Group #23766003 — subgroups

Subgroups — #23766003

Order 10112.

512 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 229 normal subgroups are shown.

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

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19098449	C₂	1	8	yes	maximal, complemented
2	#19098426	C₂	1	2	yes	maximal
3	#19098423	C₂	1	2	yes	maximal
4	C₄×C₂²⋊Dic₇₉	C₂	1	2	yes	maximal
5	C₂²×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₉	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
8	C₂×C₄×Dic₇₉	C₂²	1	4	yes	—
9	C₇₉⋊#95	C₂²	1	4	yes	—
10	C₇₉⋊#95	C₂²	1	4	yes	—
11	C₇₉⋊#95	C₂²	1	8	yes	—
12	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₇₉	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
17	C₂²×Dic₇₉	C₂³	1	8	yes	—
18	C₄⋊Dic₇₉	—	4	4	no	—
19	C₂²⋊Dic₇₉	—	2	8	no	—
20	C₄×Dic₇₉	—	4	4	no	—
21	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₇₉	—	2	4	no	—
23	C₂²×C₄×C₇₉	—	4	2	no	—
24	C₂²×C₄×C₇₉	—	4	2	no	—
25	C₂²×C₄×C₇₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₇₉	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₇₉	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
29	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
30	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
31	C₄⋊C₄×C₇₉	C₂×C₄	1	16	yes	—
32	C₂³×C₇₉	—	2	2	no	—
33	C₂³×C₇₉	—	2	2	no	—
34	C₂³×C₇₉	C₂²×C₄	1	2	yes	—
35	C₂³×C₇₉	#41	1	4	yes	—
36	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₇₉	—	2	8	no	—
38	C₂×C₄×C₇₉	—	2	4	no	—
39	C₂×C₄×C₇₉	—	2	4	no	—
40	C₂×C₄×C₇₉	—	2	4	no	—
41	C₂×C₄×C₇₉	—	4	4	no	—
42	C₂×C₄×C₇₉	—	4	4	no	—
43	C₂×C₄×C₇₉	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₇₉	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₇₉	C₂²×C₄	1	4	yes	—
46	C₂×Dic₇₉	—	2	8	no	—
47	C₂×Dic₇₉	—	2	8	no	—
48	C₂×Dic₇₉	—	2	8	no	—
49	C₂²×C₇₉	—	2	2	no	—
50	C₂²×C₇₉	—	2	2	no	—
51	C₂²×C₇₉	—	2	8	no	—
52	C₂²×C₇₉	#117	1	4	yes	—
53	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
54	C₂²×C₇₉	C₂×#41	1	2	yes	—
55	C₂²×C₇₉	H(5,𝔽₂)	1	2	yes	—
56	C₂²×C₇₉	2₋¹⁺⁴	1	2	yes	—
57	C₄×C₇₉	—	2	4	no	—
58	C₄×C₇₉	—	2	4	no	—
59	C₄×C₇₉	—	2	4	no	—
60	Dic₇₉	—	4	8	no	—
61	C₂×C₇₉	—	2	4	no	—
62	C₂×C₇₉	C₂×#117	1	1	yes	characteristic
63	C₂×C₇₉	#518	1	1	yes	characteristic
64	C₂×C₇₉	#519	1	1	yes	characteristic
65	C₂×C₇₉	#539	1	4	yes	—
66	#2313	—	79	1	no	maximal, Sylow
67	C₇₉	#2313	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₄⋊C₄	D₇₉	1	1	yes	characteristic
69	#387	—	79	8	no	—
70	#381	—	79	2	no	—
71	#380	—	79	2	no	—
72	C₄×C₂²⋊C₄	—	79	2	no	—
73	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
74	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
75	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	79	4	no	—
77	C₂×C₄⋊C₄	Dic₇₉	1	8	yes	complemented
78	C₂×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
79	C₂×C₂²⋊C₄	—	79	4	no	—
80	C₂×C₄²	—	79	4	no	—
81	#95	—	79	4	no	—
82	#95	—	79	4	no	—
83	#95	—	79	8	no	—
84	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	79	8	no	—
89	C₂²×C₄	C₂×Dic₇₉	1	4	yes	—
90	C₂²×C₄	C₂×Dic₇₉	1	4	yes	—
91	C₂²×C₄	C₂×Dic₇₉	1	4	yes	—
92	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
93	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
94	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
95	C₄⋊C₄	—	316	4	no	—
96	C₄⋊C₄	C₂×Dic₇₉	1	16	yes	—
97	C₂²⋊C₄	—	158	8	no	—
98	C₄²	—	316	4	no	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	C₂²×Dic₇₉	1	2	yes	—
102	C₂³	#11929624	1	4	yes	—
103	C₂³	C₂³×D₇₉	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	158	8	no	—
111	C₂×C₄	—	158	8	no	—
112	C₂×C₄	—	158	8	no	—
113	C₂×C₄	C₂²×Dic₇₉	1	4	yes	—
114	C₂×C₄	C₂²×Dic₇₉	1	4	yes	—
115	C₂×C₄	C₂²×Dic₇₉	1	4	yes	—
116	C₂²	—	2	8	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	#14579120	1	4	yes	—
120	C₂²	C₂³×Dic₇₉	1	1	yes	characteristic
121	C₂²	C₂×#11929624	1	2	yes	—
122	C₂²	#14579206	1	2	yes	—
123	C₂²	#14579209	1	2	yes	—
124	C₄	—	2	4	no	—
125	C₄	—	2	4	no	—
126	C₄	—	2	4	no	—
127	C₄	—	316	8	no	—
128	C₂	—	2	4	no	—
129	C₂	C₂×#14579120	1	1	yes	characteristic
130	C₂	#19099038	1	1	yes	characteristic
131	C₂	#19099041	1	1	yes	characteristic
132	C₂	#19099160	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19098449

