Finite Group Databasebeta

Subgroups — #34113284

Order 20096.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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	#23758090	C₂	1	1	yes	maximal, characteristic, complemented
2	#23758078	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757716	C₂	1	3	yes	maximal, complemented
4	#23757711	C₂	1	3	yes	maximal, complemented
5	#23757705	C₂	1	3	yes	maximal, complemented
6	C₂×#19097716	C₂	1	3	yes	maximal, complemented
7	#504×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#19097818	—	2	1	no	—
9	#19097817	—	2	1	no	—
10	#19097804	—	2	3	no	—
11	#19097799	—	2	3	no	—
12	#19097793	C₂²	1	3	yes	—
13	#19097793	C₂²	1	3	yes	complemented
14	C₂×Dic₆₂₈	C₂²	1	3	yes	complemented
15	C₂×D₁₂₅₆	C₂²	1	3	yes	—
16	Dic₁₂₅₆	C₂²	1	3	yes	complemented
17	#19097716	C₂²	1	3	yes	complemented
18	#19097716	C₂²	1	6	yes	complemented
19	#19097716	C₂²	1	1	yes	characteristic, complemented
20	D₂₅₁₂	C₂²	1	3	yes	complemented
21	#131×C₁₅₇	C₂²	1	1	yes	characteristic
22	M₃₂×C₁₅₇	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₁₅₇	C₂²	1	3	yes	complemented
24	#14578955	—	4	1	no	—
25	#14578954	—	2	3	no	—
26	#14578954	—	2	3	no	—
27	#14578953	—	4	3	no	—
28	C₂×Dic₃₁₄	—	2	3	no	—
29	Q₈×D₁₅₇	—	4	1	no	—
30	C₂×D₆₂₈	—	2	3	no	—
31	D₄×D₁₅₇	—	4	3	no	—
32	Dic₆₂₈	—	2	3	no	—
33	Dic₆₂₈	C₂³	1	3	yes	—
34	Dic₆₂₈	C₂³	1	1	yes	characteristic
35	#14578943	—	2	3	no	—
36	#14578943	—	2	3	no	—
37	D₁₂₅₆	—	2	3	no	—
38	D₁₂₅₆	C₂³	1	1	yes	characteristic
39	D₁₂₅₆	C₂³	1	3	yes	—
40	#41×C₁₅₇	D₄	1	1	yes	characteristic
41	M₁₆×C₁₅₇	D₄	1	3	yes	—
42	C₂×C₈×C₁₅₇	C₂³	1	3	yes	—
43	C₁₆×C₁₅₇	C₂³	1	1	yes	characteristic
44	C₁₆×C₁₅₇	C₂³	1	3	yes	—
45	C₂²×D₁₅₇	—	4	3	no	—
46	Dic₃₁₄	—	2	3	no	—
47	Dic₃₁₄	—	2	1	no	—
48	Dic₃₁₄	—	4	3	no	—
49	D₆₂₈	—	2	3	no	—
50	D₆₂₈	—	2	1	no	—
51	D₆₂₈	—	4	3	no	—
52	C₂²⋊D₁₅₇	—	4	3	no	—
53	C₂²⋊D₁₅₇	—	4	3	no	—
54	C₂×Dic₁₅₇	—	4	3	no	—
55	C₄×D₁₅₇	—	4	3	no	—
56	C₄×D₁₅₇	—	4	3	no	—
57	Q₈×C₁₅₇	D₈	1	1	yes	characteristic
58	D₄×C₁₅₇	D₈	1	3	yes	—
59	C₂×C₄×C₁₅₇	C₂×D₄	1	3	yes	—
60	C₈×C₁₅₇	C₂×D₄	1	3	yes	—
61	C₈×C₁₅₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₅₇	C₂×D₈	1	3	yes	—
63	C₂×D₁₅₇	—	4	1	no	—
64	C₂×D₁₅₇	—	4	3	no	—
65	C₂×D₁₅₇	—	16	3	no	—
66	C₄×C₁₅₇	C₂×D₈	1	3	yes	—
67	C₄×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
68	Dic₁₅₇	—	4	1	no	—
69	Dic₁₅₇	—	4	3	no	—
70	C₂×C₁₅₇	—	2	3	no	—
71	C₂×C₁₅₇	C₂²×D₈	1	1	yes	characteristic
72	D₁₅₇	—	8	1	no	—
73	D₁₅₇	—	8	3	no	—
74	C₁₅₇	#3416	1	1	yes	characteristic, complemented, Sylow
75	#3416	—	157	1	no	maximal, Sylow
76	#578	—	157	1	no	—
77	#576	—	157	1	no	—
78	#510	—	157	3	no	—
79	#509	—	157	3	no	—
80	#508	—	157	3	no	—
81	C₂×SD₃₂	—	157	3	no	—
