Group #34196952 — subgroups

Subgroups — #34196952

Order 20224.

686 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 235 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	#23771990	C₂	1	3	yes	maximal
2	#23763098	C₂	1	3	yes	maximal
3	C₂×#338×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
5	C₂×#14579120	C₂²	1	3	yes	—
6	#19098605	—	2	12	no	—
7	#19098601	—	2	6	no	—
8	#19098317	—	2	6	no	—
9	C₂²×M₁₆×C₇₉	C₂²	1	3	yes	—
10	#338×C₇₉	C₄	1	4	yes	complemented
11	C₂²×C₇₉⋊C₈	—	2	3	no	—
12	C₂²×C₇₉⋊C₈	—	2	3	no	—
13	C₂²×C₇₉⋊C₈	Q₈	1	2	yes	—
14	#14579120	—	2	6	no	—
15	#14579120	—	2	6	no	—
16	C₂×C₄⋊Dic₇₉	D₄	1	6	yes	—
17	C₂×C₂²⋊Dic₇₉	—	2	3	no	—
18	C₂×C₄×Dic₇₉	—	2	3	no	—
19	#14579095	—	2	12	no	—
20	#14579093	—	2	12	no	—
21	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₇₉	—	2	6	no	—
23	C₂×M₁₆×C₇₉	—	2	6	no	—
24	C₂×M₁₆×C₇₉	C₂×C₄	1	6	yes	—
25	C₂×M₁₆×C₇₉	Q₈	1	6	yes	—
26	C₂²×C₈×C₇₉	D₄	1	6	yes	—
27	C₂²×Dic₇₉	—	2	6	no	—
28	C₂×C₇₉⋊C₈	—	2	6	no	—
29	C₂×C₇₉⋊C₈	—	4	3	no	—
30	C₂×C₇₉⋊C₈	—	4	1	no	—
31	C₄⋊Dic₇₉	—	2	12	no	—
32	C₂²⋊Dic₇₉	—	4	6	no	—
33	C₄×Dic₇₉	—	4	3	no	—
34	C₄×Dic₇₉	—	4	3	no	—
35	C₂⁴×C₇₉	#41	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
38	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₇₉	C₂×Q₈	1	3	yes	—
40	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	#41	1	3	yes	—
42	M₁₆×C₇₉	—	2	12	no	—
43	M₁₆×C₇₉	—	2	12	no	—
44	C₂×C₈×C₇₉	—	2	6	no	—
45	C₂×C₈×C₇₉	—	2	6	no	—
46	C₂×C₈×C₇₉	C₄⋊C₄	1	12	yes	—
47	C₂³×C₇₉	—	4	2	no	—
48	C₂³×C₇₉	C₄×Q₈	1	1	yes	characteristic
49	C₂³×C₇₉	#121	1	3	yes	—
50	C₂³×C₇₉	#125	1	3	yes	—
51	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
52	C₂×C₄×C₇₉	C₄×D₄	1	3	yes	—
53	C₂×C₄×C₇₉	#120	1	3	yes	—
54	C₂×C₄×C₇₉	C₂²⋊Q₈	1	6	yes	—
55	C₂×C₄×C₇₉	C₂²⋊Q₈	1	3	yes	—
56	C₂×C₄×C₇₉	#123	1	3	yes	—
57	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₄⋊Q₈	1	3	yes	—
59	C₂×C₄×C₇₉	C₄⋊Q₈	1	3	yes	—
60	C₂×Dic₇₉	—	4	12	no	—
61	C₂×Dic₇₉	—	4	6	no	—
62	C₈×C₇₉	—	2	12	no	—
63	C₇₉⋊C₈	—	4	2	no	—
64	C₂²×C₇₉	—	2	6	no	—
65	C₂²×C₇₉	—	4	1	no	—
66	C₂²×C₇₉	—	4	3	no	—
67	C₂²×C₇₉	#389	1	3	yes	—
68	C₂²×C₇₉	#392	1	1	yes	characteristic
69	C₂²×C₇₉	#398	1	3	yes	—
70	C₄×C₇₉	#384	1	1	yes	characteristic
71	C₄×C₇₉	#385	1	3	yes	—
72	C₄×C₇₉	#395	1	3	yes	—
73	C₄×C₇₉	#399	1	1	yes	characteristic
74	Dic₇₉	—	8	6	no	—
75	#11337	—	79	1	no	maximal, Sylow
76	C₂×C₇₉	—	2	3	no	—
77	C₂×C₇₉	—	2	3	no	—
