Group #34199466 — subgroups

Subgroups — #34199466

Order 20224.

274 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 79 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	#23765212	C₂	1	2	yes	maximal
2	#23761435	C₂	1	4	yes	maximal, complemented
3	C₇₉×#3139	C₂	1	1	yes	maximal, characteristic
4	#19098384	—	2	2	no	—
5	#19098345	—	2	2	no	—
6	#19098341	C₂²	1	4	yes	—
7	#533×C₇₉	C₄	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₇₉	C₄	1	1	yes	characteristic
9	#491×C₇₉	C₄	1	2	yes	—
10	#491×C₇₉	C₂²	1	2	yes	—
11	#487×C₇₉	—	2	2	no	—
12	#419×C₇₉	—	2	2	no	—
13	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	#14579101	—	2	4	no	—
15	C₂²×Q₈×C₇₉	—	2	1	no	—
16	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
17	C₄⋊Q₈×C₇₉	—	2	1	no	—
18	C₄⋊Q₈×C₇₉	—	2	1	no	—
19	C₄⋊Q₈×C₇₉	—	4	1	no	—
20	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
21	#125×C₇₉	C₂×C₄	1	2	yes	—
22	C₂²⋊Q₈×C₇₉	—	4	1	no	—
23	C₄×Q₈×C₇₉	—	4	1	no	—
24	#117×C₇₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
26	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
27	#103×C₇₉	—	2	4	no	—
28	#103×C₇₉	—	2	4	no	—
29	#97×C₇₉	C₂×C₄	1	1	yes	characteristic
30	#97×C₇₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₇₉	D₄	1	2	yes	—
32	C₇₉⋊C₁₆	—	4	4	no	—
33	C₂×Q₈×C₇₉	—	2	2	no	—
34	C₂×Q₈×C₇₉	—	4	1	no	—
35	C₂×Q₈×C₇₉	—	4	4	no	—
36	C₂×Q₈×C₇₉	—	4	1	no	—
37	C₂²×C₄×C₇₉	—	2	1	no	—
38	C₂²×C₄×C₇₉	—	4	1	no	—
39	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
40	M₁₆×C₇₉	—	4	2	no	—
41	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₇₉	—	2	2	no	—
44	C₄⋊C₄×C₇₉	—	4	2	no	—
45	C₄⋊C₄×C₇₉	—	4	1	no	—
46	C₄⋊C₄×C₇₉	—	4	2	no	—
47	C₄⋊C₄×C₇₉	—	4	2	no	—
48	C₂²⋊C₄×C₇₉	—	4	1	no	—
49	C₄²×C₇₉	—	2	1	no	—
50	C₄²×C₇₉	—	4	1	no	—
51	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂³×C₇₉	#100	1	1	yes	characteristic
54	Q₈×C₇₉	—	4	4	no	—
55	Q₈×C₇₉	—	8	2	no	—
56	Q₈×C₇₉	—	8	2	no	—
57	C₂×C₄×C₇₉	—	2	1	no	—
58	C₂×C₄×C₇₉	—	2	2	no	—
59	C₂×C₄×C₇₉	—	2	1	no	—
60	C₂×C₄×C₇₉	—	4	2	no	—
61	C₂×C₄×C₇₉	—	4	2	no	—
62	C₂×C₄×C₇₉	—	4	2	no	—
63	C₂×C₄×C₇₉	—	8	2	no	—
64	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₇₉	—	2	2	no	—
67	C₈×C₇₉	—	2	2	no	—
68	C₂²×C₇₉	—	2	1	no	—
69	C₂²×C₇₉	—	4	1	no	—
70	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
71	C₄×C₇₉	—	2	2	no	—
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	—	4	1	no	—
74	C₄×C₇₉	—	8	2	no	—
75	C₄×C₇₉	—	8	2	no	—
76	C₄×C₇₉	#421	1	2	yes	—
77	#11916	—	79	1	no	maximal, Sylow
78	C₂×C₇₉	—	2	1	no	—
79	C₂×C₇₉	—	4	1	no	—
80	C₂×C₇₉	#1533	1	1	yes	characteristic
81	#3139	D₇₉	1	1	yes	characteristic
82	#2165	—	79	2	no	—
83	#1356	—	79	4	no	—
84	C₇₉	#11916	1	1	yes	characteristic, complemented, Sylow
85	#533	Dic₇₉	1	1	yes	characteristic
86	C₂×C₄⋊Q₈	Dic₇₉	1	1	yes	characteristic
87	#491	Dic₇₉	1	2	yes	—
88	#491	C₂×D₇₉	1	2	yes	—
89	#487	—	2	2	no	—
90	#419	—	2	2	no	—
91	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
92	#364	—	158	2	no	—
93	#349	—	158	2	no	—
94	#347	—	79	4	no	—
95	C₂²×Q₈	—	2	1	no	—
96	C₂×M₁₆	C₂²⋊D₇₉	1	2	yes	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	2	1	no	—
99	C₄⋊Q₈	—	4	1	no	—
100	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
101	#125	C₂×Dic₇₉	1	2	yes	—
102	C₂²⋊Q₈	—	4	1	no	—
103	C₄×Q₈	—	4	1	no	—
104	#117	—	2	1	no	—
105	C₂×C₄⋊C₄	—	4	1	no	—
106	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
107	M₃₂	—	158	4	no	—
108	#103	—	2	4	no	—
109	#103	—	2	4	no	—
110	#97	C₂×Dic₇₉	1	1	yes	characteristic
111	#97	C₂²×D₇₉	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
113	C₂×Q₈	—	2	2	no	—
114	C₂×Q₈	—	4	1	no	—
115	C₂×Q₈	—	4	4	no	—
116	C₂×Q₈	—	4	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	4	1	no	—
119	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
122	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₂²⋊C₄	—	4	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	4	1	no	—
131	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
132	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
133	C₁₆	—	316	4	no	—
134	C₂³	#14579078	1	1	yes	characteristic
135	Q₈	—	4	4	no	—
136	Q₈	—	8	2	no	—
137	Q₈	—	8	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#14579078	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	C₂×#14579078	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	2	no	—
156	C₄	—	8	2	no	—
157	C₄	#19098580	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	4	1	no	—
160	C₂	#23762024	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765212

