Group #23777514 — subgroups

Subgroups — C₂×#19099300 #23777514

Order 10112.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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	#19099300	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14579155	C₂	1	2	yes	maximal, complemented
3	C₂×#14579142	C₂	1	3	yes	maximal, complemented
4	C₂×#14579133	C₂	1	6	yes	maximal, complemented
5	C₂×#14579129	C₂	1	3	yes	maximal, complemented
6	C₂×#127×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₇₉	C₂²	1	1	yes	characteristic
8	C₂²×C₂²⋊D₇₉	C₂²	1	6	yes	—
9	C₂³×Dic₇₉	C₂²	1	3	yes	complemented
10	#14579155	C₂²	1	16	yes	complemented
11	#14579142	C₂²	1	24	yes	complemented
12	#14579133	C₂²	1	48	yes	complemented
13	#14579129	C₂²	1	24	yes	complemented
14	C₂×Dic₇₉⋊C₄	C₂²	1	6	yes	complemented
15	C₂×C₂²⋊Dic₇₉	C₂²	1	3	yes	—
16	C₂×C₂²⋊Dic₇₉	C₂²	1	3	yes	complemented
17	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	6	yes	—
18	C₂²×D₄×C₇₉	C₂²	1	3	yes	—
19	C₂²×D₄×C₇₉	C₂²	1	3	yes	complemented
20	#127×C₇₉	C₂²	1	8	yes	complemented
21	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic, complemented
22	C₂³×D₇₉	—	4	6	no	—
23	C₂³×D₇₉	C₂³	1	2	yes	—
24	C₂³×D₇₉	C₂³	1	4	yes	—
25	C₂×C₂²⋊D₇₉	—	4	12	no	—
26	C₂×C₂²⋊D₇₉	C₂³	1	24	yes	—
27	C₂²×Dic₇₉	—	4	6	no	—
28	C₂²×Dic₇₉	C₂³	1	6	yes	—
29	C₂²×Dic₇₉	C₂³	1	12	yes	complemented
30	Dic₇₉⋊C₄	C₂³	1	24	yes	complemented
31	C₂²⋊Dic₇₉	C₂³	1	12	yes	—
32	C₂²⋊Dic₇₉	C₂³	1	12	yes	complemented
33	(C₂×D₇₉)⋊C₄	C₂³	1	24	yes	—
34	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
35	C₂⁴×C₇₉	C₂³	1	3	yes	—
36	C₂×D₄×C₇₉	—	4	6	no	—
37	C₂×D₄×C₇₉	—	4	6	no	—
38	C₂×D₄×C₇₉	C₂³	1	12	yes	—
39	C₂×D₄×C₇₉	C₂³	1	12	yes	complemented
40	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
41	C₄²×C₇₉	C₂³	1	4	yes	complemented
42	C₂³×C₇₉	—	2	6	no	—
43	C₂³×C₇₉	—	2	12	no	—
44	C₂³×C₇₉	—	2	6	no	—
45	C₂³×C₇₉	C₂⁴	1	2	yes	—
46	C₂³×C₇₉	C₂⁴	1	6	yes	—
47	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
48	C₂²×D₇₉	—	2	12	no	—
49	C₂²×D₇₉	—	4	12	no	—
50	C₂²×D₇₉	—	4	16	no	—
51	C₂²×D₇₉	C₂⁴	1	4	yes	—
52	D₄×C₇₉	—	4	12	no	—
53	D₄×C₇₉	—	4	12	no	—
54	C₂×C₄×C₇₉	—	2	6	no	—
55	C₂×C₄×C₇₉	C₂⁴	1	6	yes	—
56	C₂²⋊D₇₉	—	4	24	no	—
57	C₂×Dic₇₉	—	2	12	no	—
58	C₂×Dic₇₉	—	4	12	no	—
59	C₂×Dic₇₉	C₂⁴	1	12	yes	—
60	C₂²×C₇₉	—	2	12	no	—
61	C₂²×C₇₉	—	2	6	no	—
62	C₂²×C₇₉	—	2	6	no	—
63	C₂²×C₇₉	—	4	4	no	—
64	C₂²×C₇₉	—	4	12	no	—
65	C₂²×C₇₉	H(5,𝔽₂)	1	6	yes	—
