Group #23765437 — subgroups

Subgroups — #23765437

Order 10112.

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

normal subgroups only

This group has 174 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×#14579105	C₂	1	2	yes	maximal, complemented
2	C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098364	C₂	1	2	yes	maximal, complemented
5	#493×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₅₈	—	2	1	no	—
8	#14579105	—	2	4	no	—
9	C₂×C₇₉⋊C₁₆	C₂²	1	2	yes	—
10	#14579095	C₂²	1	1	yes	characteristic
11	#14579086	—	2	1	no	—
12	C₂×D₈×C₇₉	—	2	1	no	—
13	C₂×D₈×C₇₉	C₂²	1	2	yes	—
14	#127×C₇₉	—	2	1	no	—
15	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₅₈	—	2	1	no	—
17	Dic₃₁₆	—	2	1	no	—
18	Dic₃₁₆	D₄	1	2	yes	complemented
19	C₄⋊Dic₇₉	—	2	1	no	—
20	Dic₇₉⋊C₄	—	4	1	no	—
21	C₄×Dic₇₉	—	4	1	no	—
22	C₇₉⋊C₁₆	—	2	2	no	—
23	C₂×D₄×C₇₉	—	2	2	no	—
24	C₂×D₄×C₇₉	—	4	1	no	—
25	C₂×D₄×C₇₉	—	4	1	no	—
26	D₈×C₇₉	—	2	2	no	—
27	D₈×C₇₉	—	2	2	no	—
28	D₈×C₇₉	—	4	2	no	—
29	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₄²×C₇₉	D₄	1	1	yes	characteristic
32	C₂³×C₇₉	—	4	2	no	—
33	D₄×C₇₉	—	4	2	no	—
34	D₄×C₇₉	—	4	2	no	—
35	D₄×C₇₉	—	8	2	no	—
36	D₄×C₇₉	—	8	2	no	—
37	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
38	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₇₉	—	4	1	no	—
41	C₂×Dic₇₉	—	4	1	no	—
42	Dic₁₅₈	—	2	2	no	—
43	Dic₁₅₈	—	4	1	no	—
44	C₈×C₇₉	—	2	1	no	—
45	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₈×C₇₉	#41	1	1	yes	characteristic
47	C₂²×C₇₉	—	8	2	no	—
48	C₂²×C₇₉	—	8	2	no	—
49	C₂²×C₇₉	—	8	2	no	—
50	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
51	C₄×C₇₉	—	2	1	no	—
52	C₄×C₇₉	SD₃₂	1	2	yes	complemented
53	C₄×C₇₉	#121	1	1	yes	characteristic
54	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
55	Dic₇₉	—	4	2	no	—
56	Dic₇₉	—	8	1	no	—
57	C₂×C₇₉	—	16	2	no	—
58	C₂×C₇₉	#459	1	1	yes	characteristic
59	C₂×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
60	C₂×C₇₉	#509	1	1	yes	characteristic
61	#2207	—	79	1	no	maximal, Sylow
62	C₇₉	#2207	1	1	yes	characteristic, complemented, Sylow
63	C₂×SD₃₂	—	79	2	no	—
64	#493	D₇₉	1	1	yes	characteristic
65	C₄×Q₁₆	—	79	1	no	—
66	C₄⋊C₁₆	—	79	1	no	—
67	#357	—	79	2	no	—
68	C₂×Q₁₆	—	79	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	C₂×D₇₉	1	2	yes	—
71	#127	—	2	1	no	—
72	C₄×Q₈	—	158	1	no	—
73	SD₃₂	—	158	4	no	—
74	C₂×C₁₆	—	79	2	no	—
75	#107	—	79	1	no	—
76	#103	—	158	1	no	—
77	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
78	C₂×Q₈	—	158	1	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂×D₄	—	4	1	no	—
82	Q₁₆	—	79	2	no	—
83	Q₁₆	—	158	1	no	—
84	D₈	—	2	2	no	—
85	D₈	—	2	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
88	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
89	C₄⋊C₄	—	316	1	no	—
90	C₄⋊C₄	—	158	1	no	—
91	C₄²	—	316	1	no	—
92	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
93	C₁₆	—	158	2	no	—
94	C₂³	—	4	2	no	—
95	Q₈	—	316	1	no	—
96	Q₈	—	158	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	8	2	no	—
100	D₄	—	8	2	no	—
101	C₂×C₄	—	316	1	no	—
102	C₂×C₄	—	316	1	no	—
103	C₂×C₄	#11929609	1	1	yes	characteristic
104	C₂×C₄	#11929609	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	D₄×D₇₉	1	1	yes	characteristic
108	C₈	#11929623	1	1	yes	characteristic
109	C₂²	—	8	2	no	—
110	C₂²	—	8	2	no	—
111	C₂²	—	8	2	no	—
112	C₂²	C₂×#11929609	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	316	2	no	—
115	C₄	—	632	1	no	—
116	C₄	#14579105	1	2	yes	complemented
117	C₄	#14579132	1	1	yes	characteristic
118	C₄	#14579187	1	1	yes	characteristic
119	C₂	—	16	2	no	—
120	C₂	#19098755	1	1	yes	characteristic
121	C₂	C₂×#14579105	1	1	yes	characteristic
122	C₂	#19098999	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579105

