Group #34209233 — subgroups

Subgroups — #34209233

Order 20224.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#23765508	C₂	1	1	yes	maximal, characteristic
2	#23761607	C₂	1	2	yes	maximal, complemented
3	#23761592	C₂	1	2	yes	maximal, complemented
4	#23761540	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#2161	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
7	#19098804	—	2	1	no	—
8	#19098397	C₂²	1	2	yes	—
9	#19098391	C₂²	1	1	yes	characteristic
10	#19098368	—	2	1	no	—
11	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×#108×C₇₉	—	2	1	no	—
13	#368×C₇₉	—	2	1	no	—
14	#368×C₇₉	C₂²	1	2	yes	—
15	C₂×Dic₃₁₆	—	2	1	no	—
16	#14579140	—	4	1	no	—
17	Dic₆₃₂	—	2	2	no	—
18	#14579095	—	2	1	no	—
19	#14579086	—	4	1	no	—
20	C₇₉⋊C₃₂	—	2	2	no	—
21	C₂×M₁₆×C₇₉	—	4	1	no	—
22	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
23	M₃₂×C₇₉	—	2	2	no	—
24	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
26	#108×C₇₉	—	2	2	no	—
27	#108×C₇₉	—	4	1	no	—
28	C₂×Dic₁₅₈	—	4	1	no	—
29	Dic₃₁₆	—	4	1	no	—
30	Dic₃₁₆	—	4	1	no	—
31	C₄⋊Dic₇₉	—	4	1	no	—
32	Dic₇₉⋊C₄	—	8	1	no	—
33	C₂²⋊Dic₇₉	—	8	1	no	—
34	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
35	M₁₆×C₇₉	—	4	2	no	—
36	M₁₆×C₇₉	—	8	1	no	—
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₂×C₈×C₇₉	—	8	1	no	—
40	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₉	—	2	1	no	—
43	C₁₆×C₇₉	#41	1	2	yes	—
44	C₂³×C₇₉	SD₃₂	1	1	yes	characteristic
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	—	2	1	no	—
47	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
49	C₂×Dic₇₉	—	8	1	no	—
50	C₂×Dic₇₉	—	8	1	no	—
51	Dic₁₅₈	—	8	1	no	—
52	Dic₁₅₈	—	8	1	no	—
53	C₈×C₇₉	—	2	1	no	—
54	C₈×C₇₉	—	8	2	no	—
55	C₈×C₇₉	#123	1	1	yes	characteristic
56	C₈×C₇₉	#137	1	1	yes	characteristic
57	C₂²×C₇₉	—	2	1	no	—
58	C₂²×C₇₉	—	4	1	no	—
59	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
60	C₄×C₇₉	—	2	1	no	—
61	C₄×C₇₉	#480	1	1	yes	characteristic
62	C₄×C₇₉	#509	1	1	yes	characteristic
63	Dic₇₉	—	16	1	no	—
64	Dic₇₉	—	16	1	no	—
65	#13718	—	79	1	no	maximal, Sylow
66	C₂×C₇₉	—	2	1	no	—
67	C₂×C₇₉	—	4	1	no	—
68	C₂×C₇₉	#2233	1	1	yes	characteristic
69	#2219	—	79	1	no	—
70	#2161	D₇₉	1	1	yes	characteristic
71	#1426	—	79	2	no	—
72	#1420	—	79	2	no	—
73	#1400	—	79	1	no	—
74	C₇₉	#13718	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	—	79	1	no	—
76	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
77	#467	—	158	1	no	—
78	C₂×#108	—	2	1	no	—
79	M₆₄	—	79	2	no	—
80	#368	—	2	1	no	—
81	#368	C₂×D₇₉	1	2	yes	—
82	#367	—	79	1	no	—
83	#358	—	158	1	no	—
84	C₂×Q₁₆	—	158	1	no	—
85	C₂×M₁₆	—	4	1	no	—
86	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
87	C₂²⋊Q₈	—	316	1	no	—
88	Q₃₂	—	158	2	no	—
89	M₃₂	—	2	2	no	—
90	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
91	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
92	#108	—	2	2	no	—
93	#108	—	4	1	no	—
94	#107	—	158	1	no	—
95	#103	—	316	1	no	—
96	C₃₂	—	158	2	no	—
97	C₂×Q₈	—	316	1	no	—
98	C₂²×C₄	#11929609	1	1	yes	characteristic
99	Q₁₆	—	316	1	no	—
100	Q₁₆	—	316	1	no	—
101	M₁₆	—	4	2	no	—
102	M₁₆	—	8	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	#11929609	1	1	yes	characteristic
107	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
108	C₄⋊C₄	—	316	1	no	—
109	C₄⋊C₄	—	632	1	no	—
110	C₂²⋊C₄	—	632	1	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	#11929624	1	2	yes	—
113	C₂³	#14579104	1	1	yes	characteristic
114	Q₈	—	632	1	no	—
115	Q₈	—	632	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	632	1	no	—
119	C₂×C₄	—	632	1	no	—
120	C₂×C₄	#14579104	1	1	yes	characteristic
121	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	8	2	no	—
124	C₈	#14579144	1	1	yes	characteristic
125	C₈	#14579192	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	4	1	no	—
128	C₂²	C₂×#14579104	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	1264	1	no	—
131	C₄	—	1264	1	no	—
132	C₄	#19098876	1	1	yes	characteristic
133	C₄	#19098999	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	4	1	no	—
136	C₂	#23765585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765508