82	#504	D₁₅₇	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	314	1	no	—
84	H(5,𝔽₂)	—	314	1	no	—
85	#137	—	314	3	no	—
86	AGL(1,ℤ/8)	—	314	3	no	—
87	#135	—	157	3	no	—
88	#135	—	157	3	no	—
89	C₂×Q₁₆	—	157	3	no	—
90	C₂×D₈	—	157	3	no	—
91	#131	C₂×D₁₅₇	1	1	yes	characteristic
92	Q₃₂	—	157	3	no	—
93	SD₃₂	—	157	3	no	—
94	SD₃₂	—	157	6	no	—
95	SD₃₂	—	157	1	no	—
96	D₁₆	—	157	3	no	—
97	M₃₂	C₂×D₁₅₇	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₁₅₇	1	3	yes	complemented
99	#41	—	314	3	no	—
100	#41	—	314	3	no	—
101	#41	—	628	1	no	—
102	#41	—	628	3	no	—
103	#41	D₆₂₈	1	1	yes	characteristic
104	C₂×Q₈	—	314	3	no	—
105	C₂×Q₈	—	628	1	no	—
106	C₂×D₄	—	314	3	no	—
107	C₂×D₄	—	628	3	no	—
108	Q₁₆	—	314	3	no	—
109	Q₁₆	—	157	3	no	—
110	Q₁₆	—	157	1	no	—
111	SD₁₆	—	314	3	no	—
112	SD₁₆	—	314	3	no	—
113	D₈	—	314	3	no	—
114	D₈	—	157	1	no	—
115	D₈	—	157	3	no	—
116	M₁₆	D₆₂₈	1	3	yes	—
117	C₂×C₈	C₂²×D₁₅₇	1	3	yes	—
118	C₁₆	C₂²×D₁₅₇	1	1	yes	characteristic
119	C₁₆	C₂²×D₁₅₇	1	3	yes	—
120	C₂³	—	628	3	no	—
121	Q₈	—	314	1	no	—
122	Q₈	—	314	3	no	—
123	Q₈	—	628	3	no	—
124	Q₈	D₁₂₅₆	1	1	yes	characteristic
125	D₄	—	314	3	no	—
126	D₄	—	314	1	no	—
127	D₄	—	628	3	no	—
128	D₄	—	628	3	no	—
129	D₄	—	628	3	no	—
130	D₄	D₁₂₅₆	1	3	yes	—
131	C₂×C₄	—	628	3	no	—
132	C₂×C₄	—	628	3	no	—
133	C₂×C₄	—	628	3	no	—
134	C₂×C₄	C₂×D₆₂₈	1	3	yes	—
135	C₈	C₂×D₆₂₈	1	3	yes	—
136	C₈	C₂³×D₁₅₇	1	1	yes	characteristic
137	C₂²	—	628	1	no	—
138	C₂²	—	628	3	no	—
139	C₂²	—	2512	3	no	—
140	C₂²	C₂×D₁₂₅₆	1	3	yes	—
141	C₄	—	628	1	no	—
142	C₄	—	628	3	no	—
143	C₄	C₂×D₁₂₅₆	1	3	yes	—
144	C₄	C₂²×D₆₂₈	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	1256	3	no	—
147	C₂	—	1256	1	no	—
148	C₂	C₂²×D₁₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₂₅₆
commutator, cosocle	C₈×C₁₅₇	C₂⁴
Frattini	C₈	C₂³×D₁₅₇
Fitting	#504×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂²×D₈
hypercenter	#504	D₁₅₇
hypocenter	C₁₅₇	#3416
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #23758090	C₂	semidirect
2. #23758078	C₂	semidirect
3. #23757716	C₂	semidirect
4. #23757711	C₂	semidirect
5. #23757705	C₂	semidirect
6. C₂×#19097716	C₂	semidirect
7. #504×C₁₅₇	C₂	semidirect
13. #19097793	C₂²	semidirect
14. C₂×Dic₆₂₈	C₂²	semidirect
16. Dic₁₂₅₆	C₂²	semidirect
17. #19097716	C₂²	semidirect
18. #19097716	C₂²	semidirect
19. #19097716	C₂²	semidirect
20. D₂₅₁₂	C₂²	semidirect
22. M₃₂×C₁₅₇	C₂²	semidirect
23. C₂×C₁₆×C₁₅₇	C₂²	semidirect
74. C₁₅₇	#3416	semidirect
82. #504	D₁₅₇	semidirect
97. M₃₂	C₂×D₁₅₇	semidirect
98. C₂×C₁₆	C₂×D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23758090	2
#23758078	2
#23757716	2
#23757711	2
#23757705	2
C₂×#19097716	2
#504×C₁₅₇	2
#3416	157