—	157	2	no	—
179	C₂×D₄	—	157	2	no	—
180	C₂×D₄	—	157	2	no	—
181	C₂×D₄	—	157	1	no	—
182	C₂×D₄	—	157	2	no	—
183	C₂×D₄	C₄×D₁₅₇	1	1	yes	characteristic
184	C₂²×C₄	—	157	1	no	—
185	C₂²×C₄	—	157	1	no	—
186	C₂²×C₄	—	157	1	no	—
187	C₂²×C₄	—	157	2	no	—
188	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
189	M₁₆	—	628	2	no	—
190	M₁₆	—	628	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	628	2	no	—
193	C₂×C₈	—	157	1	no	—
194	C₂×C₈	—	157	2	no	—
195	C₂×C₈	—	157	1	no	—
196	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
197	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
199	C₄⋊C₄	—	157	2	no	—
200	C₄⋊C₄	—	157	1	no	—
201	C₄⋊C₄	C₄×D₁₅₇	1	1	yes	characteristic
202	C₂²⋊C₄	—	157	2	no	—
203	C₂²⋊C₄	—	157	2	no	—
204	C₂²⋊C₄	—	157	2	no	—
205	C₂²⋊C₄	C₄×D₁₅₇	1	2	yes	—
206	C₄²	—	157	1	no	—
207	C₄²	—	157	1	no	—
208	C₄²	—	157	1	no	—
209	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
210	C₂³	—	157	2	no	—
211	C₂³	—	157	1	no	—
212	C₂³	C₂×C₄×D₁₅₇	1	2	yes	—
213	Q₈	—	628	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	314	4	no	—
216	D₄	—	314	2	no	—
217	D₄	—	628	2	no	—
218	D₄	—	628	2	no	—
219	D₄	—	628	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	314	2	no	—
223	C₂×C₄	—	314	2	no	—
224	C₂×C₄	—	314	2	no	—
225	C₂×C₄	—	314	1	no	—
226	C₂×C₄	—	314	2	no	—
227	C₂×C₄	—	314	1	no	—
228	C₂×C₄	—	157	1	no	—
229	C₂×C₄	—	157	1	no	—
230	C₂×C₄	—	157	2	no	—
231	C₂×C₄	—	157	1	no	—
232	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	314	1	no	—
240	C₈	—	314	2	no	—
241	C₈	—	314	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	314	1	no	—
246	C₂²	—	314	2	no	—
247	C₂²	—	314	1	no	—
248	C₂²	—	314	2	no	—
249	C₂²	—	314	1	no	—
250	C₂²	—	314	2	no	—
251	C₂²	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	314	1	no	—
255	C₄	—	314	2	no	—
256	C₄	—	314	1	no	—
257	C₄	—	157	2	no	—
258	C₄	#19097777	1	2	yes	—
259	C₄	#19097816	1	1	yes	characteristic
260	C₄	#19097817	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	628	1	no	—
263	C₂	—	628	2	no	—
264	C₂	#23757745	1	1	yes	characteristic
265	C₂	#131×D₁₅₇	1	1	yes	characteristic
266	C₂	C₂×#19097777	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	C₈×D₄×C₁₅₇	C₂
prime radical	#23757820	C₂
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	C₈×D₄	D₁₅₇
hypocenter	C₁₅₇	#2979
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #23757352	C₂	semidirect
3. #23757351	C₂	semidirect
4. #23757348	C₂	semidirect
5. #23757338	C₂	semidirect
6. C₈×C₂²⋊D₁₅₇	C₂	semidirect
7. #23757325	C₂	semidirect
8. #23757324	C₂	semidirect
9. #23757239	C₂	semidirect
10. #23757238	C₂	semidirect
11. C₈×D₄×C₁₅₇	C₂	semidirect
133. C₁₅₇	#2979	semidirect
141. C₈×D₄	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757820	2
#23757352	2
#23757351	2
#23757348	2
#23757338	2
C₈×C₂²⋊D₁₅₇	2
#23757325	2
#23757324	2
#23757239	2
#23757238	2
C₈×D₄×C₁₅₇	2
#2979	157