Group #34200121 — subgroups

Subgroups — #34200121

Order 20224.

200 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 87 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	#23765308	C₂	1	2	yes	maximal
2	C₈×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
3	#23761464	C₂	1	1	yes	maximal, characteristic
4	#23761380	C₂	1	2	yes	maximal
5	C₇₉×#1312	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₅₈	C₂²	1	2	yes	—
7	C₄×Dic₃₁₆	C₄	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
9	Dic₇₉⋊C₁₆	—	2	2	no	—
10	#19098338	—	2	2	no	—
11	#19098316	C₄	1	1	yes	characteristic
12	#19098295	C₄	1	2	yes	—
13	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₈²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₁₆	C₈	1	1	yes	characteristic
17	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
18	C₂×C₇₉⋊C₁₆	—	2	2	no	—
19	#14579095	C₈	1	1	yes	characteristic
20	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
21	Dic₇₉⋊C₈	—	2	2	no	—
22	#14579086	C₈	1	2	yes	—
23	C₈×Dic₇₉	—	2	2	no	—
24	C₂×C₁₆×C₇₉	—	2	1	no	—
25	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
26	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₅₈	C₂×C₈	1	2	yes	—
30	Dic₃₁₆	—	2	2	no	—
31	C₂×C₇₉⋊C₈	—	2	2	no	—
32	C₄⋊Dic₇₉	C₂×C₈	1	2	yes	—
33	Dic₇₉⋊C₄	—	2	2	no	—
34	C₄×Dic₇₉	—	2	2	no	—
35	C₇₉⋊C₁₆	—	4	2	no	—
36	C₂×C₈×C₇₉	—	2	1	no	—
37	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
41	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₇₉	—	2	2	no	—
43	C₁₆×C₇₉	—	4	1	no	—
44	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
47	C₂×Dic₇₉	—	2	2	no	—
48	C₂×Dic₇₉	—	2	2	no	—
49	Dic₁₅₈	—	2	2	no	—
50	Dic₁₅₈	—	2	2	no	—
51	C₈×C₇₉	—	2	1	no	—
52	C₈×C₇₉	—	2	1	no	—
53	C₈×C₇₉	—	4	1	no	—
54	C₈×C₇₉	M₃₂	1	2	yes	—
55	C₈×C₇₉	#137	1	2	yes	—
56	C₇₉⋊C₈	—	4	2	no	—
57	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
58	C₄×C₇₉	—	2	1	no	—
59	C₄×C₇₉	#441	1	1	yes	characteristic
60	C₄×C₇₉	#444	1	1	yes	characteristic
61	C₄×C₇₉	C₂×M₃₂	1	1	yes	characteristic
62	C₄×C₇₉	#504	1	1	yes	characteristic
63	Dic₇₉	—	4	2	no	—
64	Dic₇₉	—	4	2	no	—
65	#12061	—	79	1	no	maximal, Sylow
66	C₂×C₇₉	#1579	1	1	yes	characteristic
67	C₂×C₇₉	#2169	1	1	yes	characteristic
68	C₂×C₇₉	#2171	1	1	yes	characteristic
69	#2183	—	79	2	no	—
70	C₈×Q₁₆	—	79	1	no	—
71	#1367	—	79	1	no	—
72	Q₈⋊C₁₆	—	79	2	no	—
73	#1312	D₇₉	1	1	yes	characteristic
74	C₇₉	#12061	1	1	yes	characteristic, complemented, Sylow
75	C₈×Q₈	—	79	2	no	—
76	C₄×Q₁₆	—	79	1	no	—
77	C₄⋊C₁₆	—	79	2	no	—
78	C₄⋊C₁₆	—	158	2	no	—
79	#346	—	158	2	no	—
80	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
82	#335	—	79	1	no	—
83	Q₈⋊C₈	—	79	2	no	—
84	C₈²	C₂×D₇₉	1	1	yes	characteristic
85	C₂×Q₁₆	—	79	1	no	—
86	C₄×Q₈	—	79	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	158	2	no	—
89	C₂×C₁₆	D₃₁₆	1	2	yes	—
90	#107	—	79	1	no	—
91	C₄⋊C₈	—	79	2	no	—
92	C₄⋊C₈	—	158	2	no	—
93	#103	—	79	2	no	—
94	C₄×C₈	—	158	2	no	—
95	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
96	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
98	C₂×Q₈	—	79	2	no	—
99	Q₁₆	—	158	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	158	2	no	—
102	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
103	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
104	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
105	C₂×C₈	#11929624	1	1	yes	characteristic
106	C₄⋊C₄	—	79	2	no	—
107	C₄⋊C₄	—	158	2	no	—
108	C₄²	—	158	2	no	—
109	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	—	316	2	no	—
113	Q₈	—	158	2	no	—
114	Q₈	—	158	2	no	—
115	C₂×C₄	—	158	2	no	—
116	C₂×C₄	—	158	2	no	—
117	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
119	C₂×C₄	#14579167	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	316	2	no	—
124	C₈	#14579100	1	2	yes	—
125	C₈	#14579194	1	2	yes	—
126	C₂²	C₈×D₃₁₆	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	316	2	no	—
129	C₄	—	316	2	no	—
130	C₄	#19098671	1	1	yes	characteristic
131	C₄	#19098686	1	1	yes	characteristic
132	C₄	C₂×#14579100	1	1	yes	characteristic
133	C₄	#19098979	1	1	yes	characteristic
134	C₂	#23762220	1	1	yes	characteristic
135	C₂	#23765233	1	1	yes	characteristic
136	C₂	#23765243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765308

