Group #23761241 — subgroups

Subgroups — #23761241

Order 10112.

208 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₉⋊C₈	—	2	1	no	—
5	C₂²×C₇₉⋊C₈	—	2	1	no	—
6	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	complemented
8	C₄×C₇₉⋊C₈	—	2	1	no	—
9	C₄×C₇₉⋊C₈	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
11	C₂×C₄²×C₇₉	—	2	1	no	—
12	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
13	#95×C₇₉	—	2	1	no	—
14	C₂×C₇₉⋊C₈	—	2	4	no	—
15	C₂×C₇₉⋊C₈	—	2	2	no	—
16	C₂×C₇₉⋊C₈	—	2	2	no	—
17	C₂×C₇₉⋊C₈	—	4	1	no	—
18	C₂×C₇₉⋊C₈	—	4	1	no	—
19	C₂²×C₄×C₇₉	—	2	2	no	—
20	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
23	C₄⋊C₄×C₇₉	—	2	2	no	—
24	C₄⋊C₄×C₇₉	C₈	1	4	yes	complemented
25	C₄²×C₇₉	—	4	1	no	—
26	C₄²×C₇₉	—	4	1	no	—
27	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₇₉	D₄	1	1	yes	characteristic
30	C₄²×C₇₉	Q₈	1	1	yes	characteristic
31	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₉	—	2	2	no	—
33	C₂×C₄×C₇₉	—	2	4	no	—
34	C₂×C₄×C₇₉	—	4	2	no	—
35	C₂×C₄×C₇₉	—	4	2	no	—
36	C₂×C₄×C₇₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₇₉	M₁₆	1	2	yes	—
44	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
45	C₂×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
46	C₂×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₉	Q₁₆	1	1	yes	characteristic
48	C₇₉⋊C₈	—	2	4	no	—
49	C₇₉⋊C₈	—	4	2	no	—
50	C₂²×C₇₉	#95	1	1	yes	characteristic
51	C₂²×C₇₉	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₇₉	#102	1	1	yes	characteristic
53	C₂²×C₇₉	#103	1	1	yes	characteristic
54	C₂²×C₇₉	#104	1	2	yes	—
55	C₄×C₇₉	—	2	1	no	—
56	C₄×C₇₉	—	2	1	no	—
57	C₄×C₇₉	—	4	4	no	—
58	C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
59	C₄×C₇₉	#97	1	1	yes	characteristic
60	C₄×C₇₉	#102	1	1	yes	characteristic
61	C₄×C₇₉	#103	1	1	yes	characteristic
62	C₄×C₇₉	C₄⋊C₈	1	2	yes	—
63	C₄×C₇₉	#106	1	1	yes	characteristic
64	C₄×C₇₉	#107	1	1	yes	characteristic
65	C₂×C₇₉	D₄⋊C₈	1	2	yes	—
66	C₂×C₇₉	Q₈⋊C₈	1	2	yes	—
67	C₂×C₇₉	#336	1	1	yes	characteristic
68	C₂×C₇₉	#339	1	1	yes	characteristic
69	C₂×C₇₉	#340	1	1	yes	characteristic
70	#1279	—	79	1	no	maximal, Sylow
71	C₇₉	#1279	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄⋊C₈	—	79	1	no	—
73	C₂×C₄×C₈	—	79	1	no	—
74	C₄×C₄⋊C₄	D₇₉	1	1	yes	characteristic
75	C₂²×C₈	—	158	1	no	—
76	C₂²×C₈	—	158	1	no	—
77	C₂×C₄⋊C₄	Dic₇₉	1	2	yes	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
80	C₄⋊C₈	—	79	2	no	—
81	C₄⋊C₈	—	158	1	no	—
82	C₄×C₈	—	79	2	no	—
83	C₄×C₈	—	158	1	no	—
84	#95	—	2	1	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₂×Dic₇₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
89	C₂×C₈	—	316	1	no	—
90	C₂×C₈	—	316	1	no	—
91	C₂×C₈	—	158	2	no	—
92	C₂×C₈	—	158	4	no	—
93	C₂×C₈	—	158	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₇₉⋊C₈	1	4	yes	complemented
96	C₄²	—	4	1	no	—
97	C₄²	—	4	1	no	—
98	C₄²	Dic₁₅₈	1	1	yes	characteristic
99	C₄²	C₄×D₇₉	1	1	yes	characteristic
100	C₄²	C₄×D₇₉	1	1	yes	characteristic
101	C₄²	D₃₁₆	1	1	yes	characteristic
102	C₂³	C₂²⋊Dic₇₉	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₄×Dic₇₉	1	1	yes	characteristic
108	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
109	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
110	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
111	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
112	C₂×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
113	C₂×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
114	C₂×C₄	#11929608	1	2	yes	—
115	C₂×C₄	#11929609	1	1	yes	characteristic
116	C₂×C₄	#11929611	1	1	yes	characteristic
117	C₂×C₄	#11929612	1	1	yes	characteristic
118	C₂×C₄	#11929614	1	1	yes	characteristic
119	C₈	—	316	2	no	—
120	C₈	—	158	4	no	—
121	C₂²	C₇₉⋊#95	1	1	yes	characteristic
122	C₂²	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
123	C₂²	#14579082	1	1	yes	characteristic
124	C₂²	#14579085	1	1	yes	characteristic
125	C₂²	#14579088	1	2	yes	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	4	4	no	—
129	C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
130	C₄	#14579072	1	1	yes	characteristic
131	C₄	#14579081	1	1	yes	characteristic
132	C₄	#14579084	1	1	yes	characteristic
133	C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
134	C₄	#14579092	1	1	yes	characteristic
135	C₄	#14579094	1	1	yes	characteristic
136	C₂	D₄⋊(C₇₉⋊C₈)	1	2	yes	—
137	C₂	Q₈⋊(C₇₉⋊C₈)	1	2	yes	—
138	C₂	#19098318	1	1	yes	characteristic
139	C₂	#19098322	1	1	yes	characteristic
140	C₂	#19098325	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

