Group #34209270 — subgroups

Subgroups — #34209270

Order 20224.

250 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 83 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×#19098402	C₂	1	4	yes	maximal, complemented
2	#23765653	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765650	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₆₃₂	—	2	1	no	—
6	C₂×Dic₆₃₂	C₂²	1	2	yes	—
7	C₂×D₁₂₆₄	—	2	1	no	—
8	C₂×D₁₂₆₄	C₂²	1	2	yes	—
9	#19098943	—	2	1	no	—
10	#19098940	—	2	1	no	—
11	#19098402	—	2	8	no	—
12	C₂×C₃₂×C₇₉	C₂²	1	2	yes	—
13	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₁₆	—	2	2	no	—
15	C₂×Dic₃₁₆	—	4	1	no	—
16	C₂×D₆₃₂	—	2	2	no	—
17	C₂×D₆₃₂	—	4	1	no	—
18	#14579161	—	4	1	no	—
19	#14579158	—	4	1	no	—
20	Dic₆₃₂	—	2	4	no	—
21	Dic₆₃₂	—	4	2	no	—
22	D₁₂₆₄	—	2	4	no	—
23	D₁₂₆₄	—	4	2	no	—
24	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₃₂×C₇₉	D₄	1	4	yes	complemented
28	C₂×Dic₁₅₈	—	4	2	no	—
29	C₂×D₃₁₆	—	4	2	no	—
30	C₂×D₃₁₆	—	8	2	no	—
31	Dic₃₁₆	—	4	2	no	—
32	Dic₃₁₆	—	4	2	no	—
33	Dic₃₁₆	—	8	2	no	—
34	D₆₃₂	—	4	2	no	—
35	D₆₃₂	—	4	2	no	—
36	D₆₃₂	—	8	2	no	—
37	C₄⋊Dic₇₉	—	8	2	no	—
38	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇₉	D₈	1	1	yes	characteristic
41	C₁₆×C₇₉	D₈	1	2	yes	—
42	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
43	C₂²×D₇₉	—	8	2	no	—
44	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
45	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
46	C₂×Dic₇₉	—	8	2	no	—
47	D₃₁₆	—	8	2	no	—
48	D₃₁₆	—	8	2	no	—
49	D₃₁₆	—	16	4	no	—
50	Dic₁₅₈	—	8	2	no	—
51	Dic₁₅₈	—	8	2	no	—
52	C₈×C₇₉	D₁₆	1	2	yes	—
53	C₈×C₇₉	#127	1	1	yes	characteristic
54	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
56	C₂×D₇₉	—	16	2	no	—
57	C₂×D₇₉	—	16	4	no	—
58	C₄×C₇₉	SD₆₄	1	4	yes	complemented
59	C₄×C₇₉	#493	1	1	yes	characteristic
60	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
61	Dic₇₉	—	16	2	no	—
62	#13727	—	79	1	no	maximal, Sylow
63	C₂×C₇₉	#2246	1	1	yes	characteristic
64	C₂×C₇₉	C₂×SD₆₄	1	2	yes	—
65	D₇₉	—	32	2	no	—
66	C₂×SD₆₄	—	79	4	no	—
67	#2247	—	79	1	no	—
68	#2246	—	79	1	no	—
69	C₄×C₃₂	D₇₉	1	1	yes	characteristic, complemented
70	C₇₉	#13727	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₃₂	—	79	2	no	—
72	C₂×Q₃₂	—	158	1	no	—
73	C₂×D₁₆	—	79	2	no	—
74	C₂×D₁₆	—	158	1	no	—
75	#494	—	158	1	no	—
76	#493	—	158	1	no	—
77	SD₆₄	—	158	8	no	—
78	C₂×C₃₂	C₂×D₇₉	1	2	yes	—
79	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
80	C₂×Q₁₆	—	316	1	no	—
81	C₂×Q₁₆	—	158	2	no	—
82	C₂×D₈	—	316	1	no	—
83	C₂×D₈	—	158	2	no	—
84	C₄⋊Q₈	—	316	1	no	—
85	#127	—	316	1	no	—
86	Q₃₂	—	316	2	no	—
87	Q₃₂	—	158	4	no	—
88	D₁₆	—	316	2	no	—
89	D₁₆	—	158	4	no	—
90	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
91	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
92	C₄×C₈	D₃₁₆	1	1	yes	characteristic
93	C₃₂	D₃₁₆	1	4	yes	complemented
94	C₂×Q₈	—	316	2	no	—
95	C₂×D₄	—	316	2	no	—
96	C₂×D₄	—	632	2	no	—
97	Q₁₆	—	316	2	no	—
98	Q₁₆	—	316	2	no	—
99	Q₁₆	—	632	2	no	—
100	D₈	—	316	2	no	—
101	D₈	—	316	2	no	—
102	D₈	—	632	2	no	—
103	C₂×C₈	D₆₃₂	1	1	yes	characteristic
104	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
105	C₄⋊C₄	—	632	2	no	—
106	C₄²	D₆₃₂	1	1	yes	characteristic
107	C₁₆	D₆₃₂	1	2	yes	—
108	C₁₆	C₂×D₃₁₆	1	2	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	—	1264	4	no	—
115	C₂×C₄	—	632	2	no	—
116	C₂×C₄	D₁₂₆₄	1	2	yes	—
117	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
118	C₈	D₁₂₆₄	1	2	yes	—
119	C₈	#14579158	1	1	yes	characteristic
120	C₈	C₂×D₆₃₂	1	1	yes	characteristic
121	C₂²	—	1264	2	no	—
122	C₂²	—	1264	4	no	—
123	C₂²	C₂×D₁₂₆₄	1	1	yes	characteristic
124	C₄	—	1264	2	no	—
125	C₄	#19098402	1	4	yes	complemented
126	C₄	#19098940	1	1	yes	characteristic
127	C₄	C₂×D₁₂₆₄	1	1	yes	characteristic
128	C₂	—	2528	2	no	—
129	C₂	#23765650	1	1	yes	characteristic
130	C₂	C₂×#19098402	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098402

