Group #34200136 — subgroups

Subgroups — #34200136

Order 20224.

236 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 85 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	#23765231	C₂	1	2	yes	maximal, complemented
2	C₈×D₆₃₂	C₂	1	1	yes	maximal, characteristic
3	#23761354	C₂	1	2	yes	maximal, complemented
4	#23761318	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579100	—	2	2	no	—
7	C₄×D₆₃₂	C₄	1	1	yes	characteristic
8	C₈×D₃₁₆	C₂²	1	2	yes	—
9	(C₂×D₇₉)⋊C₁₆	—	2	2	no	—
10	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#19098316	C₄	1	1	yes	characteristic
13	#19098292	C₄	1	2	yes	—
14	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
15	C₈²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×D₆₃₂	C₈	1	1	yes	characteristic
17	C₂×C₈×D₇₉	—	2	2	no	—
18	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
19	#14579100	—	4	4	no	—
20	C₂×C₇₉⋊C₁₆	—	2	1	no	—
21	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
22	#14579095	C₈	1	1	yes	characteristic
23	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
24	#14579083	C₈	1	2	yes	—
25	(C₂×D₇₉)⋊C₈	—	2	2	no	—
26	C₂×C₁₆×C₇₉	—	2	2	no	—
27	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×D₃₁₆	C₂×C₈	1	2	yes	—
31	C₂×C₄×D₇₉	—	2	2	no	—
32	D₆₃₂	—	2	2	no	—
33	C₂×C₇₉⋊C₈	—	2	2	no	—
34	C₈×D₇₉	—	4	2	no	—
35	C₈×D₇₉	—	4	2	no	—
36	C₄⋊Dic₇₉	C₂×C₈	1	2	yes	—
37	(C₂×D₇₉)⋊C₄	—	2	2	no	—
38	C₇₉⋊C₁₆	—	2	2	no	—
39	C₇₉⋊C₁₆	—	4	1	no	—
40	C₂×C₈×C₇₉	—	2	1	no	—
41	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
45	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₇₉	—	4	2	no	—
47	C₂²×D₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
51	C₂×Dic₇₉	—	2	2	no	—
52	D₃₁₆	—	2	2	no	—
53	D₃₁₆	—	2	2	no	—
54	C₄×D₇₉	—	4	2	no	—
55	C₄×D₇₉	—	4	2	no	—
56	C₈×C₇₉	—	2	1	no	—
57	C₈×C₇₉	—	2	1	no	—
58	C₈×C₇₉	—	4	1	no	—
59	C₈×C₇₉	M₃₂	1	2	yes	complemented
60	C₈×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
61	C₇₉⋊C₈	—	4	2	no	—
62	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
63	C₂×D₇₉	—	4	2	no	—
64	C₂×D₇₉	—	4	2	no	—
65	C₂×D₇₉	—	4	2	no	—
66	C₄×C₇₉	—	2	1	no	—
67	C₄×C₇₉	#442	1	1	yes	characteristic
68	C₄×C₇₉	#444	1	1	yes	characteristic
69	C₄×C₇₉	C₂×M₃₂	1	1	yes	characteristic
70	C₄×C₇₉	#504	1	1	yes	characteristic
71	Dic₇₉	—	4	2	no	—
72	#12064	—	79	1	no	maximal, Sylow
73	C₂×C₇₉	#1580	1	1	yes	characteristic
74	C₂×C₇₉	#2169	1	1	yes	characteristic
75	C₂×C₇₉	#2171	1	1	yes	characteristic
76	D₇₉	—	8	2	no	—
77	#2169	—	79	2	no	—
78	C₈×D₈	—	79	1	no	—
79	#1367	D₇₉	1	1	yes	characteristic, complemented
80	D₄⋊C₁₆	—	79	2	no	—
81	#1312	—	79	1	no	—
82	C₇₉	#12064	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	158	2	no	—
84	C₄×D₈	—	79	1	no	—
85	C₈×D₄	—	79	2	no	—
86	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
87	C₂²⋊C₁₆	—	158	2	no	—
88	C₄×C₁₆	—	79	1	no	—
89	C₄×C₁₆	—	79	1	no	—
90	#335	—	79	1	no	—
91	D₄⋊C₈	—	79	2	no	—
92	C₈²	C₂×D₇₉	1	1	yes	characteristic
93	C₂×D₈	—	79	1	no	—
94	C₂²×C₈	—	158	2	no	—
95	C₄×D₄	—	79	2	no	—
96	M₃₂	—	316	4	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	79	2	no	—
99	C₂×C₁₆	—	158	1	no	—
100	#107	—	79	1	no	—
101	C₄⋊C₈	—	79	2	no	—
102	#102	—	79	2	no	—
103	C₂²⋊C₈	—	158	2	no	—
104	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
105	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
107	C₂×D₄	—	79	2	no	—
108	C₂²×C₄	—	158	2	no	—
109	D₈	—	158	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	316	2	no	—
112	C₂×C₈	—	316	2	no	—
113	C₂×C₈	—	158	2	no	—
114	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
115	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
116	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
117	C₂×C₈	#11929625	1	1	yes	characteristic
118	C₄⋊C₄	—	79	2	no	—
119	C₂²⋊C₄	—	158	2	no	—
120	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
121	C₁₆	—	4	2	no	—
122	C₁₆	—	316	1	no	—
123	C₁₆	—	158	2	no	—
124	C₂³	—	158	2	no	—
125	D₄	—	158	2	no	—
126	D₄	—	158	2	no	—
127	C₂×C₄	—	316	2	no	—
128	C₂×C₄	—	316	2	no	—
129	C₂×C₄	—	158	2	no	—
130	C₂×C₄	#14579123	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
132	C₂×C₄	#14579169	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	316	2	no	—
137	C₈	#14579100	1	2	yes	complemented
138	C₈	#14579188	1	2	yes	—
139	C₂²	—	316	2	no	—
140	C₂²	—	316	2	no	—
141	C₂²	—	316	2	no	—
142	C₂²	#19098628	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	316	2	no	—
145	C₄	#19098674	1	1	yes	characteristic
146	C₄	#19098684	1	1	yes	characteristic
147	C₄	C₂×#14579100	1	1	yes	characteristic
148	C₄	#19098981	1	1	yes	characteristic
149	C₂	—	632	2	no	—
150	C₂	#23762223	1	1	yes	characteristic
151	C₂	#23765231	1	1	yes	characteristic
152	C₂	#23765241	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765231

