Group #23765504 — subgroups

Subgroups — #23765504

Order 10112.

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

normal subgroups only

This group has 170 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	C₂×D₁₂₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579100	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098800	C₂	1	2	yes	maximal, complemented
4	#19098363	C₂	1	2	yes	maximal, complemented
5	#367×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₆₃₂	C₂²	1	2	yes	—
7	C₂×C₈×D₇₉	C₂²	1	1	yes	characteristic
8	#14579136	—	2	2	no	—
9	D₁₂₆₄	—	2	2	no	—
10	#14579100	—	2	2	no	—
11	C₂×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14579081	—	2	2	no	—
13	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	#107×C₇₉	C₂²	1	2	yes	—
15	C₂×D₃₁₆	—	2	2	no	—
16	C₂×D₃₁₆	—	4	2	no	—
17	C₂×C₄×D₇₉	D₄	1	1	yes	characteristic
18	D₆₃₂	—	2	2	no	—
19	D₆₃₂	—	2	2	no	—
20	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
21	C₈×D₇₉	—	2	1	no	—
22	C₈×D₇₉	—	2	1	no	—
23	(C₂×D₇₉)⋊C₄	—	4	2	no	—
24	C₇₉⋊C₁₆	—	2	1	no	—
25	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₇₉	—	2	2	no	—
27	C₁₆×C₇₉	D₄	1	2	yes	complemented
28	C₂²×D₇₉	—	4	2	no	—
29	C₂²×D₇₉	D₈	1	1	yes	characteristic
30	C₂×C₄×C₇₉	—	4	2	no	—
31	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₇₉	D₈	1	1	yes	characteristic
33	D₃₁₆	—	4	2	no	—
34	D₃₁₆	—	4	2	no	—
35	D₃₁₆	—	8	4	no	—
36	C₄×D₇₉	—	2	1	no	—
37	C₄×D₇₉	—	2	1	no	—
38	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₈×C₇₉	#41	1	1	yes	characteristic
40	C₇₉⋊C₈	—	2	1	no	—
41	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
42	C₂×D₇₉	—	2	1	no	—
43	C₂×D₇₉	—	2	2	no	—
44	C₂×D₇₉	—	8	2	no	—
45	C₂×D₇₉	—	8	2	no	—
46	C₂×D₇₉	—	8	2	no	—
47	C₄×C₇₉	—	8	2	no	—
48	C₄×C₇₉	#121	1	1	yes	characteristic
49	C₄×C₇₉	#135	1	1	yes	characteristic
50	Dic₇₉	—	2	1	no	—
51	C₂×C₇₉	#466	1	1	yes	characteristic
52	C₂×C₇₉	#509	1	2	yes	—
53	D₇₉	—	4	1	no	—
54	D₇₉	—	16	2	no	—
55	#2218	—	79	1	no	maximal, Sylow
56	C₇₉	#2218	1	1	yes	characteristic, complemented, Sylow
57	C₂×D₁₆	—	79	1	no	—
58	C₂×M₃₂	—	79	1	no	—
59	#466	—	79	2	no	—
60	#367	D₇₉	1	1	yes	characteristic, complemented
61	#357	—	79	2	no	—
62	C₂×D₈	—	79	2	no	—
63	C₂²×C₈	—	79	1	no	—
64	#121	—	158	2	no	—
65	D₁₆	—	158	2	no	—
66	M₃₂	—	158	2	no	—
67	C₂×C₁₆	—	79	1	no	—
68	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
69	#107	C₂×D₇₉	1	2	yes	—
70	#102	—	158	2	no	—
71	C₂×D₄	—	316	2	no	—
72	C₂×D₄	—	158	2	no	—
73	C₂²×C₄	—	79	1	no	—
74	D₈	—	158	2	no	—
75	D₈	—	158	2	no	—
76	C₂×C₈	—	79	1	no	—
77	C₂×C₈	—	158	1	no	—
78	C₂×C₈	—	158	1	no	—
79	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
80	C₄⋊C₄	—	2	2	no	—
81	C₂²⋊C₄	—	316	2	no	—
82	C₁₆	—	158	1	no	—
83	C₁₆	D₃₁₆	1	2	yes	complemented
84	C₂³	—	316	2	no	—
85	C₂³	—	79	1	no	—
86	D₄	—	316	2	no	—
87	D₄	—	316	2	no	—
88	D₄	—	632	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	79	1	no	—
91	C₂×C₄	—	158	1	no	—
92	C₂×C₄	—	158	1	no	—
93	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
94	C₈	—	158	1	no	—
95	C₈	C₂×D₃₁₆	1	1	yes	characteristic
96	C₈	#11929625	1	1	yes	characteristic
97	C₂²	—	632	2	no	—
98	C₂²	—	632	2	no	—
99	C₂²	—	632	2	no	—
100	C₂²	—	158	2	no	—
101	C₂²	—	158	1	no	—
102	C₂²	D₈×D₇₉	1	1	yes	characteristic
103	C₄	—	8	2	no	—
104	C₄	—	158	1	no	—
105	C₄	#14579136	1	1	yes	characteristic
106	C₄	#14579184	1	1	yes	characteristic
107	C₂	—	316	1	no	—
108	C₂	—	1264	2	no	—
109	C₂	#19098800	1	1	yes	characteristic
110	C₂	#19099000	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₂₆₄

