Group #34204632 — subgroups

Subgroups — #34204632

Order 20224.

302 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 135 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	#23765156	C₂	1	1	yes	maximal, characteristic
2	#23763086	C₂	1	1	yes	maximal, characteristic
3	#23761481	C₂	1	4	yes	maximal
4	C₇₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	C₂²	1	2	yes	—
6	C₂×#14579072	C₄	1	1	yes	characteristic
7	C₂×#14579071	C₄	1	2	yes	—
8	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
9	#19098409	C₄	1	1	yes	characteristic
10	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
11	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₉	—	2	2	no	—
14	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
16	#14579101	—	2	4	no	—
17	C₂×C₇₉⋊C₁₆	—	2	2	no	—
18	#14579072	C₈	1	4	yes	—
19	#14579071	—	2	4	no	—
20	C₈×Dic₇₉	—	2	2	no	—
21	C₈×Dic₇₉	—	2	2	no	—
22	C₇₉⋊#95	—	2	2	no	—
23	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₇₉	—	2	4	no	—
27	C₂×C₁₆×C₇₉	—	2	2	no	—
28	C₄×C₈×C₇₉	D₄	1	2	yes	—
29	C₄×C₈×C₇₉	Q₈	1	2	yes	—
30	C₂²×Dic₇₉	—	2	2	no	—
31	C₂×C₇₉⋊C₈	—	2	2	no	—
32	C₂×C₇₉⋊C₈	—	2	2	no	—
33	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
34	C₄×Dic₇₉	—	2	2	no	—
35	C₄×Dic₇₉	M₁₆	1	4	yes	—
36	C₇₉⋊C₁₆	—	4	2	no	—
37	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	—	2	2	no	—
40	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₇₉	#41	1	2	yes	—
44	C₂×C₈×C₇₉	#41	1	2	yes	—
45	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
46	C₄²×C₇₉	C₂×C₈	1	2	yes	—
47	C₁₆×C₇₉	—	4	2	no	—
48	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	4	2	no	—
50	C₂×C₄×C₇₉	C₄⋊C₈	1	4	yes	—
51	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#117	1	2	yes	—
53	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
55	C₂×Dic₇₉	—	2	4	no	—
56	C₂×Dic₇₉	—	4	4	no	—
57	C₈×C₇₉	—	2	2	no	—
58	C₈×C₇₉	#124	1	1	yes	characteristic
59	C₈×C₇₉	#124	1	1	yes	characteristic
60	C₈×C₇₉	#125	1	2	yes	—
61	C₇₉⋊C₈	—	2	4	no	—
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂²×C₇₉	#432	1	2	yes	—
65	C₄×C₇₉	—	2	4	no	—
66	C₄×C₇₉	#382	1	1	yes	characteristic
67	C₄×C₇₉	#423	1	1	yes	characteristic
68	C₄×C₇₉	#431	1	2	yes	—
69	Dic₇₉	—	4	4	no	—
70	#12884	—	79	1	no	maximal, Sylow
71	C₂×C₇₉	—	2	2	no	—
72	C₂×C₇₉	#1831	1	1	yes	characteristic
73	C₂×C₇₉	#2153	1	2	yes	—
74	#2150	—	79	1	no	—
75	#2150	D₇₉	1	1	yes	characteristic
76	#1726	—	79	1	no	—
77	#1376	—	79	4	no	—
78	C₇₉	#12884	1	1	yes	characteristic, complemented, Sylow
79	C₂×M₃₂	—	79	2	no	—
80	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
81	C₂×#97	—	79	1	no	—
82	C₂×#97	—	79	2	no	—
83	C₂×C₄×C₈	—	79	2	no	—
84	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
85	#376	—	79	1	no	—
86	C₄⋊C₁₆	—	2	2	no	—
87	C₄⋊C₁₆	—	158	2	no	—
88	C₂²×C₈	—	79	2	no	—
89	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
91	C₂×C₄²	—	79	2	no	—
92	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
93	M₃₂	—	2	4	no	—
94	M₃₂	—	158	4	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	158	2	no	—
97	#97	—	79	4	no	—
98	#97	—	158	4	no	—
99	C₄×C₈	—	158	2	no	—
100	C₄×C₈	—	158	2	no	—
101	C₄×C₈	Dic₁₅₈	1	2	yes	—
102	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
103	#95	—	158	2	no	—
104	C₂²×C₄	—	158	2	no	—
105	C₂²×C₄	#11929607	1	2	yes	—
106	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	79	4	no	—
109	C₂×C₈	—	158	2	no	—
110	C₂×C₈	—	158	2	no	—
111	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
112	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
113	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
114	C₂×C₈	#11929623	1	2	yes	—
115	C₂×C₈	#11929625	1	2	yes	—
116	C₄²	—	79	4	no	—
117	C₄²	—	158	2	no	—
118	C₄²	Dic₇₉⋊C₄	1	2	yes	—
119	C₄²	C₈×D₇₉	1	2	yes	—
120	C₁₆	—	4	2	no	—
121	C₁₆	—	316	2	no	—
122	C₂³	C₂×#11929607	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	316	4	no	—
125	C₂×C₄	—	158	4	no	—
126	C₂×C₄	Dic₇₉⋊C₈	1	4	yes	—
127	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
128	C₂×C₄	#14579119	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
130	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	—	158	4	no	—
133	C₈	#14579148	1	1	yes	characteristic
134	C₈	#14579150	1	1	yes	characteristic
135	C₈	#14579152	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
138	C₂²	#19098619	1	2	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	316	4	no	—
141	C₄	#19098428	1	1	yes	characteristic
142	C₄	#19098586	1	1	yes	characteristic
143	C₄	#19098615	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#23763516	1	1	yes	characteristic
146	C₂	#23765165	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765156

