Group #34209177 — subgroups

Subgroups — #34209177

Order 20224.

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

normal subgroups only

This group has 158 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	#23765677	C₂	1	1	yes	maximal, characteristic
2	#23765258	C₂	1	1	yes	maximal, characteristic
3	#23761601	C₂	1	1	yes	maximal, characteristic
4	Dic₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
5	#23761582	C₂	1	1	yes	maximal, characteristic
6	#23761580	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1423	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
9	#19098964	—	2	1	no	—
10	#19098657	—	2	1	no	—
11	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
12	#19098390	—	2	1	no	—
13	#19098389	C₂²	1	1	yes	characteristic
14	#19098366	—	2	1	no	—
15	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
17	#366×C₇₉	C₂²	1	1	yes	characteristic
18	#366×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₁₆	—	2	1	no	—
20	#14579152	—	4	1	no	—
21	#14579126	—	4	1	no	—
22	Dic₆₃₂	—	2	1	no	—
23	Dic₆₃₂	Q₈	1	2	yes	—
24	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
25	#14579095	—	2	1	no	—
26	#14579092	—	4	1	no	—
27	#14579084	—	4	1	no	—
28	C₈×Dic₇₉	D₄	1	1	yes	characteristic
29	C₇₉⋊C₃₂	—	2	1	no	—
30	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
31	#107×C₇₉	—	2	1	no	—
32	#107×C₇₉	—	2	1	no	—
33	C₃₂×C₇₉	—	2	1	no	—
34	C₂×Dic₁₅₈	—	4	1	no	—
35	Dic₃₁₆	—	2	2	no	—
36	Dic₃₁₆	—	4	1	no	—
37	C₂×C₇₉⋊C₈	D₈	1	1	yes	characteristic
38	C₄⋊Dic₇₉	—	4	1	no	—
39	C₄⋊Dic₇₉	—	8	1	no	—
40	Dic₇₉⋊C₄	—	8	1	no	—
41	Dic₇₉⋊C₄	—	8	1	no	—
42	C₄×Dic₇₉	—	8	1	no	—
43	C₄×Dic₇₉	D₈	1	1	yes	characteristic
44	C₇₉⋊C₁₆	—	2	1	no	—
45	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₇₉	—	4	1	no	—
47	C₄⋊C₄×C₇₉	—	4	1	no	—
48	C₁₆×C₇₉	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₇₉	#41	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	8	1	no	—
51	C₂×C₄×C₇₉	—	8	1	no	—
52	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₇₉	—	8	1	no	—
54	C₂×Dic₇₉	—	8	1	no	—
55	C₂×Dic₇₉	D₁₆	1	1	yes	characteristic
56	C₂×Dic₇₉	D₁₆	1	1	yes	characteristic
57	Dic₁₅₈	—	4	2	no	—
58	Dic₁₅₈	—	8	1	no	—
59	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₈×C₇₉	#137	1	1	yes	characteristic
61	C₇₉⋊C₈	—	2	1	no	—
62	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
63	C₄×C₇₉	—	16	1	no	—
64	C₄×C₇₉	—	16	1	no	—
65	C₄×C₇₉	#474	1	1	yes	characteristic
66	C₄×C₇₉	#510	1	1	yes	characteristic
67	Dic₇₉	—	2	1	no	—
68	Dic₇₉	—	2	1	no	—
69	Dic₇₉	—	8	2	no	—
70	Dic₇₉	—	16	1	no	—
71	#13708	—	79	1	no	maximal, Sylow
72	C₂×C₇₉	#2224	1	1	yes	characteristic
73	C₂×C₇₉	#2262	1	1	yes	characteristic
74	C₂×C₇₉	#2264	1	1	yes	characteristic
75	#2253	—	79	1	no	—
76	C₄×Q₃₂	—	79	1	no	—
77	#1424	—	79	1	no	—
78	#1423	D₇₉	1	1	yes	characteristic
79	C₄⋊C₃₂	—	79	1	no	—
80	#1416	—	79	1	no	—
81	#1416	—	79	1	no	—
82	C₇₉	#13708	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₃₂	—	79	1	no	—
84	#499	—	158	1	no	—
85	C₄×Q₁₆	—	158	1	no	—
86	C₂×C₃₂	—	79	1	no	—
87	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
88	#367	—	158	1	no	—
89	#366	—	79	1	no	—
90	#366	C₂×D₇₉	1	1	yes	characteristic
91	#366	C₂×D₇₉	1	1	yes	characteristic
92	#358	—	158	1	no	—
93	C₄×C₁₆	—	79	1	no	—
94	C₂×Q₁₆	—	158	1	no	—
95	#125	—	316	1	no	—
96	C₄×Q₈	—	316	1	no	—
97	Q₃₂	—	79	2	no	—
98	Q₃₂	—	158	1	no	—
99	C₂×C₁₆	—	79	1	no	—
100	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	2	1	no	—
103	#107	—	158	1	no	—
104	#106	—	316	1	no	—
105	#103	—	316	1	no	—
106	C₄×C₈	—	79	1	no	—
107	C₃₂	—	2	1	no	—
108	C₃₂	—	158	1	no	—
109	C₂×Q₈	—	316	1	no	—
110	Q₁₆	—	316	1	no	—
111	Q₁₆	—	158	2	no	—
112	C₂×C₈	—	79	1	no	—
113	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	316	1	no	—
117	C₄⋊C₄	—	632	1	no	—
118	C₄⋊C₄	—	632	1	no	—
119	C₄⋊C₄	—	632	1	no	—
120	C₄²	—	79	1	no	—
121	C₄²	—	632	1	no	—
122	C₁₆	—	158	1	no	—
123	C₁₆	Q₈×D₇₉	1	1	yes	characteristic
124	C₁₆	#11929624	1	1	yes	characteristic
125	Q₈	—	316	2	no	—
126	Q₈	—	632	1	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	79	1	no	—
130	C₂×C₄	—	79	1	no	—
131	C₂×C₄	—	632	1	no	—
132	C₂×C₄	—	632	1	no	—
133	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
134	C₈	—	158	1	no	—
135	C₈	#14579139	1	1	yes	characteristic
136	C₈	#14579193	1	1	yes	characteristic
137	C₂²	D₁₆×D₇₉	1	1	yes	characteristic
138	C₄	—	16	1	no	—
139	C₄	—	16	1	no	—
140	C₄	—	632	2	no	—
141	C₄	—	158	1	no	—
142	C₄	—	158	1	no	—
143	C₄	—	1264	1	no	—
144	C₄	#19098842	1	1	yes	characteristic
145	C₄	#19099005	1	1	yes	characteristic
146	C₂	#23765536	1	1	yes	characteristic
147	C₂	#23765706	1	1	yes	characteristic
148	C₂	#23765719	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765677

