Group #23771652 — subgroups

Subgroups — #23771652

Order 10112.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19098476	C₂	1	6	yes	maximal, complemented
2	#19098475	C₂	1	3	yes	maximal, complemented
3	#19098474	C₂	1	2	yes	maximal, complemented
4	#19098418	C₂	1	3	yes	maximal
5	#393×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₇₉	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₅₈	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊Dic₇₉	C₂²	1	3	yes	—
9	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	6	yes	—
10	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	6	yes	—
11	C₇₉⋊#95	C₂²	1	6	yes	complemented
12	C₇₉⋊#95	C₂²	1	3	yes	complemented
13	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊C₄×C₇₉	C₂²	1	3	yes	—
15	#95×C₇₉	C₂²	1	3	yes	complemented
16	C₂³×D₇₉	—	4	3	no	—
17	C₂³×D₇₉	—	4	3	no	—
18	C₂³×D₇₉	Q₈	1	4	yes	complemented
19	C₂³×D₇₉	C₂³	1	2	yes	—
20	C₂×Dic₁₅₈	—	4	6	no	—
21	C₂×Dic₁₅₈	—	4	3	no	—
22	C₂²×Dic₇₉	C₂³	1	6	yes	—
23	C₂²⋊Dic₇₉	—	4	3	no	—
24	(C₂×D₇₉)⋊C₄	—	4	6	no	—
25	(C₂×D₇₉)⋊C₄	—	4	6	no	—
26	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₇₉	—	4	3	no	—
28	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
29	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
30	C₂²⋊C₄×C₇₉	—	4	3	no	—
31	C₂³×C₇₉	—	2	3	no	—
32	C₂³×C₇₉	—	2	3	no	—
33	C₂³×C₇₉	C₂×Q₈	1	2	yes	—
34	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
35	C₂²×D₇₉	—	2	6	no	—
36	C₂²×D₇₉	—	2	6	no	—
37	C₂²×D₇₉	—	4	12	no	—
38	C₂²×D₇₉	—	4	12	no	—
39	C₂²×D₇₉	—	4	4	no	—
40	C₂²×D₇₉	C₂×Q₈	1	4	yes	—
41	Q₈×C₇₉	—	4	4	no	—
42	C₂×C₄×C₇₉	—	2	6	no	—
43	C₂×C₄×C₇₉	—	2	6	no	—
44	C₂×C₄×C₇₉	—	2	3	no	—
45	C₂×C₄×C₇₉	—	2	3	no	—
46	C₂×Dic₇₉	—	2	6	no	—
47	C₂×Dic₇₉	—	2	6	no	—
48	C₂×Dic₇₉	—	2	6	no	—
49	Dic₁₅₈	—	4	12	no	—
50	C₂²×C₇₉	—	2	6	no	—
51	C₂²×C₇₉	—	4	3	no	—
52	C₂²×C₇₉	—	4	1	no	—
53	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
54	C₂²×C₇₉	H(5,𝔽₂)	1	3	yes	—
55	C₂²×C₇₉	H(5,𝔽₂)	1	3	yes	—
56	C₂×D₇₉	—	2	12	no	—
57	C₂×D₇₉	—	4	16	no	—
58	C₂×D₇₉	—	4	6	no	—
59	C₂×D₇₉	—	4	2	no	—
60	C₄×C₇₉	—	4	3	no	—
61	C₄×C₇₉	—	4	3	no	—
62	Dic₇₉	—	4	6	no	—
63	C₂×C₇₉	—	4	2	no	—
64	C₂×C₇₉	#543	1	3	yes	—
65	C₂×C₇₉	H(3,𝔽₄)	1	3	yes	—
66	C₂×C₇₉	H(3,𝔽₄)	1	1	yes	characteristic
67	D₇₉	—	4	4	no	—
68	#2840	—	79	1	no	maximal, Sylow
69	C₇₉	#2840	1	1	yes	characteristic, complemented, Sylow
70	#393	—	79	2	no	—
71	#393	—	79	3	no	—
72	#393	—	79	6	no	—
73	#393	D₇₉	1	1	yes	characteristic, complemented
74	#379	—	79	3	no	—
75	C₂⁵	—	79	1	no	—
76	C₂²×Q₈	—	79	3	no	—
77	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊C₄	—	79	3	no	—
79	C₂×C₂²⋊C₄	—	79	6	no	—
80	C₂×C₂²⋊C₄	—	79	6	no	—
81	C₂×C₂²⋊C₄	C₂×D₇₉	1	3	yes	—
82	#95	—	79	6	no	—
83	#95	—	79	3	no	—
84	#95	C₂×D₇₉	1	3	yes	complemented
85	C₂⁴	—	316	3	no	—
86	C₂⁴	—	316	3	no	—
87	C₂⁴	—	79	4	no	—
88	C₂⁴	—	79	2	no	—
89	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
90	C₂×Q₈	—	4	3	no	—
91	C₂×Q₈	—	316	3	no	—
92	C₂×Q₈	—	316	6	no	—
93	C₂²×C₄	—	79	6	no	—
94	C₂²×C₄	C₂²×D₇₉	1	3	yes	—
95	C₂²×C₄	C₂²×D₇₉	1	3	yes	—
96	C₂²⋊C₄	—	4	3	no	—
97	C₂²⋊C₄	—	316	6	no	—
98	C₂²⋊C₄	—	316	6	no	—
99	C₂²⋊C₄	—	316	3	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	2	3	no	—
102	C₂³	—	316	12	no	—
103	C₂³	—	316	12	no	—
104	C₂³	—	316	4	no	—
105	C₂³	—	79	4	no	—
106	C₂³	—	158	6	no	—
107	C₂³	—	158	6	no	—
108	C₂³	Q₈×D₇₉	1	2	yes	—
109	C₂³	C₂³×D₇₉	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	316	12	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	158	6	no	—
117	C₂×C₄	—	158	6	no	—
118	C₂×C₄	—	158	6	no	—
119	C₂²	—	2	6	no	—
120	C₂²	—	4	3	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	316	16	no	—
123	C₂²	—	316	6	no	—
124	C₂²	—	316	2	no	—
125	C₂²	—	158	12	no	—
126	C₂²	C₂×Q₈×D₇₉	1	1	yes	characteristic
127	C₂²	#14579206	1	3	yes	—
128	C₂²	#14579207	1	3	yes	—
129	C₄	—	4	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	316	6	no	—
132	C₂	—	4	2	no	—
133	C₂	—	316	4	no	—
134	C₂	#19099188	1	3	yes	—
135	C₂	#19099305	1	3	yes	—
136	C₂	#19099306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098476

