Group #34188238 — subgroups

Subgroups — #34188238

Order 20224.

406 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 159 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	#23763572	C₂	1	1	yes	maximal, characteristic
2	#23761855	C₂	1	4	yes	maximal
3	C₂×#19098284	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579097	—	2	2	no	—
6	#19098587	C₂²	1	2	yes	—
7	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
8	Q₈⋊(C₇₉⋊C₈)	—	2	8	no	—
9	#19098284	—	2	2	no	—
10	#533×C₇₉	C₂²	1	2	yes	—
11	#500×C₇₉	C₄	1	2	yes	—
12	#499×C₇₉	—	2	2	no	—
13	#498×C₇₉	C₄	1	2	yes	—
14	#427×C₇₉	—	2	2	no	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#11929608	—	2	2	no	—
17	C₂²×C₇₉⋊C₈	—	2	2	no	—
18	#14579097	—	4	4	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
20	C₄×C₇₉⋊C₈	—	2	2	no	—
21	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
22	C₂²×C₈×C₇₉	D₄	1	2	yes	—
23	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
24	#125×C₇₉	—	2	2	no	—
25	C₂²⋊Q₈×C₇₉	—	2	4	no	—
26	C₄×Q₈×C₇₉	—	2	4	no	—
27	#117×C₇₉	—	2	2	no	—
28	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
29	#107×C₇₉	—	2	4	no	—
30	#106×C₇₉	—	2	4	no	—
31	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₉	D₄	1	2	yes	—
34	#11929608	—	4	4	no	—
35	C₂×C₇₉⋊C₈	—	2	4	no	—
36	C₂×C₇₉⋊C₈	—	4	4	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₂×Q₈×C₇₉	—	2	4	no	—
39	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	#41	1	2	yes	—
41	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
44	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
45	C₄⋊C₄×C₇₉	—	2	4	no	—
46	C₄⋊C₄×C₇₉	—	2	4	no	—
47	C₄⋊C₄×C₇₉	—	4	4	no	—
48	C₂²⋊C₄×C₇₉	—	4	2	no	—
49	C₄²×C₇₉	—	4	2	no	—
50	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₇₉	C₂×D₄	1	2	yes	—
53	C₂³×C₇₉	#124	1	1	yes	characteristic
54	Q₈×C₇₉	—	4	8	no	—
55	C₂×C₄×C₇₉	—	2	4	no	—
56	C₂×C₄×C₇₉	—	4	4	no	—
57	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₇₉	#121	1	2	yes	—
60	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
63	C₈×C₇₉	#103	1	4	yes	—
64	C₈×C₇₉	#103	1	4	yes	—
65	C₇₉⋊C₈	—	4	4	no	—
66	C₇₉⋊C₈	—	8	2	no	—
67	C₂²×C₇₉	—	2	2	no	—
68	C₂²×C₇₉	#390	1	1	yes	characteristic
69	C₂²×C₇₉	#487	1	2	yes	—
70	C₄×C₇₉	—	2	2	no	—
71	C₄×C₇₉	—	8	4	no	—
72	C₄×C₇₉	C₂×#103	1	2	yes	—
73	C₄×C₇₉	#444	1	4	yes	—
74	C₄×C₇₉	#492	1	1	yes	characteristic
75	C₄×C₇₉	#494	1	1	yes	characteristic
76	#9861	—	79	1	no	maximal, Sylow
77	C₂×C₇₉	—	2	2	no	—
78	C₂×C₇₉	#1956	1	1	yes	characteristic
79	C₂×C₇₉	#1975	1	2	yes	—
80	#3160	D₇₉	1	1	yes	characteristic
81	#1848	—	79	1	no	—
82	#1493	—	79	4	no	—
83	C₂×#322	—	79	1	no	—
84	C₇₉	#9861	1	1	yes	characteristic, complemented, Sylow
85	#533	C₂×D₇₉	1	2	yes	—
86	#500	Dic₇₉	1	2	yes	—
87	#499	—	2	2	no	—
88	#498	Dic₇₉	1	2	yes	—
89	C₂×#108	—	158	2	no	—
90	#427	—	2	2	no	—
91	#423	—	79	2	no	—
92	C₂×C₄×C₈	—	79	2	no	—
93	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
94	Q₈⋊C₈	—	158	8	no	—
95	#322	—	158	2	no	—
96	C₂×M₁₆	—	158	2	no	—
97	C₂²×C₈	—	158	2	no	—
98	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
99	C₄⋊Q₈	C₂×Dic₇₉	1	4	yes	—
100	#125	—	2	2	no	—
101	C₂²⋊Q₈	—	2	4	no	—
102	C₄×Q₈	—	2	4	no	—
103	#117	—	2	2	no	—
104	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
105	#108	—	316	4	no	—
106	#107	—	2	4	no	—
107	#106	—	2	4	no	—
108	C₄⋊C₈	—	158	4	no	—
109	C₄×C₈	—	79	4	no	—
110	C₄×C₈	—	158	2	no	—
111	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
114	C₂×Q₈	—	2	4	no	—
115	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
116	C₂²×C₄	#11929623	1	2	yes	—
117	M₁₆	—	316	4	no	—
118	C₂×C₈	—	316	4	no	—
119	C₂×C₈	—	316	2	no	—
120	C₂×C₈	—	158	4	no	—
121	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
122	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
123	C₂×C₈	#11929611	1	4	yes	—
124	C₂×C₈	#11929614	1	4	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
131	C₄²	D₄×D₇₉	1	2	yes	—
132	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
133	C₂³	#14579148	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
138	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
139	C₂×C₄	#14579132	1	2	yes	—
140	C₂×C₄	#14579157	1	1	yes	characteristic
141	C₂×C₄	C₂×#11929611	1	2	yes	—
142	C₂×C₄	C₂×#11929614	1	2	yes	—
143	C₈	—	316	4	no	—
144	C₈	—	632	2	no	—
145	C₈	#14579085	1	4	yes	—
146	C₈	#14579085	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#19098466	1	1	yes	characteristic
149	C₂²	#19098913	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂×#14579085	1	2	yes	—
153	C₄	#19098685	1	4	yes	—
154	C₄	#19098937	1	1	yes	characteristic
155	C₄	#19098942	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#23764203	1	1	yes	characteristic
158	C₂	#23764294	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763572

