Finite Group Databasebeta

Subgroups — #34115358

Order 20096.

178 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758292	C₂	1	1	yes	maximal, characteristic
2	#23758291	C₂	1	1	yes	maximal, characteristic
3	#11929441⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11929441⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#23758114	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11929441	C₂	1	1	yes	maximal, characteristic, complemented
7	#23757325	C₂	1	1	yes	maximal, characteristic
8	#19097838	C₄	1	2	yes	—
9	C₄⋊#11929441	C₄	1	2	yes	—
10	#19097828	C₄	1	2	yes	—
11	#19097828	C₄	1	2	yes	—
12	C₄×#11929441	C₂²	1	1	yes	characteristic
13	C₄×#11929441	C₂²	1	1	yes	characteristic
14	C₄×#11929441	C₂²	1	1	yes	characteristic
15	C₄×#11929441	C₂²	1	1	yes	characteristic
16	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
17	C₄⋊#11929434	C₄	1	1	yes	characteristic
18	#19097683	C₄	1	1	yes	characteristic
19	#19097683	C₄	1	1	yes	characteristic
20	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
21	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₅₇	C₄	1	1	yes	characteristic
23	C₂×#11929441	C₂×C₄	1	2	yes	—
24	C₂×#11929441	C₂×C₄	1	2	yes	—
25	C₂×#11929441	C₂×C₄	1	2	yes	—
26	C₂×#11929441	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
30	C₂×#11929434	C₂×C₄	1	1	yes	characteristic
31	C₂×#11929434	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
38	#11929441	—	2	2	no	—
39	#11929441	—	2	2	no	—
40	#11929441	—	2	2	no	—
41	#11929441	—	2	2	no	—
42	C₂²×D₁₅₇	C₄²	1	1	yes	characteristic
43	C₂×Dic₁₅₇	C₄²	1	1	yes	characteristic
44	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
46	C₄×D₁₅₇	—	2	1	no	—
47	C₄×D₁₅₇	—	2	1	no	—
48	C₄×D₁₅₇	—	2	2	no	—
49	C₄×D₁₅₇	—	2	2	no	—
50	#11929434	—	2	1	no	—
51	#11929434	—	2	1	no	—
52	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
53	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
54	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₁₅₇	—	2	1	no	—
56	C₈×C₁₅₇	—	2	1	no	—
57	C₂²×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
58	C₂×D₁₅₇	—	2	1	no	—
59	C₂×D₁₅₇	—	2	1	no	—
60	C₂×D₁₅₇	—	2	1	no	—
61	C₄×C₁₅₇	—	2	1	no	—
62	C₄×C₁₅₇	—	2	1	no	—
63	C₄×C₁₅₇	#131	1	1	yes	characteristic
64	C₄×C₁₅₇	#131	1	1	yes	characteristic
65	Dic₁₅₇	—	2	1	no	—
66	Dic₁₅₇	#131	1	2	yes	—
67	Dic₁₅₇	#131	1	2	yes	—
68	C₂×C₁₅₇	#405	1	1	yes	characteristic
69	C₂×C₁₅₇	#405	1	1	yes	characteristic
70	C₂×C₁₅₇	#405	1	1	yes	characteristic
71	D₁₅₇	—	4	1	no	—
72	C₁₅₇	#1454	1	1	yes	characteristic, complemented, Sylow
73	#1454	—	157	1	no	maximal, Sylow
74	#431	—	157	1	no	—
75	#431	—	157	1	no	—
76	#431	—	157	1	no	—
77	#322	—	157	1	no	—
78	#322	—	157	1	no	—
79	#322	—	157	1	no	—
80	C₈²	—	157	1	no	—
81	C₂×C₄²	—	157	1	no	—
82	C₄⋊C₈	—	157	2	no	—
83	C₄⋊C₈	—	157	2	no	—
84	C₄⋊C₈	—	157	1	no	—
85	C₄⋊C₈	#10597845	1	1	yes	characteristic
86	C₂²⋊C₈	—	157	2	no	—
87	C₂²⋊C₈	—	157	2	no	—
88	C₂²⋊C₈	—	157	1	no	—
89	C₂²⋊C₈	—	157	1	no	—
90	C₄×C₈	—	157	1	no	—
91	C₄×C₈	—	157	1	no	—
92	C₄×C₈	—	157	1	no	—
93	C₄×C₈	—	157	1	no	—
94	C₄×C₈	—	157	1	no	—
95	C₄×C₈	—	157	1	no	—
96	C₂²×C₄	—	157	1	no	—
97	C₂²×C₄	—	157	1	no	—
98	C₂²×C₄	—	157	1	no	—
99	C₂×C₈	—	157	2	no	—
100	C₂×C₈	—	157	2	no	—
101	C₂×C₈	—	157	2	no	—
102	C₂×C₈	—	157	2	no	—
103	C₂×C₈	—	157	1	no	—
104	C₂×C₈	—	157	1	no	—
105	C₂×C₈	C₂×#10597845	1	1	yes	characteristic
106	C₂×C₈	C₂×#10597845	1	1	yes	characteristic
107	C₄²	—	157	1	no	—
108	C₄²	—	157	1	no	—
109	C₄²	—	157	1	no	—
110	C₄²	C₂×#10597845	1	1	yes	characteristic
111	C₂³	—	157	1	no	—
112	C₂×C₄	—	314	2	no	—
113	C₂×C₄	—	314	2	no	—
114	C₂×C₄	—	314	1	no	—
115	C₂×C₄	—	314	1	no	—
116	C₂×C₄	—	157	1	no	—
117	C₂×C₄	—	157	1	no	—
118	C₂×C₄	—	157	1	no	—
119	C₂×C₄	C₄×#10597845	1	1	yes	characteristic
120	C₂×C₄	C₄×#10597845	1	1	yes	characteristic
121	C₂×C₄	C₂²×#10597845	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	314	2	no	—
125	C₈	—	314	2	no	—
126	C₈	—	314	2	no	—
127	C₈	—	314	2	no	—
128	C₈	—	314	1	no	—
129	C₈	—	314	1	no	—
130	C₂²	—	314	1	no	—
131	C₂²	—	314	1	no	—
132	C₂²	—	314	1	no	—
133	C₂²	C₂×C₄×#10597845	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	314	1	no	—
137	C₄	—	157	2	no	—
138	C₄	—	157	2	no	—
139	C₄	#19097858	1	1	yes	characteristic
140	C₄	#19097859	1	1	yes	characteristic
141	C₂	—	628	1	no	—
142	C₂	#23758236	1	1	yes	characteristic
143	C₂	#23758238	1	1	yes	characteristic
144	C₂	#23758239	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10597845
commutator, socle	C₂²×C₁₅₇	C₂×C₄²
Fitting	C₄⋊C₈×C₁₅₇	C₄
prime radical	C₂²×D₁₅₇	C₄²
cosocle	C₄×Dic₁₅₇	C₂³
hypercenter	C₄⋊C₈	#10597845
hypocenter	C₁₅₇	#1454
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #11929441⋊C₈	C₂	semidirect
4. #11929441⋊C₈	C₂	semidirect
5. #23758114	C₂	semidirect
6. C₈×#11929441	C₂	semidirect
72. C₁₅₇	#1454	semidirect

Maximal subgroups

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

subgroup	index
#23758292	2
#23758291	2
#11929441⋊C₈	2
#23758114	2
C₈×#11929441	2
#23757325	2
#1454	157