Group #23773736 — subgroups

Subgroups — C₂×#19098929 #23773736

Order 10112.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14579158	C₂	1	1	yes	maximal, characteristic
2	#19098929	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14579081	C₂	1	4	yes	maximal, complemented
4	C₂×#14579072	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₁₆	—	2	2	no	—
7	C₂²×D₃₁₆	C₂²	1	2	yes	—
8	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
9	#14579158	—	2	2	no	—
10	#14579158	C₂²	1	4	yes	—
11	#14579081	C₂²	1	16	yes	complemented
12	#14579072	C₂²	1	4	yes	complemented
13	#125×C₇₉	—	2	2	no	—
14	#125×C₇₉	C₂²	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
16	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
18	C₂³×D₇₉	—	2	2	no	—
19	C₂×D₃₁₆	—	2	2	no	—
20	C₂×D₃₁₆	—	2	8	no	—
21	C₂×D₃₁₆	—	2	8	no	—
22	C₂×D₃₁₆	—	4	8	no	—
23	C₂×D₃₁₆	C₂³	1	4	yes	—
24	C₂×C₇₉⋊C₈	—	2	4	no	—
25	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
26	C₂²×C₄×C₇₉	—	2	2	no	—
27	C₂²×C₄×C₇₉	D₄	1	2	yes	—
28	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₉	—	2	8	no	—
30	C₄⋊C₄×C₇₉	—	2	2	no	—
31	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
32	C₄²×C₇₉	D₄	1	2	yes	—
33	C₄²×C₇₉	C₂³	1	2	yes	—
34	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×D₇₉	—	2	4	no	—
36	C₂²×D₇₉	—	4	4	no	—
37	C₂²×D₇₉	—	4	8	no	—
38	C₂×C₄×C₇₉	—	2	4	no	—
39	C₂×C₄×C₇₉	—	2	4	no	—
40	C₂×C₄×C₇₉	—	4	4	no	—
41	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	#41	1	4	yes	—
44	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
45	D₃₁₆	—	2	8	no	—
46	D₃₁₆	—	4	16	no	—
47	D₃₁₆	—	4	4	no	—
48	C₇₉⋊C₈	—	2	4	no	—
49	C₂²×C₇₉	#124	1	2	yes	—
50	C₂²×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
51	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₇₉	—	4	4	no	—
53	C₂×D₇₉	—	4	8	no	—
54	C₂×D₇₉	—	8	8	no	—
55	C₄×C₇₉	—	2	4	no	—
56	C₄×C₇₉	—	4	4	no	—
57	C₄×C₇₉	#124	1	2	yes	—
58	C₄×C₇₉	C₂×#41	1	2	yes	—
59	C₂×C₇₉	#490	1	4	yes	complemented
60	C₂×C₇₉	C₂×#124	1	1	yes	characteristic
61	C₂×C₇₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
62	D₇₉	—	8	4	no	—
63	C₂×#490	—	79	1	no	maximal, Sylow
64	C₇₉	C₂×#490	1	1	yes	characteristic, complemented, Sylow
65	C₂×#127	—	79	1	no	—
66	C₂×#125	D₇₉	1	1	yes	characteristic, complemented
67	#490	—	79	8	no	—
68	C₂×#102	—	79	4	no	—
69	C₂×#97	—	79	1	no	—
70	C₂²×D₄	—	79	2	no	—
71	C₂²×D₄	—	158	2	no	—
72	C₂²×C₈	—	79	2	no	—
73	#127	—	79	4	no	—
74	#127	—	158	2	no	—
75	#125	—	2	2	no	—
76	#125	C₂×D₇₉	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
79	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
80	#102	—	79	16	no	—
81	#97	—	79	4	no	—
82	C₂⁴	—	158	2	no	—
83	C₂×D₄	—	316	8	no	—
84	C₂×D₄	—	79	4	no	—
85	C₂×D₄	—	158	8	no	—
86	C₂×D₄	—	158	2	no	—
87	C₂×D₄	—	158	8	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
90	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
91	C₂×C₈	—	79	4	no	—
92	C₂×C₈	—	158	4	no	—
93	C₄⋊C₄	—	2	8	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
96	C₄²	C₂²⋊D₇₉	1	2	yes	—
97	C₄²	C₂²×D₇₉	1	2	yes	—
98	C₂³	—	316	4	no	—
99	C₂³	—	316	8	no	—
100	C₂³	—	158	4	no	—
101	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
102	D₄	—	316	16	no	—
103	D₄	—	316	4	no	—
104	D₄	—	158	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	C₂×C₂²⋊D₇₉	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂×C₄	#11929625	1	4	yes	—
111	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
112	C₈	—	158	4	no	—
113	C₂²	—	316	4	no	—
114	C₂²	—	316	8	no	—
115	C₂²	—	632	8	no	—
116	C₂²	#14579150	1	2	yes	—
117	C₂²	#14579186	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	4	4	no	—
121	C₄	#14579150	1	2	yes	—
122	C₄	C₂×#11929625	1	2	yes	—
123	C₂	—	632	4	no	—
124	C₂	#19098929	1	4	yes	direct product factor
125	C₂	C₂×#14579150	1	1	yes	characteristic
126	C₂	C₂×#14579186	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579158

