Group #23765555 — subgroups

Subgroups — #23765555

Order 10112.

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

normal subgroups only

This group has 136 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.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19098964	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#19098390	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098389	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098364	C₂	1	1	yes	maximal, characteristic
7	#357×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#14579152	—	2	1	no	—
9	D₄×Dic₇₉	—	2	1	no	—
10	C₂×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14579095	C₂²	1	1	yes	characteristic
12	#14579095	C₂²	1	1	yes	characteristic
13	#14579092	—	2	1	no	—
14	#14579082	—	2	1	no	—
15	C₈×Dic₇₉	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	#107×C₇₉	C₂²	1	1	yes	characteristic
19	C₂²×Dic₇₉	—	4	1	no	—
20	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊Dic₇₉	—	2	1	no	—
22	C₄⋊Dic₇₉	—	2	1	no	—
23	C₄⋊Dic₇₉	—	4	1	no	—
24	Dic₇₉⋊C₄	—	4	1	no	—
25	C₂²⋊Dic₇₉	—	4	1	no	—
26	C₄×Dic₇₉	D₄	1	1	yes	characteristic
27	C₇₉⋊C₁₆	—	2	1	no	—
28	C₂×D₄×C₇₉	—	2	1	no	—
29	D₈×C₇₉	—	2	1	no	—
30	D₈×C₇₉	Q₈	1	2	yes	—
31	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₇₉	—	2	1	no	—
33	C₁₆×C₇₉	—	2	1	no	—
34	C₂³×C₇₉	—	4	1	no	—
35	D₄×C₇₉	—	2	2	no	—
36	D₄×C₇₉	—	4	1	no	—
37	C₂×C₄×C₇₉	—	4	1	no	—
38	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₇₉	—	4	1	no	—
40	C₂×Dic₇₉	—	4	1	no	—
41	C₂×Dic₇₉	—	8	2	no	—
42	C₂×Dic₇₉	D₈	1	1	yes	characteristic
43	C₂×Dic₇₉	D₈	1	1	yes	characteristic
44	C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₇₉	#41	1	1	yes	characteristic
46	C₇₉⋊C₈	—	2	1	no	—
47	C₂²×C₇₉	—	4	2	no	—
48	C₂²×C₇₉	—	8	1	no	—
49	C₂²×C₇₉	—	8	1	no	—
50	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
51	C₄×C₇₉	—	8	1	no	—
52	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₇₉	#137	1	1	yes	characteristic
54	Dic₇₉	—	2	1	no	—
55	Dic₇₉	—	2	1	no	—
56	Dic₇₉	—	8	1	no	—
57	Dic₇₉	—	8	1	no	—
58	C₂×C₇₉	—	8	2	no	—
59	C₂×C₇₉	#474	1	1	yes	characteristic
60	C₂×C₇₉	#508	1	1	yes	characteristic
61	C₂×C₇₉	#509	1	1	yes	characteristic
62	#2228	—	79	1	no	maximal, Sylow
63	C₇₉	#2228	1	1	yes	characteristic, complemented, Sylow
64	#499	—	79	1	no	—
65	C₄×D₈	—	79	1	no	—
66	#367	—	79	1	no	—
67	#366	—	79	1	no	—
68	C₄⋊C₁₆	—	79	1	no	—
69	#357	—	79	1	no	—
70	#357	D₇₉	1	1	yes	characteristic
71	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
72	#125	—	158	1	no	—
73	C₄×D₄	—	158	1	no	—
74	C₂×C₁₆	—	79	1	no	—
75	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
76	#107	—	79	1	no	—
77	#107	—	79	1	no	—
78	#107	C₂×D₇₉	1	1	yes	characteristic
79	#106	—	158	1	no	—
80	#102	—	158	1	no	—
81	C₄×C₈	—	79	1	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂²×C₄	—	316	1	no	—
84	D₈	—	2	1	no	—
85	D₈	Dic₁₅₈	1	2	yes	—
86	C₂×C₈	—	79	1	no	—
87	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	316	1	no	—
90	C₄⋊C₄	—	316	1	no	—
91	C₄⋊C₄	—	158	1	no	—
92	C₄⋊C₄	—	158	1	no	—
93	C₂²⋊C₄	—	316	1	no	—
94	C₄²	—	79	1	no	—
95	C₁₆	—	2	1	no	—
96	C₁₆	—	158	1	no	—
97	C₂³	—	4	1	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	316	1	no	—
102	C₂×C₄	—	316	1	no	—
103	C₂×C₄	—	79	1	no	—
104	C₂×C₄	—	79	1	no	—
105	C₂×C₄	—	632	2	no	—
106	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
107	C₈	—	158	1	no	—
108	C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
109	C₈	#11929623	1	1	yes	characteristic
110	C₂²	—	4	2	no	—
111	C₂²	—	8	1	no	—
112	C₂²	—	8	1	no	—
113	C₂²	D₈×D₇₉	1	1	yes	characteristic
114	C₄	—	8	1	no	—
115	C₄	—	632	1	no	—
116	C₄	—	632	1	no	—
117	C₄	—	158	1	no	—
118	C₄	—	158	1	no	—
119	C₄	#14579138	1	1	yes	characteristic
120	C₄	#14579190	1	1	yes	characteristic
121	C₂	—	8	2	no	—
122	C₂	#19098841	1	1	yes	characteristic
123	C₂	#19098992	1	1	yes	characteristic
124	C₂	#19099000	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098964

