Group #34187774 — subgroups

Subgroups — #34187774

Order 20224.

558 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 127 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	#23763538	C₂	1	1	yes	maximal, characteristic, complemented
2	#23762068	C₂	1	4	yes	maximal, complemented
3	#23761692	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#493×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19098616	C₂²	1	2	yes	—
6	#19098587	C₂²	1	2	yes	—
7	#19098328	—	2	2	no	—
8	#19098307	—	2	8	no	—
9	#19098285	—	2	2	no	—
10	C₂²×D₈×C₇₉	—	2	2	no	—
11	C₂×#127×C₇₉	C₂²	1	2	yes	—
12	#493×C₇₉	—	2	2	no	—
13	#493×C₇₉	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#11929608	—	2	2	no	—
16	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
17	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
18	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
19	C₄×C₇₉⋊C₈	—	2	2	no	—
20	C₂²×D₄×C₇₉	—	2	2	no	—
21	C₂²×D₄×C₇₉	—	2	4	no	—
22	C₂×D₈×C₇₉	—	2	8	no	—
23	C₂×D₈×C₇₉	—	4	8	no	—
24	C₂²×C₈×C₇₉	D₄	1	2	yes	—
25	#127×C₇₉	—	2	2	no	—
26	#127×C₇₉	—	2	4	no	—
27	#127×C₇₉	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₉	D₄	1	2	yes	—
31	#11929608	—	4	4	no	—
32	C₂×C₇₉⋊C₈	—	2	4	no	—
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₂⁴×C₇₉	—	4	2	no	—
35	C₂×D₄×C₇₉	—	2	4	no	—
36	C₂×D₄×C₇₉	—	2	8	no	—
37	C₂×D₄×C₇₉	—	4	4	no	—
38	C₂×D₄×C₇₉	—	4	2	no	—
39	C₂×D₄×C₇₉	—	4	8	no	—
40	C₂×D₄×C₇₉	—	8	8	no	—
41	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	#41	1	2	yes	—
43	D₈×C₇₉	—	4	16	no	—
44	C₂×C₈×C₇₉	—	2	4	no	—
45	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
46	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₇₉	C₂×D₄	1	2	yes	—
49	C₂³×C₇₉	—	4	4	no	—
50	C₂³×C₇₉	—	8	8	no	—
51	C₂³×C₇₉	—	8	4	no	—
52	C₂³×C₇₉	#124	1	1	yes	characteristic
53	D₄×C₇₉	—	4	8	no	—
54	D₄×C₇₉	—	8	8	no	—
55	D₄×C₇₉	—	8	8	no	—
56	D₄×C₇₉	—	8	4	no	—
57	C₂×C₄×C₇₉	—	2	4	no	—
58	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₇₉	#121	1	2	yes	—
61	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
62	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
63	C₈×C₇₉	—	2	4	no	—
64	C₇₉⋊C₈	—	4	4	no	—
65	C₇₉⋊C₈	—	8	2	no	—
66	C₂²×C₇₉	—	2	2	no	—
67	C₂²×C₇₉	—	8	4	no	—
68	C₂²×C₇₉	—	8	8	no	—
69	C₂²×C₇₉	—	16	8	no	—
70	C₂²×C₇₉	#390	1	1	yes	characteristic
71	C₂²×C₇₉	#490	1	2	yes	—
72	C₄×C₇₉	—	2	2	no	—
73	C₄×C₇₉	#418	1	2	yes	—
74	C₄×C₇₉	#472	1	4	yes	—
75	C₄×C₇₉	#496	1	2	yes	—
76	#9780	—	79	1	no	maximal, Sylow
77	C₂×C₇₉	—	2	2	no	—
78	C₂×C₇₉	—	16	4	no	—
79	C₂×C₇₉	#1961	1	1	yes	characteristic
80	C₂×C₇₉	#1978	1	2	yes	—
81	C₂×#493	D₇₉	1	1	yes	characteristic
82	#1837	—	79	1	no	—
83	#1545	—	79	4	no	—
84	#1452	—	79	1	no	—
85	C₇₉	#9780	1	1	yes	characteristic, complemented, Sylow
86	C₂²×D₈	—	2	2	no	—
87	C₂×#127	C₂×D₇₉	1	2	yes	—
88	#493	—	2	2	no	—
89	#493	Dic₇₉	1	4	yes	—
90	#431	—	79	2	no	—
91	#423	—	79	2	no	—
92	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
93	#341	—	158	2	no	—
94	#331	—	158	8	no	—
95	#322	—	158	2	no	—
96	C₂²×D₄	—	2	2	no	—
97	C₂²×D₄	—	2	4	no	—
98	C₂×D₈	—	2	8	no	—
99	C₂×D₈	—	4	8	no	—
100	C₂×M₁₆	—	158	2	no	—
101	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
102	#127	—	2	4	no	—
103	#127	—	2	2	no	—
104	#127	C₂×Dic₇₉	1	4	yes	—
105	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
106	C₄⋊C₈	—	79	4	no	—
107	C₄⋊C₈	—	158	4	no	—
108	C₂²⋊C₈	—	158	2	no	—
109	C₄×C₈	—	158	2	no	—
110	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
111	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
112	C₂⁴	—	4	2	no	—
113	C₂×D₄	—	2	4	no	—
114	C₂×D₄	—	2	8	no	—
115	C₂×D₄	—	4	4	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	—	4	8	no	—
118	C₂×D₄	—	8	8	no	—
119	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
120	C₂²×C₄	#11929623	1	2	yes	—
121	D₈	—	4	16	no	—
122	M₁₆	—	316	4	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	316	2	no	—
125	C₂×C₈	—	158	4	no	—
126	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
127	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
128	C₄²	D₄×D₇₉	1	2	yes	—
129	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂³	—	4	4	no	—
131	C₂³	—	8	8	no	—
132	C₂³	—	8	4	no	—
133	C₂³	#14579148	1	1	yes	characteristic
134	D₄	—	4	8	no	—
135	D₄	—	8	8	no	—
136	D₄	—	8	4	no	—
137	D₄	—	8	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
140	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
141	C₂×C₄	#14579132	1	2	yes	—
142	C₂×C₄	#14579157	1	1	yes	characteristic
143	C₂×C₄	#14579187	1	4	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	316	4	no	—
146	C₈	—	632	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	8	4	no	—
149	C₂²	—	8	8	no	—
150	C₂²	—	16	8	no	—
151	C₂²	#19098466	1	1	yes	characteristic
152	C₂²	#19098927	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	#19098568	1	2	yes	—
155	C₄	#19098830	1	4	yes	—
156	C₄	#19098950	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	16	4	no	—
159	C₂	#23764225	1	1	yes	characteristic
160	C₂	#23764311	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763538