C₂

yes

maximal

#23761435

C₂

yes

maximal, complemented

C₇₉×#3139

C₂

yes

maximal, characteristic

#19098384

—

#19098345

—

#19098341

C₂²

yes

—

#533×C₇₉

C₄

yes

characteristic

C₂×C₄⋊Q₈×C₇₉

C₄

yes

characteristic

#491×C₇₉

C₄

yes

—

#491×C₇₉

C₂²

yes

—

#487×C₇₉

—

#419×C₇₉

—

C₄×M₁₆×C₇₉

C₂²

yes

characteristic

#14579101

—

C₂²×Q₈×C₇₉

—

C₂×M₁₆×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

#125×C₇₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

#103×C₇₉

—

#103×C₇₉

—

#97×C₇₉

C₂×C₄

yes

characteristic

#97×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₂³×C₇₉

#100

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#100

yes

characteristic

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#100

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#421

yes

—

#11916

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1533

yes

characteristic

#3139

D₇₉

yes

characteristic

#2165

—

#1356

—

C₇₉

#11916

yes

characteristic, complemented, Sylow

#533

Dic₇₉

yes

characteristic

C₂×C₄⋊Q₈

Dic₇₉

yes

characteristic

#491

Dic₇₉

yes

—

#491

C₂×D₇₉

yes

—

#487

—

#419

—

C₄×M₁₆

C₂×D₇₉

yes

characteristic

#364

—

158

—

#349

—

158

—

#347

—

C₂²×Q₈

—

C₂×M₁₆

C₂²⋊D₇₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

C₂×Dic₇₉

yes

—

101

#125

C₂×Dic₇₉

yes

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

#117

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

107

M₃₂

—

158

—

108

#103

—

109

#103

—

110

#97

C₂×Dic₇₉

yes

characteristic

111

#97

C₂²×D₇₉

yes

characteristic

112

C₄×C₈

C₂²⋊D₇₉

yes

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

120

M₁₆

—

121

C₂×C₈

C₂²⋊Dic₇₉

yes

—

122

C₂×C₈

C₂×C₂²⋊D₇₉

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²⋊Dic₇₉

yes

characteristic

132

C₄²

C₂²×Dic₇₉

yes

characteristic

133

C₁₆

—

316

—

134

C₂³

#14579078

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#14579078

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

147

C₈

—

148

C₈

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂×#14579078

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#19098580

yes

—

158

C₂

—

159

C₂

—

160

C₂

#23762024

yes

characteristic

	subgroup	G/H
center	C₂	#23762024
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	#97	C₂²×D₇₉
Fitting	C₇₉×#3139	C₂
socle	C₂×C₇₉	#1533
prime radical	C₂³×C₇₉	#100
cosocle	#97×C₇₉	C₂³
hypercenter	#3139	D₇₉
hypocenter	C₇₉	#11916
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

subgroup

G/H

center

C₂

#23762024

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

#97

C₂²×D₇₉

Fitting

C₇₉×#3139

C₂

C₂×C₇₉

C₂³×C₇₉

#97×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

N	H ≅ G/N	type
2. #23761435	C₂	semidirect
84. C₇₉	#11916	semidirect

H ≅ G/N

type

2. #23761435

C₂

semidirect

84. C₇₉

#11916

semidirect

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

subgroup	index
#23765212	2
#23761435	2
C₇₉×#3139	2
#11916	79

subgroup

index

#23765212

#23761435

C₇₉×#3139

#11916

Subgroups — #34199466

Conjugacy classes of subgroups

Complements

Maximal subgroups