Group #23765436 — subgroups

Subgroups — #23765436

Order 10112.

186 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₂×D₁₂₆₄	C₂	1	2	yes	maximal, complemented
2	#19098940	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098365	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₆₃₂	—	2	1	no	—
7	C₂×D₆₃₂	C₂²	1	2	yes	—
8	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
9	#14579158	—	2	1	no	—
10	C₄×D₃₁₆	—	2	1	no	—
11	D₁₂₆₄	—	2	4	no	—
12	#14579095	C₂²	1	1	yes	characteristic
13	#14579083	—	2	1	no	—
14	C₂×C₁₆×C₇₉	C₂²	1	2	yes	—
15	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×D₃₁₆	—	2	1	no	—
17	C₂×D₃₁₆	—	2	2	no	—
18	C₂×D₃₁₆	—	4	1	no	—
19	C₂×D₃₁₆	—	4	1	no	—
20	C₂×C₄×D₇₉	—	4	1	no	—
21	D₆₃₂	—	2	2	no	—
22	D₆₃₂	—	2	1	no	—
23	D₆₃₂	—	2	2	no	—
24	D₆₃₂	—	4	2	no	—
25	D₆₃₂	D₄	1	2	yes	complemented
26	C₄⋊Dic₇₉	—	2	1	no	—
27	(C₂×D₇₉)⋊C₄	—	4	1	no	—
28	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₄²×C₇₉	D₄	1	1	yes	characteristic
31	C₁₆×C₇₉	—	2	2	no	—
32	C₂²×D₇₉	—	4	1	no	—
33	C₂²×D₇₉	—	4	2	no	—
34	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
35	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
36	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₇₉	—	4	1	no	—
38	D₃₁₆	—	2	2	no	—
39	D₃₁₆	—	4	2	no	—
40	D₃₁₆	—	4	1	no	—
41	D₃₁₆	—	4	2	no	—
42	D₃₁₆	—	8	2	no	—
43	D₃₁₆	—	8	2	no	—
44	C₄×D₇₉	—	8	2	no	—
45	C₈×C₇₉	—	2	1	no	—
46	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₈×C₇₉	#41	1	1	yes	characteristic
48	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
49	C₂×D₇₉	—	4	2	no	—
50	C₂×D₇₉	—	8	1	no	—
51	C₂×D₇₉	—	8	1	no	—
52	C₂×D₇₉	—	8	2	no	—
53	C₂×D₇₉	—	8	2	no	—
54	C₂×D₇₉	—	8	2	no	—
55	C₄×C₇₉	—	2	1	no	—
56	C₄×C₇₉	D₁₆	1	2	yes	complemented
57	C₄×C₇₉	#121	1	1	yes	characteristic
58	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₇₉	—	8	1	no	—
60	C₂×C₇₉	#459	1	1	yes	characteristic
61	C₂×C₇₉	C₂×D₁₆	1	1	yes	characteristic
62	C₂×C₇₉	#509	1	1	yes	characteristic
63	D₇₉	—	8	2	no	—
64	D₇₉	—	16	2	no	—
65	#2206	—	79	1	no	maximal, Sylow
66	C₇₉	#2206	1	1	yes	characteristic, complemented, Sylow
67	C₂×D₁₆	—	79	2	no	—
68	#493	—	79	1	no	—
69	C₄×D₈	—	79	1	no	—
70	C₄⋊C₁₆	D₇₉	1	1	yes	characteristic, complemented
71	#357	—	79	2	no	—
72	C₂×D₈	—	79	1	no	—
73	C₂×D₈	—	79	2	no	—
74	C₂×D₈	—	158	1	no	—
75	#127	—	158	1	no	—
76	C₄×D₄	—	158	1	no	—
77	D₁₆	—	158	4	no	—
78	C₂×C₁₆	C₂×D₇₉	1	2	yes	—
79	#107	—	79	1	no	—
80	#102	—	158	1	no	—
81	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
82	C₂×D₄	—	316	1	no	—
83	C₂×D₄	—	316	1	no	—
84	C₂×D₄	—	158	2	no	—
85	C₂×D₄	—	158	1	no	—
86	C₂²×C₄	—	316	1	no	—
87	D₈	—	316	2	no	—
88	D₈	—	79	2	no	—
89	D₈	—	158	2	no	—
90	D₈	—	158	2	no	—
91	D₈	—	158	1	no	—
92	C₂×C₈	D₃₁₆	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
94	C₄⋊C₄	—	158	1	no	—
95	C₂²⋊C₄	—	316	1	no	—
96	C₄²	D₃₁₆	1	1	yes	characteristic
97	C₁₆	—	2	2	no	—
98	C₂³	—	316	1	no	—
99	C₂³	—	316	2	no	—
100	D₄	—	316	2	no	—
101	D₄	—	316	1	no	—
102	D₄	—	316	2	no	—
103	D₄	—	632	2	no	—
104	D₄	—	632	2	no	—
105	D₄	—	158	2	no	—
106	C₂×C₄	—	316	1	no	—
107	C₂×C₄	—	632	2	no	—
108	C₂×C₄	D₆₃₂	1	1	yes	characteristic
109	C₂×C₄	D₆₃₂	1	1	yes	characteristic
110	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
111	C₈	—	2	1	no	—
112	C₈	D₄×D₇₉	1	1	yes	characteristic
113	C₈	#11929625	1	1	yes	characteristic
114	C₂²	—	316	2	no	—
115	C₂²	—	632	1	no	—
116	C₂²	—	632	1	no	—
117	C₂²	—	632	2	no	—
118	C₂²	—	632	2	no	—
119	C₂²	—	632	2	no	—
120	C₂²	C₂×D₆₃₂	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	632	1	no	—
123	C₄	D₁₂₆₄	1	2	yes	complemented
124	C₄	#14579136	1	1	yes	characteristic
125	C₄	#14579189	1	1	yes	characteristic
126	C₂	—	632	2	no	—
127	C₂	—	1264	2	no	—
128	C₂	#19098759	1	1	yes	characteristic
129	C₂	C₂×D₁₂₆₄	1	1	yes	characteristic
130	C₂	#19099001	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₂₆₄

