Group #30676297 — subgroups

Subgroups — C₂×#22553678 #30676297

Order 16768.

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×#16862243	C₂	1	1	yes	maximal, characteristic
2	#22553678	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16862166	C₂	1	4	yes	maximal, complemented
4	C₂×#16862157	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₂₄	—	2	2	no	—
7	C₂²×D₅₂₄	C₂²	1	2	yes	—
8	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
9	#16862243	—	2	2	no	—
10	#16862243	C₂²	1	4	yes	—
11	#16862166	C₂²	1	16	yes	complemented
12	#16862157	C₂²	1	4	yes	complemented
13	#125×C₁₃₁	—	2	2	no	—
14	#125×C₁₃₁	C₂²	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
16	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂³×D₁₃₁	—	2	2	no	—
19	C₂×D₅₂₄	—	2	2	no	—
20	C₂×D₅₂₄	—	2	8	no	—
21	C₂×D₅₂₄	—	2	8	no	—
22	C₂×D₅₂₄	—	4	8	no	—
23	C₂×D₅₂₄	C₂³	1	4	yes	—
24	C₂×C₁₃₁⋊C₈	—	2	4	no	—
25	C₂×C₁₃₁⋊C₈	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₃₁	—	2	2	no	—
27	C₂²×C₄×C₁₃₁	D₄	1	2	yes	—
28	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₃₁	—	2	8	no	—
30	C₄⋊C₄×C₁₃₁	—	2	2	no	—
31	C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
32	C₄²×C₁₃₁	D₄	1	2	yes	—
33	C₄²×C₁₃₁	C₂³	1	2	yes	—
34	C₂²×D₁₃₁	—	2	4	no	—
35	C₂²×D₁₃₁	—	4	4	no	—
36	C₂²×D₁₃₁	—	4	8	no	—
37	D₅₂₄	—	2	8	no	—
38	D₅₂₄	—	4	16	no	—
39	D₅₂₄	—	4	4	no	—
40	C₁₃₁⋊C₈	—	2	4	no	—
41	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₁	—	2	4	no	—
43	C₂×C₄×C₁₃₁	—	2	4	no	—
44	C₂×C₄×C₁₃₁	—	4	4	no	—
45	C₂×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₁	#41	1	4	yes	—
48	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
49	C₂²×C₁₃₁	#124	1	2	yes	—
50	C₂²×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
51	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₁₃₁	—	4	4	no	—
53	C₂×D₁₃₁	—	4	8	no	—
54	C₂×D₁₃₁	—	8	8	no	—
55	C₄×C₁₃₁	—	2	4	no	—
56	C₄×C₁₃₁	—	4	4	no	—
57	C₄×C₁₃₁	#124	1	2	yes	—
58	C₄×C₁₃₁	C₂×#41	1	2	yes	—
59	C₂×C₁₃₁	#490	1	4	yes	complemented
60	C₂×C₁₃₁	C₂×#124	1	1	yes	characteristic
61	C₂×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
62	D₁₃₁	—	8	4	no	—
63	C₁₃₁	C₂×#490	1	1	yes	characteristic, complemented, Sylow
64	C₂×#490	—	131	1	no	maximal, Sylow
65	C₂×#127	—	131	1	no	—
66	C₂×#125	D₁₃₁	1	1	yes	characteristic, complemented
67	#490	—	131	8	no	—
68	C₂×#102	—	131	4	no	—
69	C₂×#97	—	131	1	no	—
70	C₂²×D₄	—	262	2	no	—
71	C₂²×D₄	—	131	2	no	—
72	C₂²×C₈	—	131	2	no	—
73	#127	—	262	2	no	—
74	#127	—	131	4	no	—
75	#125	—	2	2	no	—
76	#125	C₂×D₁₃₁	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂×D₁₃₁	1	2	yes	—
79	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
80	#102	—	131	16	no	—
81	#97	—	131	4	no	—
82	C₂⁴	—	262	2	no	—
83	C₂×D₄	—	262	2	no	—
84	C₂×D₄	—	262	8	no	—
85	C₂×D₄	—	262	8	no	—
86	C₂×D₄	—	524	8	no	—
87	C₂×D₄	—	131	4	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂²⋊D₁₃₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
91	C₂×C₈	—	262	4	no	—
92	C₂×C₈	—	131	4	no	—
93	C₄⋊C₄	—	2	8	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₂²×D₁₃₁	1	4	yes	—
96	C₄²	C₂²⋊D₁₃₁	1	2	yes	—
97	C₄²	C₂²×D₁₃₁	1	2	yes	—
98	C₂³	—	262	4	no	—
99	C₂³	—	524	4	no	—
100	C₂³	—	524	8	no	—
101	C₂³	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
102	D₄	—	262	8	no	—
103	D₄	—	524	16	no	—
104	D₄	—	524	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
110	C₂×C₄	#14523514	1	4	yes	—
111	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
112	C₈	—	262	4	no	—
113	C₂²	—	524	4	no	—
114	C₂²	—	524	8	no	—
115	C₂²	—	1048	8	no	—
116	C₂²	#16862235	1	2	yes	—
117	C₂²	#16862271	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₁₃₁	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	4	4	no	—
121	C₄	#16862235	1	2	yes	—
122	C₄	C₂×#14523514	1	2	yes	—
123	C₂	—	1048	4	no	—
124	C₂	#22553678	1	4	yes	direct product factor
125	C₂	C₂×#16862235	1	1	yes	characteristic
126	C₂	C₂×#16862271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16862243