78	C₂×C₇₉	#1444	1	1	yes	characteristic
79	C₂×C₇₉	#2017	1	2	yes	—
80	#2911	—	79	3	no	—
81	C₂×#338	D₇₉	1	1	yes	characteristic
82	#1729	—	79	3	no	—
83	C₇₉	#11337	1	1	yes	characteristic, complemented, Sylow
84	C₂²×M₁₆	C₂×D₇₉	1	3	yes	—
85	C₂³×C₈	—	79	1	no	—
86	C₂×#117	—	79	3	no	—
87	#428	—	158	12	no	—
88	#427	—	158	6	no	—
89	#338	Dic₇₉	1	4	yes	complemented
90	#336	—	158	6	no	—
91	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
92	C₂×M₁₆	—	2	6	no	—
93	C₂×M₁₆	—	2	6	no	—
94	C₂×M₁₆	Dic₁₅₈	1	6	yes	—
95	C₂×M₁₆	C₂×Dic₇₉	1	6	yes	—
96	C₂²×C₈	—	79	2	no	—
97	C₂²×C₈	—	158	3	no	—
98	C₂²×C₈	—	158	3	no	—
99	C₂²×C₈	D₃₁₆	1	6	yes	—
100	#117	—	158	6	no	—
101	#117	—	158	6	no	—
102	C₂×C₄⋊C₄	—	79	6	no	—
103	C₂×C₂²⋊C₄	—	158	3	no	—
104	C₂×C₄²	—	158	3	no	—
105	#107	—	158	12	no	—
106	#106	—	158	12	no	—
107	C₂⁴	#11929625	1	1	yes	characteristic
108	C₂²×C₄	—	158	6	no	—
109	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
110	C₂²×C₄	D₄×D₇₉	1	3	yes	—
111	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
112	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₅₈	1	3	yes	—
114	C₂²×C₄	#11929623	1	3	yes	—
115	M₁₆	—	2	12	no	—
116	M₁₆	—	2	12	no	—
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	2	6	no	—
119	C₂×C₈	—	316	3	no	—
120	C₂×C₈	—	316	1	no	—
121	C₂×C₈	—	158	6	no	—
122	C₂×C₈	C₄⋊Dic₇₉	1	12	yes	—
123	C₄⋊C₄	—	158	12	no	—
124	C₂²⋊C₄	—	316	6	no	—
125	C₄²	—	316	3	no	—
126	C₄²	—	316	3	no	—
127	C₂³	—	4	2	no	—
128	C₂³	Q₈×Dic₇₉	1	1	yes	characteristic
129	C₂³	#14579136	1	3	yes	—
130	C₂³	#14579152	1	3	yes	—
131	C₂×C₄	—	316	12	no	—
132	C₂×C₄	—	316	6	no	—
133	C₂×C₄	C₂×C₄⋊Dic₇₉	1	3	yes	—
134	C₂×C₄	D₄×Dic₇₉	1	3	yes	—
135	C₂×C₄	#14579131	1	3	yes	—
136	C₂×C₄	#14579138	1	6	yes	—
137	C₂×C₄	#14579141	1	3	yes	—
138	C₂×C₄	#14579145	1	3	yes	—
139	C₂×C₄	#14579158	1	1	yes	characteristic
140	C₂×C₄	#14579160	1	3	yes	—
141	C₂×C₄	#14579161	1	3	yes	—
142	C₈	—	2	12	no	—
143	C₈	—	316	2	no	—
144	C₂²	—	2	6	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	3	no	—
147	C₂²	#19098462	1	3	yes	—
148	C₂²	#19098473	1	1	yes	characteristic
149	C₂²	#19098495	1	3	yes	—
150	C₄	—	632	6	no	—
151	C₄	#19098433	1	1	yes	characteristic
152	C₄	#19098438	1	3	yes	—
153	C₄	#19098484	1	3	yes	—
154	C₄	#19098500	1	1	yes	characteristic
155	C₂	—	2	3	no	—
156	C₂	—	2	3	no	—
157	C₂	#23761665	1	1	yes	characteristic
158	C₂	#23764505	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23771990

