Group #34174906 — subgroups

Subgroups — #34174906

Order 20224.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	#23761340	C₂	1	2	yes	maximal
2	C₇₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
4	C₂×C₄⋊C₈×C₇₉	—	2	2	no	—
5	C₂×C₄⋊C₈×C₇₉	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
7	C₄³×C₇₉	C₄	1	1	yes	characteristic
8	C₂×C₇₉⋊C₁₆	—	4	2	no	—
9	C₂²×C₈×C₇₉	—	2	1	no	—
10	C₂²×C₈×C₇₉	D₄	1	2	yes	—
11	C₂×C₄²×C₇₉	—	2	1	no	—
12	C₂×C₄²×C₇₉	—	2	1	no	—
13	C₂×C₄²×C₇₉	C₈	1	1	yes	characteristic
14	C₂×C₄²×C₇₉	C₈	1	1	yes	characteristic
15	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₄⋊C₈×C₇₉	—	4	1	no	—
17	C₄⋊C₈×C₇₉	—	4	1	no	—
18	C₄⋊C₈×C₇₉	—	4	4	no	—
19	C₇₉⋊C₁₆	—	8	2	no	—
20	C₂²×C₄×C₇₉	—	2	1	no	—
21	C₂²×C₄×C₇₉	—	2	1	no	—
22	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
25	C₂×C₈×C₇₉	—	2	2	no	—
26	C₂×C₈×C₇₉	—	4	2	no	—
27	C₂×C₈×C₇₉	—	4	2	no	—
28	C₂×C₈×C₇₉	—	4	1	no	—
29	C₄²×C₇₉	—	2	2	no	—
30	C₄²×C₇₉	—	2	2	no	—
31	C₄²×C₇₉	—	2	1	no	—
32	C₄²×C₇₉	—	2	1	no	—
33	C₄²×C₇₉	—	2	1	no	—
34	C₄²×C₇₉	—	4	2	no	—
35	C₄²×C₇₉	—	4	1	no	—
36	C₄²×C₇₉	C₁₆	1	2	yes	complemented
37	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₄×C₇₉	—	2	2	no	—
39	C₂×C₄×C₇₉	—	2	2	no	—
40	C₂×C₄×C₇₉	—	2	1	no	—
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	—	2	1	no	—
43	C₂×C₄×C₇₉	—	2	1	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	—	2	1	no	—
47	C₂×C₄×C₇₉	—	4	1	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	—	4	1	no	—
50	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₄×C₇₉	M₃₂	1	1	yes	characteristic
54	C₈×C₇₉	—	4	2	no	—
55	C₈×C₇₉	—	8	1	no	—
56	C₂²×C₇₉	—	2	1	no	—
57	C₂²×C₇₉	—	2	1	no	—
58	C₂²×C₇₉	#324	1	1	yes	characteristic
59	C₂²×C₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
60	C₂²×C₇₉	#349	1	1	yes	characteristic
61	C₄×C₇₉	—	2	4	no	—
62	C₄×C₇₉	—	2	1	no	—
63	C₄×C₇₉	—	2	1	no	—
64	C₄×C₇₉	—	2	1	no	—
65	C₄×C₇₉	—	4	2	no	—
66	C₄×C₇₉	—	4	1	no	—
67	C₄×C₇₉	#353	1	1	yes	characteristic
68	C₄×C₇₉	#356	1	1	yes	characteristic
69	#7063	—	79	1	no	maximal, Sylow
70	C₂×C₇₉	—	2	1	no	—
71	C₂×C₇₉	—	2	1	no	—
72	C₂×C₇₉	#1322	1	1	yes	characteristic
73	C₂×C₇₉	#1325	1	1	yes	characteristic
74	C₂×C₇₉	#1327	1	1	yes	characteristic
75	#1842	D₇₉	1	1	yes	characteristic
76	#1322	—	79	2	no	—
77	C₇₉	#7063	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄⋊C₈	—	2	2	no	—
79	C₂×C₄⋊C₈	Dic₇₉	1	1	yes	characteristic
80	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
81	C₄³	Dic₇₉	1	1	yes	characteristic
82	C₂²⋊C₁₆	—	158	2	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	—	2	1	no	—
87	C₂×C₄²	C₇₉⋊C₈	1	1	yes	characteristic
88	C₂×C₄²	C₇₉⋊C₈	1	1	yes	characteristic
89	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
90	C₂×C₁₆	—	316	2	no	—
91	C₄⋊C₈	—	4	1	no	—
92	C₄⋊C₈	—	4	1	no	—
93	C₄⋊C₈	—	4	4	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
98	C₂²×C₄	#11929608	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	2	1	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	4	1	no	—
109	C₄²	—	4	2	no	—
110	C₄²	C₇₉⋊C₁₆	1	2	yes	complemented
111	C₁₆	—	632	2	no	—
112	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	#14579076	1	1	yes	characteristic
126	C₂×C₄	#14579080	1	1	yes	characteristic
127	C₂×C₄	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
128	C₂×C₄	#14579101	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₈	—	8	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#19098289	1	1	yes	characteristic
134	C₂²	C₂²⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
135	C₂²	#19098345	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	—	4	1	no	—
142	C₄	#19098356	1	1	yes	characteristic
143	C₄	#19098362	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#23761340	1	1	yes	characteristic
147	C₂	#23761346	1	1	yes	characteristic
148	C₂	#23761351	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23761340

