Group #23764136 — subgroups

Subgroups — #23764136

Order 10112.

240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 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×#14579095	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579071	C₂	1	1	yes	maximal, characteristic
3	#19098459	C₂	1	2	yes	maximal
4	#19098325	C₂	1	2	yes	maximal
5	C₂×#106×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
8	C₂×Dic₇₉⋊C₄	—	2	2	no	—
9	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
10	#14579095	C₄	1	4	yes	complemented
11	#14579071	—	2	2	no	—
12	C₇₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
15	#106×C₇₉	—	2	2	no	—
16	C₂²×Dic₇₉	—	2	2	no	—
17	C₂²×Dic₇₉	D₄	1	2	yes	—
18	C₂×C₇₉⋊C₈	—	2	1	no	—
19	C₂×C₇₉⋊C₈	—	2	1	no	—
20	C₂×C₇₉⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₇₉	—	2	2	no	—
22	C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
23	Dic₇₉⋊C₄	—	4	4	no	—
24	C₄×Dic₇₉	—	2	1	no	—
25	C₄×Dic₇₉	—	2	1	no	—
26	C₂²×C₄×C₇₉	—	2	2	no	—
27	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₉	D₄	1	2	yes	—
30	C₂×C₈×C₇₉	Q₈	1	2	yes	—
31	C₄⋊C₄×C₇₉	—	2	2	no	—
32	C₄⋊C₄×C₇₉	—	2	2	no	—
33	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₉	—	4	2	no	—
35	C₂×C₄×C₇₉	—	4	2	no	—
36	C₂×C₄×C₇₉	—	4	2	no	—
37	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
43	C₂×Dic₇₉	—	2	4	no	—
44	C₂×Dic₇₉	—	2	2	no	—
45	C₂×Dic₇₉	—	2	2	no	—
46	C₂×Dic₇₉	—	2	2	no	—
47	C₂×Dic₇₉	—	4	2	no	—
48	C₂×Dic₇₉	—	4	2	no	—
49	C₈×C₇₉	C₄⋊C₄	1	4	yes	complemented
50	C₇₉⋊C₈	—	2	2	no	—
51	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
52	C₂²×C₇₉	#121	1	1	yes	characteristic
53	C₂²×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₇₉	#137	1	2	yes	—
56	C₄×C₇₉	—	8	2	no	—
57	C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
59	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₇₉	#125	1	1	yes	characteristic
61	Dic₇₉	—	4	4	no	—
62	Dic₇₉	—	4	2	no	—
63	C₂×C₇₉	#389	1	1	yes	characteristic
64	C₂×C₇₉	#441	1	1	yes	characteristic
65	C₂×C₇₉	#442	1	1	yes	characteristic
66	C₂×C₇₉	#468	1	2	yes	—
67	C₂×C₇₉	#501	1	2	yes	—
68	#1943	—	79	1	no	maximal, Sylow
69	C₇₉	#1943	1	1	yes	characteristic, complemented, Sylow
70	C₂×#107	—	79	1	no	—
71	C₂×#106	D₇₉	1	1	yes	characteristic
72	C₂×#97	—	79	1	no	—
73	#389	—	79	2	no	—
74	#340	—	79	2	no	—
75	C₂²×C₈	—	79	1	no	—
76	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	79	2	no	—
78	C₂×C₄⋊C₄	—	158	2	no	—
79	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
80	C₂×C₄²	—	79	1	no	—
81	#107	—	79	4	no	—
82	#106	—	2	2	no	—
83	#97	—	158	2	no	—
84	#95	—	158	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	79	2	no	—
87	C₂²×C₄	—	158	2	no	—
88	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
89	C₂×C₈	—	79	2	no	—
90	C₂×C₈	—	158	1	no	—
91	C₂×C₈	—	158	1	no	—
92	C₂×C₈	Dic₁₅₈	1	2	yes	—
93	C₂×C₈	C₄×D₇₉	1	2	yes	—
94	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	316	4	no	—
98	C₄⋊C₄	—	79	4	no	—
99	C₄⋊C₄	—	158	2	no	—
100	C₄²	—	158	1	no	—
101	C₄²	—	158	1	no	—
102	C₂³	D₄×D₇₉	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	316	2	no	—
107	C₂×C₄	—	316	2	no	—
108	C₂×C₄	—	158	4	no	—
109	C₂×C₄	—	158	2	no	—
110	C₂×C₄	—	158	2	no	—
111	C₂×C₄	—	158	2	no	—
112	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
116	C₂×C₄	#11929623	1	1	yes	characteristic
117	C₂×C₄	#11929625	1	1	yes	characteristic
118	C₈	—	158	2	no	—
119	C₈	Dic₇₉⋊C₄	1	4	yes	complemented
120	C₂²	#14579123	1	1	yes	characteristic
121	C₂²	#14579132	1	1	yes	characteristic
122	C₂²	#14579160	1	1	yes	characteristic
123	C₂²	#14579188	1	2	yes	—
124	C₂²	#14579190	1	2	yes	—
125	C₄	—	8	2	no	—
126	C₄	—	316	4	no	—
127	C₄	—	316	2	no	—
128	C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
129	C₄	#14579126	1	1	yes	characteristic
130	C₄	#14579137	1	1	yes	characteristic
131	C₄	#14579152	1	1	yes	characteristic
132	C₂	#19098459	1	1	yes	characteristic
133	C₂	#19098669	1	1	yes	characteristic
134	C₂	#19098674	1	1	yes	characteristic
135	C₂	#19098806	1	2	yes	—
136	C₂	#19098970	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579095