C₂

yes

maximal

#23763098

C₂

yes

maximal

C₂×#338×C₇₉

C₂

yes

maximal, characteristic

C₂³×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×#14579120

C₂²

yes

—

#19098605

—

#19098601

—

#19098317

—

C₂²×M₁₆×C₇₉

C₂²

yes

—

#338×C₇₉

C₄

yes

complemented

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

Q₈

yes

—

#14579120

—

#14579120

—

C₂×C₄⋊Dic₇₉

D₄

yes

—

C₂×C₂²⋊Dic₇₉

—

C₂×C₄×Dic₇₉

—

#14579095

—

#14579093

—

C₂³×C₄×C₇₉

C₂³

yes

characteristic

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂×M₁₆×C₇₉

Q₈

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂⁴×C₇₉

#41

yes

characteristic

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×Q₈

yes

—

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

C₄×Q₈

yes

characteristic

C₂³×C₇₉

#121

yes

—

C₂³×C₇₉

#125

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#120

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#389

yes

—

C₂²×C₇₉

#392

yes

characteristic

C₂²×C₇₉

#398

yes

—

C₄×C₇₉

#384

yes

characteristic

C₄×C₇₉

#385

yes

—

C₄×C₇₉

#395

yes

—

C₄×C₇₉

#399

yes

characteristic

Dic₇₉

—

#11337

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1444

yes

characteristic

C₂×C₇₉

#2017

yes

—

#2911

—

C₂×#338

D₇₉

yes

characteristic

#1729

—

C₇₉

#11337

yes

characteristic, complemented, Sylow

C₂²×M₁₆

C₂×D₇₉

yes

—

C₂³×C₈

—

C₂×#117

—

#428

—

158

—

#427

—

158

—

#338

Dic₇₉

yes

complemented

#336

—

158

—

C₂³×C₄

C₂²×D₇₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

Dic₁₅₈

yes

—

C₂×M₁₆

C₂×Dic₇₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

158

—

C₂²×C₈

—

158

—

C₂²×C₈

D₃₁₆

yes

—

100

#117

—

158

—

101

#117

—

158

—

102

C₂×C₄⋊C₄

—

103

C₂×C₂²⋊C₄

—

158

—

104

C₂×C₄²

—

158

—

105

#107

—

158

—

106

#106

—

158

—

107

C₂⁴

#11929625

yes

characteristic

108

C₂²×C₄

—

158

—

109

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

110

C₂²×C₄

D₄×D₇₉

yes

—

111

C₂²×C₄

C₂×D₃₁₆

yes

—

112

C₂²×C₄

Q₈×D₇₉

yes

characteristic

113

C₂²×C₄

C₂×Dic₁₅₈

yes

—

114

C₂²×C₄

#11929623

yes

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

316

—

120

C₂×C₈

—

316

—

121

C₂×C₈

—

158

—

122

C₂×C₈

C₄⋊Dic₇₉

yes

—

123

C₄⋊C₄

—

158

—

124

C₂²⋊C₄

—

316

—

125

C₄²

—

316

—

126

C₄²

—

316

—

127

C₂³

—

128

C₂³

Q₈×Dic₇₉

yes

characteristic

129

C₂³

#14579136

yes

—

130

C₂³

#14579152

yes

—

131

C₂×C₄

—

316

—

132

C₂×C₄

—

316

—

133

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

—

134

C₂×C₄

D₄×Dic₇₉

yes

—

135

C₂×C₄

#14579131

yes

—

136

C₂×C₄

#14579138

yes

—

137

C₂×C₄

#14579141

yes

—

138

C₂×C₄

#14579145

yes

—

139

C₂×C₄

#14579158

yes

characteristic

140

C₂×C₄

#14579160

yes

—

141

C₂×C₄

#14579161

yes

—

142

C₈

—

143

C₈

—

316

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#19098462

yes

—

148

C₂²

#19098473

yes

characteristic

149

C₂²

#19098495

yes

—

150

C₄

—

632

—

151

C₄

#19098433

yes

characteristic

152

C₄

#19098438

yes

—

153

C₄

#19098484

yes

—

154

C₄

#19098500

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#23761665

yes

characteristic

158

C₂

#23764505

yes

—

	subgroup	G/H
center	C₂×C₄	#14579158
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×#338×C₇₉	C₂
socle	C₂²×C₇₉	#392
prime radical	C₂⁴×C₇₉	#41
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#338	D₇₉
hypocenter	C₇₉	#11337
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

subgroup

G/H

center

C₂×C₄

#14579158

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₇₉

Fitting

C₂×#338×C₇₉

C₂

C₂²×C₇₉

C₂⁴×C₇₉

C₂³×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

N	H ≅ G/N	type
10. #338×C₇₉	C₄	semidirect
83. C₇₉	#11337	semidirect
89. #338	Dic₇₉	semidirect

H ≅ G/N

type

10. #338×C₇₉

C₄

semidirect

83. C₇₉

#11337

semidirect

89. #338

Dic₇₉

semidirect

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

subgroup	index
#23771990	2
#23763098	2
C₂×#338×C₇₉	2
#11337	79

subgroup

index

#23771990

#23763098

C₂×#338×C₇₉

#11337

Subgroups — #34196952

Conjugacy classes of subgroups

Complements

Maximal subgroups