Group #23776989 — subgroups

Subgroups — C₂×#19099299 #23776989

Order 10112.

1240 conjugacy classes of subgroups; only the 501 normal subgroups are shown.

normal subgroups only

This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 501 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	#19099299	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14579148	C₂	1	4	yes	maximal, complemented
3	C₂×#14579142	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Dic₁₅₈	C₂	1	2	yes	maximal, complemented
5	C₂×D₄×Dic₇₉	C₂	1	4	yes	maximal, complemented
6	C₂×#127×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
8	C₂³×Dic₇₉	C₂²	1	4	yes	—
9	#14579148	C₂²	1	32	yes	complemented
10	#14579142	C₂²	1	32	yes	complemented
11	C₄×Dic₁₅₈	C₂²	1	16	yes	complemented
12	D₄×Dic₇₉	C₂²	1	32	yes	complemented
13	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
14	C₂×Dic₇₉⋊C₄	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₇₉	C₂²	1	8	yes	—
16	C₂×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
17	C₂×C₄×Dic₇₉	C₂²	1	4	yes	—
18	C₂²×D₄×C₇₉	C₂²	1	4	yes	—
19	C₂²×D₄×C₇₉	C₂²	1	2	yes	—
20	#127×C₇₉	C₂²	1	8	yes	—
21	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₅₈	—	2	8	no	—
23	C₂×Dic₁₅₈	C₂³	1	8	yes	—
24	C₂²×Dic₇₉	—	4	8	no	—
25	C₂²×Dic₇₉	C₂³	1	4	yes	—
26	C₂²×Dic₇₉	C₂³	1	4	yes	—
27	C₂²×Dic₇₉	C₂³	1	16	yes	—
28	C₄⋊Dic₇₉	C₂³	1	8	yes	—
29	Dic₇₉⋊C₄	C₂³	1	16	yes	—
30	C₂²⋊Dic₇₉	C₂³	1	32	yes	—
31	C₂²⋊Dic₇₉	C₂³	1	16	yes	—
32	C₄×Dic₇₉	C₂³	1	16	yes	—
33	C₂⁴×C₇₉	C₂³	1	4	yes	—
34	C₂×D₄×C₇₉	—	2	16	no	—
35	C₂×D₄×C₇₉	—	4	4	no	—
36	C₂×D₄×C₇₉	C₂³	1	16	yes	—
37	C₂×D₄×C₇₉	C₂³	1	8	yes	—
38	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
40	C₄²×C₇₉	C₂³	1	4	yes	—
41	C₂³×C₇₉	—	2	16	no	—
42	C₂³×C₇₉	—	2	8	no	—
43	C₂³×C₇₉	C₂⁴	1	8	yes	—
44	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
45	D₄×C₇₉	—	2	32	no	—
46	D₄×C₇₉	—	4	8	no	—
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	#41	1	8	yes	—
49	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
50	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
51	C₂×Dic₇₉	—	2	8	no	—
52	C₂×Dic₇₉	—	2	8	no	—
53	C₂×Dic₇₉	—	4	16	no	—
54	C₂×Dic₇₉	C₂⁴	1	8	yes	—
55	C₂×Dic₇₉	C₂⁴	1	8	yes	—
56	Dic₁₅₈	—	2	16	no	—
57	C₂²×C₇₉	—	2	16	no	—
58	C₂²×C₇₉	—	2	8	no	—
59	C₂²×C₇₉	—	4	16	no	—
