Group #23772327 — subgroups

Subgroups — C₂×#19098693 #23772327

Order 10112.

400 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 221 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×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
2	#19098693	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₉⋊C₈	C₂	1	2	yes	maximal
5	C₂×#14579071	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₅₈	C₄	1	1	yes	characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
9	C₄×Dic₁₅₈	C₄	1	4	yes	—
10	C₄×Dic₁₅₈	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₇₉	C₄	1	1	yes	characteristic
12	C₂×Dic₇₉⋊C₄	C₄	1	2	yes	—
13	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
14	C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
15	Dic₇₉⋊C₈	C₂²	1	8	yes	—
16	#14579071	C₂²	1	8	yes	—
17	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₇₉	C₂²	1	4	yes	—
21	C₂×Dic₁₅₈	—	2	4	no	—
22	C₂×Dic₁₅₈	C₂×C₄	1	4	yes	—
23	C₂²×Dic₇₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₇₉	C₂×C₄	1	2	yes	—
25	C₂×C₇₉⋊C₈	—	2	4	no	—
26	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
27	C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
28	Dic₇₉⋊C₄	C₂×C₄	1	8	yes	—
29	C₄×Dic₇₉	C₂×C₄	1	4	yes	—
30	C₄×Dic₇₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₇₉	—	2	2	no	—
35	C₂×C₈×C₇₉	Q₈	1	4	yes	complemented
36	C₂×C₈×C₇₉	C₂³	1	2	yes	—
37	C₂×C₈×C₇₉	C₂³	1	2	yes	—
38	C₄²×C₇₉	C₂×C₄	1	2	yes	—
39	C₄²×C₇₉	C₂³	1	2	yes	—
40	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	M₁₆	1	4	yes	—
43	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₇₉	#41	1	2	yes	—
48	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₇₉	—	2	4	no	—
50	C₂×Dic₇₉	—	2	4	no	—
51	C₂×Dic₇₉	C₂²×C₄	1	4	yes	—
52	C₂×Dic₇₉	C₂²×C₄	1	4	yes	—
53	Dic₁₅₈	—	2	8	no	—
54	C₈×C₇₉	—	2	2	no	—
55	C₈×C₇₉	C₂×Q₈	1	4	yes	complemented
56	C₇₉⋊C₈	—	2	4	no	—
57	C₂²×C₇₉	C₄×Q₈	1	2	yes	—
58	C₂²×C₇₉	C₂×M₁₆	1	2	yes	—
59	C₂²×C₇₉	#131	1	2	yes	—
60	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₇₉	—	2	2	no	—
62	C₄×C₇₉	C₄×Q₈	1	2	yes	—
63	C₄×C₇₉	C₂×M₁₆	1	4	yes	—
64	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
66	Dic₇₉	—	2	4	no	—
67	Dic₇₉	—	2	4	no	—
68	C₂×C₇₉	#446	1	4	yes	complemented
69	C₂×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
71	C₂×C₇₉	C₂×#131	1	1	yes	characteristic
72	C₂×#446	—	79	1	no	maximal, Sylow
73	C₇₉	C₂×#446	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄×Q₈	—	79	1	no	—
75	#446	—	79	8	no	—
76	C₂×C₄⋊C₈	—	79	2	no	—
77	C₂×C₄⋊C₈	—	79	1	no	—
78	C₂×#97	—	79	2	no	—
79	C₂×C₄×C₈	D₇₉	1	1	yes	characteristic
80	C₂²×Q₈	—	79	1	no	—
81	C₂²×C₈	—	79	2	no	—
82	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
84	C₄×Q₈	—	79	4	no	—
85	C₄×Q₈	—	79	4	no	—
86	C₂×C₄⋊C₄	—	79	2	no	—
87	C₂×C₄⋊C₄	—	79	1	no	—
88	C₂×C₄²	—	79	2	no	—
89	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
90	C₄⋊C₈	—	79	8	no	—
91	C₄⋊C₈	—	79	4	no	—
92	#97	—	79	8	no	—
93	C₄×C₈	C₂×D₇₉	1	4	yes	—
94	C₂×Q₈	—	79	4	no	—
95	C₂×Q₈	—	158	4	no	—
96	C₂²×C₄	—	79	2	no	—
97	C₂²×C₄	—	79	2	no	—
98	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	79	4	no	—
103	C₂×C₈	—	158	4	no	—
104	C₂×C₈	Dic₁₅₈	1	4	yes	complemented
105	C₂×C₈	C₂²×D₇₉	1	2	yes	—
106	C₂×C₈	C₂²×D₇₉	1	2	yes	—
107	C₄⋊C₄	—	79	8	no	—
108	C₄⋊C₄	—	79	4	no	—
109	C₄²	—	79	4	no	—
110	C₄²	—	79	4	no	—
111	C₄²	C₄×D₇₉	1	2	yes	—
112	C₄²	C₂²×D₇₉	1	2	yes	—
113	C₂³	C₂×C₄×D₇₉	1	1	yes	characteristic
114	Q₈	—	158	8	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	79	4	no	—
117	C₂×C₄	—	79	4	no	—
118	C₂×C₄	—	158	4	no	—
119	C₂×C₄	—	158	4	no	—
120	C₂×C₄	#11929607	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
124	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
125	C₂×C₄	#11929624	1	2	yes	—
126	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	158	4	no	—
129	C₈	C₂×Dic₁₅₈	1	4	yes	complemented
130	C₂²	C₄×Dic₁₅₈	1	2	yes	—
131	C₂²	C₂×#11929607	1	2	yes	—
132	C₂²	#14579167	1	2	yes	—
133	C₂²	C₂²×C₄×D₇₉	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	158	4	no	—
136	C₄	—	158	4	no	—
137	C₄	C₄×Dic₁₅₈	1	2	yes	—
138	C₄	C₂×#11929607	1	4	yes	—
139	C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
140	C₄	C₂×#11929624	1	1	yes	characteristic
141	C₂	#19098693	1	4	yes	direct product factor
142	C₂	C₂×C₄×Dic₁₅₈	1	1	yes	characteristic
143	C₂	C₂²×#11929607	1	1	yes	characteristic
144	C₂	C₂×#14579167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₅₈

