Group #34203273 — subgroups

Subgroups — #34203273

Order 20224.

358 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
2	#23763086	C₂	1	1	yes	maximal, characteristic
3	#23761477	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14579072	C₄	1	1	yes	characteristic
7	C₂×#14579071	C₄	1	2	yes	—
8	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
9	#19098409	C₄	1	1	yes	characteristic
10	#19098339	C₄	1	4	yes	complemented
11	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₉	C₄	1	4	yes	complemented
14	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
16	C₂×C₇₉⋊C₁₆	—	2	4	no	—
17	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
18	#14579072	C₈	1	4	yes	—
19	#14579071	—	2	4	no	—
20	C₈×Dic₇₉	—	2	4	no	—
21	C₇₉⋊#95	C₈	1	4	yes	—
22	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	—	2	4	no	—
26	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
29	C₂²×Dic₇₉	C₂×C₈	1	4	yes	—
30	C₂×C₇₉⋊C₈	—	2	2	no	—
31	C₂×C₇₉⋊C₈	—	2	2	no	—
32	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
33	C₄×Dic₇₉	—	2	2	no	—
34	C₄×Dic₇₉	—	2	2	no	—
35	C₇₉⋊C₁₆	—	2	4	no	—
36	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₄²	1	2	yes	—
39	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
40	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	#41	1	4	yes	—
43	C₄²×C₇₉	C₄²	1	2	yes	—
44	C₄²×C₇₉	C₂×C₈	1	2	yes	—
45	C₁₆×C₇₉	—	2	4	no	—
46	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	—	2	4	no	—
48	C₂×C₄×C₇₉	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#117	1	2	yes	—
51	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
53	C₂×Dic₇₉	—	2	4	no	—
54	C₂×Dic₇₉	—	2	8	no	—
55	C₈×C₇₉	#97	1	4	yes	—
56	C₈×C₇₉	#117	1	2	yes	—
57	C₈×C₇₉	#117	1	2	yes	—
58	C₇₉⋊C₈	—	2	4	no	—
59	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
60	C₂²×C₇₉	#431	1	2	yes	—
61	C₂²×C₇₉	#504	1	4	yes	—
62	C₄×C₇₉	—	2	4	no	—
63	C₄×C₇₉	#376	1	1	yes	characteristic
64	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
65	C₄×C₇₉	#431	1	2	yes	—
66	Dic₇₉	—	4	4	no	—
67	#12592	—	79	1	no	maximal, Sylow
68	C₂×C₇₉	#1744	1	1	yes	characteristic
69	C₂×C₇₉	#2108	1	2	yes	—
70	C₂×C₇₉	#2164	1	4	yes	—
71	C₂×#346	—	79	1	no	—
72	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
73	#1726	—	79	1	no	—
74	#1374	—	79	4	no	—
75	C₇₉	#12592	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₁₆	—	79	2	no	—
77	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
78	C₂×#97	—	79	1	no	—
79	C₂×#97	—	79	2	no	—
80	C₂×C₄×C₈	—	79	2	no	—
81	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
82	#376	—	79	1	no	—
83	#346	—	79	4	no	—
84	C₄×C₁₆	Dic₇₉	1	4	yes	complemented
85	C₂²×C₈	—	79	2	no	—
86	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
88	C₂×C₄²	—	79	2	no	—
89	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
90	C₂×C₁₆	—	2	4	no	—
91	C₂×C₁₆	—	79	4	no	—
92	C₂×C₁₆	—	158	4	no	—
93	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
94	#97	—	79	4	no	—
95	#97	—	158	4	no	—
96	C₄×C₈	—	158	4	no	—
97	C₄×C₈	C₄×D₇₉	1	2	yes	—
98	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
99	#95	—	79	4	no	—
100	C₂²×C₄	—	79	4	no	—
101	C₂²×C₄	C₈×D₇₉	1	2	yes	—
102	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
103	C₂×C₈	—	79	4	no	—
104	C₂×C₈	—	158	2	no	—
105	C₂×C₈	—	158	2	no	—
106	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
107	C₂×C₈	#11929607	1	4	yes	—
108	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
109	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
110	C₂×C₈	#11929624	1	4	yes	—
111	C₄²	—	158	2	no	—
112	C₄²	—	158	2	no	—
113	C₄²	C₄×Dic₇₉	1	2	yes	—
114	C₄²	C₈×D₇₉	1	2	yes	—
115	C₁₆	—	2	4	no	—
116	C₁₆	—	158	4	no	—
117	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	158	8	no	—
120	C₂×C₄	—	158	4	no	—
121	C₂×C₄	C₈×Dic₇₉	1	4	yes	—
122	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
123	C₂×C₄	#14579118	1	2	yes	—
124	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
125	C₂×C₄	C₂×#11929607	1	2	yes	—
126	C₈	—	158	4	no	—
127	C₈	#14579071	1	4	yes	—
128	C₈	#14579118	1	2	yes	—
129	C₈	#14579120	1	2	yes	—
130	C₂²	C₂×C₈×Dic₇₉	1	1	yes	characteristic
131	C₂²	#19098614	1	2	yes	—
132	C₂²	#19098979	1	4	yes	—
133	C₄	—	2	4	no	—
134	C₄	—	316	4	no	—
135	C₄	#19098409	1	1	yes	characteristic
136	C₄	C₂×#14579071	1	1	yes	characteristic
137	C₄	#19098614	1	2	yes	—
138	C₂	#23763145	1	1	yes	characteristic
139	C₂	#23764996	1	2	yes	—
140	C₂	#23765205	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098339