60	C₂²×C₇₉	C₂×#41	1	4	yes	—
61	C₂²×C₇₉	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₇₉	C₂⁵	1	1	yes	characteristic
63	C₄×C₇₉	—	2	2	no	—
64	C₄×C₇₉	C₂×#41	1	8	yes	—
65	Dic₇₉	—	2	8	no	—
66	Dic₇₉	—	2	8	no	—
67	C₂×C₇₉	—	4	8	no	—
68	C₂×C₇₉	#559	1	4	yes	complemented
69	C₂×C₇₉	C₂²×#41	1	2	yes	—
70	C₂×C₇₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×#559	—	79	1	no	maximal, Sylow
72	C₇₉	C₂×#559	1	1	yes	characteristic, complemented, Sylow
73	#559	—	79	16	no	—
74	C₂×#127	D₇₉	1	1	yes	characteristic
75	C₂×#124	—	79	4	no	—
76	C₂×#123	—	79	4	no	—
77	C₂×C₄×Q₈	—	79	2	no	—
78	C₂×C₄×D₄	—	79	4	no	—
79	C₂²×Q₈	—	79	2	no	—
80	C₂²×D₄	C₂×D₇₉	1	4	yes	—
81	C₂²×D₄	C₂×D₇₉	1	2	yes	—
82	C₂³×C₄	—	79	4	no	—
83	#127	C₂×D₇₉	1	8	yes	—
84	#124	—	79	32	no	—
85	#123	—	79	32	no	—
86	C₄×Q₈	—	79	16	no	—
87	C₄×D₄	—	79	32	no	—
88	C₂×C₄⋊C₄	—	79	2	no	—
89	C₂×C₄⋊C₄	—	79	4	no	—
90	C₂×C₂²⋊C₄	—	79	8	no	—
91	C₂×C₂²⋊C₄	—	79	4	no	—
92	C₂×C₄²	—	79	4	no	—
93	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
94	C₂⁴	C₂²×D₇₉	1	4	yes	—
95	C₂×Q₈	—	79	8	no	—
96	C₂×Q₈	—	158	8	no	—
97	C₂×D₄	—	2	16	no	—
98	C₂×D₄	—	4	4	no	—
99	C₂×D₄	C₂²×D₇₉	1	16	yes	—
100	C₂×D₄	C₂²×D₇₉	1	8	yes	—
101	C₂²×C₄	—	316	8	no	—
102	C₂²×C₄	—	79	16	no	—
103	C₂²×C₄	—	79	4	no	—
104	C₂²×C₄	—	79	4	no	—
105	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
107	C₄⋊C₄	—	79	16	no	—
108	C₄⋊C₄	—	79	8	no	—
109	C₂²⋊C₄	—	79	32	no	—
110	C₂²⋊C₄	—	79	16	no	—
111	C₄²	—	79	16	no	—
112	C₄²	C₂²×D₇₉	1	4	yes	—
113	C₂³	—	2	16	no	—
114	C₂³	—	2	8	no	—
115	C₂³	C₂³×D₇₉	1	8	yes	—
116	C₂³	C₂³×D₇₉	1	1	yes	characteristic
117	Q₈	—	158	16	no	—
118	D₄	—	2	32	no	—
119	D₄	—	4	8	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	316	16	no	—
122	C₂×C₄	—	79	8	no	—
123	C₂×C₄	—	79	8	no	—
124	C₂×C₄	—	158	8	no	—
125	C₂×C₄	—	158	8	no	—
126	C₂×C₄	#11929623	1	8	yes	—
127	C₂×C₄	C₂³×D₇₉	1	2	yes	—
128	C₂×C₄	C₂³×D₇₉	1	4	yes	—
129	C₂²	—	2	8	no	—
130	C₂²	—	2	16	no	—
131	C₂²	—	4	16	no	—
132	C₂²	C₂×#11929623	1	4	yes	—
133	C₂²	#14579206	1	2	yes	—
134	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	158	8	no	—
137	C₄	—	158	8	no	—
138	C₄	C₂×#11929623	1	8	yes	—
139	C₂	—	4	8	no	—
140	C₂	#19099299	1	4	yes	direct product factor
141	C₂	C₂²×#11929623	1	2	yes	—
142	C₂	C₂×#14579206	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099299