C₂

yes

maximal, characteristic

#23761855

C₂

yes

maximal

C₂×#19098284

C₂

yes

maximal, characteristic

C₇₉×#3160

C₂

yes

maximal, characteristic

C₂×#14579097

—

#19098587

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

Q₈⋊(C₇₉⋊C₈)

—

#19098284

—

#533×C₇₉

C₂²

yes

—

#500×C₇₉

C₄

yes

—

#499×C₇₉

—

#498×C₇₉

C₄

yes

—

#427×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₂²×C₇₉⋊C₈

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

#125×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#106×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#124

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×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

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₈×C₇₉

#103

yes

—

C₈×C₇₉

#103

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#487

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#103

yes

—

C₄×C₇₉

#444

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

#1956

yes

characteristic

C₂×C₇₉

#1975

yes

—

#3160

D₇₉

yes

—

—

—

C₇₉

#9861

yes

characteristic, complemented, Sylow

#533

C₂×D₇₉

yes

—

#500

Dic₇₉

yes

—

#499

—

#498

Dic₇₉

yes

—

C₂×#108

—

158

—

#427

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

Q₈⋊C₈

—

158

—

#322

—

158

—

C₂×M₁₆

—

158

—

C₂²×C₈

—

158

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₄⋊Q₈

C₂×Dic₇₉

yes

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄²

C₂²×D₇₉

yes

characteristic

105

#108

—

316

—

106

#107

—

107

#106

—

108

C₄⋊C₈

—

158

—

109

C₄×C₈

—

110

C₄×C₈

—

158

—

111

C₄×C₈

C₂×Dic₇₉

yes

characteristic

112

C₄×C₈

C₂×Dic₇₉

yes

characteristic

113

C₄×C₈

C₂²⋊D₇₉

yes

—

114

C₂×Q₈

—

115

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

116

C₂²×C₄

#11929623

yes

—

117

M₁₆

—

316

—

118

C₂×C₈

—

316

—

119

C₂×C₈

—

316

—

120

C₂×C₈

—

158

—

121

C₂×C₈

C₂²⋊Dic₇₉

yes

—

122

C₂×C₈

C₂²⋊Dic₇₉

yes

—

123

C₂×C₈

#11929611

yes

—

124

C₂×C₈

#11929614

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

C₂²×Dic₇₉

yes

characteristic

131

C₄²

D₄×D₇₉

yes

—

132

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

133

C₂³

#14579148

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

138

C₂×C₄

D₄×Dic₇₉

yes

—

139

C₂×C₄

#14579132

yes

—

140

C₂×C₄

#14579157

yes

characteristic

141

C₂×C₄

C₂×#11929611

yes

—

142

C₂×C₄

C₂×#11929614

yes

—

143

C₈

—

316

—

144

C₈

—

632

—

145

C₈

#14579085

yes

—

146

C₈

#14579085

yes

—

147

C₂²

—

148

C₂²

#19098466

yes

characteristic

149

C₂²

#19098913

yes

—

150

C₄

—

151

C₄

—

152

C₄

C₂×#14579085

yes

—

153

C₄

#19098685

yes

—

154

C₄

#19098937

yes

characteristic

155

C₄

#19098942

yes

characteristic

156

C₂

—

157

C₂

#23764203

yes

characteristic

158

C₂

#23764294

yes

—

	subgroup	G/H
center	C₂×C₄	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3160	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#124
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3160	D₇₉
hypocenter	C₇₉	#9861
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#14579157

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3160

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
84. C₇₉	#9861	semidirect

H ≅ G/N

type

84. C₇₉

#9861

semidirect

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

subgroup	index
#23763572	2
#23761855	2
C₂×#19098284	2
C₇₉×#3160	2
#9861	79

subgroup

index

C₇₉×#3160

Subgroups — #34188238

Conjugacy classes of subgroups

Complements

Maximal subgroups