Finite Group Databasebeta

Subgroups — #34099721

Order 20096.

140 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 49 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	#23757625	C₂	1	1	yes	maximal, characteristic
2	#23757565	C₂	1	1	yes	maximal, characteristic
3	#23757559	C₂	1	1	yes	maximal, characteristic
4	#23757400	C₂	1	1	yes	maximal, characteristic
5	#23757023	C₂	1	1	yes	maximal, characteristic
6	#23757020	C₂	1	1	yes	maximal, characteristic
7	#333×C₁₅₇	C₂	1	1	yes	maximal, characteristic
8	#19097771	C₂²	1	1	yes	characteristic
9	C₄×Dic₃₁₄	C₂²	1	1	yes	characteristic
10	#19097704	—	2	1	no	—
11	#19097703	—	2	1	no	—
12	C₄⋊#11929434	C₂²	1	1	yes	characteristic
13	#19097700	—	2	1	no	—
14	#19097696	—	2	1	no	—
15	#19097694	—	2	1	no	—
16	#19097694	—	2	1	no	—
17	#19097694	—	2	1	no	—
18	#19097681	—	2	1	no	—
19	C₄×#11929434	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₁₄	—	2	1	no	—
24	C₂×Dic₃₁₄	D₄	1	1	yes	characteristic
25	C₂×#11929434	—	2	1	no	—
26	C₂×#11929434	D₄	1	1	yes	characteristic
27	C₂×#11929434	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₅₇	—	2	1	no	—
29	C₄⋊Dic₁₅₇	—	2	1	no	—
30	C₄⋊Dic₁₅₇	D₄	1	1	yes	characteristic
31	#14578933	—	2	1	no	—
32	C₄×Dic₁₅₇	—	2	1	no	—
33	C₂×Q₈×C₁₅₇	—	2	1	no	—
34	C₂×C₈×C₁₅₇	—	2	1	no	—
35	C₂×C₈×C₁₅₇	—	2	1	no	—
36	C₄⋊C₄×C₁₅₇	—	2	1	no	—
37	C₄⋊C₄×C₁₅₇	—	2	1	no	—
38	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
39	Dic₃₁₄	—	2	1	no	—
40	Dic₃₁₄	—	2	1	no	—
41	Dic₃₁₄	—	4	1	no	—
42	Dic₃₁₄	—	4	1	no	—
43	C₂×Dic₁₅₇	—	2	1	no	—
44	C₂×Dic₁₅₇	—	2	1	no	—
45	C₂×Dic₁₅₇	—	4	1	no	—
46	#11929434	—	2	1	no	—
47	#11929434	—	2	1	no	—
48	#11929434	—	4	1	no	—
49	Q₈×C₁₅₇	—	8	1	no	—
50	C₂×C₄×C₁₅₇	—	4	1	no	—
51	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
54	C₈×C₁₅₇	—	4	1	no	—
55	C₈×C₁₅₇	—	4	1	no	—
56	C₂²×C₁₅₇	#121	1	1	yes	characteristic
57	C₄×C₁₅₇	—	2	1	no	—
58	C₄×C₁₅₇	—	8	1	no	—
59	C₄×C₁₅₇	#135	1	1	yes	characteristic
60	C₄×C₁₅₇	#137	1	1	yes	characteristic
61	C₄×C₁₅₇	#137	1	1	yes	characteristic
62	C₄×C₁₅₇	#137	1	1	yes	characteristic
63	Dic₁₅₇	—	4	1	no	—
64	Dic₁₅₇	—	4	1	no	—
65	Dic₁₅₇	—	8	1	no	—
66	C₂×C₁₅₇	#464	1	1	yes	characteristic
67	C₂×C₁₅₇	#470	1	1	yes	characteristic
68	C₂×C₁₅₇	#471	1	1	yes	characteristic
69	C₁₅₇	#1704	1	1	yes	characteristic, complemented, Sylow
70	#1704	—	157	1	no	maximal, Sylow
71	#487	—	157	1	no	—
72	#477	—	157	1	no	—
73	#475	—	157	1	no	—
74	#446	—	157	1	no	—
75	#334	—	157	1	no	—
76	#333	D₁₅₇	1	1	yes	characteristic
77	#332	—	157	1	no	—
78	C₄⋊Q₈	—	157	1	no	—
79	C₄⋊Q₈	C₂×D₁₅₇	1	1	yes	characteristic
80	C₄×Q₈	—	157	1	no	—
81	#107	—	314	1	no	—
82	#106	—	314	1	no	—
83	C₄⋊C₈	—	314	1	no	—
84	C₄⋊C₈	—	157	1	no	—
85	C₄⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
87	#103	—	314	1	no	—
88	#103	—	314	1	no	—
89	#103	—	314	1	no	—
90	#103	—	314	1	no	—
91	#97	—	314	1	no	—
92	C₄×C₈	—	157	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	314	1	no	—
95	C₂×Q₈	—	157	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	314	1	no	—
99	C₂×C₈	—	157	1	no	—
100	C₂×C₈	—	157	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	314	1	no	—
104	C₄⋊C₄	—	314	1	no	—
105	C₄⋊C₄	—	314	1	no	—
106	C₄⋊C₄	—	157	1	no	—
107	C₄²	—	314	1	no	—
108	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
109	Q₈	—	8	1	no	—
110	Q₈	—	314	1	no	—
111	Q₈	—	314	1	no	—
112	Q₈	—	628	1	no	—
113	Q₈	—	628	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	314	1	no	—
116	C₂×C₄	—	314	1	no	—
117	C₂×C₄	—	628	1	no	—
118	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
119	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
120	C₂×C₄	#14578954	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	314	1	no	—
124	C₈	—	314	1	no	—
125	C₈	—	628	1	no	—
126	C₂²	#19097744	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	628	1	no	—
130	C₄	—	628	1	no	—
131	C₄	—	1256	1	no	—
132	C₄	#19097792	1	1	yes	characteristic
133	C₄	#19097800	1	1	yes	characteristic
134	C₄	#19097803	1	1	yes	characteristic
135	C₄	#19097803	1	1	yes	characteristic
136	C₂	#23757498	1	1	yes	characteristic
137	C₂	#23757530	1	1	yes	characteristic
138	C₂	#23757535	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19097744
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#333×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#121
hypercenter	#333	D₁₅₇
hypocenter	C₁₅₇	#1704
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

Complements

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

N	H ≅ G/N	type
69. C₁₅₇	#1704	semidirect

Maximal subgroups

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

subgroup	index
#23757625	2
#23757565	2
#23757559	2
#23757400	2
#23757023	2
#23757020	2
#333×C₁₅₇	2
#1704	157