Group #23764239 — subgroups

Subgroups — #23764239

Order 10112.

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

normal subgroups only

This group has 396 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×#14579158	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579077	C₂	1	4	yes	maximal, complemented
3	C₄×#11929608	C₂	1	1	yes	maximal, characteristic, complemented
4	#423×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₁₆	C₄	1	4	yes	—
6	C₂²×D₃₁₆	C₂²	1	2	yes	—
7	C₂×#11929608	C₂²	1	2	yes	—
8	#14579158	—	2	2	no	—
9	#14579158	—	2	1	no	—
10	#14579158	—	2	1	no	—
11	#14579077	—	2	8	no	—
12	#14579072	—	2	1	no	—
13	C₄×C₇₉⋊C₈	—	2	1	no	—
14	C₂×M₁₆×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₉	—	2	2	no	—
17	C₂³×D₇₉	C₂×C₄	1	4	yes	—
18	C₂×D₃₁₆	—	2	4	no	—
19	C₂×D₃₁₆	—	2	4	no	—
20	C₂×D₃₁₆	—	2	4	no	—
21	C₂×D₃₁₆	—	2	2	no	—
22	C₂×D₃₁₆	—	2	2	no	—
23	C₂×D₃₁₆	—	2	4	no	—
24	C₂×D₃₁₆	—	4	8	no	—
25	#11929608	—	2	2	no	—
26	#11929608	D₄	1	4	yes	complemented
27	C₂×C₇₉⋊C₈	—	2	2	no	—
28	C₂²×C₄×C₇₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
30	M₁₆×C₇₉	—	2	4	no	—
31	C₂×C₈×C₇₉	—	2	2	no	—
32	C₄²×C₇₉	D₄	1	2	yes	—
33	C₄²×C₇₉	D₄	1	1	yes	characteristic
34	C₄²×C₇₉	D₄	1	1	yes	characteristic
35	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₇₉	—	2	4	no	—
37	C₂²×D₇₉	—	2	8	no	—
38	C₂²×D₇₉	—	4	8	no	—
39	C₂×C₄×C₇₉	—	2	4	no	—
40	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	#41	1	2	yes	—
46	D₃₁₆	—	4	8	no	—
47	D₃₁₆	—	4	8	no	—
48	D₃₁₆	—	4	4	no	—
49	D₃₁₆	—	4	2	no	—
50	D₃₁₆	—	4	2	no	—
51	C₈×C₇₉	—	4	2	no	—
52	C₇₉⋊C₈	—	2	4	no	—
53	C₂²×C₇₉	—	2	2	no	—
54	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₇₉	C₄×D₄	1	2	yes	—
56	C₂×D₇₉	—	4	4	no	—
57	C₂×D₇₉	—	4	8	no	—
58	C₂×D₇₉	—	8	8	no	—
59	C₄×C₇₉	—	2	2	no	—
60	C₄×C₇₉	#100	1	4	yes	complemented
61	C₄×C₇₉	#121	1	2	yes	—
62	C₄×C₇₉	#124	1	1	yes	characteristic
63	C₄×C₇₉	#127	1	1	yes	characteristic
64	C₂×C₇₉	—	2	2	no	—
65	C₂×C₇₉	#390	1	1	yes	characteristic
66	C₂×C₇₉	C₂×#100	1	2	yes	—
67	D₇₉	—	8	4	no	—
68	#1965	—	79	1	no	maximal, Sylow
69	C₇₉	#1965	1	1	yes	characteristic, complemented, Sylow
70	C₂×#127	—	79	1	no	—
71	#423	D₇₉	1	1	yes	characteristic, complemented
72	C₂×#100	—	79	4	no	—
73	C₄×M₁₆	—	79	1	no	—
74	C₂²×D₄	—	79	2	no	—
75	C₂²×D₄	—	79	4	no	—
76	C₂×M₁₆	—	79	2	no	—
77	C₂×M₁₆	C₂×D₇₉	1	2	yes	—
78	#127	—	158	2	no	—
79	#127	—	158	1	no	—
80	#127	—	158	1	no	—
81	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	#100	—	158	8	no	—
84	#97	—	158	1	no	—
85	C₄×C₈	—	158	1	no	—
86	C₂⁴	—	79	4	no	—
87	C₂×D₄	—	316	8	no	—
88	C₂×D₄	—	158	4	no	—
89	C₂×D₄	—	158	4	no	—
90	C₂×D₄	—	158	4	no	—
91	C₂×D₄	—	158	2	no	—
92	C₂×D₄	—	158	2	no	—
93	C₂×D₄	—	158	4	no	—
94	C₂²×C₄	C₄×D₇₉	1	2	yes	—
95	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
96	M₁₆	—	2	4	no	—
97	M₁₆	—	79	4	no	—
98	M₁₆	—	158	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	158	2	no	—
101	C₄²	D₃₁₆	1	1	yes	characteristic
102	C₄²	D₃₁₆	1	1	yes	characteristic
103	C₄²	C₂²⋊D₇₉	1	2	yes	—
104	C₂³	—	316	8	no	—
105	C₂³	—	158	4	no	—
106	C₂³	—	158	8	no	—
107	C₂³	C₂×C₄×D₇₉	1	1	yes	characteristic
108	D₄	—	316	8	no	—
109	D₄	—	316	8	no	—
110	D₄	—	316	4	no	—
111	D₄	—	316	2	no	—
112	D₄	—	316	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
115	C₂×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
116	C₂×C₄	D₄×D₇₉	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
118	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
119	C₂×C₄	#11929625	1	2	yes	—
120	C₈	—	4	2	no	—
121	C₈	—	158	4	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	316	4	no	—
124	C₂²	—	316	8	no	—
125	C₂²	—	632	8	no	—
126	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
127	C₂²	#14579123	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	#14579077	1	4	yes	complemented
130	C₄	#14579136	1	2	yes	—
131	C₄	#14579150	1	1	yes	characteristic
132	C₄	#14579157	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	632	4	no	—
135	C₂	#19098464	1	1	yes	characteristic
136	C₂	C₂×#14579077	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579158

