Group #34202965 — subgroups

Subgroups — #34202965

Order 20224.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	#23765629	C₂	1	2	yes	maximal, complemented
2	#23765625	C₂	1	2	yes	maximal
3	#23763006	C₂	1	1	yes	maximal, characteristic
4	C₈×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098943	C₂²	1	1	yes	characteristic
7	#19098940	C₂²	1	1	yes	characteristic
8	#19098910	—	2	2	no	—
9	#19098366	—	2	4	no	—
10	#19098363	—	2	4	no	—
11	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
12	#500×C₇₉	C₂²	1	2	yes	—
13	#487×C₇₉	—	2	2	no	—
14	C₈²×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₆	—	2	2	no	—
16	C₂×D₆₃₂	—	2	2	no	—
17	#14579161	—	2	1	no	—
18	#14579158	—	2	1	no	—
19	C₂×C₇₉⋊C₁₆	D₄	1	4	yes	—
20	#14579083	—	4	4	no	—
21	C₄⋊Q₈×C₇₉	—	2	2	no	—
22	#107×C₇₉	—	2	2	no	—
23	#107×C₇₉	—	2	2	no	—
24	#103×C₇₉	—	4	4	no	—
25	C₄×C₈×C₇₉	D₄	1	2	yes	—
26	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₅₈	—	4	1	no	—
28	C₂×D₃₁₆	—	4	1	no	—
29	C₂×D₃₁₆	—	4	2	no	—
30	Dic₃₁₆	—	4	4	no	—
31	D₆₃₂	—	4	4	no	—
32	C₄⋊Dic₇₉	—	4	2	no	—
33	C₇₉⋊C₁₆	—	2	4	no	—
34	C₂×Q₈×C₇₉	—	4	2	no	—
35	C₂×C₈×C₇₉	D₈	1	4	yes	—
36	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
37	C₄⋊C₄×C₇₉	—	4	2	no	—
38	C₄⋊C₄×C₇₉	—	4	2	no	—
39	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₇₉	—	8	1	no	—
41	Q₈×C₇₉	—	8	4	no	—
42	C₂×C₄×C₇₉	—	8	2	no	—
43	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
45	C₂×Dic₇₉	—	8	1	no	—
46	D₃₁₆	—	8	2	no	—
47	D₃₁₆	—	8	4	no	—
48	Dic₁₅₈	—	8	2	no	—
49	C₈×C₇₉	—	2	4	no	—
50	C₈×C₇₉	#135	1	4	yes	—
51	C₂²×C₇₉	#493	1	1	yes	characteristic
52	C₂×D₇₉	—	16	1	no	—
53	C₂×D₇₉	—	16	2	no	—
54	C₄×C₇₉	—	16	2	no	—
55	C₄×C₇₉	#495	1	2	yes	—
56	C₄×C₇₉	#508	1	4	yes	—
57	Dic₇₉	—	16	1	no	—
58	#12511	—	79	1	no	maximal, Sylow
59	C₂×C₇₉	#1709	1	1	yes	characteristic
60	C₂×C₇₉	#2249	1	2	yes	—
61	D₇₉	—	32	1	no	—
62	#2241	—	79	2	no	—
63	#2240	—	79	2	no	—
64	#1710	D₇₉	1	1	yes	characteristic, complemented
65	#1709	—	79	1	no	—
66	C₈×C₁₆	—	79	1	no	—
67	C₇₉	#12511	1	1	yes	characteristic, complemented, Sylow
68	#500	C₂×D₇₉	1	2	yes	—
69	#494	—	79	1	no	—
70	#493	—	79	1	no	—
71	#487	—	2	2	no	—
72	#486	—	158	2	no	—
73	#358	—	158	4	no	—
74	#357	—	158	4	no	—
75	C₄×C₁₆	—	79	2	no	—
76	C₈²	C₂×D₇₉	1	1	yes	characteristic
77	C₂×Q₁₆	—	158	2	no	—
78	C₂×D₈	—	158	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	158	1	no	—
81	#127	—	158	1	no	—
82	C₂×C₁₆	—	79	4	no	—
83	#107	—	2	2	no	—
84	#107	—	2	2	no	—
85	#103	—	4	4	no	—
86	#102	—	316	4	no	—
87	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
88	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	316	1	no	—
91	C₂×D₄	—	316	1	no	—
92	C₂×D₄	—	316	2	no	—
93	Q₁₆	—	316	4	no	—
94	D₈	—	316	4	no	—
95	C₂×C₈	#11929609	1	4	yes	—
96	C₂×C₈	D₄×D₇₉	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	316	2	no	—
100	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
101	C₁₆	—	158	4	no	—
102	C₂³	—	632	1	no	—
103	Q₈	—	8	4	no	—
104	Q₈	—	632	2	no	—
105	D₄	—	632	2	no	—
106	D₄	—	632	4	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	632	1	no	—
109	C₂×C₄	#14579157	1	1	yes	characteristic
110	C₂×C₄	C₂×#11929609	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#14579184	1	4	yes	—
113	C₂²	—	1264	1	no	—
114	C₂²	—	1264	2	no	—
115	C₂²	#19098939	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	1264	1	no	—
118	C₄	#19098948	1	2	yes	—
119	C₄	#19098993	1	4	yes	—
120	C₂	—	2528	1	no	—
121	C₂	#23763007	1	1	yes	characteristic
122	C₂	#23765659	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765629