C₂

yes

maximal, characteristic

#22553678

C₂

yes

maximal, direct product factor

C₂×#16862166

C₂

yes

maximal, complemented

C₂×#16862157

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₂₄

—

C₂²×D₅₂₄

C₂²

yes

—

C₂²×C₁₃₁⋊C₈

C₂²

yes

—

#16862243

—

#16862243

C₂²

yes

—

#16862166

C₂²

yes

complemented

#16862157

C₂²

yes

complemented

#125×C₁₃₁

—

#125×C₁₃₁

C₂²

yes

complemented

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₂³×D₁₃₁

—

C₂×D₅₂₄

—

C₂×D₅₂₄

—

C₂×D₅₂₄

—

C₂×D₅₂₄

—

C₂×D₅₂₄

C₂³

yes

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

D₄

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

C₂³

yes

—

C₄²×C₁₃₁

D₄

yes

—

C₄²×C₁₃₁

C₂³

yes

—

C₂²×D₁₃₁

—

C₂²×D₁₃₁

—

C₂²×D₁₃₁

—

D₅₂₄

—

D₅₂₄

—

D₅₂₄

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

#41

yes

—

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₂²×C₁₃₁

#124

yes

—

C₂²×C₁₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃₁

C₂²×D₄

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#124

yes

—

C₄×C₁₃₁

C₂×#41

yes

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

—

D₁₃₁

—

C₁₃₁

C₂×#490

yes

characteristic, complemented, Sylow

C₂×#490

—

131

maximal, Sylow

C₂×#127

—

131

—

C₂×#125

D₁₃₁

yes

characteristic, complemented

#490

—

131

—

C₂×#102

—

131

—

C₂×#97

—

131

—

C₂²×D₄

—

262

—

C₂²×D₄

—

131

—

C₂²×C₈

—

131

—

#127

—

262

—

#127

—

131

—

#125

—

#125

C₂×D₁₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃₁

yes

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

#102

—

131

—

#97

—

131

—

C₂⁴

—

262

—

C₂×D₄

—

262

—

C₂×D₄

—

262

—

C₂×D₄

—

262

—

C₂×D₄

—

524

—

C₂×D₄

—

131

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃₁

yes

—

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

C₂×C₈

—

262

—

C₂×C₈

—

131

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₁₃₁

yes

—

C₄²

C₂²⋊D₁₃₁

yes

—

C₄²

C₂²×D₁₃₁

yes

—

C₂³

—

262

—

C₂³

—

524

—

100

C₂³

—

524

—

101

C₂³

C₂×C₂²⋊D₁₃₁

yes

characteristic

102

D₄

—

262

—

103

D₄

—

524

—

104

D₄

—

524

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

110

C₂×C₄

#14523514

yes

—

111

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

112

C₈

—

262

—

113

C₂²

—

524

—

114

C₂²

—

524

—

115

C₂²

—

1048

—

116

C₂²

#16862235

yes

—

117

C₂²

#16862271

yes

—

118

C₂²

C₂²×C₂²⋊D₁₃₁

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#16862235

yes

—

122

C₄

C₂×#14523514

yes

—

123

C₂

—

1048

—

124

C₂

#22553678

yes

direct product factor

125

C₂

C₂×#16862235

yes

characteristic

126

C₂

C₂×#16862271

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	C₂×#125×C₁₃₁	C₂
socle	C₂³×C₁₃₁	C₂×D₄
prime radical	C₂×#16862243	C₂
hypercenter	C₂×#125	D₁₃₁
hypocenter	C₁₃₁	C₂×#490
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₁

commutator, cosocle

C₂×C₄×C₁₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₁

Fitting

C₂×#125×C₁₃₁

C₂

socle

C₂³×C₁₃₁

C₂×D₄

C₂

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #22553678	C₂	direct (N × H)
3. C₂×#16862166	C₂	semidirect
4. C₂×#16862157	C₂	semidirect
5. C₂×#125×C₁₃₁	C₂	semidirect
11. #16862166	C₂²	semidirect
12. #16862157	C₂²	semidirect
14. #125×C₁₃₁	C₂²	semidirect
59. C₂×C₁₃₁	#490	semidirect
63. C₁₃₁	C₂×#490	semidirect
66. C₂×#125	D₁₃₁	semidirect
76. #125	C₂×D₁₃₁	semidirect
124. C₂	#22553678	direct (N × H)

H ≅ G/N

type

2. #22553678

C₂

direct (N × H)

3. C₂×#16862166

C₂

semidirect

4. C₂×#16862157

C₂

semidirect

5. C₂×#125×C₁₃₁

C₂

semidirect

11. #16862166

C₂²

semidirect

12. #16862157

C₂²

semidirect

14. #125×C₁₃₁

C₂²

semidirect

59. C₂×C₁₃₁

#490

semidirect

63. C₁₃₁

C₂×#490

semidirect

66. C₂×#125

D₁₃₁

semidirect

76. #125

C₂×D₁₃₁

semidirect

124. C₂

#22553678

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16862243	2
#22553678	2
C₂×#16862166	2
C₂×#16862157	2
C₂×#125×C₁₃₁	2
C₂×#490	131

subgroup

index

C₂×#125×C₁₃₁

131

Subgroups — C₂×#22553678 #30676297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22553678 #30676297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22553678 #30676297