C₂

yes

maximal, characteristic

#23763086

C₂

yes

maximal, characteristic

#23761477

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

—

C₂×#14579072

C₄

yes

characteristic

C₂×#14579071

C₄

yes

—

C₂×C₈×Dic₇₉

C₂²

yes

—

#19098409

C₄

yes

characteristic

#19098339

C₄

yes

complemented

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₄

yes

complemented

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

#14579072

C₈

yes

—

#14579071

—

C₈×Dic₇₉

—

C₇₉⋊#95

C₈

yes

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₂²×Dic₇₉

C₂×C₈

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₈

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×C₈

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₄²

yes

—

C₂×C₈×C₇₉

M₁₆

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

C₄²

yes

—

C₄²×C₇₉

C₂×C₈

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×C₈

yes

—

C₂×C₄×C₇₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

C₂×M₁₆

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

#97

yes

—

C₈×C₇₉

#117

yes

—

C₈×C₇₉

#117

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×C₄×C₈

yes

characteristic

C₂²×C₇₉

#431

yes

—

C₂²×C₇₉

#504

yes

—

C₄×C₇₉

—

C₄×C₇₉

#376

yes

characteristic

C₄×C₇₉

C₂×#97

yes

characteristic

C₄×C₇₉

#431

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2164

yes

—

C₂×#346

—

C₂×C₄×C₁₆

D₇₉

yes

characteristic

#1726

—

#1374

—

C₇₉

#12592

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₉

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#376

—

#346

—

C₄×C₁₆

Dic₇₉

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄×D₇₉

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₇₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

158

—

C₂×C₁₆

C₂×Dic₇₉

yes

—

#97

—

#97

—

158

—

C₄×C₈

—

158

—

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

#95

—

100

C₂²×C₄

—

101

C₂²×C₄

C₈×D₇₉

yes

—

102

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

158

—

105

C₂×C₈

—

158

—

106

C₂×C₈

C₄×Dic₇₉

yes

—

107

C₂×C₈

#11929607

yes

—

108

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

109

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

110

C₂×C₈

#11929624

yes

—

111

C₄²

—

158

—

112

C₄²

—

158

—

113

C₄²

C₄×Dic₇₉

yes

—

114

C₄²

C₈×D₇₉

yes

—

115

C₁₆

—

116

C₁₆

—

158

—

117

C₂³

C₂×C₈×D₇₉

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

158

—

120

C₂×C₄

—

158

—

121

C₂×C₄

C₈×Dic₇₉

yes

—

122

C₂×C₄

C₂×C₄×Dic₇₉

yes

characteristic

123

C₂×C₄

#14579118

yes

—

124

C₂×C₄

C₂×C₈×D₇₉

yes

characteristic

125

C₂×C₄

C₂×#11929607

yes

—

126

C₈

—

158

—

127

C₈

#14579071

yes

—

128

C₈

#14579118

yes

—

129

C₈

#14579120

yes

—

130

C₂²

C₂×C₈×Dic₇₉

yes

characteristic

131

C₂²

#19098614

yes

—

132

C₂²

#19098979

yes

—

133

C₄

—

134

C₄

—

316

—

135

C₄

#19098409

yes

characteristic

136

C₄

C₂×#14579071

yes

characteristic

137

C₄

#19098614

yes

—

138

C₂

#23763145

yes

characteristic

139

C₂

#23764996

yes

—

140

C₂

#23765205

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#12592
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₉

commutator

C₂²×C₇₉

C₂×C₄×C₈

Fitting

C₂×C₄×C₁₆×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂²×C₈

cosocle

C₂²×C₈×C₇₉

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
10. #19098339	C₄	semidirect
13. C₄×C₁₆×C₇₉	C₄	semidirect
75. C₇₉	#12592	semidirect
84. C₄×C₁₆	Dic₇₉	semidirect

H ≅ G/N

type

10. #19098339

C₄

semidirect

13. C₄×C₁₆×C₇₉

C₄

semidirect

75. C₇₉

#12592

semidirect

84. C₄×C₁₆

Dic₇₉

semidirect

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

subgroup	index
C₂×#19098339	2
#23763086	2
#23761477	2
C₂×C₄×C₁₆×C₇₉	2
#12592	79

subgroup

index

C₂×C₄×C₁₆×C₇₉

Subgroups — #34203273

Conjugacy classes of subgroups

Complements

Maximal subgroups