Group #46830919 — subgroups

Subgroups — #46830919

Order 34624.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#31772620	C₂	1	1	yes	maximal, characteristic, complemented
2	#31772616	C₂	1	1	yes	maximal, characteristic, complemented
3	#31772612	C₂	1	2	yes	maximal, complemented
4	#31772610	C₂	1	2	yes	maximal, complemented
5	#31772605	C₂	1	2	yes	maximal, complemented
6	#31772604	C₂	1	2	yes	maximal, complemented
7	#31772601	C₂	1	2	yes	maximal, complemented
8	#31772600	C₂	1	2	yes	maximal, complemented
9	#124×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₀₈₂	C₂²	1	1	yes	characteristic
11	C₂×D₂₁₆₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₅₄₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₅₄₁	C₂²	1	2	yes	—
14	C₂²×Dic₅₄₁	C₂²	1	2	yes	—
15	C₂×C₄×D₅₄₁	C₂²	1	2	yes	—
16	C₄⋊Dic₅₄₁	C₂²	1	2	yes	—
17	C₄⋊Dic₅₄₁	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₅₄₁	C₂²	1	1	yes	characteristic
19	#22575815	C₂²	1	2	yes	—
20	#22575815	C₂²	1	2	yes	—
21	#22575815	C₂²	1	2	yes	—
22	C₂²⋊Dic₅₄₁	C₂²	1	2	yes	—
23	#22575813	C₂²	1	2	yes	—
24	#22575813	C₂²	1	2	yes	—
25	#22575813	C₂²	1	2	yes	—
26	C₂×Q₈×C₅₄₁	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₅₄₁	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₅₄₁	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₅₄₁	C₂²	1	2	yes	—
30	C₄²×C₅₄₁	C₂²	1	1	yes	characteristic
31	C₂²×D₅₄₁	C₂³	1	2	yes	—
32	Dic₁₀₈₂	—	4	1	no	—
33	D₂₁₆₄	—	4	1	no	—
34	C₂²⋊D₅₄₁	—	4	2	no	—
35	C₂²⋊D₅₄₁	—	4	2	no	—
36	C₂×Dic₅₄₁	—	4	2	no	—
37	C₂×Dic₅₄₁	C₂³	1	2	yes	—
38	C₂×Dic₅₄₁	C₂³	1	2	yes	—
39	C₂×Dic₅₄₁	C₂³	1	2	yes	—
40	C₄×D₅₄₁	—	4	2	no	—
41	C₂³×C₅₄₁	C₂³	1	2	yes	—
42	Q₈×C₅₄₁	—	4	1	no	—
43	D₄×C₅₄₁	—	4	1	no	—
44	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₅₄₁	C₂³	1	2	yes	—
48	C₂²×C₅₄₁	—	2	2	no	—
49	C₂²×C₅₄₁	—	2	2	no	—
50	C₂²×C₅₄₁	—	2	2	no	—
51	C₂²×C₅₄₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₅₄₁	—	2	2	no	—
53	C₂×D₅₄₁	—	2	2	no	—
54	C₂×D₅₄₁	—	2	2	no	—
55	C₄×C₅₄₁	—	2	2	no	—
56	C₄×C₅₄₁	—	2	1	no	—
57	C₄×C₅₄₁	—	2	1	no	—
58	C₄×C₅₄₁	—	2	1	no	—
59	Dic₅₄₁	—	2	2	no	—
60	Dic₅₄₁	—	2	2	no	—
61	Dic₅₄₁	—	2	2	no	—
62	C₂×C₅₄₁	—	4	2	no	—
63	C₂×C₅₄₁	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₅₄₁	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₅₄₁	2₋¹⁺⁴	1	1	yes	characteristic
66	D₅₄₁	—	4	2	no	—
67	C₅₄₁	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	541	1	no	maximal, Sylow
69	#125	—	541	1	no	—
70	#124	—	541	1	no	—
71	#124	D₅₄₁	1	1	yes	characteristic, complemented
72	#123	—	541	2	no	—
73	#123	—	541	2	no	—
74	C₂²⋊Q₈	—	541	2	no	—
75	C₂²⋊Q₈	—	541	2	no	—
76	#121	—	541	2	no	—
77	#121	—	541	2	no	—
78	C₂×Q₈	—	541	1	no	—
79	C₂×Q₈	C₂×D₅₄₁	1	1	yes	characteristic
80	C₂×D₄	—	541	2	no	—
81	C₂×D₄	—	541	2	no	—
82	C₂×D₄	—	541	1	no	—
83	C₂×D₄	C₂×D₅₄₁	1	1	yes	characteristic
84	C₂²×C₄	—	541	2	no	—
85	C₂²×C₄	—	541	2	no	—
86	C₄⋊C₄	—	541	2	no	—
87	C₄⋊C₄	—	541	2	no	—
88	C₄⋊C₄	—	541	2	no	—
89	C₄⋊C₄	—	541	2	no	—
90	C₄⋊C₄	—	541	1	no	—
91	C₄⋊C₄	—	541	1	no	—
92	C₂²⋊C₄	—	541	2	no	—
93	C₂²⋊C₄	—	541	2	no	—
94	C₂²⋊C₄	—	541	2	no	—
95	C₂²⋊C₄	—	541	2	no	—
96	C₂²⋊C₄	C₂×D₅₄₁	1	2	yes	—
97	C₂²⋊C₄	C₂×D₅₄₁	1	2	yes	—
98	C₄²	C₂×D₅₄₁	1	1	yes	characteristic
99	C₂³	—	541	2	no	—
100	C₂³	C₂²×D₅₄₁	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	2164	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	2164	1	no	—
105	D₄	—	2164	2	no	—
106	D₄	—	2164	2	no	—
107	C₂×C₄	—	541	2	no	—
108	C₂×C₄	—	541	2	no	—
109	C₂×C₄	—	541	2	no	—
110	C₂×C₄	—	2164	2	no	—
111	C₂×C₄	—	2164	2	no	—
112	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₅₄₁	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	1082	2	no	—
120	C₂²	—	1082	2	no	—
121	C₂²	—	1082	2	no	—
122	C₂²	C₂³×D₅₄₁	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	1082	2	no	—
128	C₄	—	1082	2	no	—
129	C₄	—	1082	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	2164	2	no	—
132	C₂	#31772673	1	1	yes	characteristic
133	C₂	#31772674	1	1	yes	characteristic
134	C₂	#31772675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31772620