C₂

yes

maximal, characteristic

#23761607

C₂

yes

maximal, complemented

#23761592

C₂

yes

maximal, complemented

#23761540

C₂

yes

maximal, characteristic

C₇₉×#2161

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

C₂²

yes

characteristic

#19098804

—

#19098397

C₂²

yes

—

#19098391

C₂²

yes

characteristic

#19098368

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂×#108×C₇₉

—

#368×C₇₉

—

#368×C₇₉

C₂²

yes

—

C₂×Dic₃₁₆

—

#14579140

—

Dic₆₃₂

—

#14579095

—

#14579086

—

C₇₉⋊C₃₂

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#108×C₇₉

—

#108×C₇₉

—

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂²×C₄×C₇₉

D₈

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

#41

yes

—

C₂³×C₇₉

SD₃₂

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

SD₃₂

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×SD₃₂

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#13718

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

D₇₉

yes

—

—

—

C₇₉

#13718

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×M₃₂

C₂×D₇₉

yes

characteristic

#467

—

158

—

C₂×#108

—

M₆₄

—

#368

—

#368

C₂×D₇₉

yes

—

#367

—

#358

—

158

—

C₂×Q₁₆

—

158

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

C₂²⋊Q₈

—

316

—

Q₃₂

—

158

—

M₃₂

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

—

—

—

158

—

#103

—

316

—

C₃₂

—

158

—

C₂×Q₈

—

316

—

C₂²×C₄

#11929609

yes

characteristic

Q₁₆

—

316

—

100

Q₁₆

—

316

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

#11929609

yes

characteristic

107

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

108

C₄⋊C₄

—

316

—

109

C₄⋊C₄

—

632

—

110

C₂²⋊C₄

—

632

—

111

C₁₆

—

112

C₁₆

#11929624

yes

—

113

C₂³

#14579104

yes

characteristic

114

Q₈

—

632

—

115

Q₈

—

632

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

632

—

119

C₂×C₄

—

632

—

120

C₂×C₄

#14579104

yes

characteristic

121

C₂×C₄

C₂×#11929609

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

#14579144

yes

characteristic

125

C₈

#14579192

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

C₂×#14579104

yes

characteristic

129

C₄

—

130

C₄

—

1264

—

131

C₄

—

1264

—

132

C₄

#19098876

yes

characteristic

133

C₄

#19098999

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#23765585

yes

characteristic

	subgroup	G/H
center	C₂	#23765585
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#2161	C₂
socle	C₂×C₇₉	#2233
prime radical	C₂³×C₇₉	SD₃₂
hypercenter	#2161	D₇₉
hypocenter	C₇₉	#13718
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2161

subgroup

G/H

center

C₂

#23765585

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×#2161

C₂

socle

C₂×C₇₉

#2233

prime radical

C₂³×C₇₉

SD₃₂

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2161

N	H ≅ G/N	type
2. #23761607	C₂	semidirect
3. #23761592	C₂	semidirect
74. C₇₉	#13718	semidirect

H ≅ G/N

type

2. #23761607

C₂

semidirect

3. #23761592

C₂

semidirect

74. C₇₉

#13718

semidirect

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

subgroup	index
#23765508	2
#23761607	2
#23761592	2
#23761540	2
C₇₉×#2161	2
#13718	79

subgroup

index

C₇₉×#2161

Subgroups — #34209233

Conjugacy classes of subgroups

Complements

Maximal subgroups