66	C₂²×C₇₉	C₂⁵	1	1	yes	characteristic
67	C₂×D₇₉	—	2	12	no	—
68	C₂×D₇₉	—	4	16	no	—
69	C₂×D₇₉	—	4	8	no	—
70	C₄×C₇₉	—	2	6	no	—
71	Dic₇₉	—	2	12	no	—
72	C₂×C₇₉	—	4	2	no	—
73	C₂×C₇₉	—	4	6	no	—
74	C₂×C₇₉	#560	1	4	yes	complemented
75	C₂×C₇₉	C₂×H(5,𝔽₂)	1	3	yes	—
76	D₇₉	—	4	4	no	—
77	C₂×#560	—	79	1	no	maximal, Sylow
78	C₇₉	C₂×#560	1	1	yes	characteristic, complemented, Sylow
79	#560	—	79	16	no	—
80	C₂×#127	D₇₉	1	1	yes	characteristic, complemented
81	C₂×#126	—	79	2	no	—
82	C₂×#123	—	79	3	no	—
83	C₂×#121	—	79	6	no	—
84	C₂×#120	—	79	3	no	—
85	C₂⁵	—	79	1	no	—
86	C₂²×D₄	—	79	6	no	—
87	C₂²×D₄	C₂×D₇₉	1	3	yes	—
88	C₂²×D₄	C₂×D₇₉	1	3	yes	complemented
89	C₂³×C₄	—	79	3	no	—
90	#127	C₂×D₇₉	1	8	yes	complemented
91	#126	—	79	16	no	—
92	#123	—	79	24	no	—
93	#121	—	79	48	no	—
94	#120	—	79	24	no	—
95	C₂×C₄⋊C₄	—	79	6	no	—
96	C₂×C₂²⋊C₄	—	79	3	no	—
97	C₂×C₂²⋊C₄	—	79	3	no	—
98	C₂×C₂²⋊C₄	—	79	6	no	—
99	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic, complemented
100	C₂⁴	—	316	6	no	—
101	C₂⁴	—	79	4	no	—
102	C₂⁴	—	79	2	no	—
103	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
104	C₂⁴	C₂²×D₇₉	1	3	yes	—
105	C₂×D₄	—	4	6	no	—
106	C₂×D₄	—	4	6	no	—
107	C₂×D₄	—	316	12	no	—
108	C₂×D₄	—	79	24	no	—
109	C₂×D₄	C₂²×D₇₉	1	12	yes	—
110	C₂×D₄	C₂²×D₇₉	1	12	yes	complemented
111	C₂²×C₄	—	316	6	no	—
112	C₂²×C₄	—	79	12	no	—
113	C₂²×C₄	—	79	6	no	—
114	C₂²×C₄	C₂²×D₇₉	1	3	yes	—
115	C₄⋊C₄	—	79	24	no	—
116	C₂²⋊C₄	—	79	12	no	—
117	C₂²⋊C₄	—	79	12	no	—
118	C₂²⋊C₄	—	79	24	no	—
119	C₄²	C₂²×D₇₉	1	4	yes	complemented
120	C₂³	—	2	6	no	—
121	C₂³	—	2	12	no	—
122	C₂³	—	2	6	no	—
123	C₂³	—	316	12	no	—
124	C₂³	—	316	16	no	—
125	C₂³	—	79	4	no	—
126	C₂³	—	158	12	no	—
127	C₂³	C₂³×D₇₉	1	2	yes	—
128	C₂³	C₂³×D₇₉	1	6	yes	—
129	C₂³	C₂³×D₇₉	1	1	yes	characteristic
130	D₄	—	4	12	no	—
131	D₄	—	4	12	no	—
132	D₄	—	316	24	no	—
133	C₂×C₄	—	2	6	no	—
134	C₂×C₄	—	316	12	no	—
135	C₂×C₄	—	79	12	no	—
136	C₂×C₄	—	158	12	no	—
137	C₂×C₄	C₂³×D₇₉	1	6	yes	—
138	C₂²	—	2	6	no	—
139	C₂²	—	2	12	no	—
140	C₂²	—	2	6	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	12	no	—
143	C₂²	—	316	16	no	—
144	C₂²	—	316	8	no	—
145	C₂²	—	158	12	no	—
146	C₂²	#14579206	1	6	yes	—
147	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	—	158	12	no	—
150	C₂	—	4	2	no	—
151	C₂	—	4	6	no	—
152	C₂	—	316	4	no	—
153	C₂	#19099300	1	4	yes	direct product factor
154	C₂	C₂×#14579206	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099300

