Finite Group Databasebeta

Subgroups — #34115708

Order 20096.

150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 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	#23758295	C₂	1	1	yes	maximal, characteristic
2	#23758292	C₂	1	1	yes	maximal, characteristic
3	#23758137	C₂	1	1	yes	maximal, characteristic, complemented
4	#23758136	C₂	1	1	yes	maximal, characteristic, complemented
5	#23758135	C₂	1	1	yes	maximal, characteristic, complemented
6	#23758134	C₂	1	1	yes	maximal, characteristic, complemented
7	#23757325	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#11929441	C₂²	1	1	yes	characteristic
9	C₄⋊#11929441	C₂²	1	1	yes	characteristic
10	C₄⋊#11929441	C₂²	1	1	yes	characteristic
11	C₄⋊#11929441	C₂²	1	1	yes	characteristic
12	#19097828	—	2	1	no	—
13	#19097828	—	2	1	no	—
14	#19097826	—	2	1	no	—
15	C₄×#11929441	—	2	1	no	—
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	—	2	1	no	—
24	C₂×#11929441	—	2	1	no	—
25	C₂×#11929441	—	2	1	no	—
26	C₂×#11929441	—	2	1	no	—
27	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
29	C₂×C₄×D₁₅₇	Q₈	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	D₄	1	1	yes	characteristic
33	C₄×Dic₁₅₇	Q₈	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	—	4	1	no	—
39	#11929441	—	4	1	no	—
40	#11929441	—	4	1	no	—
41	#11929441	—	4	1	no	—
42	C₂²×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₁₅₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₁₅₇	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₁₅₇	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₁₅₇	—	2	1	no	—
47	C₄×D₁₅₇	—	2	1	no	—
48	C₄×D₁₅₇	—	2	1	no	—
49	C₄×D₁₅₇	—	2	1	no	—
50	C₄×D₁₅₇	—	4	1	no	—
51	#11929434	—	2	1	no	—
52	#11929434	—	2	1	no	—
53	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
56	C₈×C₁₅₇	—	2	1	no	—
57	C₈×C₁₅₇	—	2	1	no	—
58	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×D₁₅₇	—	2	1	no	—
60	C₂×D₁₅₇	—	2	1	no	—
61	C₂×D₁₅₇	—	2	1	no	—
62	C₄×C₁₅₇	—	2	1	no	—
63	C₄×C₁₅₇	—	2	1	no	—
64	C₄×C₁₅₇	#131	1	1	yes	characteristic
65	C₄×C₁₅₇	#131	1	1	yes	characteristic
66	Dic₁₅₇	—	2	1	no	—
67	Dic₁₅₇	—	2	1	no	—
68	Dic₁₅₇	#135	1	1	yes	characteristic
69	Dic₁₅₇	#135	1	1	yes	characteristic
70	C₂×C₁₅₇	#424	1	1	yes	characteristic
71	C₂×C₁₅₇	#427	1	1	yes	characteristic
72	C₂×C₁₅₇	#430	1	1	yes	characteristic
73	D₁₅₇	—	4	1	no	—
74	C₁₅₇	#1572	1	1	yes	characteristic, complemented, Sylow
75	#1572	—	157	1	no	maximal, Sylow
76	#432	—	157	1	no	—
77	#431	—	157	1	no	—
78	#431	—	157	1	no	—
79	#335	—	157	1	no	—
80	#335	—	157	1	no	—
81	#334	—	157	1	no	—
82	#334	—	157	1	no	—
83	C₂×C₄²	—	157	1	no	—
84	C₄⋊C₈	—	157	1	no	—
85	C₄⋊C₈	—	157	1	no	—
86	C₄⋊C₈	—	157	1	no	—
87	C₄⋊C₈	—	157	1	no	—
88	C₄⋊C₈	—	157	1	no	—
89	C₄⋊C₈	#10597845	1	1	yes	characteristic
90	C₂²⋊C₈	—	314	1	no	—
91	C₂²⋊C₈	—	314	1	no	—
92	C₂²⋊C₈	—	157	1	no	—
93	C₂²⋊C₈	—	157	1	no	—
94	#97	—	314	1	no	—
95	C₄×C₈	—	314	1	no	—
96	C₄×C₈	—	157	1	no	—
97	C₄×C₈	—	157	1	no	—
98	C₂²×C₄	—	157	1	no	—
99	C₂²×C₄	—	157	1	no	—
100	C₂²×C₄	—	157	1	no	—
101	C₂×C₈	—	314	1	no	—
102	C₂×C₈	—	314	1	no	—
103	C₂×C₈	—	314	1	no	—
104	C₂×C₈	—	314	1	no	—
105	C₂×C₈	—	157	1	no	—
106	C₂×C₈	—	157	1	no	—
107	C₂×C₈	C₂×#10597845	1	1	yes	characteristic
108	C₂×C₈	C₂×#10597845	1	1	yes	characteristic
109	C₄²	—	157	1	no	—
110	C₄²	—	157	1	no	—
111	C₄²	—	157	1	no	—
112	C₄²	C₂×#10597845	1	1	yes	characteristic
113	C₂³	—	157	1	no	—
114	C₂×C₄	—	314	1	no	—
115	C₂×C₄	—	314	1	no	—
116	C₂×C₄	—	314	1	no	—
117	C₂×C₄	—	314	1	no	—
118	C₂×C₄	—	628	1	no	—
119	C₂×C₄	—	157	1	no	—
120	C₂×C₄	—	157	1	no	—
121	C₂×C₄	—	157	1	no	—
122	C₂×C₄	C₄⋊#10597845	1	1	yes	characteristic
123	C₂×C₄	C₄⋊#10597845	1	1	yes	characteristic
124	C₂×C₄	C₂²×#10597845	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	314	1	no	—
128	C₈	—	314	1	no	—
129	C₈	—	628	1	no	—
130	C₈	—	628	1	no	—
131	C₈	—	628	1	no	—
132	C₈	—	628	1	no	—
133	C₂²	—	314	1	no	—
134	C₂²	—	314	1	no	—
135	C₂²	—	314	1	no	—
136	C₂²	C₂×C₄⋊#10597845	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	314	1	no	—
140	C₄	—	314	1	no	—
141	C₄	—	157	1	no	—
142	C₄	—	157	1	no	—
143	C₄	#19097858	1	1	yes	characteristic
144	C₄	#19097859	1	1	yes	characteristic
145	C₂	—	628	1	no	—
146	C₂	#23758280	1	1	yes	characteristic
147	C₂	#23758283	1	1	yes	characteristic
148	C₂	#23758288	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10597845
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10597845
Fitting	C₄⋊C₈×C₁₅₇	C₄
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₅₇	C₄⋊C₄
cosocle	C₄×Dic₁₅₇	C₂³
hypercenter	C₄⋊C₈	#10597845
hypocenter	C₁₅₇	#1572
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #23758137	C₂	semidirect
4. #23758136	C₂	semidirect
5. #23758135	C₂	semidirect
6. #23758134	C₂	semidirect
74. C₁₅₇	#1572	semidirect

Maximal subgroups

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

subgroup	index
#23758295	2
#23758292	2
#23758137	2
#23758136	2
#23758135	2
#23758134	2
#23757325	2
#1572	157