Group #34202927 — subgroups

Subgroups — #34202927

Order 20224.

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

normal subgroups only

This group has 262 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	#23765675	C₂	1	1	yes	maximal, characteristic
2	#23765653	C₂	1	1	yes	maximal, characteristic
3	#23765628	C₂	1	2	yes	maximal
4	#23762997	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₂	—	2	2	no	—
7	#19098968	C₂²	1	2	yes	—
8	#19098961	—	2	2	no	—
9	#19098943	C₂²	1	2	yes	—
10	#19098914	—	2	2	no	—
11	#19098389	—	2	2	no	—
12	#19098368	—	2	4	no	—
13	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₈²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₁₆	—	2	2	no	—
17	C₂×Dic₃₁₆	—	2	2	no	—
18	#14579161	—	2	2	no	—
19	#14579161	—	2	2	no	—
20	Dic₆₃₂	—	4	4	no	—
21	#14579095	—	2	2	no	—
22	#14579095	—	2	2	no	—
23	#14579093	—	4	4	no	—
24	#14579086	—	4	4	no	—
25	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₅₈	—	4	2	no	—
31	C₂×Dic₁₅₈	—	4	2	no	—
32	Dic₃₁₆	—	4	4	no	—
33	Dic₃₁₆	—	4	4	no	—
34	C₄⋊Dic₇₉	—	4	2	no	—
35	C₄⋊Dic₇₉	—	4	2	no	—
36	C₄⋊Dic₇₉	—	4	2	no	—
37	C₄⋊Dic₇₉	—	4	2	no	—
38	C₂×C₈×C₇₉	D₈	1	2	yes	—
39	C₂×C₈×C₇₉	D₈	1	2	yes	—
40	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₉	—	2	2	no	—
44	C₁₆×C₇₉	SD₁₆	1	4	yes	—
45	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₇₉	—	8	2	no	—
49	C₂×Dic₇₉	—	8	2	no	—
50	Dic₁₅₈	—	8	2	no	—
51	Dic₁₅₈	—	8	2	no	—
52	Dic₁₅₈	—	8	4	no	—
53	C₈×C₇₉	—	2	2	no	—
54	C₈×C₇₉	Q₃₂	1	4	yes	complemented
55	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
56	C₈×C₇₉	#135	1	2	yes	—
57	C₂²×C₇₉	#493	1	1	yes	characteristic
58	C₄×C₇₉	#492	1	1	yes	characteristic
59	C₄×C₇₉	#495	1	1	yes	characteristic
60	C₄×C₇₉	C₂×Q₃₂	1	2	yes	—
61	C₄×C₇₉	#508	1	2	yes	—
62	Dic₇₉	—	16	2	no	—
63	Dic₇₉	—	16	2	no	—
64	#12504	—	79	1	no	maximal, Sylow
65	C₂×C₇₉	#1706	1	1	yes	characteristic
66	C₂×C₇₉	#2247	1	1	yes	characteristic
67	C₂×C₇₉	#2249	1	1	yes	characteristic
68	#2252	—	79	1	no	—
69	#2247	—	79	1	no	—
70	#2241	—	79	2	no	—
71	#1707	—	79	2	no	—
72	C₈×C₁₆	D₇₉	1	1	yes	characteristic
73	C₇₉	#12504	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	—	158	2	no	—
75	#500	—	79	2	no	—
76	#498	—	158	2	no	—
77	#494	—	79	2	no	—
78	#487	—	158	2	no	—
79	#366	—	158	2	no	—
80	#358	—	158	4	no	—
81	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
82	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
83	C₈²	C₂×D₇₉	1	1	yes	characteristic
84	C₂×Q₁₆	—	158	2	no	—
85	C₂×Q₁₆	—	158	2	no	—
86	C₄⋊Q₈	—	158	2	no	—
87	C₄⋊Q₈	—	158	2	no	—
88	Q₃₂	—	316	4	no	—
89	C₂×C₁₆	D₃₁₆	1	2	yes	—
90	C₂×C₁₆	D₃₁₆	1	2	yes	—
91	#107	—	158	2	no	—
92	#107	—	158	2	no	—
93	#106	—	316	4	no	—
94	#103	—	316	4	no	—
95	C₄×C₈	D₃₁₆	1	1	yes	characteristic
96	C₄×C₈	D₃₁₆	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
98	C₂×Q₈	—	316	2	no	—
99	C₂×Q₈	—	316	2	no	—
100	Q₁₆	—	316	4	no	—
101	Q₁₆	—	316	4	no	—
102	C₂×C₈	D₆₃₂	1	2	yes	—
103	C₂×C₈	D₆₃₂	1	2	yes	—
104	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
105	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
106	C₄⋊C₄	—	316	2	no	—
107	C₄⋊C₄	—	316	2	no	—
108	C₄⋊C₄	—	316	2	no	—
109	C₄⋊C₄	—	316	2	no	—
110	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	#11929613	1	4	yes	—
113	Q₈	—	632	2	no	—
114	Q₈	—	632	2	no	—
115	Q₈	—	632	4	no	—
116	C₂×C₄	—	632	2	no	—
117	C₂×C₄	—	632	2	no	—
118	C₂×C₄	#14579158	1	1	yes	characteristic
119	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
120	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	Dic₆₃₂	1	4	yes	complemented
123	C₈	C₂×#11929613	1	2	yes	—
124	C₈	#14579183	1	2	yes	—
125	C₂²	#19098940	1	1	yes	characteristic
126	C₄	—	1264	2	no	—
127	C₄	—	1264	2	no	—
128	C₄	#19098936	1	1	yes	characteristic
129	C₄	#19098947	1	1	yes	characteristic
130	C₄	C₂×Dic₆₃₂	1	2	yes	—
131	C₄	#19098995	1	2	yes	—
132	C₂	#23762992	1	1	yes	characteristic
133	C₂	#23765653	1	1	yes	characteristic
134	C₂	#23765661	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765675