C₂

yes

maximal, characteristic, complemented

C₂×#14579100

C₂

yes

maximal, characteristic, complemented

#19098800

C₂

yes

maximal, complemented

#19098363

C₂

yes

maximal, complemented

#367×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₆₃₂

C₂²

yes

—

C₂×C₈×D₇₉

C₂²

yes

characteristic

#14579136

—

D₁₂₆₄

—

#14579100

—

C₂×C₇₉⋊C₁₆

C₂²

yes

characteristic

#14579081

—

C₂×C₁₆×C₇₉

C₂²

yes

characteristic

#107×C₇₉

C₂²

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₄×D₇₉

D₄

yes

characteristic

D₆₃₂

—

D₆₃₂

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₈×D₇₉

—

C₈×D₇₉

—

(C₂×D₇₉)⋊C₄

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

D₄

yes

complemented

C₂²×D₇₉

—

C₂²×D₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

D₈

yes

characteristic

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

C₂×D₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

#466

yes

characteristic

C₂×C₇₉

#509

yes

—

D₇₉

—

D₇₉

—

#2218

—

maximal, Sylow

C₇₉

#2218

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×M₃₂

—

#466

—

#367

D₇₉

yes

characteristic, complemented

#357

—

C₂×D₈

—

C₂²×C₈

—

#121

—

158

—

D₁₆

—

158

—

M₃₂

—

158

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₉

yes

characteristic

#107

C₂×D₇₉

yes

—

#102

—

158

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

C₂²×C₄

—

D₈

—

158

—

D₈

—

158

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

316

—

C₁₆

—

158

—

C₁₆

D₃₁₆

yes

complemented

C₂³

—

316

—

C₂³

—

D₄

—

316

—

D₄

—

316

—

D₄

—

632

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

158

—

C₂×C₄

—

158

—

C₂×C₄

D₄×D₇₉

yes

characteristic

C₈

—

158

—

C₈

C₂×D₃₁₆

yes

characteristic

C₈

#11929625

yes

characteristic

C₂²

—

632

—

C₂²

—

632

—

C₂²

—

632

—

100

C₂²

—

158

—

101

C₂²

—

158

—

102

C₂²

D₈×D₇₉

yes

characteristic

103

C₄

—

104

C₄

—

158

—

105

C₄

#14579136

yes

characteristic

106

C₄

#14579184

yes

characteristic

107

C₂

—

316

—

108

C₂

—

1264

—

109

C₂

#19098800

yes

characteristic

110

C₂

#19099000

yes

—

	subgroup	G/H
center	C₂²	D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#367×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
hypercenter	#367	D₇₉
hypocenter	C₇₉	#2218
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

D₈×D₇₉

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#367×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
1. C₂×D₁₂₆₄	C₂	semidirect
2. C₂×#14579100	C₂	semidirect
3. #19098800	C₂	semidirect
4. #19098363	C₂	semidirect
5. #367×C₇₉	C₂	semidirect
27. C₁₆×C₇₉	D₄	semidirect
56. C₇₉	#2218	semidirect
60. #367	D₇₉	semidirect
83. C₁₆	D₃₁₆	semidirect

H ≅ G/N

type

1. C₂×D₁₂₆₄

C₂

semidirect

2. C₂×#14579100

C₂

semidirect

3. #19098800

C₂

semidirect

4. #19098363

C₂

semidirect

5. #367×C₇₉

C₂

semidirect

27. C₁₆×C₇₉

D₄

semidirect

56. C₇₉

#2218

semidirect

60. #367

D₇₉

semidirect

83. C₁₆

D₃₁₆

semidirect

Subgroups — #23765504

Conjugacy classes of subgroups

Complements