Group #30649066 — subgroups

Subgroups — C₂×#22550242 #30649066

Order 16256.

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×#16861206	C₂	1	1	yes	maximal, characteristic
2	#22550242	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16861129	C₂	1	4	yes	maximal, complemented
4	C₂×#16861120	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₀₈	—	2	2	no	—
7	C₂²×D₅₀₈	C₂²	1	2	yes	—
8	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	#16861206	—	2	2	no	—
10	#16861206	C₂²	1	4	yes	—
11	#16861129	C₂²	1	16	yes	complemented
12	#16861120	C₂²	1	4	yes	complemented
13	#125×C₁₂₇	—	2	2	no	—
14	#125×C₁₂₇	C₂²	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂³×D₁₂₇	—	2	2	no	—
19	C₂×D₅₀₈	—	2	2	no	—
20	C₂×D₅₀₈	—	2	8	no	—
21	C₂×D₅₀₈	—	2	8	no	—
22	C₂×D₅₀₈	—	4	8	no	—
23	C₂×D₅₀₈	C₂³	1	4	yes	—
24	C₂×C₁₂₇⋊C₈	—	2	4	no	—
25	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₂₇	—	2	2	no	—
27	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
28	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₂₇	—	2	8	no	—
30	C₄⋊C₄×C₁₂₇	—	2	2	no	—
31	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
32	C₄²×C₁₂₇	D₄	1	2	yes	—
33	C₄²×C₁₂₇	C₂³	1	2	yes	—
34	C₂²×D₁₂₇	—	2	4	no	—
35	C₂²×D₁₂₇	—	4	4	no	—
36	C₂²×D₁₂₇	—	4	8	no	—
37	D₅₀₈	—	2	8	no	—
38	D₅₀₈	—	4	16	no	—
39	D₅₀₈	—	4	4	no	—
40	C₁₂₇⋊C₈	—	2	4	no	—
41	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	—	2	4	no	—
43	C₂×C₄×C₁₂₇	—	2	4	no	—
44	C₂×C₄×C₁₂₇	—	4	4	no	—
45	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	#41	1	4	yes	—
48	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
49	C₂²×C₁₂₇	#124	1	2	yes	—
50	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
51	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₁₂₇	—	4	4	no	—
53	C₂×D₁₂₇	—	4	8	no	—
54	C₂×D₁₂₇	—	8	8	no	—
55	C₄×C₁₂₇	—	2	4	no	—
56	C₄×C₁₂₇	—	4	4	no	—
57	C₄×C₁₂₇	#124	1	2	yes	—
58	C₄×C₁₂₇	C₂×#41	1	2	yes	—
59	C₂×C₁₂₇	#490	1	4	yes	complemented
60	C₂×C₁₂₇	C₂×#124	1	1	yes	characteristic
61	C₂×C₁₂₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
62	D₁₂₇	—	8	4	no	—
63	C₂×#490	—	127	1	no	maximal, Sylow
64	C₁₂₇	C₂×#490	1	1	yes	characteristic, complemented, Sylow
65	C₂×#127	—	127	1	no	—
66	C₂×#125	D₁₂₇	1	1	yes	characteristic, complemented
67	#490	—	127	8	no	—
68	C₂×#102	—	127	4	no	—
69	C₂×#97	—	127	1	no	—
70	C₂²×D₄	—	127	2	no	—
71	C₂²×D₄	—	254	2	no	—
72	C₂²×C₈	—	127	2	no	—
73	#127	—	127	4	no	—
74	#127	—	254	2	no	—
75	#125	—	2	2	no	—
76	#125	C₂×D₁₂₇	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
79	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
80	#102	—	127	16	no	—
81	#97	—	127	4	no	—
82	C₂⁴	—	254	2	no	—
83	C₂×D₄	—	127	4	no	—
84	C₂×D₄	—	508	8	no	—
85	C₂×D₄	—	254	8	no	—
86	C₂×D₄	—	254	2	no	—
87	C₂×D₄	—	254	8	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₂×C₈	—	127	4	no	—
92	C₂×C₈	—	254	4	no	—
93	C₄⋊C₄	—	2	8	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
96	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
97	C₄²	C₂²×D₁₂₇	1	2	yes	—
98	C₂³	—	508	4	no	—
99	C₂³	—	508	8	no	—
100	C₂³	—	254	4	no	—
101	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
102	D₄	—	508	16	no	—
103	D₄	—	508	4	no	—
104	D₄	—	254	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⋊D₁₂₇	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
110	C₂×C₄	#14523249	1	4	yes	—
111	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
112	C₈	—	254	4	no	—
113	C₂²	—	1016	8	no	—
114	C₂²	—	508	4	no	—
115	C₂²	—	508	8	no	—
116	C₂²	#16861198	1	2	yes	—
117	C₂²	#16861234	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	4	4	no	—
121	C₄	#16861198	1	2	yes	—
122	C₄	C₂×#14523249	1	2	yes	—
123	C₂	—	1016	4	no	—
124	C₂	#22550242	1	4	yes	direct product factor
125	C₂	C₂×#16861198	1	1	yes	characteristic
126	C₂	C₂×#16861234	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861206

