Group #34202992 — subgroups

Subgroups — #34202992

Order 20224.

330 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 99 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	#23765656	C₂	1	2	yes	maximal, complemented
2	#23765653	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765650	C₂	1	1	yes	maximal, characteristic
4	#23763006	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₃₂	—	2	2	no	—
7	C₂×#14579106	—	2	4	no	—
8	C₂×D₁₂₆₄	—	2	2	no	—
9	#19098943	C₂²	1	2	yes	—
10	#19098940	C₂²	1	2	yes	—
11	#19098910	—	2	4	no	—
12	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₈²×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₆	—	2	2	no	—
16	C₂×Dic₃₁₆	—	2	2	no	—
17	C₂×D₆₃₂	—	2	2	no	—
18	C₂×D₆₃₂	—	2	2	no	—
19	#14579161	—	2	2	no	—
20	#14579158	—	2	2	no	—
21	Dic₆₃₂	—	4	4	no	—
22	#14579106	—	4	8	no	—
23	D₁₂₆₄	—	4	4	no	—
24	#14579083	—	4	8	no	—
25	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
27	C₄×C₈×C₇₉	D₄	1	2	yes	—
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₅₈	—	4	2	no	—
30	C₂×D₃₁₆	—	4	2	no	—
31	C₂×D₃₁₆	—	4	4	no	—
32	Dic₃₁₆	—	4	4	no	—
33	Dic₃₁₆	—	4	4	no	—
34	D₆₃₂	—	4	4	no	—
35	D₆₃₂	—	4	4	no	—
36	C₄⋊Dic₇₉	—	4	4	no	—
37	C₂×C₈×C₇₉	D₈	1	4	yes	—
38	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₉	D₈	1	4	yes	—
42	C₁₆×C₇₉	D₈	1	4	yes	—
43	C₂²×D₇₉	—	8	2	no	—
44	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
46	C₂×Dic₇₉	—	8	2	no	—
47	D₃₁₆	—	8	2	no	—
48	D₃₁₆	—	8	2	no	—
49	D₃₁₆	—	8	8	no	—
50	Dic₁₅₈	—	8	2	no	—
51	Dic₁₅₈	—	8	2	no	—
52	C₈×C₇₉	—	2	4	no	—
53	C₈×C₇₉	C₂×D₈	1	2	yes	—
54	C₈×C₇₉	C₂×D₈	1	2	yes	—
55	C₂²×C₇₉	#493	1	1	yes	characteristic
56	C₂×D₇₉	—	16	2	no	—
57	C₂×D₇₉	—	16	4	no	—
58	C₄×C₇₉	#493	1	1	yes	characteristic
59	C₄×C₇₉	#493	1	1	yes	characteristic
60	C₄×C₇₉	#508	1	4	yes	—
61	Dic₇₉	—	16	2	no	—
62	#12518	—	79	1	no	maximal, Sylow
63	C₂×C₇₉	#1712	1	1	yes	characteristic
64	C₂×C₇₉	#2249	1	2	yes	—
65	D₇₉	—	32	2	no	—
66	#2248	—	79	2	no	—
67	#2247	—	79	1	no	—
68	#2246	—	79	1	no	—
69	#1709	—	79	2	no	—
70	C₈×C₁₆	D₇₉	1	1	yes	characteristic, complemented
71	C₇₉	#12518	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	158	2	no	—
73	C₂×SD₃₂	—	158	4	no	—
74	C₂×D₁₆	—	158	2	no	—
75	#494	—	79	2	no	—
76	#493	—	79	2	no	—
77	#486	—	158	4	no	—
78	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
79	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
80	C₈²	C₂×D₇₉	1	1	yes	characteristic
81	C₂×Q₁₆	—	158	2	no	—
82	C₂×Q₁₆	—	158	2	no	—
83	C₂×D₈	—	158	2	no	—
84	C₂×D₈	—	158	2	no	—
85	C₄⋊Q₈	—	158	2	no	—
86	#127	—	158	2	no	—
87	Q₃₂	—	316	4	no	—
88	SD₃₂	—	316	8	no	—
89	D₁₆	—	316	4	no	—
90	C₂×C₁₆	D₃₁₆	1	2	yes	—
91	C₂×C₁₆	D₃₁₆	1	2	yes	—
92	#102	—	316	8	no	—
93	C₄×C₈	D₃₁₆	1	2	yes	—
94	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
95	C₂×Q₈	—	316	2	no	—
96	C₂×D₄	—	316	2	no	—
97	C₂×D₄	—	316	4	no	—
98	Q₁₆	—	316	4	no	—
99	Q₁₆	—	316	4	no	—
100	D₈	—	316	4	no	—
101	D₈	—	316	4	no	—
102	C₂×C₈	D₆₃₂	1	4	yes	—
103	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
104	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
105	C₄⋊C₄	—	316	4	no	—
106	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
107	C₁₆	D₆₃₂	1	4	yes	—
108	C₁₆	D₆₃₂	1	4	yes	—
109	C₂³	—	632	2	no	—
110	Q₈	—	632	2	no	—
111	Q₈	—	632	2	no	—
112	D₄	—	632	2	no	—
113	D₄	—	632	2	no	—
114	D₄	—	632	8	no	—
115	C₂×C₄	—	632	2	no	—
116	C₂×C₄	#14579158	1	1	yes	characteristic
117	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
118	C₈	—	2	4	no	—
119	C₈	C₂×D₆₃₂	1	2	yes	—
120	C₈	C₂×D₆₃₂	1	2	yes	—
121	C₂²	—	1264	2	no	—
122	C₂²	—	1264	4	no	—
123	C₂²	#19098940	1	1	yes	characteristic
124	C₄	—	1264	2	no	—
125	C₄	#19098940	1	1	yes	characteristic
126	C₄	#19098940	1	1	yes	characteristic
127	C₄	#19098995	1	4	yes	—
128	C₂	—	2528	2	no	—
129	C₂	#23763015	1	1	yes	characteristic
130	C₂	#23765661	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765656

