Subgroups — #34108837

Order 20096.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14578937	C₂	1	1	yes	maximal, characteristic, complemented
2	#23757780	C₂	1	1	yes	maximal, characteristic
3	#23757349	C₂	1	4	yes	maximal, complemented
4	#23757344	C₂	1	4	yes	maximal, complemented
5	#23757301	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757240	C₂	1	1	yes	maximal, characteristic, complemented
7	#23757236	C₂	1	1	yes	maximal, characteristic, complemented
8	#23757228	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
11	C₂×#14578938	C₂²	1	1	yes	characteristic
12	C₂×#14578937	—	2	4	no	—
13	C₂×#14578937	C₂²	1	2	yes	—
14	C₂×#14578937	C₂²	1	2	yes	—
15	C₂²×#11929434	C₂²	1	1	yes	characteristic
16	#19097752	C₄	1	2	yes	—
17	#19097747	C₄	1	2	yes	—
18	#19097742	C₄	1	2	yes	—
19	#19097739	C₄	1	2	yes	—
20	C₄×C₂²⋊D₁₅₇	C₂²	1	4	yes	—
21	#19097730	C₂²	1	1	yes	characteristic
22	#19097700	C₂²	1	4	yes	complemented
23	C₂²⋊#11929434	C₂²	1	4	yes	complemented
24	#19097683	C₂²	1	4	yes	—
25	#19097681	C₂²	1	2	yes	complemented
26	C₈×Dic₁₅₇	C₂²	1	2	yes	complemented
27	C₂×#41×C₁₅₇	C₂²	1	1	yes	characteristic
28	#131×C₁₅₇	—	2	4	no	—
29	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	complemented
31	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	complemented
33	C₂³×D₁₅₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₅₇	C₂×C₄	1	4	yes	—
35	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₅₇	—	2	2	no	—
37	C₂×C₄×D₁₅₇	—	2	2	no	—
38	C₂×C₄×D₁₅₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₅₇	C₂³	1	2	yes	—
40	#14578938	—	2	2	no	—
41	#14578937	—	2	4	no	—
42	#14578937	—	2	4	no	—
43	C₂×#11929434	—	2	2	no	—
44	C₂×#11929434	C₂³	1	2	yes	—
45	C₂×#11929434	C₂³	1	2	yes	—
46	C₄⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
47	#14578933	C₂×C₄	1	4	yes	—
48	C₂²⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
49	C₂²⋊Dic₁₅₇	C₂×C₄	1	4	yes	—
50	#14578931	C₂×C₄	1	4	yes	—
51	C₄×Dic₁₅₇	C₂³	1	2	yes	—
52	#41×C₁₅₇	—	2	2	no	—
53	#41×C₁₅₇	—	2	2	no	—
54	C₂×Q₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
55	C₂×D₄×C₁₅₇	C₂×C₄	1	2	yes	—
56	C₂×D₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
59	M₁₆×C₁₅₇	—	2	2	no	—
60	M₁₆×C₁₅₇	—	2	4	no	—
61	C₂×C₈×C₁₅₇	—	2	4	no	—
62	C₂×C₈×C₁₅₇	D₄	1	4	yes	complemented
63	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
64	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
66	C₂²×D₁₅₇	—	2	2	no	—
67	C₂²×D₁₅₇	—	4	2	no	—
68	C₂²×D₁₅₇	C₂²×C₄	1	2	yes	—
69	C₂²⋊D₁₅₇	—	4	4	no	—
70	C₂×Dic₁₅₇	—	2	2	no	—
71	C₂×Dic₁₅₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₅₇	C₂²×C₄	1	4	yes	—
73	C₄×D₁₅₇	—	2	4	no	—
74	C₄×D₁₅₇	—	2	2	no	—
75	C₄×D₁₅₇	—	2	2	no	—
76	#11929434	—	2	2	no	—
77	#11929434	—	2	2	no	—
78	C₂³×C₁₅₇	C₂²×C₄	1	2	yes	—
79	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
80	Q₈×C₁₅₇	—	2	2	no	—
81	D₄×C₁₅₇	—	2	4	no	—
82	D₄×C₁₅₇	—	2	2	no	—
83	C₂×C₄×C₁₅₇	—	2	2	no	—
84	C₂×C₄×C₁₅₇	—	2	2	no	—
85	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₅₇	#41	1	2	yes	—
89	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₅₇	—	2	2	no	—
91	C₈×C₁₅₇	C₂×D₄	1	2	yes	—
92	C₈×C₁₅₇	C₂×D₄	1	2	yes	—
93	C₂²×C₁₅₇	—	2	2	no	—
94	C₂²×C₁₅₇	—	2	2	no	—
95	C₂²×C₁₅₇	—	2	4	no	—
96	C₂²×C₁₅₇	C₄×D₄	1	2	yes	—
97	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₅₇	—	2	2	no	—
99	C₂×D₁₅₇	—	2	2	no	—
100	C₂×D₁₅₇	—	2	2	no	—
101	C₂×D₁₅₇	—	4	4	no	—
102	C₄×C₁₅₇	—	2	2	no	—
103	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
104	C₄×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
106	Dic₁₅₇	—	2	2	no	—
107	Dic₁₅₇	—	2	4	no	—
108	C₂×C₁₅₇	—	2	2	no	—
109	C₂×C₁₅₇	—	4	2	no	—
110	C₂×C₁₅₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₁₅₇	#568	1	2	yes	—
112	D₁₅₇	—	4	2	no	—
113	C₁₅₇	#2930	1	1	yes	characteristic, complemented, Sylow
114	#2930	—	157	1	no	maximal, Sylow