C₂

yes

maximal, direct product factor

C₂×#14579155

C₂

yes

maximal, complemented

C₂×#14579142

C₂

yes

maximal, complemented

C₂×#14579133

C₂

yes

maximal, complemented

C₂×#14579129

C₂

yes

maximal, complemented

C₂×#127×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₇₉

C₂²

yes

characteristic

C₂²×C₂²⋊D₇₉

C₂²

yes

—

C₂³×Dic₇₉

C₂²

yes

complemented

#14579155

C₂²

yes

complemented

#14579142

C₂²

yes

complemented

#14579133

C₂²

yes

complemented

#14579129

C₂²

yes

complemented

C₂×Dic₇₉⋊C₄

C₂²

yes

complemented

C₂×C₂²⋊Dic₇₉

C₂²

yes

—

C₂×C₂²⋊Dic₇₉

C₂²

yes

complemented

C₂×(C₂×D₇₉)⋊C₄

C₂²

yes

—

C₂²×D₄×C₇₉

C₂²

yes

—

C₂²×D₄×C₇₉

C₂²

yes

complemented

#127×C₇₉

C₂²

yes

complemented

C₂×C₄²×C₇₉

C₂²

yes

characteristic, complemented

C₂³×D₇₉

—

C₂³×D₇₉

C₂³

yes

—

C₂³×D₇₉

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₂²⋊D₇₉

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

complemented

Dic₇₉⋊C₄

C₂³

yes

complemented

C₂²⋊Dic₇₉

C₂³

yes

—

C₂²⋊Dic₇₉

C₂³

yes

complemented

(C₂×D₇₉)⋊C₄

C₂³

yes

—

C₂⁴×C₇₉

C₂³

yes

characteristic

C₂⁴×C₇₉

C₂³

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

C₂×D₄×C₇₉

C₂³

yes

complemented

C₂²×C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂³

yes

complemented

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂⁴

yes

—

C₂³×C₇₉

C₂⁴

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

C₂⁴

yes

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂⁴

yes

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂⁴

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₂²×C₇₉

C₂⁵

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#560

yes

complemented

C₂×C₇₉

C₂×H(5,𝔽₂)

yes

—

D₇₉

—

C₂×#560

—

maximal, Sylow

C₇₉

C₂×#560

yes

characteristic, complemented, Sylow

#560

—

C₂×#127

D₇₉

yes

characteristic, complemented

—

—

—

—

C₂⁵

—

C₂²×D₄

—

C₂²×D₄

C₂×D₇₉

yes

—

C₂²×D₄

C₂×D₇₉

yes

complemented

C₂³×C₄

—

#127

C₂×D₇₉

yes

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇₉

yes

characteristic, complemented

100

C₂⁴

—

316

—

101

C₂⁴

—

102

C₂⁴

—

103

C₂⁴

C₂²×D₇₉

yes

characteristic

104

C₂⁴

C₂²×D₇₉

yes

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

316

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₇₉

yes

—

110

C₂×D₄

C₂²×D₇₉

yes

complemented

111

C₂²×C₄

—

316

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²×D₇₉

yes

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

C₂²×D₇₉

yes

complemented

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

316

—

124

C₂³

—

316

—

125

C₂³

—

126

C₂³

—

158

—

127

C₂³

C₂³×D₇₉

yes

—

128

C₂³

C₂³×D₇₉

yes

—

129

C₂³

C₂³×D₇₉

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

316

—

133

C₂×C₄

—

134

C₂×C₄

—

316

—

135

C₂×C₄

—

136

C₂×C₄

—

158

—

137

C₂×C₄

C₂³×D₇₉

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

316

—

144

C₂²

—

316

—

145

C₂²

—

158

—

146

C₂²

#14579206

yes

—

147

C₂²

C₂⁴×D₇₉

yes

characteristic

148

C₄

—

149

C₄

—

158

—

150

C₂

—

151

C₂

—

152

C₂

—

316

—

153

C₂

#19099300

yes

direct product factor

154

C₂

C₂×#14579206

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁵
Frattini	C₂²	C₂⁴×D₇₉
Fitting	C₂×#127×C₇₉	C₂
socle	C₂³×C₇₉	C₂⁴
hypercenter	C₂×#127	D₇₉
hypocenter	C₇₉	C₂×#560
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