C₂

yes

maximal, complemented

#19098426

C₂

yes

maximal

#19098423

C₂

yes

maximal

C₄×C₂²⋊Dic₇₉

C₂

yes

maximal

C₂²×C₄⋊C₄×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₂²⋊Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂⁴×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂²×C₄

yes

—

C₂³×C₇₉

#41

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#117

yes

—

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₂²×C₇₉

C₂×#41

yes

—

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₂²×C₇₉

2₋¹⁺⁴

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#2313

—

maximal, Sylow

C₇₉

#2313

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

D₇₉

yes

—

—

—

C₄×C₂²⋊C₄

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

C₂³×C₄

C₂×D₇₉

yes

characteristic

C₂³×C₄

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₇₉

yes

complemented

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

—

C₂⁴

C₂²×D₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₄⋊C₄

—

316

—

C₄⋊C₄

C₂×Dic₇₉

yes

—

C₂²⋊C₄

—

158

—

C₄²

—

316

—

C₂³

—

100

C₂³

—

101

C₂³

C₂²×Dic₇₉

yes

—

102

C₂³

#11929624

yes

—

103

C₂³

C₂³×D₇₉

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

158

—

111

C₂×C₄

—

158

—

112

C₂×C₄

—

158

—

113

C₂×C₄

C₂²×Dic₇₉

yes

—

114

C₂×C₄

C₂²×Dic₇₉

yes

—

115

C₂×C₄

C₂²×Dic₇₉

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#14579120

yes

—

120

C₂²

C₂³×Dic₇₉

yes

characteristic

121

C₂²

C₂×#11929624

yes

—

122

C₂²

#14579206

yes

—

123

C₂²

#14579209

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

316

—

128

C₂

—

129

C₂

C₂×#14579120

yes

characteristic

130

C₂

#19099038

yes

characteristic

131

C₂

#19099041

yes

characteristic

132

C₂

#19099160

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₇₉	C₂
socle, cosocle	C₂³×C₇₉	C₂⁴
prime radical	C₂⁴×C₇₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#2313
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator

C₂²×C₇₉

C₂³×C₄

Fitting

C₂²×C₄⋊C₄×C₇₉

C₂

socle, cosocle

C₂³×C₇₉

C₂⁴

prime radical

C₂⁴×C₇₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
1. #19098449	C₂	semidirect
15. C₂×C₄⋊C₄×C₇₉	C₄	semidirect
67. C₇₉	#2313	semidirect
77. C₂×C₄⋊C₄	Dic₇₉	semidirect

H ≅ G/N

type

1. #19098449

C₂

semidirect

15. C₂×C₄⋊C₄×C₇₉

C₄

semidirect

67. C₇₉

#2313

semidirect

77. C₂×C₄⋊C₄

Dic₇₉

semidirect

Subgroups — #23766003

Conjugacy classes of subgroups

Complements