C₂

yes

maximal, complemented

C₄×Dic₃₁₆

C₂

yes

maximal, characteristic, complemented

C₄⋊(C₇₉⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#19098364

C₂

yes

maximal, complemented

#493×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₄×Dic₁₅₈

—

#14579105

—

C₂×C₇₉⋊C₁₆

C₂²

yes

—

#14579095

C₂²

yes

characteristic

#14579086

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

C₂²

yes

—

#127×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

D₄

yes

complemented

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

SD₃₂

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#459

yes

characteristic

C₂×C₇₉

C₂×SD₃₂

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

#493

D₇₉

yes

characteristic

C₄×Q₁₆

—

C₄⋊C₁₆

—

#357

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

C₂×D₇₉

yes

—

#127

—

C₄×Q₈

—

158

—

SD₃₂

—

158

—

C₂×C₁₆

—

#107

—

#103

—

158

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×Q₈

—

158

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

158

—

D₈

—

D₈

—

D₈

—

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

316

—

C₄⋊C₄

—

158

—

C₄²

—

316

—

C₄²

C₂²⋊D₇₉

yes

characteristic

C₁₆

—

158

—

C₂³

—

Q₈

—

316

—

Q₈

—

158

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

C₂×C₄

—

316

—

102

C₂×C₄

—

316

—

103

C₂×C₄

#11929609

yes

characteristic

104

C₂×C₄

#11929609

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

106

C₈

—

107

C₈

D₄×D₇₉

yes

characteristic

108

C₈

#11929623

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂×#11929609

yes

characteristic

113

C₄

—

114

C₄

—

316

—

115

C₄

—

632

—

116

C₄

#14579105

yes

complemented

117

C₄

#14579132

yes

characteristic

118

C₄

#14579187

yes

characteristic

119

C₂

—

120

C₂

#19098755

yes

characteristic

121

C₂

C₂×#14579105

yes

characteristic

122

C₂

#19098999

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting, prime radical	#493×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
hypercenter	#493	D₇₉
hypocenter	C₇₉	#2207
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂²

C₂×#11929609

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting, prime radical

#493×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. C₂×#14579105	C₂	semidirect
2. C₄×Dic₃₁₆	C₂	semidirect
3. C₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
4. #19098364	C₂	semidirect
18. Dic₃₁₆	D₄	semidirect
52. C₄×C₇₉	SD₃₂	semidirect
62. C₇₉	#2207	semidirect
116. C₄	#14579105	semidirect

H ≅ G/N

type

1. C₂×#14579105

C₂

semidirect

2. C₄×Dic₃₁₆

C₂

semidirect

3. C₄⋊(C₇₉⋊C₁₆)

C₂

semidirect

4. #19098364

C₂

semidirect

18. Dic₃₁₆

D₄

semidirect

52. C₄×C₇₉

SD₃₂

semidirect

62. C₇₉

#2207

semidirect

116. C₄

#14579105

semidirect

Subgroups — #23765437

Conjugacy classes of subgroups

Complements