C₂×#127×C₇₉

C₂

socle

C₂³×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. #19099300	C₂	direct (N × H)
2. C₂×#14579155	C₂	semidirect
3. C₂×#14579142	C₂	semidirect
4. C₂×#14579133	C₂	semidirect
5. C₂×#14579129	C₂	semidirect
6. C₂×#127×C₇₉	C₂	semidirect
9. C₂³×Dic₇₉	C₂²	semidirect
10. #14579155	C₂²	semidirect
11. #14579142	C₂²	semidirect
12. #14579133	C₂²	semidirect
13. #14579129	C₂²	semidirect
14. C₂×Dic₇₉⋊C₄	C₂²	semidirect
16. C₂×C₂²⋊Dic₇₉	C₂²	semidirect
19. C₂²×D₄×C₇₉	C₂²	semidirect
20. #127×C₇₉	C₂²	semidirect
21. C₂×C₄²×C₇₉	C₂²	semidirect
29. C₂²×Dic₇₉	C₂³	semidirect
30. Dic₇₉⋊C₄	C₂³	semidirect
32. C₂²⋊Dic₇₉	C₂³	semidirect
39. C₂×D₄×C₇₉	C₂³	semidirect
41. C₄²×C₇₉	C₂³	semidirect
74. C₂×C₇₉	#560	semidirect
78. C₇₉	C₂×#560	semidirect
80. C₂×#127	D₇₉	semidirect
88. C₂²×D₄	C₂×D₇₉	semidirect
90. #127	C₂×D₇₉	semidirect
99. C₂×C₄²	C₂×D₇₉	semidirect
110. C₂×D₄	C₂²×D₇₉	semidirect
119. C₄²	C₂²×D₇₉	semidirect
153. C₂	#19099300	direct (N × H)

H ≅ G/N

type

1. #19099300

C₂

direct (N × H)

2. C₂×#14579155

C₂

semidirect

3. C₂×#14579142

C₂

semidirect

4. C₂×#14579133

C₂

semidirect

5. C₂×#14579129

C₂

semidirect

6. C₂×#127×C₇₉

C₂

semidirect

9. C₂³×Dic₇₉

C₂²

semidirect

10. #14579155

C₂²

semidirect

11. #14579142

C₂²

semidirect

12. #14579133

C₂²

semidirect

13. #14579129

C₂²

semidirect

14. C₂×Dic₇₉⋊C₄

C₂²

semidirect

16. C₂×C₂²⋊Dic₇₉

C₂²

semidirect

19. C₂²×D₄×C₇₉

C₂²

semidirect

20. #127×C₇₉

C₂²

semidirect

21. C₂×C₄²×C₇₉

C₂²

semidirect

29. C₂²×Dic₇₉

C₂³

semidirect

30. Dic₇₉⋊C₄

C₂³

semidirect

32. C₂²⋊Dic₇₉

C₂³

semidirect

39. C₂×D₄×C₇₉

C₂³

semidirect

41. C₄²×C₇₉

C₂³

semidirect

74. C₂×C₇₉

#560

semidirect

78. C₇₉

C₂×#560

semidirect

80. C₂×#127

D₇₉

semidirect

88. C₂²×D₄

C₂×D₇₉

semidirect

90. #127

C₂×D₇₉

semidirect

99. C₂×C₄²

C₂×D₇₉

semidirect

110. C₂×D₄

C₂²×D₇₉

semidirect

119. C₄²

C₂²×D₇₉

semidirect

153. C₂

#19099300

direct (N × H)

Subgroups — C₂×#19099300 #23777514

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19099300 #23777514

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19099300 #23777514