C₂

yes

maximal, complemented

#23765653

C₂

yes

maximal, characteristic, complemented

#23765650

C₂

yes

maximal, characteristic

#23763006

C₂

yes

maximal, complemented

C₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₆₃₂

—

C₂×#14579106

—

C₂×D₁₂₆₄

—

#19098943

C₂²

yes

—

#19098940

C₂²

yes

—

#19098910

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

#14579161

—

#14579158

—

Dic₆₃₂

—

#14579106

—

D₁₂₆₄

—

#14579083

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

D₈

yes

—

C₁₆×C₇₉

D₈

yes

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₈

yes

—

C₈×C₇₉

C₂×D₈

yes

—

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

—

—

—

—

C₈×C₁₆

D₇₉

yes

characteristic, complemented

C₇₉

#12518

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

158

—

C₂×SD₃₂

—

158

—

C₂×D₁₆

—

158

—

#494

—

#493

—

#486

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

158

—

#127

—

158

—

Q₃₂

—

316

—

SD₃₂

—

316

—

D₁₆

—

316

—

C₂×C₁₆

D₃₁₆

yes

—

C₂×C₁₆

D₃₁₆

yes

—

#102

—

316

—

C₄×C₈

D₃₁₆

yes

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

Q₁₆

—

316

—

Q₁₆

—

316

—

100

D₈

—

316

—

101

D₈

—

316

—

102

C₂×C₈

D₆₃₂

yes

—

103

C₂×C₈

C₂×D₃₁₆

yes

characteristic

104

C₂×C₈

C₂×D₃₁₆

yes

characteristic

105

C₄⋊C₄

—

316

—

106

C₄²

C₂×D₃₁₆

yes

characteristic

107

C₁₆

D₆₃₂

yes

—

108

C₁₆

D₆₃₂

yes

—

109

C₂³

—

632

—

110

Q₈

—

632

—

111

Q₈

—

632

—

112

D₄

—

632

—

113

D₄

—

632

—

114

D₄

—

632

—

115

C₂×C₄

—

632

—

116

C₂×C₄

#14579158

yes

characteristic

117

C₂×C₄

C₂×D₆₃₂

yes

—

118

C₈

—

119

C₈

C₂×D₆₃₂

yes

—

120

C₈

C₂×D₆₃₂

yes

—

121

C₂²

—

1264

—

122

C₂²

—

1264

—

123

C₂²

#19098940

yes

characteristic

124

C₄

—

1264

—

125

C₄

#19098940

yes

characteristic

126

C₄

#19098940

yes

characteristic

127

C₄

#19098995

yes

—

128

C₂

—

2528

—

129

C₂

#23763015

yes

characteristic

130

C₂

#23765661

yes

—

	subgroup	G/H
center	C₂²	#19098940
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₈×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#493
prime radical	#23765650	C₂
hypercenter	C₈×C₁₆	D₇₉
hypocenter	C₇₉	#12518
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#19098940

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₈×C₁₆×C₇₉

C₂

C₂²×C₇₉

C₂

C₈×C₁₆

D₇₉

hypocenter

C₇₉

#12518

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #23765656	C₂	semidirect
2. #23765653	C₂	semidirect
4. #23763006	C₂	semidirect
5. C₈×C₁₆×C₇₉	C₂	semidirect
70. C₈×C₁₆	D₇₉	semidirect
71. C₇₉	#12518	semidirect

H ≅ G/N

type

1. #23765656

C₂

semidirect

2. #23765653

C₂

semidirect

4. #23763006

C₂

semidirect

5. C₈×C₁₆×C₇₉

C₂

semidirect

70. C₈×C₁₆

D₇₉

semidirect

71. C₇₉

#12518

semidirect

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

subgroup	index
#23765656	2
#23765653	2
#23765650	2
#23763006	2
C₈×C₁₆×C₇₉	2
#12518	79

subgroup

index

C₈×C₁₆×C₇₉

Subgroups — #34202992

Conjugacy classes of subgroups

Complements

Maximal subgroups