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₉

C₂

yes

maximal, characteristic

#19098390

C₂

yes

maximal, characteristic, complemented

#19098389

C₂

yes

maximal, characteristic, complemented

Dic₇₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#19098364

C₂

yes

maximal, characteristic

#357×C₇₉

C₂

yes

maximal, characteristic

#14579152

—

D₄×Dic₇₉

—

C₂×C₇₉⋊C₁₆

C₂²

yes

characteristic

#14579095

C₂²

yes

characteristic

#14579095

C₂²

yes

characteristic

#14579092

—

#14579082

—

C₈×Dic₇₉

C₂²

yes

characteristic

C₂×D₈×C₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₇₉

C₂²

yes

characteristic

#107×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

D₄

yes

characteristic

C₇₉⋊C₁₆

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

Q₈

yes

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₈

yes

characteristic

C₂×Dic₇₉

D₈

yes

characteristic

C₈×C₇₉

C₂×Q₈

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

#137

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#499

—

C₄×D₈

—

#367

—

#366

—

C₄⋊C₁₆

—

#357

—

#357

D₇₉

yes

characteristic

C₂×D₈

C₂×D₇₉

yes

characteristic

#125

—

158

—

C₄×D₄

—

158

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₉

yes

—

—

C₂×D₇₉

yes

characteristic

#106

—

158

—

#102

—

158

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

316

—

D₈

—

D₈

Dic₁₅₈

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

158

—

C₄⋊C₄

—

158

—

C₂²⋊C₄

—

316

—

C₄²

—

C₁₆

—

C₁₆

—

158

—

C₂³

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

316

—

102

C₂×C₄

—

316

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

632

—

106

C₂×C₄

D₄×D₇₉

yes

characteristic

107

C₈

—

158

—

108

C₈

C₂×Dic₁₅₈

yes

characteristic

109

C₈

#11929623

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

D₈×D₇₉

yes

characteristic

114

C₄

—

115

C₄

—

632

—

116

C₄

—

632

—

117

C₄

—

158

—

118

C₄

—

158

—

119

C₄

#14579138

yes

characteristic

120

C₄

#14579190

yes

characteristic

121

C₂

—

122

C₂

#19098841

yes

characteristic

123

C₂

#19098992

yes

characteristic

124

C₂

#19099000

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#357×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
prime radical	C₂×D₈×C₇₉	C₂²
hypercenter	#357	D₇₉
hypocenter	C₇₉	#2228
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₉

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#357×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

prime radical

C₂×D₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #19098964	C₂	semidirect
3. #19098390	C₂	semidirect
4. #19098389	C₂	semidirect
5. Dic₇₉⋊C₁₆	C₂	semidirect
63. C₇₉	#2228	semidirect

H ≅ G/N

type

1. #19098964

C₂

semidirect

3. #19098390

C₂

semidirect

4. #19098389

C₂

semidirect

5. Dic₇₉⋊C₁₆

C₂

semidirect

63. C₇₉

#2228

semidirect

Subgroups — #23765555

Conjugacy classes of subgroups

Complements