C₂

yes

maximal, characteristic, complemented

#23762068

C₂

yes

maximal, complemented

#23761692

C₂

yes

maximal, characteristic, complemented

C₂×#493×C₇₉

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

#19098587

C₂²

yes

—

#19098328

—

#19098307

—

#19098285

—

C₂²×D₈×C₇₉

—

C₂×#127×C₇₉

C₂²

yes

—

#493×C₇₉

—

#493×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂²×D₄×C₇₉

—

C₂²×D₄×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

#127×C₇₉

—

#127×C₇₉

—

#127×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

D₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#124

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#490

yes

—

C₄×C₇₉

—

C₄×C₇₉

#418

yes

—

C₄×C₇₉

#472

yes

—

C₄×C₇₉

#496

yes

—

#9780

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1961

yes

characteristic

C₂×C₇₉

#1978

yes

—

C₂×#493

D₇₉

yes

—

—

—

C₇₉

#9780

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×#127

C₂×D₇₉

yes

—

#493

—

#493

Dic₇₉

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#341

—

158

—

#331

—

158

—

#322

—

158

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

100

C₂×M₁₆

—

158

—

101

C₂²×C₈

C₂²⋊D₇₉

yes

—

102

#127

—

103

#127

—

104

#127

C₂×Dic₇₉

yes

—

105

C₂×C₄²

C₂²×D₇₉

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

158

—

108

C₂²⋊C₈

—

158

—

109

C₄×C₈

—

158

—

110

C₄×C₈

C₂×Dic₇₉

yes

—

111

C₄×C₈

C₂²⋊D₇₉

yes

—

112

C₂⁴

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

120

C₂²×C₄

#11929623

yes

—

121

D₈

—

122

M₁₆

—

316

—

123

C₂×C₈

—

124

C₂×C₈

—

316

—

125

C₂×C₈

—

158

—

126

C₂×C₈

C₂²⋊Dic₇₉

yes

—

127

C₄²

C₂²×Dic₇₉

yes

characteristic

128

C₄²

D₄×D₇₉

yes

—

129

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

#14579148

yes

characteristic

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

140

C₂×C₄

D₄×Dic₇₉

yes

—

141

C₂×C₄

#14579132

yes

—

142

C₂×C₄

#14579157

yes

characteristic

143

C₂×C₄

#14579187

yes

—

144

C₈

—

145

C₈

—

316

—

146

C₈

—

632

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#19098466

yes

characteristic

152

C₂²

#19098927

yes

—

153

C₄

—

154

C₄

#19098568

yes

—

155

C₄

#19098830

yes

—

156

C₄

#19098950

yes

—

157

C₂

—

158

C₂

—

159

C₂

#23764225

yes

characteristic

160

C₂

#23764311

yes

—

	subgroup	G/H
center	C₂²	#19098466
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting, prime radical	C₂×#493×C₇₉	C₂
socle	C₂²×C₇₉	#390
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#493	D₇₉
hypocenter	C₇₉	#9780
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂²

#19098466

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting, prime radical

C₂×#493×C₇₉

C₂

socle

C₂²×C₇₉

#390

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #23763538	C₂	semidirect
2. #23762068	C₂	semidirect
3. #23761692	C₂	semidirect
85. C₇₉	#9780	semidirect

H ≅ G/N

type

1. #23763538

C₂

semidirect

2. #23762068

C₂

semidirect

3. #23761692

C₂

semidirect

85. C₇₉

#9780

semidirect

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

subgroup	index
#23763538	2
#23762068	2
#23761692	2
C₂×#493×C₇₉	2
#9780	79

subgroup

index

C₂×#493×C₇₉

Subgroups — #34187774

Conjugacy classes of subgroups

Complements

Maximal subgroups