Group #23764388 — subgroups

Subgroups — #23764388

Order 10112.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14579161	C₂	1	1	yes	maximal, characteristic
2	C₄×#11929608	C₂	1	1	yes	maximal, characteristic
3	#19098331	C₂	1	4	yes	maximal
4	#423×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₅₈	C₄	1	2	yes	—
6	C₂×#11929608	C₂²	1	2	yes	—
7	#14579161	—	2	1	no	—
8	#14579161	—	2	1	no	—
9	#14579161	—	2	1	no	—
10	#14579161	—	2	1	no	—
11	C₂×C₄⋊Dic₇₉	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
13	#14579072	—	2	1	no	—
14	C₄×C₇₉⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₉	—	2	1	no	—
18	C₄⋊C₈×C₇₉	—	2	1	no	—
19	C₂×Dic₁₅₈	—	2	2	no	—
20	C₂×Dic₁₅₈	—	2	2	no	—
21	C₂×Dic₁₅₈	—	4	4	no	—
22	C₂²×Dic₇₉	C₂×C₄	1	4	yes	—
23	#11929608	—	2	2	no	—
24	#11929608	Q₈	1	4	yes	complemented
25	C₂×C₇₉⋊C₈	—	2	2	no	—
26	C₄⋊Dic₇₉	—	2	2	no	—
27	C₄⋊Dic₇₉	—	2	2	no	—
28	C₄⋊Dic₇₉	—	2	2	no	—
29	C₄⋊Dic₇₉	—	2	2	no	—
30	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₇₉	—	2	4	no	—
34	C₂×C₈×C₇₉	—	2	2	no	—
35	C₄²×C₇₉	D₄	1	1	yes	characteristic
36	C₄²×C₇₉	D₄	1	1	yes	characteristic
37	C₄²×C₇₉	D₄	1	1	yes	characteristic
38	C₄²×C₇₉	D₄	1	1	yes	characteristic
39	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₉	—	2	2	no	—
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₇₉	#41	1	2	yes	—
50	C₂×Dic₇₉	—	2	4	no	—
51	C₂×Dic₇₉	—	4	4	no	—
52	Dic₁₅₈	—	4	4	no	—
53	Dic₁₅₈	—	4	4	no	—
54	C₈×C₇₉	—	4	2	no	—
55	C₇₉⋊C₈	—	2	4	no	—
56	C₂²×C₇₉	—	2	1	no	—
57	C₂²×C₇₉	—	2	1	no	—
58	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₇₉	C₄×Q₈	1	2	yes	—
60	C₄×C₇₉	—	2	1	no	—
61	C₄×C₇₉	—	2	1	no	—
62	C₄×C₇₉	#100	1	2	yes	—
63	C₄×C₇₉	#101	1	2	yes	—
64	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₇₉	#124	1	1	yes	characteristic
66	C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₇₉	—	4	4	no	—
68	C₂×C₇₉	—	2	2	no	—
69	C₂×C₇₉	#391	1	1	yes	characteristic
70	C₂×C₇₉	C₂×#100	1	1	yes	characteristic
71	C₂×C₇₉	C₂×#101	1	1	yes	characteristic
72	#1997	—	79	1	no	maximal, Sylow
73	C₇₉	#1997	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	79	1	no	—
75	#423	D₇₉	1	1	yes	characteristic
76	C₄×M₁₆	—	79	1	no	—
77	#343	—	79	4	no	—
78	C₂²×Q₈	—	79	2	no	—
79	C₂×M₁₆	—	79	2	no	—
80	C₂×M₁₆	C₂×D₇₉	1	2	yes	—
81	C₄⋊Q₈	—	158	1	no	—
82	C₄⋊Q₈	—	158	1	no	—
83	C₄⋊Q₈	—	158	1	no	—
84	C₄⋊Q₈	—	158	1	no	—
85	C₂×C₄⋊C₄	—	79	2	no	—
86	C₂×C₄⋊C₄	—	79	2	no	—
87	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	158	1	no	—
91	C₄×C₈	—	158	1	no	—
92	C₂×Q₈	—	316	4	no	—
93	C₂×Q₈	—	158	2	no	—
94	C₂×Q₈	—	158	2	no	—
95	C₂²×C₄	—	79	4	no	—
96	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	79	4	no	—
101	M₁₆	—	158	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	158	2	no	—
104	C₄⋊C₄	—	158	2	no	—
105	C₄⋊C₄	—	158	2	no	—
106	C₄⋊C₄	—	158	2	no	—
107	C₄⋊C₄	—	158	2	no	—
108	C₄²	D₃₁₆	1	1	yes	characteristic
109	C₄²	D₃₁₆	1	1	yes	characteristic
110	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
111	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₇₉	1	1	yes	characteristic
113	Q₈	—	316	4	no	—
114	Q₈	—	316	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	316	4	no	—
118	C₂×C₄	—	158	4	no	—
119	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
124	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₇₉	1	2	yes	—
126	C₂×C₄	#11929623	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₈	—	158	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
132	C₂²	#14579126	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	316	4	no	—
136	C₄	#14579077	1	2	yes	—
137	C₄	#14579079	1	2	yes	—
138	C₄	#14579141	1	2	yes	—
139	C₄	#14579148	1	1	yes	characteristic
140	C₄	#14579159	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#19098468	1	1	yes	characteristic
143	C₂	C₂×#14579077	1	1	yes	characteristic
144	C₂	C₂×#14579079	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic

