Subgroups — #34109261

Order 20096.

398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757404	C₂	1	2	yes	maximal, complemented
2	#23757345	C₂	1	4	yes	maximal, complemented
3	D₄×#11929434	C₂	1	2	yes	maximal, complemented
4	#23757330	C₂	1	2	yes	maximal, complemented
5	#23757326	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757297	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊#11929434	C₂	1	2	yes	maximal, complemented
8	C₄×#41×C₁₅₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14578938	C₂²	1	2	yes	—
10	C₂²×#11929434	C₂²	1	4	yes	—
11	C₄⋊#11929434	C₂²	1	1	yes	characteristic
12	C₄⋊#11929434	C₂²	1	1	yes	characteristic
13	C₄⋊#11929434	C₂²	1	2	yes	—
14	C₄⋊#11929434	C₂²	1	2	yes	—
15	C₄⋊#11929434	C₂²	1	4	yes	—
16	C₂²⋊#11929434	C₂²	1	2	yes	—
17	C₂²⋊#11929434	C₂²	1	4	yes	—
18	#19097682	C₂²	1	4	yes	—
19	C₄×#11929434	C₂²	1	2	yes	—
20	C₂×#41×C₁₅₇	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₅₇	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₅₇	C₄	1	2	yes	—
24	C₄×D₄×C₁₅₇	C₄	1	1	yes	characteristic
25	C₄×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₅₇	C₂²	1	2	yes	—
27	#117×C₁₅₇	C₄	1	2	yes	—
28	#117×C₁₅₇	C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₅₇	C₂²	1	2	yes	—
31	#14578938	—	4	2	no	—
32	C₂×#11929434	—	4	4	no	—
33	C₂×#11929434	C₂³	1	2	yes	—
34	C₂×#11929434	C₂³	1	2	yes	—
35	C₂×#11929434	C₂³	1	4	yes	—
36	#41×C₁₅₇	—	2	4	no	—
37	C₂×Q₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₅₇	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
53	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
55	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
56	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
58	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
59	C₄²×C₁₅₇	C₂³	1	2	yes	—
60	#11929434	—	2	2	no	—
61	#11929434	—	2	4	no	—
62	#11929434	—	2	2	no	—
63	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
64	C₂³×C₁₅₇	C₂²×C₄	1	2	yes	—
65	Q₈×C₁₅₇	—	2	2	no	—
66	D₄×C₁₅₇	—	2	4	no	—
67	D₄×C₁₅₇	M₁₆	1	4	yes	complemented
68	C₂×C₄×C₁₅₇	—	2	4	no	—
69	C₂×C₄×C₁₅₇	—	2	2	no	—
70	C₂×C₄×C₁₅₇	—	2	4	no	—
71	C₂×C₄×C₁₅₇	—	2	2	no	—
72	C₂×C₄×C₁₅₇	—	2	2	no	—
73	C₂×C₄×C₁₅₇	—	2	1	no	—
74	C₂×C₄×C₁₅₇	—	2	2	no	—
75	C₂×C₄×C₁₅₇	—	2	1	no	—
76	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
85	C₂²×C₁₅₇	—	2	2	no	—
86	C₂²×C₁₅₇	—	2	2	no	—
87	C₂²×C₁₅₇	—	2	1	no	—
88	C₂²×C₁₅₇	—	2	1	no	—
89	C₂²×C₁₅₇	—	2	1	no	—
90	C₂²×C₁₅₇	C₂×M₁₆	1	4	yes	—
91	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₁₅₇	—	2	2	no	—
93	C₄×C₁₅₇	—	2	1	no	—
94	C₄×C₁₅₇	—	2	1	no	—
95	C₄×C₁₅₇	—	2	2	no	—
96	C₄×C₁₅₇	—	2	2	no	—
97	C₄×C₁₅₇	—	2	1	no	—
98	C₄×C₁₅₇	C₂×M₁₆	1	2	yes	—
99	C₄×C₁₅₇	#131	1	2	yes	—
100	C₄×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
101	C₄×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×C₁₅₇	—	2	4	no	—
103	C₂×C₁₅₇	—	4	1	no	—
104	C₂×C₁₅₇	#520	1	1	yes	characteristic
105	C₂×C₁₅₇	C₂²×M₁₆	1	1	yes	characteristic
106	C₂×C₁₅₇	C₂×#131	1	1	yes	characteristic
107	C₁₅₇	#2990	1	1	yes	characteristic, complemented, Sylow
108	#2990	—	157	1	no	maximal, Sylow
109	C₄×#41	D₁₅₇	1	1	yes	characteristic
110	#446	—	157	2	no	—
111	#435	—	157	4	no	—
112	C₈×D₄	—	157	2	no	—
113	#432	—	157	2	no	—
114	#431	—	157	1	no	—
115	#423	—	157	1	no	—
116	C₂×C₄⋊C₈	—	157	2	no	—
117	C₂×#41	Dic₁₅₇	1	1	yes	characteristic