C₂

yes

maximal, characteristic

#23763086

C₂

yes

maximal, characteristic

#23761481

C₂

yes

maximal

C₇₉×#2150

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579072

C₄

yes

characteristic

C₂×#14579071

C₄

yes

—

C₂×C₈×Dic₇₉

C₂²

yes

—

#19098409

C₄

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

—

C₂×M₃₂×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

#14579072

C₈

yes

—

#14579071

—

C₈×Dic₇₉

—

C₈×Dic₇₉

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₈

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

M₁₆

yes

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

M₁₆

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

C₂×C₈

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₈

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

C₂×M₁₆

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₉

#432

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#12884

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1831

yes

characteristic

C₂×C₇₉

#2153

yes

—

#2150

—

#2150

D₇₉

yes

characteristic

#1726

—

#1376

—

C₇₉

#12884

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₉

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

158

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₉

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₇₉

yes

characteristic

M₃₂

—

M₃₂

—

158

—

C₂×C₁₆

—

C₂×C₁₆

—

158

—

#97

—

#97

—

158

—

C₄×C₈

—

158

—

100

C₄×C₈

—

158

—

101

C₄×C₈

Dic₁₅₈

yes

—

102

C₄×C₈

C₂²⋊D₇₉

yes

—

103

#95

—

158

—

104

C₂²×C₄

—

158

—

105

C₂²×C₄

#11929607

yes

—

106

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

158

—

110

C₂×C₈

—

158

—

111

C₂×C₈

Dic₇₉⋊C₄

yes

—

112

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

113

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

114

C₂×C₈

#11929623

yes

—

115

C₂×C₈

#11929625

yes

—

116

C₄²

—

117

C₄²

—

158

—

118

C₄²

Dic₇₉⋊C₄

yes

—

119

C₄²

C₈×D₇₉

yes

—

120

C₁₆

—

121

C₁₆

—

316

—

122

C₂³

C₂×#11929607

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

316

—

125

C₂×C₄

—

158

—

126

C₂×C₄

Dic₇₉⋊C₈

yes

—

127

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

128

C₂×C₄

#14579119

yes

—

129

C₂×C₄

C₂×C₈×D₇₉

yes

characteristic

130

C₂×C₄

M₁₆×D₇₉

yes

—

131

C₈

—

132

C₈

—

158

—

133

C₈

#14579148

yes

characteristic

134

C₈

#14579150

yes

characteristic

135

C₈

#14579152

yes

—

136

C₂²

—

137

C₂²

C₂×Dic₇₉⋊C₈

yes

characteristic

138

C₂²

#19098619

yes

—

139

C₄

—

140

C₄

—

316

—

141

C₄

#19098428

yes

characteristic

142

C₄

#19098586

yes

characteristic

143

C₄

#19098615

yes

—

144

C₂

—

145

C₂

#23763516

yes

characteristic

146

C₂

#23765165

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2150	C₂
socle	C₂²×C₇₉	C₂×C₄⋊C₈
prime radical	C₂³×C₇₉	C₂×M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2150	D₇₉
hypocenter	C₇₉	#12884
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2150

C₂

socle

C₂²×C₇₉

C₂×C₄⋊C₈

prime radical

C₂³×C₇₉

C₂×M₁₆

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
78. C₇₉	#12884	semidirect

H ≅ G/N

type

78. C₇₉

#12884

semidirect

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

subgroup	index
#23765156	2
#23763086	2
#23761481	2
C₇₉×#2150	2
#12884	79

subgroup

index

C₇₉×#2150

Subgroups — #34204632

Conjugacy classes of subgroups

Complements

Maximal subgroups