C₂

yes

maximal, complemented

#19098940

C₂

yes

maximal, characteristic, complemented

C₄×D₆₃₂

C₂

yes

maximal, characteristic, complemented

#19098365

C₂

yes

maximal, complemented

C₄⋊C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₆₃₂

—

C₂×D₆₃₂

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

characteristic

#14579158

—

C₄×D₃₁₆

—

D₁₂₆₄

—

#14579095

C₂²

yes

characteristic

#14579083

—

C₂×C₁₆×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₄×D₇₉

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

D₄

yes

complemented

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₁₆×C₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

D₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

#459

yes

characteristic

C₂×C₇₉

C₂×D₁₆

yes

characteristic

C₂×C₇₉

#509

yes

characteristic

D₇₉

—

D₇₉

—

#2206

—

maximal, Sylow

C₇₉

#2206

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

#493

—

C₄×D₈

—

C₄⋊C₁₆

D₇₉

yes

characteristic, complemented

#357

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

158

—

#127

—

158

—

C₄×D₄

—

158

—

D₁₆

—

158

—

C₂×C₁₆

C₂×D₇₉

yes

—

#107

—

#102

—

158

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×D₄

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂²×C₄

—

316

—

D₈

—

316

—

D₈

—

D₈

—

158

—

D₈

—

158

—

D₈

—

158

—

C₂×C₈

D₃₁₆

yes

characteristic

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

158

—

C₂²⋊C₄

—

316

—

C₄²

D₃₁₆

yes

characteristic

C₁₆

—

C₂³

—

316

—

C₂³

—

316

—

100

D₄

—

316

—

101

D₄

—

316

—

102

D₄

—

316

—

103

D₄

—

632

—

104

D₄

—

632

—

105

D₄

—

158

—

106

C₂×C₄

—

316

—

107

C₂×C₄

—

632

—

108

C₂×C₄

D₆₃₂

yes

characteristic

109

C₂×C₄

D₆₃₂

yes

characteristic

110

C₂×C₄

C₂×D₃₁₆

yes

characteristic

111

C₈

—

112

C₈

D₄×D₇₉

yes

characteristic

113

C₈

#11929625

yes

characteristic

114

C₂²

—

316

—

115

C₂²

—

632

—

116

C₂²

—

632

—

117

C₂²

—

632

—

118

C₂²

—

632

—

119

C₂²

—

632

—

120

C₂²

C₂×D₆₃₂

yes

characteristic

121

C₄

—

122

C₄

—

632

—

123

C₄

D₁₂₆₄

yes

complemented

124

C₄

#14579136

yes

characteristic

125

C₄

#14579189

yes

characteristic

126

C₂

—

632

—

127

C₂

—

1264

—

128

C₂

#19098759

yes

characteristic

129

C₂

C₂×D₁₂₆₄

yes

characteristic

130

C₂

#19099001

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	C₄⋊C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
hypercenter	C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#2206
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₆₃₂

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

C₄⋊C₁₆×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

hypercenter

C₄⋊C₁₆

D₇₉

hypocenter

C₇₉

#2206

LCS

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
1. C₂×D₁₂₆₄	C₂	semidirect
2. #19098940	C₂	semidirect
3. C₄×D₆₃₂	C₂	semidirect
4. #19098365	C₂	semidirect
5. C₄⋊C₁₆×C₇₉	C₂	semidirect
25. D₆₃₂	D₄	semidirect
56. C₄×C₇₉	D₁₆	semidirect
66. C₇₉	#2206	semidirect
70. C₄⋊C₁₆	D₇₉	semidirect
123. C₄	D₁₂₆₄	semidirect

H ≅ G/N

type

1. C₂×D₁₂₆₄

C₂

semidirect

2. #19098940

C₂

semidirect

3. C₄×D₆₃₂

C₂

semidirect

4. #19098365

C₂

semidirect

5. C₄⋊C₁₆×C₇₉

C₂

semidirect

25. D₆₃₂

D₄

semidirect

56. C₄×C₇₉

D₁₆

semidirect

66. C₇₉

#2206

semidirect

70. C₄⋊C₁₆

D₇₉

semidirect

123. C₄

D₁₂₆₄

semidirect

Subgroups — #23765436

Conjugacy classes of subgroups

Complements