C₂

yes

maximal, characteristic

C₂×#14579077

C₂

yes

maximal, complemented

C₄×#11929608

C₂

yes

maximal, characteristic, complemented

#423×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₁₆

C₄

yes

—

C₂²×D₃₁₆

C₂²

yes

—

C₂×#11929608

C₂²

yes

—

—

—

—

—

—

C₄×C₇₉⋊C₈

—

C₂×M₁₆×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂³×D₇₉

C₂×C₄

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

#11929608

—

#11929608

D₄

yes

complemented

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₉

C₄×D₄

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#100

yes

complemented

C₄×C₇₉

#121

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

#390

yes

characteristic

C₂×C₇₉

C₂×#100

yes

—

D₇₉

—

#1965

—

maximal, Sylow

C₇₉

#1965

yes

characteristic, complemented, Sylow

C₂×#127

—

#423

D₇₉

yes

characteristic, complemented

C₂×#100

—

C₄×M₁₆

—

C₂²×D₄

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

—

#127

—

158

—

#127

—

158

—

#127

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

#100

—

158

—

#97

—

158

—

C₄×C₈

—

158

—

C₂⁴

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂²×C₄

C₄×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

158

—

C₂×C₈

—

100

C₂×C₈

—

158

—

101

C₄²

D₃₁₆

yes

characteristic

102

C₄²

D₃₁₆

yes

characteristic

103

C₄²

C₂²⋊D₇₉

yes

—

104

C₂³

—

316

—

105

C₂³

—

158

—

106

C₂³

—

158

—

107

C₂³

C₂×C₄×D₇₉

yes

characteristic

108

D₄

—

316

—

109

D₄

—

316

—

110

D₄

—

316

—

111

D₄

—

316

—

112

D₄

—

316

—

113

C₂×C₄

—

114

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

115

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

116

C₂×C₄

D₄×D₇₉

yes

—

117

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

118

C₂×C₄

C₂×D₃₁₆

yes

characteristic

119

C₂×C₄

#11929625

yes

—

120

C₈

—

121

C₈

—

158

—

122

C₂²

—

123

C₂²

—

316

—

124

C₂²

—

316

—

125

C₂²

—

632

—

126

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

127

C₂²

#14579123

yes

—

128

C₄

—

129

C₄

#14579077

yes

complemented

130

C₄

#14579136

yes

—

131

C₄

#14579150

yes

characteristic

132

C₄

#14579157

yes

characteristic

133

C₂

—

134

C₂

—

632

—

135

C₂

#19098464

yes

characteristic

136

C₂

C₂×#14579077

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂³×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#423×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂×#14579158	C₂
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#423	D₇₉
hypocenter	C₇₉	#1965
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₉)⋊C₄

commutator

C₂³×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#423×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

prime radical

C₂×#14579158

C₂

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#14579077	C₂	semidirect
3. C₄×#11929608	C₂	semidirect
4. #423×C₇₉	C₂	semidirect
26. #11929608	D₄	semidirect
60. C₄×C₇₉	#100	semidirect
69. C₇₉	#1965	semidirect
71. #423	D₇₉	semidirect
129. C₄	#14579077	semidirect

H ≅ G/N

type

2. C₂×#14579077

C₂

semidirect

3. C₄×#11929608

C₂

semidirect

4. #423×C₇₉

C₂

semidirect

26. #11929608

D₄

semidirect

60. C₄×C₇₉

#100

semidirect

69. C₇₉

#1965

semidirect

71. #423

D₇₉

semidirect

129. C₄

#14579077

semidirect

Subgroups — #23764239

Conjugacy classes of subgroups

Complements