C₄

yes

complemented

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₉

C₄

yes

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#95×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₈

yes

complemented

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

Q₈

yes

characteristic

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄²

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₈

yes

—

C₂×C₄×C₇₉

M₁₆

yes

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

characteristic

C₂×C₄×C₇₉

Q₁₆

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#95

yes

characteristic

C₂²×C₇₉

C₂²⋊C₈

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×C₈

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₄⋊C₈

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

D₄⋊C₈

yes

—

C₂×C₇₉

Q₈⋊C₈

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

D₇₉

yes

characteristic

C₂²×C₈

—

158

—

C₂²×C₈

—

158

—

C₂×C₄⋊C₄

Dic₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

158

—

C₄×C₈

—

C₄×C₈

—

158

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

—

316

—

C₂×C₈

—

316

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

C₇₉⋊C₈

yes

complemented

C₄²

—

C₄²

—

C₄²

Dic₁₅₈

yes

characteristic

C₄²

C₄×D₇₉

yes

characteristic

100

C₄²

C₄×D₇₉

yes

characteristic

101

C₄²

D₃₁₆

yes

characteristic

102

C₂³

C₂²⋊Dic₇₉

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×Dic₇₉

yes

characteristic

108

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

109

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

110

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

111

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

112

C₂×C₄

C₄⋊Dic₇₉

yes

characteristic

113

C₂×C₄

C₂×C₇₉⋊C₈

yes

—

114

C₂×C₄

#11929608

yes

—

115

C₂×C₄

#11929609

yes

characteristic

116

C₂×C₄

#11929611

yes

characteristic

117

C₂×C₄

#11929612

yes

characteristic

118

C₂×C₄

#11929614

yes

characteristic

119

C₈

—

316

—

120

C₈

—

158

—

121

C₂²

C₇₉⋊#95

yes

characteristic

122

C₂²

C₂²⋊(C₇₉⋊C₈)

yes

—

123

C₂²

#14579082

yes

characteristic

124

C₂²

#14579085

yes

characteristic

125

C₂²

#14579088

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₄×C₇₉⋊C₈

yes

characteristic

130

C₄

#14579072

yes

characteristic

131

C₄

#14579081

yes

characteristic

132

C₄

#14579084

yes

characteristic

133

C₄

C₄⋊(C₇₉⋊C₈)

yes

—

134

C₄

#14579092

yes

characteristic

135

C₄

#14579094

yes

characteristic

136

C₂

D₄⋊(C₇₉⋊C₈)

yes

—

137

C₂

Q₈⋊(C₇₉⋊C₈)

yes

—

138

C₂

#19098318

yes

characteristic

139

C₂

#19098322

yes

characteristic

140

C₂

#19098325

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₄×C₈
Frattini	C₂×C₄²	C₂×D₇₉
Fitting	C₄×C₄⋊C₄×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂²
hypercenter	C₄×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#1279
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₇₉

commutator

C₄×C₇₉

C₄×C₈

Frattini

C₂×C₄²

C₂×D₇₉

Fitting

C₄×C₄⋊C₄×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂²⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂²

hypercenter

C₄×C₄⋊C₄

D₇₉

hypocenter

C₇₉

#1279

LCS

G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
7. C₄⋊(C₇₉⋊C₈)	C₄	semidirect
9. C₄×C₇₉⋊C₈	C₄	semidirect
24. C₄⋊C₄×C₇₉	C₈	semidirect
71. C₇₉	#1279	semidirect
95. C₄⋊C₄	C₇₉⋊C₈	semidirect

H ≅ G/N

type

7. C₄⋊(C₇₉⋊C₈)

C₄

semidirect

9. C₄×C₇₉⋊C₈

C₄

semidirect

24. C₄⋊C₄×C₇₉

C₈

semidirect

71. C₇₉

#1279

semidirect

95. C₄⋊C₄

C₇₉⋊C₈

semidirect

Subgroups — #23761241

Conjugacy classes of subgroups

Complements