C₂

yes

maximal

C₈×Dic₃₁₆

C₂

yes

maximal, characteristic

#23761464

C₂

yes

maximal, characteristic

#23761380

C₂

yes

maximal

C₇₉×#1312

C₂

yes

maximal, characteristic

C₈×Dic₁₅₈

C₂²

yes

—

C₄×Dic₃₁₆

C₄

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

C₂²

yes

—

Dic₇₉⋊C₁₆

—

#19098338

—

#19098316

C₄

yes

characteristic

#19098295

C₄

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

C₈

yes

characteristic

C₄×Dic₁₅₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₁₆

—

#14579095

C₈

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂×C₄

yes

—

Dic₇₉⋊C₈

—

#14579086

C₈

yes

—

C₈×Dic₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

C₂×C₈

yes

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

C₂×C₈

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

#131

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

M₃₂

yes

—

C₈×C₇₉

#137

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₈×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×M₃₂

yes

characteristic

C₄×C₇₉

#504

yes

characteristic

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

D₇₉

yes

characteristic

C₇₉

#12061

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

158

—

#346

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

158

—

C₂×C₁₆

D₃₁₆

yes

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

158

—

#103

—

C₄×C₈

—

158

—

C₄×C₈

C₄×D₇₉

yes

characteristic

C₄×C₈

C₄×D₇₉

yes

characteristic

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

Q₁₆

—

158

—

100

C₂×C₈

—

101

C₂×C₈

—

158

—

102

C₂×C₈

C₈×D₇₉

yes

characteristic

103

C₂×C₈

C₈×D₇₉

yes

characteristic

104

C₂×C₈

C₂×D₃₁₆

yes

characteristic

105

C₂×C₈

#11929624

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

158

—

108

C₄²

—

158

—

109

C₄²

C₂×C₄×D₇₉

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

316

—

113

Q₈

—

158

—

114

Q₈

—

158

—

115

C₂×C₄

—

158

—

116

C₂×C₄

—

158

—

117

C₂×C₄

C₄×D₃₁₆

yes

characteristic

118

C₂×C₄

C₂×C₈×D₇₉

yes

characteristic

119

C₂×C₄

#14579167

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

316

—

124

C₈

#14579100

yes

—

125

C₈

#14579194

yes

—

126

C₂²

C₈×D₃₁₆

yes

characteristic

127

C₄

—

128

C₄

—

316

—

129

C₄

—

316

—

130

C₄

#19098671

yes

characteristic

131

C₄

#19098686

yes

characteristic

132

C₄

C₂×#14579100

yes

characteristic

133

C₄

#19098979

yes

characteristic

134

C₂

#23762220

yes

characteristic

135

C₂

#23765233

yes

characteristic

136

C₂

#23765243

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1312	C₂
socle, prime radical	C₂²×C₇₉	C₈×D₄
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1312	D₇₉
hypocenter	C₇₉	#12061
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

subgroup

G/H

center

C₂×C₈

C₂×D₃₁₆

commutator

C₂×C₄×C₇₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1312

C₂

socle, prime radical

C₂²×C₇₉

C₈×D₄

cosocle

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

N	H ≅ G/N	type
74. C₇₉	#12061	semidirect

H ≅ G/N

type

74. C₇₉

#12061

semidirect

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

subgroup	index
#23765308	2
C₈×Dic₃₁₆	2
#23761464	2
#23761380	2
C₇₉×#1312	2
#12061	79

subgroup

index

C₈×Dic₃₁₆

C₇₉×#1312

Subgroups — #34200121

Conjugacy classes of subgroups

Complements

Maximal subgroups