C₂

yes

maximal, characteristic

#19098929

C₂

yes

maximal, direct product factor

C₂×#14579081

C₂

yes

maximal, complemented

C₂×#14579072

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₁₆

—

C₂²×D₃₁₆

C₂²

yes

—

C₂²×C₇₉⋊C₈

C₂²

yes

—

#14579158

—

#14579158

C₂²

yes

—

#14579081

C₂²

yes

complemented

#14579072

C₂²

yes

complemented

#125×C₇₉

—

#125×C₇₉

C₂²

yes

complemented

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₇₉⋊C₈

—

C₂²×C₇₉

#124

yes

—

C₂²×C₇₉

AGL(1,ℤ/8)

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#124

yes

—

C₄×C₇₉

C₂×#41

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

C₂×#490

—

maximal, Sylow

C₇₉

C₂×#490

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#125

D₇₉

yes

characteristic, complemented

#490

—

C₂×#102

—

C₂×#97

—

C₂²×D₄

—

C₂²×D₄

—

158

—

C₂²×C₈

—

#127

—

#127

—

158

—

#125

—

#125

C₂×D₇₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#102

—

#97

—

C₂⁴

—

158

—

C₂×D₄

—

316

—

C₂×D₄

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₇₉

yes

—

C₄²

C₂²⋊D₇₉

yes

—

C₄²

C₂²×D₇₉

yes

—

C₂³

—

316

—

C₂³

—

316

—

100

C₂³

—

158

—

101

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

102

D₄

—

316

—

103

D₄

—

316

—

104

D₄

—

158

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

109

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

110

C₂×C₄

#11929625

yes

—

111

C₂×C₄

C₂³×D₇₉

yes

characteristic

112

C₈

—

158

—

113

C₂²

—

316

—

114

C₂²

—

316

—

115

C₂²

—

632

—

116

C₂²

#14579150

yes

—

117

C₂²

#14579186

yes

—

118

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#14579150

yes

—

122

C₄

C₂×#11929625

yes

—

123

C₂

—

632

—

124

C₂

#19098929

yes

direct product factor

125

C₂

C₂×#14579150

yes

characteristic

126

C₂

C₂×#14579186

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×#125×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
prime radical	C₂×#14579158	C₂
hypercenter	C₂×#125	D₇₉
hypocenter	C₇₉	C₂×#490
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×#125×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

C₂

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #19098929	C₂	direct (N × H)
3. C₂×#14579081	C₂	semidirect
4. C₂×#14579072	C₂	semidirect
5. C₂×#125×C₇₉	C₂	semidirect
11. #14579081	C₂²	semidirect
12. #14579072	C₂²	semidirect
14. #125×C₇₉	C₂²	semidirect
59. C₂×C₇₉	#490	semidirect
64. C₇₉	C₂×#490	semidirect
66. C₂×#125	D₇₉	semidirect
76. #125	C₂×D₇₉	semidirect
124. C₂	#19098929	direct (N × H)

H ≅ G/N

type

2. #19098929

C₂

direct (N × H)

3. C₂×#14579081

C₂

semidirect

4. C₂×#14579072

C₂

semidirect

5. C₂×#125×C₇₉

C₂

semidirect

11. #14579081

C₂²

semidirect

12. #14579072

C₂²

semidirect

14. #125×C₇₉

C₂²

semidirect

59. C₂×C₇₉

#490

semidirect

64. C₇₉

C₂×#490

semidirect

66. C₂×#125

D₇₉

semidirect

76. #125

C₂×D₇₉

semidirect

124. C₂

#19098929

direct (N × H)

Subgroups — C₂×#19098929 #23773736

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098929 #23773736

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098929 #23773736