C₂

yes

maximal, complemented

C₈×D₆₃₂

C₂

yes

maximal, characteristic

#23761354

C₂

yes

maximal, complemented

#23761318

C₂

yes

maximal, characteristic, complemented

C₇₉×#1367

C₂

yes

maximal, characteristic, complemented

C₂×#14579100

—

C₄×D₆₃₂

C₄

yes

characteristic

C₈×D₃₁₆

C₂²

yes

—

(C₂×D₇₉)⋊C₁₆

—

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098316

C₄

yes

characteristic

#19098292

C₄

yes

—

C₄⋊C₁₆×C₇₉

C₂²

yes

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×D₆₃₂

C₈

yes

characteristic

C₂×C₈×D₇₉

—

C₄×D₃₁₆

C₂×C₄

yes

—

#14579100

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

#14579095

C₈

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂×C₄

yes

—

#14579083

C₈

yes

—

(C₂×D₇₉)⋊C₈

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×D₃₁₆

C₂×C₈

yes

—

C₂×C₄×D₇₉

—

D₆₃₂

—

C₂×C₇₉⋊C₈

—

C₈×D₇₉

—

C₈×D₇₉

—

C₄⋊Dic₇₉

C₂×C₈

yes

—

(C₂×D₇₉)⋊C₄

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

#131

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

M₃₂

yes

complemented

C₈×C₇₉

AGL(1,ℤ/8)