C₂

yes

maximal, complemented

#23765653

C₂

yes

maximal, characteristic, complemented

#23765650

C₂

yes

maximal, characteristic

C₄×C₃₂×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₆₃₂

—

C₂×Dic₆₃₂

C₂²

yes

—

C₂×D₁₂₆₄

—

C₂×D₁₂₆₄

C₂²

yes

—

#19098943

—

#19098940

—

#19098402

—

C₂×C₃₂×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

#14579161

—

#14579158

—

Dic₆₃₂

—

Dic₆₃₂

—

D₁₂₆₄

—

D₁₂₆₄

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₃₂×C₇₉

D₄

yes

complemented

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

D₈

yes

—

C₁₆×C₇₉

C₂×D₄

yes

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

D₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

D₁₆

yes

—

C₈×C₇₉

#127

yes

characteristic

C₈×C₇₉

C₂×D₈

yes

characteristic

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

SD₆₄

yes

complemented

C₄×C₇₉

#493

yes

characteristic

C₄×C₇₉

C₂×D₁₆

yes

characteristic

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

C₂×SD₆₄

yes

—

D₇₉

—

C₂×SD₆₄

—

#2247

—

#2246

—

C₄×C₃₂

D₇₉

yes

characteristic, complemented

C₇₉

#13727

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

158

—

C₂×D₁₆

—

C₂×D₁₆

—

158

—

#494

—

158

—

#493

—

158

—

SD₆₄

—

158

—

C₂×C₃₂

C₂×D₇₉

yes

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

316

—

C₂×Q₁₆

—

158

—

C₂×D₈

—

316

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

316

—

#127

—

316

—

Q₃₂

—

316

—

Q₃₂

—

158

—

D₁₆

—

316

—

D₁₆

—

158

—

C₂×C₁₆

D₃₁₆

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

C₄×C₈

D₃₁₆

yes

characteristic

C₃₂

D₃₁₆

yes

complemented

C₂×Q₈

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

632

—

Q₁₆

—

316

—

Q₁₆

—

316

—

Q₁₆

—

632

—

100

D₈

—

316

—

101

D₈

—

316

—

102

D₈

—

632

—

103

C₂×C₈

D₆₃₂

yes

characteristic

104

C₂×C₈

C₂×D₃₁₆

yes

characteristic

105

C₄⋊C₄

—

632

—

106

C₄²

D₆₃₂

yes

characteristic

107

C₁₆

D₆₃₂

yes

—

108

C₁₆

C₂×D₃₁₆

yes

—

109

C₂³

—

632

—

110

Q₈

—

632

—

111

Q₈

—

632

—

112

D₄

—

632

—

113

D₄

—

632

—

114

D₄

—

1264

—

115

C₂×C₄

—

632

—

116

C₂×C₄

D₁₂₆₄

yes

—

117

C₂×C₄

C₂×D₆₃₂

yes

characteristic

118

C₈

D₁₂₆₄

yes

—

119

C₈

#14579158

yes

characteristic

120

C₈

C₂×D₆₃₂

yes

characteristic

121

C₂²

—

1264

—

122

C₂²

—

1264

—

123

C₂²

C₂×D₁₂₆₄

yes

characteristic

124

C₄

—

1264

—

125

C₄

#19098402

yes

complemented

126

C₄

#19098940

yes

characteristic

127

C₄

C₂×D₁₂₆₄

yes

characteristic

128

C₂

—

2528

—

129

C₂

#23765650

yes

characteristic

130

C₂

C₂×#19098402

yes

—

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₄
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₄×C₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₁₆
prime radical	#23765650	C₂
hypercenter	C₄×C₃₂	D₇₉
hypocenter	C₇₉	#13727
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂²

C₂×D₁₂₆₄

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₄×C₃₂×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₁₆

prime radical

#23765650

C₂

hypercenter

C₄×C₃₂

D₇₉

hypocenter

C₇₉

#13727

LCS

G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
1. C₂×#19098402	C₂	semidirect
2. #23765653	C₂	semidirect
4. C₄×C₃₂×C₇₉	C₂	semidirect
27. C₃₂×C₇₉	D₄	semidirect
58. C₄×C₇₉	SD₆₄	semidirect
69. C₄×C₃₂	D₇₉	semidirect
70. C₇₉	#13727	semidirect
93. C₃₂	D₃₁₆	semidirect
125. C₄	#19098402	semidirect

H ≅ G/N

type

1. C₂×#19098402

C₂

semidirect

2. #23765653

C₂

semidirect

4. C₄×C₃₂×C₇₉

C₂

semidirect

27. C₃₂×C₇₉

D₄

semidirect

58. C₄×C₇₉

SD₆₄

semidirect

69. C₄×C₃₂

D₇₉

semidirect

70. C₇₉

#13727

semidirect

93. C₃₂

D₃₁₆

semidirect

125. C₄

#19098402

semidirect

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

subgroup	index
C₂×#19098402	2
#23765653	2
#23765650	2
C₄×C₃₂×C₇₉	2
#13727	79

subgroup

index

C₄×C₃₂×C₇₉

Subgroups — #34209270

Conjugacy classes of subgroups

Complements

Maximal subgroups