C₂

yes

maximal, characteristic

C₂×#14579071

C₂

yes

maximal, characteristic

#19098459

C₂

yes

maximal

#19098325

C₂

yes

maximal

C₂×#106×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579095

C₄

yes

complemented

#14579071

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

#106×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

Q₈

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

Q₈

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

C₄⋊C₄

yes

complemented

C₇₉⋊C₈

—

C₂²×C₇₉

C₄×D₄

yes

characteristic

C₂²×C₇₉

#121

yes

characteristic

C₂²×C₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₇₉

AGL(1,ℤ/8)

yes

—

C₂²×C₇₉

#137

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

C₄×Q₈

yes

characteristic

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

#125

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#501

yes

—

#1943

—

maximal, Sylow

C₇₉

#1943

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

D₇₉

yes

characteristic

C₂×#97

—

#389

—

#340

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

—

#107

—

#106

—

#97

—

158

—

#95

—

158

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₂×C₈

Dic₁₅₈

yes

—

C₂×C₈

C₄×D₇₉

yes

—

C₂×C₈

C₂²⋊D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

C₄⋊C₄

—

158

—

100

C₄²

—

158

—

101

C₄²

—

158

—

102

C₂³

D₄×D₇₉

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

316

—

107

C₂×C₄

—

316

—

108

C₂×C₄

—

158

—

109

C₂×C₄

—

158

—

110

C₂×C₄

—

158

—

111

C₂×C₄

—

158

—

112

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

114

C₂×C₄

Q₈×D₇₉

yes

characteristic

115

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

116

C₂×C₄

#11929623

yes

characteristic

117

C₂×C₄

#11929625

yes

characteristic

118

C₈

—

158

—

119

C₈

Dic₇₉⋊C₄

yes

complemented

120

C₂²

#14579123

yes

characteristic

121

C₂²

#14579132

yes

characteristic

122

C₂²

#14579160

yes

characteristic

123

C₂²

#14579188

yes

—

124

C₂²

#14579190

yes

—

125

C₄

—

126

C₄

—

316

—

127

C₄

—

316

—

128

C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

129

C₄

#14579126

yes

characteristic

130

C₄

#14579137

yes

characteristic

131

C₄

#14579152

yes

characteristic

132

C₂

#19098459

yes

characteristic

133

C₂

#19098669

yes

characteristic

134

C₂

#19098674

yes

characteristic

135

C₂

#19098806

yes

—

136

C₂

#19098970

yes

—

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×#106×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×#106	D₇₉
hypocenter	C₇₉	#1943
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×#106×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
10. #14579095	C₄	semidirect
49. C₈×C₇₉	C₄⋊C₄	semidirect
69. C₇₉	#1943	semidirect
119. C₈	Dic₇₉⋊C₄	semidirect

H ≅ G/N

type

10. #14579095

C₄

semidirect

49. C₈×C₇₉

C₄⋊C₄

semidirect

69. C₇₉

#1943

semidirect

119. C₈

Dic₇₉⋊C₄

semidirect

Subgroups — #23764136

Conjugacy classes of subgroups

Complements