118	C₂×M₁₆	—	157	2	no	—
119	C₂²×C₈	—	157	4	no	—
120	C₄×Q₈	Dic₁₅₇	1	1	yes	characteristic
121	C₄×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
122	C₄×D₄	Dic₁₅₇	1	2	yes	—
123	C₄×D₄	Dic₁₅₇	1	1	yes	characteristic
124	C₄×D₄	C₂×D₁₅₇	1	1	yes	characteristic
125	C₄×D₄	C₂×D₁₅₇	1	2	yes	—
126	#117	Dic₁₅₇	1	1	yes	characteristic
127	#117	Dic₁₅₇	1	2	yes	—
128	C₂×C₄²	C₂×D₁₅₇	1	2	yes	—
129	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
130	C₄⋊C₈	—	157	1	no	—
131	C₄⋊C₈	—	157	2	no	—
132	C₄⋊C₈	—	157	4	no	—
133	C₄⋊C₈	—	157	2	no	—
134	C₄⋊C₈	—	157	1	no	—
135	C₂²⋊C₈	—	157	2	no	—
136	C₂²⋊C₈	—	157	4	no	—
137	#97	—	157	4	no	—
138	C₄×C₈	—	157	2	no	—
139	#41	—	2	4	no	—
140	C₂×Q₈	C₂×Dic₁₅₇	1	1	yes	characteristic
141	C₂×D₄	C₂×Dic₁₅₇	1	2	yes	—
142	C₂×D₄	C₂×Dic₁₅₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
144	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
147	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
148	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
149	M₁₆	—	628	2	no	—
150	C₂×C₈	—	628	4	no	—
151	C₂×C₈	—	157	2	no	—
152	C₂×C₈	—	157	2	no	—
153	C₂×C₈	—	157	4	no	—
154	C₄⋊C₄	C₂×Dic₁₅₇	1	2	yes	—
155	C₄⋊C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
156	C₄⋊C₄	C₂×Dic₁₅₇	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
158	C₂²⋊C₄	C₂×Dic₁₅₇	1	2	yes	—
159	C₂²⋊C₄	C₂×Dic₁₅₇	1	2	yes	—
160	C₂²⋊C₄	C₂×Dic₁₅₇	1	2	yes	—
161	C₄²	C₂×Dic₁₅₇	1	2	yes	—
162	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
163	C₄²	C₂×Dic₁₅₇	1	2	yes	—
164	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
165	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
166	C₄²	C₂²×D₁₅₇	1	2	yes	—
167	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
168	C₂³	C₂²×Dic₁₅₇	1	1	yes	characteristic
169	C₂³	C₂²×Dic₁₅₇	1	2	yes	—
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	#14578938	1	4	yes	complemented
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₁₅₇	1	2	yes	—
183	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₁₅₇	1	2	yes	—
186	C₂×C₄	C₂²×Dic₁₅₇	1	2	yes	—
187	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
190	C₈	—	314	2	no	—
191	C₈	—	314	4	no	—
192	C₈	—	314	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	C₂×#14578938	1	4	yes	—
199	C₂²	C₂³×Dic₁₅₇	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×#14578938	1	2	yes	—
207	C₄	#19097778	1	2	yes	—
208	C₄	#19097817	1	1	yes	characteristic
209	C₄	#19097818	1	1	yes	characteristic
210	C₂	—	2	4	no	—
211	C₂	—	4	1	no	—
212	C₂	#23757756	1	1	yes	characteristic
213	C₂	C₂²×#14578938	1	1	yes	characteristic
214	C₂	C₂×#19097778	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	C₄×#41×C₁₅₇	C₂
prime radical	C₂×#41×C₁₅₇	C₄
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	C₄×#41	D₁₅₇
hypocenter	C₁₅₇	#2990
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #23757404	C₂	semidirect
2. #23757345	C₂	semidirect
3. D₄×#11929434	C₂	semidirect
4. #23757330	C₂	semidirect
5. #23757326	C₂	semidirect
6. #23757297	C₂	semidirect
7. C₂×C₄⋊#11929434	C₂	semidirect
67. D₄×C₁₅₇	M₁₆	semidirect
107. C₁₅₇	#2990	semidirect
172. D₄	#14578938	semidirect

Maximal subgroups

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

subgroup	index
#23757404	2
#23757345	2
D₄×#11929434	2
#23757330	2
#23757326	2
#23757297	2
C₂×C₄⋊#11929434	2
C₄×#41×C₁₅₇	2
#2990	157