C₂

yes

maximal, characteristic

#19098693

C₂

yes

maximal, direct product factor

C₂×C₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₇₉⋊C₈

C₂

yes

maximal

C₂×#14579071

C₂

yes

maximal

C₂×C₄×C₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

C₄×Dic₁₅₈

C₄

yes

—

C₄×Dic₁₅₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₄

yes

characteristic

C₂×Dic₇₉⋊C₄

C₄

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

Dic₇₉⋊C₈

C₂²

yes

—

#14579071

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂²×Dic₇₉

C₂×C₄

yes

—

C₂²×Dic₇₉

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

C₂×C₄

yes

—

C₄×Dic₇₉

C₂×C₄

yes

—

C₄×Dic₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

Q₈

yes

complemented

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

M₁₆

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂²×C₄

yes

—

C₂×Dic₇₉

C₂²×C₄

yes

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×Q₈

yes

complemented

C₇₉⋊C₈

—

C₂²×C₇₉

C₄×Q₈

yes

—

C₂²×C₇₉

C₂×M₁₆

yes

—

C₂²×C₇₉

#131

yes

—

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₄×Q₈

yes

—

C₄×C₇₉

C₂×M₁₆

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

C₂×#41

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#446

yes

complemented

C₂×C₇₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₇₉

C₂²×M₁₆

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#446

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

D₇₉

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

C₂×D₇₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₄×D₇₉

yes

characteristic

100

C₂²×C₄

C₂²×D₇₉

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

158

—

104

C₂×C₈

Dic₁₅₈

yes

complemented

105

C₂×C₈

C₂²×D₇₉

yes

—

106

C₂×C₈

C₂²×D₇₉

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₄×D₇₉

yes

—

112

C₄²

C₂²×D₇₉

yes

—

113

C₂³

C₂×C₄×D₇₉

yes

characteristic

114

Q₈

—

158

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

158

—

119

C₂×C₄

—

158

—

120

C₂×C₄

#11929607

yes

—

121

C₂×C₄

C₂×C₄×D₇₉

yes

—

122

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

123

C₂×C₄

C₂×C₄×D₇₉

yes

—

124

C₂×C₄

C₂×Dic₁₅₈

yes

—

125

C₂×C₄

#11929624

yes

—

126

C₂×C₄

C₂³×D₇₉

yes

characteristic

127

C₈

—

128

C₈

—

158

—

129

C₈

C₂×Dic₁₅₈

yes

complemented

130

C₂²

C₄×Dic₁₅₈

yes

—

131

C₂²

C₂×#11929607

yes

—

132

C₂²

#14579167

yes

—

133

C₂²

C₂²×C₄×D₇₉

yes

characteristic

134

C₄

—

135

C₄

—

158

—

136

C₄

—

158

—

137

C₄

C₄×Dic₁₅₈

yes

—

138

C₄

C₂×#11929607

yes

—

139

C₄

C₂²×Dic₁₅₈

yes

characteristic

140

C₄

C₂×#11929624

yes

characteristic

141

C₂

#19098693

yes

direct product factor

142

C₂

C₂×C₄×Dic₁₅₈

yes

characteristic

143

C₂

C₂²×#11929607

yes

characteristic

144

C₂

C₂×#14579167

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×C₄×C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×C₄
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₇₉
hypocenter	C₇₉	C₂×#446
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₇₉

commutator

C₂²×C₇₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×C₄×C₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂²×C₄

cosocle

C₂×C₄×C₇₉

C₂⁴

hypercenter

C₂×C₄×C₈

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #19098693	C₂	direct (N × H)
35. C₂×C₈×C₇₉	Q₈	semidirect
55. C₈×C₇₉	C₂×Q₈	semidirect
68. C₂×C₇₉	#446	semidirect
73. C₇₉	C₂×#446	semidirect
104. C₂×C₈	Dic₁₅₈	semidirect
129. C₈	C₂×Dic₁₅₈	semidirect
141. C₂	#19098693	direct (N × H)

H ≅ G/N

type

2. #19098693

C₂

direct (N × H)

35. C₂×C₈×C₇₉

Q₈

semidirect

55. C₈×C₇₉

C₂×Q₈

semidirect

68. C₂×C₇₉

#446

semidirect

73. C₇₉

C₂×#446

semidirect

104. C₂×C₈

Dic₁₅₈

semidirect

129. C₈

C₂×Dic₁₅₈

semidirect

141. C₂

#19098693

direct (N × H)

Subgroups — C₂×#19098693 #23772327

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098693 #23772327

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098693 #23772327