C₄×#11929608

C₂

yes

maximal, characteristic

#19098331

C₂

yes

maximal

#423×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₄

yes

—

C₂×#11929608

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₇₉

C₄

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₂×M₁₆×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

C₂×C₄

yes

—

#11929608

—

#11929608

Q₈

yes

complemented

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₉

C₄×Q₈

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#100

yes

—

C₄×C₇₉

#101

yes

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

C₄×C₇₉

#124

yes

characteristic

C₄×C₇₉

C₄⋊Q₈

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#423

D₇₉

yes

characteristic

C₄×M₁₆

—

#343

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

158

—

C₄×C₈

—

158

—

C₂×Q₈

—

316

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₂²×D₇₉

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

158

—

102

C₂×C₈

—

103

C₂×C₈

—

158

—

104

C₄⋊C₄

—

158

—

105

C₄⋊C₄

—

158

—

106

C₄⋊C₄

—

158

—

107

C₄⋊C₄

—

158

—

108

C₄²

D₃₁₆

yes

characteristic

109

C₄²

D₃₁₆

yes

characteristic

110

C₄²

C₂²⋊D₇₉

yes

characteristic

111

C₄²

C₂²⋊D₇₉

yes

characteristic

112

C₂³

C₂×C₄×D₇₉

yes

characteristic

113

Q₈

—

316

—

114

Q₈

—

316

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

316

—

118

C₂×C₄

—

158

—

119

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

124

C₂×C₄

C₂×D₃₁₆

yes

characteristic

125

C₂×C₄

Q₈×D₇₉

yes

—

126

C₂×C₄

#11929623

yes

—

127

C₈

—

128

C₈

—

158

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

132

C₂²

#14579126

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

316

—

136

C₄

#14579077

yes

—

137

C₄

#14579079

yes

—

138

C₄

#14579141

yes

—

139

C₄

#14579148

yes

characteristic

140

C₄

#14579159

yes

characteristic

141

C₂

—

142

C₂

#19098468

yes

characteristic

143

C₂

C₂×#14579077

yes

characteristic

144

C₂

C₂×#14579079

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₉)⋊C₄
commutator, prime radical	C₂³×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#423×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#423	D₇₉
hypocenter	C₇₉	#1997
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₉)⋊C₄

commutator, prime radical

C₂³×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#423×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
24. #11929608	Q₈	semidirect
73. C₇₉	#1997	semidirect

H ≅ G/N

type

24. #11929608

Q₈

semidirect

73. C₇₉

#1997

semidirect

Subgroups — #23764388

Conjugacy classes of subgroups

Complements