C₂

yes

maximal

C₇₉×#1842

C₂

yes

maximal, characteristic

C₂²⋊(C₇₉⋊C₁₆)

—

C₂×C₄⋊C₈×C₇₉

—

C₂×C₄⋊C₈×C₇₉

C₄

yes

characteristic

C₂×C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄³×C₇₉

C₄

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₈

yes

characteristic

C₂×C₄²×C₇₉

C₈

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₉

M₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₁₆

yes

complemented

C₂³×C₇₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₉

M₃₂

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#324

yes

characteristic

C₂²×C₇₉

C₂²⋊C₁₆

yes

characteristic

C₂²×C₇₉

#349

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

#1322

—

C₇₉

#7063

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

Dic₇₉

yes

characteristic

C₂×C₄⋊C₈

C₂×D₇₉

yes

characteristic

C₄³

Dic₇₉

yes

characteristic

C₂²⋊C₁₆

—

158

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₇₉⋊C₈

yes

characteristic

C₂×C₄²

C₇₉⋊C₈

yes

characteristic

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

—

316

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

C₂²×C₄

C₂×C₇₉⋊C₈

yes

characteristic

C₂²×C₄

#11929608

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₇₉⋊C₁₆

yes

complemented

111

C₁₆

—

632

—

112

C₂³

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#14579076

yes

characteristic

126

C₂×C₄

#14579080

yes

characteristic

127

C₂×C₄

C₂×C₇₉⋊C₁₆

yes

characteristic

128

C₂×C₄

#14579101

yes

characteristic

129

C₈

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

#19098289

yes

characteristic

134

C₂²

C₂²⋊(C₇₉⋊C₁₆)

yes

characteristic

135

C₂²

#19098345

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#19098356

yes

characteristic

143

C₄

#19098362

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#23761340

yes

characteristic

147

C₂

#23761346

yes

characteristic

148

C₂

#23761351

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579076
commutator	C₂×C₄×C₇₉	C₂×C₁₆
Frattini	C₂×C₄⋊C₈	C₂×D₇₉
Fitting	C₇₉×#1842	C₂
socle	C₂²×C₇₉	#324
prime radical	C₂³×C₇₉	C₂²⋊C₈
cosocle	C₂×C₄⋊C₈×C₇₉	C₂²
hypercenter	#1842	D₇₉
hypocenter	C₇₉	#7063
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1842

subgroup

G/H

center

C₂×C₄

#14579076

commutator

C₂×C₄×C₇₉

C₂×C₁₆

Frattini

C₂×C₄⋊C₈

C₂×D₇₉

Fitting

C₇₉×#1842

C₂

socle

C₂²×C₇₉

#324

prime radical

C₂³×C₇₉

C₂²⋊C₈

cosocle

C₂×C₄⋊C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1842

N	H ≅ G/N	type
36. C₄²×C₇₉	C₁₆	semidirect
77. C₇₉	#7063	semidirect
110. C₄²	C₇₉⋊C₁₆	semidirect

H ≅ G/N

type

36. C₄²×C₇₉

C₁₆

semidirect

77. C₇₉

#7063

semidirect

110. C₄²

C₇₉⋊C₁₆

semidirect

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

subgroup	index
#23761340	2
C₇₉×#1842	2
#7063	79

subgroup

index

#23761340

C₇₉×#1842

#7063

Subgroups — #34174906

Conjugacy classes of subgroups

Complements

Maximal subgroups