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₈×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×M₃₂

yes

characteristic

C₄×C₇₉

#504

yes

characteristic

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#2169

—

C₈×D₈

—

#1367

D₇₉

yes

characteristic, complemented

D₄⋊C₁₆

—

#1312

—

C₇₉

#12064

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

158

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

C₂×D₇₉

yes

—

C₂²⋊C₁₆

—

158

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×D₈

—

C₂²×C₈

—

158

—

C₄×D₄

—

M₃₂

—

316

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

158

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

C₂²⋊C₈

—

158

—

104

C₄×C₈

C₄×D₇₉

yes

characteristic

105

C₄×C₈

C₄×D₇₉

yes

characteristic

106

C₄×C₈

C₂²×D₇₉

yes

characteristic

107

C₂×D₄

—

108

C₂²×C₄

—

158

—

109

D₈

—

158

—

110

C₂×C₈

—

111

C₂×C₈

—

316

—

112

C₂×C₈

—

316

—

113

C₂×C₈

—

158

—

114

C₂×C₈

C₈×D₇₉

yes

characteristic

115

C₂×C₈

C₈×D₇₉

yes

characteristic

116

C₂×C₈

D₄×D₇₉

yes

characteristic

117

C₂×C₈

#11929625

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

158

—

120

C₄²

C₂×C₄×D₇₉

yes

characteristic

121

C₁₆

—

122

C₁₆

—

316

—

123

C₁₆

—

158

—

124

C₂³

—

158

—

125

D₄

—

158

—

126

D₄

—

158

—

127

C₂×C₄

—

316

—

128

C₂×C₄

—

316

—

129

C₂×C₄

—

158

—

130

C₂×C₄

#14579123

yes

characteristic

131

C₂×C₄

C₂×C₈×D₇₉

yes

characteristic

132

C₂×C₄

#14579169

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

316

—

137

C₈

#14579100

yes

complemented

138

C₈

#14579188

yes

—

139

C₂²

—

316

—

140

C₂²

—

316

—

141

C₂²

—

316

—

142

C₂²

#19098628

yes

characteristic

143

C₄

—

144

C₄

—

316

—

145

C₄

#19098674

yes

characteristic

146

C₄

#19098684

yes

characteristic

147

C₄

C₂×#14579100

yes

characteristic

148

C₄

#19098981

yes

characteristic

149

C₂

—

632

—

150

C₂

#23762223

yes

characteristic

151

C₂

#23765231

yes

characteristic

152

C₂

#23765241

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1367	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×D₆₃₂	C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1367	D₇₉
hypocenter	C₇₉	#12064
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1367

C₂

socle

C₂²×C₇₉

C₈×D₄

prime radical

C₂×D₆₃₂

C₈

cosocle

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
1. #23765231	C₂	semidirect
3. #23761354	C₂	semidirect
4. #23761318	C₂	semidirect
5. C₇₉×#1367	C₂	semidirect
59. C₈×C₇₉	M₃₂	semidirect
79. #1367	D₇₉	semidirect
82. C₇₉	#12064	semidirect
137. C₈	#14579100	semidirect

H ≅ G/N

type

1. #23765231

C₂

semidirect

3. #23761354

C₂

semidirect

4. #23761318

C₂

semidirect

5. C₇₉×#1367

C₂

semidirect

59. C₈×C₇₉

M₃₂

semidirect

79. #1367

D₇₉

semidirect

82. C₇₉

#12064

semidirect

137. C₈

#14579100

semidirect

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

subgroup	index
#23765231	2
C₈×D₆₃₂	2
#23761354	2
#23761318	2
C₇₉×#1367	2
#12064	79

subgroup

index

C₈×D₆₃₂

C₇₉×#1367

Subgroups — #34200136

Conjugacy classes of subgroups

Complements

Maximal subgroups