C₂

yes

maximal, characteristic

#22550242

C₂

yes

maximal, direct product factor

C₂×#16861129

C₂

yes

maximal, complemented

C₂×#16861120

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₀₈

—

C₂²×D₅₀₈

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

#16861206

—

#16861206

C₂²

yes

—

#16861129

C₂²

yes

complemented

#16861120

C₂²

yes

complemented

#125×C₁₂₇

—

#125×C₁₂₇

C₂²

yes

complemented

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂³×D₁₂₇

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

#124

yes

—

C₂²×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#124

yes

—

C₄×C₁₂₇

C₂×#41

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

D₁₂₇

—

C₂×#490

—

127

maximal, Sylow

C₁₂₇

C₂×#490

yes

characteristic, complemented, Sylow

C₂×#127

—

127

—

C₂×#125

D₁₂₇

yes

characteristic, complemented

#490

—

127

—

C₂×#102

—

127

—

C₂×#97

—

127

—

C₂²×D₄

—

127

—

C₂²×D₄

—

254

—

C₂²×C₈

—

127

—

#127

—

127

—

#127

—

254

—

#125

—

#125

C₂×D₁₂₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#102

—

127

—

#97

—

127

—

C₂⁴

—

254

—

C₂×D₄

—

127

—

C₂×D₄

—

508

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₁₂₇

yes

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₄²

C₂²×D₁₂₇

yes

—

C₂³

—

508

—

C₂³

—

508

—

100

C₂³

—

254

—

101

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

102

D₄

—

508

—

103

D₄

—

508

—

104

D₄

—

254

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

110

C₂×C₄

#14523249

yes

—

111

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

112

C₈

—

254

—

113

C₂²

—

1016

—

114

C₂²

—

508

—

115

C₂²

—

508

—

116

C₂²

#16861198

yes

—

117

C₂²

#16861234

yes

—

118

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#16861198

yes

—

122

C₄

C₂×#14523249

yes

—

123

C₂

—

1016

—

124

C₂

#22550242

yes

direct product factor

125

C₂

C₂×#16861198

yes

characteristic

126

C₂

C₂×#16861234

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#125×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×D₄
prime radical	C₂×#16861206	C₂
hypercenter	C₂×#125	D₁₂₇
hypocenter	C₁₂₇	C₂×#490
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×#125×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂×D₄

C₂

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #22550242	C₂	direct (N × H)
3. C₂×#16861129	C₂	semidirect
4. C₂×#16861120	C₂	semidirect
5. C₂×#125×C₁₂₇	C₂	semidirect
11. #16861129	C₂²	semidirect
12. #16861120	C₂²	semidirect
14. #125×C₁₂₇	C₂²	semidirect
59. C₂×C₁₂₇	#490	semidirect
64. C₁₂₇	C₂×#490	semidirect
66. C₂×#125	D₁₂₇	semidirect
76. #125	C₂×D₁₂₇	semidirect
124. C₂	#22550242	direct (N × H)

H ≅ G/N

type

2. #22550242

C₂

direct (N × H)

3. C₂×#16861129

C₂

semidirect

4. C₂×#16861120

C₂

semidirect

5. C₂×#125×C₁₂₇

C₂

semidirect

11. #16861129

C₂²

semidirect

12. #16861120

C₂²

semidirect

14. #125×C₁₂₇

C₂²

semidirect

59. C₂×C₁₂₇

#490

semidirect

64. C₁₂₇

C₂×#490

semidirect

66. C₂×#125

D₁₂₇

semidirect

76. #125

C₂×D₁₂₇

semidirect

124. C₂

#22550242

direct (N × H)

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

subgroup	index
C₂×#16861206	2
#22550242	2
C₂×#16861129	2
C₂×#16861120	2
C₂×#125×C₁₂₇	2
C₂×#490	127

subgroup

index

C₂×#125×C₁₂₇

127

Subgroups — C₂×#22550242 #30649066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22550242 #30649066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550242 #30649066