115	C₂×#131	D₁₅₇	1	1	yes	characteristic, complemented
116	C₂²×M₁₆	—	157	1	no	—
117	#525	—	157	1	no	—
118	#436	—	157	4	no	—
119	#435	—	157	4	no	—
120	#424	—	157	1	no	—
121	#408	—	157	1	no	—
122	#407	—	157	1	no	—
123	#405	—	157	1	no	—
124	C₂×#41	C₂×D₁₅₇	1	1	yes	characteristic
125	C₂³×C₄	—	157	1	no	—
126	#131	—	2	4	no	—
127	C₂×M₁₆	—	314	4	no	—
128	C₂×M₁₆	—	157	1	no	—
129	C₂×M₁₆	—	157	2	no	—
130	C₂×M₁₆	—	157	2	no	—
131	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
132	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	complemented
133	C₂²×C₈	—	157	1	no	—
134	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₁₅₇	1	2	yes	complemented
136	#123	—	157	2	no	—
137	C₂²⋊Q₈	—	157	2	no	—
138	#121	—	157	2	no	—
139	#120	—	157	2	no	—
140	C₄×D₄	—	157	4	no	—
141	#117	—	157	1	no	—
142	C₄⋊C₈	—	157	4	no	—
143	C₂²⋊C₈	—	157	4	no	—
144	C₂²⋊C₈	—	157	4	no	—
145	#97	—	157	2	no	—
146	C₄×C₈	—	157	2	no	—
147	C₂⁴	—	157	1	no	—
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₄×D₁₅₇	1	1	yes	characteristic
151	C₂×D₄	—	157	4	no	—
152	C₂×D₄	C₄×D₁₅₇	1	2	yes	—
153	C₂×D₄	C₄×D₁₅₇	1	1	yes	characteristic
154	C₂²×C₄	—	314	2	no	—
155	C₂²×C₄	—	314	2	no	—
156	C₂²×C₄	—	157	1	no	—
157	C₂²×C₄	—	157	2	no	—
158	C₂²×C₄	—	157	2	no	—
159	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	314	2	no	—
164	M₁₆	—	314	4	no	—
165	M₁₆	—	314	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	314	2	no	—
168	C₂×C₈	—	157	2	no	—
169	C₂×C₈	—	157	2	no	—
170	C₂×C₈	C₂²⋊D₁₅₇	1	4	yes	complemented
171	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
172	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
174	C₄⋊C₄	—	157	2	no	—
175	C₄⋊C₄	—	157	4	no	—
176	C₂²⋊C₄	—	157	2	no	—
177	C₂²⋊C₄	—	157	4	no	—
178	C₂²⋊C₄	—	157	4	no	—
179	C₄²	—	157	2	no	—
180	C₂³	—	314	2	no	—
181	C₂³	—	628	2	no	—
182	C₂³	—	157	2	no	—
183	C₂³	C₂×C₄×D₁₅₇	1	2	yes	—
184	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	628	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	314	2	no	—
192	C₂×C₄	—	314	4	no	—
193	C₂×C₄	—	314	2	no	—
194	C₂×C₄	—	314	2	no	—
195	C₂×C₄	—	157	4	no	—
196	C₂×C₄	—	157	2	no	—
197	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
200	C₂×C₄	#14578954	1	2	yes	—
201	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	314	2	no	—
204	C₈	—	314	2	no	—
205	C₈	C₂×C₂²⋊D₁₅₇	1	2	yes	—
206	C₈	C₂×C₂²⋊D₁₅₇	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	314	2	no	—
211	C₂²	—	314	2	no	—
212	C₂²	—	314	2	no	—
213	C₂²	—	628	4	no	—
214	C₂²	C₄×C₂²⋊D₁₅₇	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	314	2	no	—
218	C₄	—	314	4	no	—
219	C₄	C₄×C₂²⋊D₁₅₇	1	2	yes	—
220	C₄	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
221	C₄	C₂×#14578954	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	628	2	no	—
225	C₂	C₂×C₄×C₂²⋊D₁₅₇	1	1	yes	characteristic
226	C₂	#23758038	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	C₂×#131×C₁₅₇	C₂
prime radical	#23757780	C₂
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	C₂×#131	D₁₅₇
hypocenter	C₁₅₇	#2930
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#14578937	C₂	semidirect
3. #23757349	C₂	semidirect
4. #23757344	C₂	semidirect
5. #23757301	C₂	semidirect
6. #23757240	C₂	semidirect
7. #23757236	C₂	semidirect
8. #23757228	C₂	semidirect
9. C₂×#131×C₁₅₇	C₂	semidirect
22. #19097700	C₂²	semidirect
23. C₂²⋊#11929434	C₂²	semidirect
25. #19097681	C₂²	semidirect
26. C₈×Dic₁₅₇	C₂²	semidirect
30. C₂×M₁₆×C₁₅₇	C₂²	semidirect
32. C₂²×C₈×C₁₅₇	C₂²	semidirect
62. C₂×C₈×C₁₅₇	D₄	semidirect
113. C₁₅₇	#2930	semidirect
115. C₂×#131	D₁₅₇	semidirect
132. C₂×M₁₆	C₂×D₁₅₇	semidirect
135. C₂²×C₈	C₂×D₁₅₇	semidirect
170. C₂×C₈	C₂²⋊D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14578937	2
#23757780	2
#23757349	2
#23757344	2
#23757301	2
#23757240	2
#23757236	2
#23757228	2
C₂×#131×C₁₅₇	2
#2930	157