C₂

yes

maximal, complemented

#19098475

C₂

yes

maximal, complemented

#19098474

C₂

yes

maximal, complemented

#19098418

C₂

yes

maximal

#393×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₇₉

C₂²

yes

characteristic

C₂²×Dic₁₅₈

C₂²

yes

complemented

C₂×C₂²⋊Dic₇₉

C₂²

yes

—

C₂×(C₂×D₇₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₇₉)⋊C₄

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

complemented

C₇₉⋊#95

C₂²

yes

complemented

C₂²×Q₈×C₇₉

C₂²

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₇₉

C₂²

yes

—

#95×C₇₉

C₂²

yes

complemented

C₂³×D₇₉

—

C₂³×D₇₉

—

C₂³×D₇₉

Q₈

yes

complemented

C₂³×D₇₉

C₂³

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₂⁴×C₇₉

C₂³

yes

characteristic

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

C₂×Q₈

yes

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×Q₈

yes

characteristic

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#543

yes

—

C₂×C₇₉

H(3,𝔽₄)

yes

—

C₂×C₇₉

H(3,𝔽₄)

yes

characteristic

D₇₉

—

#2840

—

maximal, Sylow

C₇₉

#2840

yes

characteristic, complemented, Sylow

—

—

—

D₇₉

yes

characteristic, complemented

#379

—

C₂⁵

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₇₉

yes

—

#95

—

#95

—

#95

C₂×D₇₉

yes

complemented

C₂⁴

—

316

—

C₂⁴

—

316

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

316

—

C₂×Q₈

—

316

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

316

—

C₂²⋊C₄

—

316

—

C₂²⋊C₄

—

316

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

316

—

103

C₂³

—

316

—

104

C₂³

—

316

—

105

C₂³

—

106

C₂³

—

158

—

107

C₂³

—

158

—

108

C₂³

Q₈×D₇₉

yes

—

109

C₂³

C₂³×D₇₉

yes

characteristic

110

Q₈

—

111

Q₈

—

316

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

158

—

117

C₂×C₄

—

158

—

118

C₂×C₄

—

158

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

316

—

123

C₂²

—

316

—

124

C₂²

—

316

—

125

C₂²

—

158

—

126

C₂²

C₂×Q₈×D₇₉

yes

characteristic

127

C₂²

#14579206

yes

—

128

C₂²

#14579207

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

316

—

132

C₂

—

133

C₂

—

316

—

134

C₂

#19099188

yes

—

135

C₂

#19099305

yes

—

136

C₂

#19099306

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁴
Fitting	#393×C₇₉	C₂
prime radical	C₂⁴×D₇₉	C₂²
hypercenter	#393	D₇₉
hypocenter	C₇₉	#2840
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator, socle, cosocle

C₂³×C₇₉

C₂⁴

Fitting

#393×C₇₉

C₂

prime radical

C₂⁴×D₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
1. #19098476	C₂	semidirect
2. #19098475	C₂	semidirect
3. #19098474	C₂	semidirect
5. #393×C₇₉	C₂	semidirect
7. C₂²×Dic₁₅₈	C₂²	semidirect
11. C₇₉⋊#95	C₂²	semidirect
12. C₇₉⋊#95	C₂²	semidirect
13. C₂²×Q₈×C₇₉	C₂²	semidirect
15. #95×C₇₉	C₂²	semidirect
18. C₂³×D₇₉	Q₈	semidirect
69. C₇₉	#2840	semidirect
73. #393	D₇₉	semidirect
77. C₂²×Q₈	C₂×D₇₉	semidirect
84. #95	C₂×D₇₉	semidirect

H ≅ G/N

type

1. #19098476

C₂

semidirect

2. #19098475

C₂

semidirect

3. #19098474

C₂

semidirect

5. #393×C₇₉

C₂

semidirect

7. C₂²×Dic₁₅₈

C₂²

semidirect

11. C₇₉⋊#95

C₂²

semidirect

12. C₇₉⋊#95

C₂²

semidirect

13. C₂²×Q₈×C₇₉

C₂²

semidirect

15. #95×C₇₉

C₂²

semidirect

18. C₂³×D₇₉

Q₈

semidirect

69. C₇₉

#2840

semidirect

73. #393

D₇₉

semidirect

77. C₂²×Q₈

C₂×D₇₉

semidirect

84. #95

C₂×D₇₉

semidirect

Subgroups — #23771652

Conjugacy classes of subgroups

Complements