C₂

yes

maximal, complemented

#23765625

C₂

yes

maximal

#23763006

C₂

yes

maximal, characteristic

C₈×C₇₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₇₉×#1710

C₂

yes

maximal, characteristic, complemented

#19098943

C₂²

yes

characteristic

#19098940

C₂²

yes

—

—

—

C₄×C₇₉⋊C₁₆

C₂²

yes

—

#500×C₇₉

C₂²

yes

—

#487×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×D₆₃₂

—

#14579161

—

#14579158

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

#14579083

—

C₄⋊Q₈×C₇₉

—

#107×C₇₉

—

#107×C₇₉

—

#103×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

Dic₃₁₆

—

D₆₃₂

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

#135

yes

—

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#495

yes

—

C₄×C₇₉

#508

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

#2241

—

#2240

—

#1710

D₇₉

yes

characteristic, complemented

#1709

—

C₈×C₁₆

—

C₇₉

#12511

yes

characteristic, complemented, Sylow

#500

C₂×D₇₉

yes

—

—

—

—

—

158

—

#358

—

158

—

#357

—

158

—

C₄×C₁₆

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

158

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

#127

—

158

—

C₂×C₁₆

—

—

—

—

—

316

—

C₄×C₈

C₂²⋊D₇₉

yes

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

Q₁₆

—

316

—

D₈

—

316

—

C₂×C₈

#11929609

yes

—

C₂×C₈

D₄×D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

316

—

100

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

101

C₁₆

—

158

—

102

C₂³

—

632

—

103

Q₈

—

104

Q₈

—

632

—

105

D₄

—

632

—

106

D₄

—

632

—

107

C₂×C₄

—

108

C₂×C₄

—

632

—

109

C₂×C₄

#14579157

yes

characteristic

110

C₂×C₄

C₂×#11929609

yes

—

111

C₈

—

112

C₈

#14579184

yes

—

113

C₂²

—

1264

—

114

C₂²

—

1264

—

115

C₂²

#19098939

yes

characteristic

116

C₄

—

117

C₄

—

1264

—

118

C₄

#19098948

yes

—

119

C₄

#19098993

yes

—

120

C₂

—

2528

—

121

C₂

#23763007

yes

characteristic

122

C₂

#23765659

yes

—

	subgroup	G/H
center	C₂²	#19098939
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1710	C₂
socle	C₂²×C₇₉	#493
prime radical	#19098940	C₂²
hypercenter	#1710	D₇₉
hypocenter	C₇₉	#12511
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#19098939

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1710

C₂

C₂²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #23765629	C₂	semidirect
4. C₈×C₇₉⋊C₁₆	C₂	semidirect
5. C₇₉×#1710	C₂	semidirect
64. #1710	D₇₉	semidirect
67. C₇₉	#12511	semidirect

H ≅ G/N

type

1. #23765629

C₂

semidirect

4. C₈×C₇₉⋊C₁₆

C₂

semidirect

5. C₇₉×#1710

C₂

semidirect

64. #1710

D₇₉

semidirect

67. C₇₉

#12511

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23765629	2
#23765625	2
#23763006	2
C₈×C₇₉⋊C₁₆	2
C₇₉×#1710	2
#12511	79

subgroup

index

#23765629

#23765625

#23763006

C₈×C₇₉⋊C₁₆

C₇₉×#1710

#12511

Subgroups — #34202965

Conjugacy classes of subgroups

Complements

Maximal subgroups