C₂

yes

maximal, characteristic

#23765258

C₂

yes

maximal, characteristic

#23761601

C₂

yes

maximal, characteristic

Dic₇₉⋊C₃₂

C₂

yes

maximal, characteristic

#23761582

C₂

yes

maximal, characteristic

#23761580

C₂

yes

maximal, characteristic

C₇₉×#1423

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

C₂²

yes

characteristic

#19098964

—

#19098657

—

C₂×C₇₉⋊C₃₂

C₂²

yes

characteristic

#19098390

—

#19098389

C₂²

yes

characteristic

#19098366

—

C₁₆×Dic₇₉

C₂²

yes

characteristic

C₂×C₃₂×C₇₉

C₂²

yes

characteristic

#366×C₇₉

C₂²

yes

characteristic

#366×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

#14579152

—

#14579126

—

Dic₆₃₂

—

Dic₆₃₂

Q₈

yes

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

—

—

C₈×Dic₇₉

D₄

yes

characteristic

C₇₉⋊C₃₂

—

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#107×C₇₉

—

C₃₂×C₇₉

—

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₈

yes

characteristic

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

C₂×Q₈

yes

characteristic

C₁₆×C₇₉

#41

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₁₆

yes

characteristic

C₂×Dic₇₉

D₁₆

yes

characteristic

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂²⋊Q₈

yes

characteristic

C₈×C₇₉

#137

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₄×Q₃₂

—

#1424

—

#1423

D₇₉

yes

characteristic

C₄⋊C₃₂

—

#1416

—

#1416

—

C₇₉

#13708

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#499

—

158

—

C₄×Q₁₆

—

158

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₉

yes

characteristic

#367

—

158

—

#366

—

#366

C₂×D₇₉

yes

characteristic

#366

C₂×D₇₉

yes

characteristic

#358

—

158

—

C₄×C₁₆

—

C₂×Q₁₆

—

158

—

#125

—

316

—

C₄×Q₈

—

316

—

Q₃₂

—

Q₃₂

—

158

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

101

#107

—

102

#107

—

103

#107

—

158

—

104

#106

—

316

—

105

#103

—

316

—

106

C₄×C₈

—

107

C₃₂

—

108

C₃₂

—

158

—

109

C₂×Q₈

—

316

—

110

Q₁₆

—

316

—

111

Q₁₆

—

158

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₇₉

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

316

—

117

C₄⋊C₄

—

632

—

118

C₄⋊C₄

—

632

—

119

C₄⋊C₄

—

632

—

120

C₄²

—

121

C₄²

—

632

—

122

C₁₆

—

158

—

123

C₁₆

Q₈×D₇₉

yes

characteristic

124

C₁₆

#11929624

yes

characteristic

125

Q₈

—

316

—

126

Q₈

—

632

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

632

—

132

C₂×C₄

—

632

—

133

C₂×C₄

D₈×D₇₉

yes

characteristic

134

C₈

—

158

—

135

C₈

#14579139

yes

characteristic

136

C₈

#14579193

yes

characteristic

137

C₂²

D₁₆×D₇₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

632

—

141

C₄

—

158

—

142

C₄

—

158

—

143

C₄

—

1264

—

144

C₄

#19098842

yes

characteristic

145

C₄

#19099005

yes

characteristic

146

C₂

#23765536

yes

characteristic

147

C₂

#23765706

yes

characteristic

148

C₂

#23765719

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₉
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#1423	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₁₆
hypercenter	#1423	D₇₉
hypocenter	C₇₉	#13708
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

subgroup

G/H

center

C₂²

D₁₆×D₇₉

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×#1423

C₂

socle, prime radical

C₂²×C₇₉

C₂×D₁₆

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

N	H ≅ G/N	type
82. C₇₉	#13708	semidirect

H ≅ G/N

type

82. C₇₉

#13708

semidirect

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

subgroup	index
#23765677	2
#23765258	2
#23761601	2
Dic₇₉⋊C₃₂	2
#23761582	2
#23761580	2
C₇₉×#1423	2
#13708	79

subgroup

index

Dic₇₉⋊C₃₂

C₇₉×#1423

Subgroups — #34209177

Conjugacy classes of subgroups

Complements

Maximal subgroups