C₂

yes

maximal, direct product factor

C₂×#14579148

C₂

yes

maximal, complemented

C₂×#14579142

C₂

yes

maximal, complemented

C₂×C₄×Dic₁₅₈

C₂

yes

maximal, complemented

C₂×D₄×Dic₇₉

C₂

yes

maximal, complemented

C₂×#127×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

C₂³×Dic₇₉

C₂²

yes

—

#14579148

C₂²

yes

complemented

#14579142

C₂²

yes

complemented

C₄×Dic₁₅₈

C₂²

yes

complemented

D₄×Dic₇₉

C₂²

yes

complemented

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₇₉

C₂²

yes

—

C₂×C₂²⋊Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₂²×D₄×C₇₉

C₂²

yes

—

C₂²×D₄×C₇₉

C₂²

yes

—

#127×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

C₂³

yes

—

C₂²⋊Dic₇₉

C₂³

yes

—

C₂²⋊Dic₇₉

C₂³

yes

—

C₄×Dic₇₉

C₂³

yes

—

C₂⁴×C₇₉

C₂³

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

C₂×D₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂⁴

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂⁴

yes

—

C₂×Dic₇₉

C₂⁴

yes

—

Dic₁₅₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#41

yes

—

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₂²×C₇₉

C₂⁵

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂×#41

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#559

yes

complemented

C₂×C₇₉

C₂²×#41

yes

—

C₂×C₇₉

C₂×H(5,𝔽₂)

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#559

—

C₂×#127

D₇₉

yes

characteristic

C₂×#124

—

C₂×#123

—

C₂×C₄×Q₈

—

C₂×C₄×D₄

—

C₂²×Q₈

—

C₂²×D₄

C₂×D₇₉

yes

—

C₂²×D₄

C₂×D₇₉

yes

—

C₂³×C₄

—

#127

C₂×D₇₉

yes

—

#124

—

#123

—

C₄×Q₈

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₂⁴

C₂²×D₇₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₇₉

yes

—

100

C₂×D₄

C₂²×D₇₉

yes

—

101

C₂²×C₄

—

316

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₇₉

yes

characteristic

106

C₂²×C₄

C₂²×D₇₉

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

C₂²×D₇₉

yes

—

113

C₂³

—

114

C₂³

—

115

C₂³

C₂³×D₇₉

yes

—

116

C₂³

C₂³×D₇₉

yes

characteristic

117

Q₈

—

158

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

316

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

158

—

125

C₂×C₄

—

158

—

126

C₂×C₄

#11929623

yes

—

127

C₂×C₄

C₂³×D₇₉

yes

—

128

C₂×C₄

C₂³×D₇₉

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#11929623

yes

—

133

C₂²

#14579206

yes

—

134

C₂²

C₂⁴×D₇₉

yes

characteristic

135

C₄

—

136

C₄

—

158

—

137

C₄

—

158

—

138

C₄

C₂×#11929623

yes

—

139

C₂

—

140

C₂

#19099299

yes

direct product factor

141

C₂

C₂²×#11929623

yes

—

142

C₂

C₂×#14579206

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁵
Frattini	C₂²	C₂⁴×D₇₉
Fitting, prime radical	C₂×#127×C₇₉	C₂
socle	C₂³×C₇₉	C₂⁴
hypercenter	C₂×#127	D₇₉
hypocenter	C₇₉	C₂×#559
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂³×D₇₉

commutator, cosocle

C₂²×C₇₉

C₂⁵

Frattini

C₂²

C₂⁴×D₇₉

Fitting, prime radical

C₂×#127×C₇₉

C₂

socle

C₂³×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. #19099299	C₂	direct (N × H)
2. C₂×#14579148	C₂	semidirect
3. C₂×#14579142	C₂	semidirect
4. C₂×C₄×Dic₁₅₈	C₂	semidirect
5. C₂×D₄×Dic₇₉	C₂	semidirect
9. #14579148	C₂²	semidirect
10. #14579142	C₂²	semidirect
11. C₄×Dic₁₅₈	C₂²	semidirect
12. D₄×Dic₇₉	C₂²	semidirect
68. C₂×C₇₉	#559	semidirect
72. C₇₉	C₂×#559	semidirect
140. C₂	#19099299	direct (N × H)

H ≅ G/N

type

1. #19099299

C₂

direct (N × H)

2. C₂×#14579148

C₂

semidirect

3. C₂×#14579142

C₂

semidirect

4. C₂×C₄×Dic₁₅₈

C₂

semidirect

5. C₂×D₄×Dic₇₉

C₂

semidirect

9. #14579148

C₂²

semidirect

10. #14579142

C₂²

semidirect

11. C₄×Dic₁₅₈

C₂²

semidirect

12. D₄×Dic₇₉

C₂²

semidirect

68. C₂×C₇₉

#559

semidirect

72. C₇₉

C₂×#559

semidirect

140. C₂

#19099299

direct (N × H)

Subgroups — C₂×#19099299 #23776989

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19099299 #23776989

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19099299 #23776989