C₂

yes

maximal, characteristic

#23765653

C₂

yes

maximal, characteristic

#23765628

C₂

yes

maximal

#23762997

C₂

yes

maximal

C₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

—

#19098968

C₂²

yes

—

#19098961

—

#19098943

C₂²

yes

—

#19098914

—

#19098389

—

#19098368

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

#14579161

—

#14579161

—

Dic₆₃₂

—

—

—

—

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

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

—

C₁₆×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

Q₃₂

yes

complemented

C₈×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#135

yes

—

C₂²×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×Q₃₂

yes

—

C₄×C₇₉

#508

yes

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

—

—

C₈×C₁₆

D₇₉

yes

characteristic

C₇₉

#12504

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

158

—

#500

—

#498

—

158

—

#494

—

#487

—

158

—

#366

—

158

—

#358

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

Q₃₂

—

316

—

C₂×C₁₆

D₃₁₆

yes

—

C₂×C₁₆

D₃₁₆

yes

—

#107

—

158

—

#107

—

158

—

#106

—

316

—

#103

—

316

—

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

316

—

C₂×Q₈

—

316

—

100

Q₁₆

—

316

—

101

Q₁₆

—

316

—

102

C₂×C₈

D₆₃₂

yes

—

103

C₂×C₈

D₆₃₂

yes

—

104

C₂×C₈

C₂×D₃₁₆

yes

characteristic

105

C₂×C₈

C₂×D₃₁₆

yes

characteristic

106

C₄⋊C₄

—

316

—

107

C₄⋊C₄

—

316

—

108

C₄⋊C₄

—

316

—

109

C₄⋊C₄

—

316

—

110

C₄²

C₂×D₃₁₆

yes

characteristic

111

C₁₆

—

112

C₁₆

#11929613

yes

—

113

Q₈

—

632

—

114

Q₈

—

632

—

115

Q₈

—

632

—

116

C₂×C₄

—

632

—

117

C₂×C₄

—

632

—

118

C₂×C₄

#14579158

yes

characteristic

119

C₂×C₄

C₂×D₆₃₂

yes

characteristic

120

C₂×C₄

C₂×D₆₃₂

yes

characteristic

121

C₈

—

122

C₈

Dic₆₃₂

yes

complemented

123

C₈

C₂×#11929613

yes

—

124

C₈

#14579183

yes

—

125

C₂²

#19098940

yes

characteristic

126

C₄

—

1264

—

127

C₄

—

1264

—

128

C₄

#19098936

yes

characteristic

129

C₄

#19098947

yes

characteristic

130

C₄

C₂×Dic₆₃₂

yes

—

131

C₄

#19098995

yes

—

132

C₂

#23762992

yes

characteristic

133

C₂

#23765653

yes

characteristic

134

C₂

#23765661

yes

characteristic

	subgroup	G/H
center	C₂²	#19098940
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₈×C₁₆×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	#493
hypercenter	C₈×C₁₆	D₇₉
hypocenter	C₇₉	#12504
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

socle, prime radical

C₂²×C₇₉

#493

hypercenter

C₈×C₁₆

D₇₉

hypocenter

C₇₉

#12504

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
54. C₈×C₇₉	Q₃₂	semidirect
73. C₇₉	#12504	semidirect
122. C₈	Dic₆₃₂	semidirect

H ≅ G/N

type

54. C₈×C₇₉

Q₃₂

semidirect

73. C₇₉

#12504

semidirect

122. C₈

Dic₆₃₂

semidirect

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

subgroup	index
#23765675	2
#23765653	2
#23765628	2
#23762997	2
C₈×C₁₆×C₇₉	2
#12504	79

subgroup

index

C₈×C₁₆×C₇₉

Subgroups — #34202927

Conjugacy classes of subgroups

Complements

Maximal subgroups