C₂

yes

maximal, characteristic, complemented

#31772616

C₂

yes

maximal, characteristic, complemented

#31772612

C₂

yes

maximal, complemented

#31772610

C₂

yes

maximal, complemented

#31772605

C₂

yes

maximal, complemented

#31772604

C₂

yes

maximal, complemented

#31772601

C₂

yes

maximal, complemented

#31772600

C₂

yes

maximal, complemented

#124×C₅₄₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₀₈₂

C₂²

yes

characteristic

C₂×D₂₁₆₄

C₂²

yes

characteristic

C₂×C₂²⋊D₅₄₁

C₂²

yes

—

C₂×C₂²⋊D₅₄₁

C₂²

yes

—

C₂²×Dic₅₄₁

C₂²

yes

—

C₂×C₄×D₅₄₁

C₂²

yes

—

C₄⋊Dic₅₄₁

C₂²

yes

—

C₄⋊Dic₅₄₁

C₂²

yes

characteristic

C₄⋊Dic₅₄₁

C₂²

yes

characteristic

#22575815

C₂²

yes

—

#22575815

C₂²

yes

—

#22575815

C₂²

yes

—

C₂²⋊Dic₅₄₁

C₂²

yes

—

#22575813

C₂²

yes

—

#22575813

C₂²

yes

—

#22575813

C₂²

yes

—

C₂×Q₈×C₅₄₁

C₂²

yes

characteristic

C₂×D₄×C₅₄₁

C₂²

yes

characteristic

C₂²⋊C₄×C₅₄₁

C₂²

yes

—

C₂²⋊C₄×C₅₄₁

C₂²

yes

—

C₄²×C₅₄₁

C₂²

yes

characteristic

C₂²×D₅₄₁

C₂³

yes

—

Dic₁₀₈₂

—

D₂₁₆₄

—

C₂²⋊D₅₄₁

—

C₂²⋊D₅₄₁

—

C₂×Dic₅₄₁

—

C₂×Dic₅₄₁

C₂³

yes

—

C₂×Dic₅₄₁

C₂³

yes

—

C₂×Dic₅₄₁

C₂³

yes

—

C₄×D₅₄₁

—

C₂³×C₅₄₁

C₂³

yes

—

Q₈×C₅₄₁

—

D₄×C₅₄₁

—

C₂×C₄×C₅₄₁

C₂³

yes

characteristic

C₂×C₄×C₅₄₁

C₂³

yes

characteristic

C₂×C₄×C₅₄₁

C₂³

yes

characteristic

C₂×C₄×C₅₄₁

C₂³

yes

—

C₂²×C₅₄₁

—

C₂²×C₅₄₁

—

C₂²×C₅₄₁

—

C₂²×C₅₄₁

C₂⁴

yes

characteristic

C₂×D₅₄₁

—

C₂×D₅₄₁

—

C₂×D₅₄₁

—

C₄×C₅₄₁

—

C₄×C₅₄₁

—

C₄×C₅₄₁

—

C₄×C₅₄₁

—

Dic₅₄₁

—

Dic₅₄₁

—

Dic₅₄₁

—

C₂×C₅₄₁

—

C₂×C₅₄₁

yes

C₂×C₅₄₁

yes

C₂×C₅₄₁

2₋¹⁺⁴

yes

characteristic

D₅₄₁

—

C₅₄₁

#562

yes

characteristic, complemented, Sylow

#562

—

541

maximal, Sylow

#125

—

541

—

#124

—

541

—

#124

D₅₄₁

yes

characteristic, complemented

#123

—

541

—

#123

—

541

—

C₂²⋊Q₈

—

541

—

C₂²⋊Q₈

—

541

—

#121

—

541

—

#121

—

541

—

C₂×Q₈

—

541

—

C₂×Q₈

C₂×D₅₄₁

yes

characteristic

C₂×D₄

—

541

—

C₂×D₄

—

541

—

C₂×D₄

—

541

—

C₂×D₄

C₂×D₅₄₁

yes

characteristic

C₂²×C₄

—

541

—

C₂²×C₄

—

541

—

C₄⋊C₄

—

541

—

C₄⋊C₄

—

541

—

C₄⋊C₄

—

541

—

C₄⋊C₄

—

541

—

C₄⋊C₄

—

541

—

C₄⋊C₄

—

541

—

C₂²⋊C₄

—

541

—

C₂²⋊C₄

—

541

—

C₂²⋊C₄

—

541

—

C₂²⋊C₄

—

541

—

C₂²⋊C₄

C₂×D₅₄₁

yes

—

C₂²⋊C₄

C₂×D₅₄₁

yes

—

C₄²

C₂×D₅₄₁

yes

characteristic

C₂³

—

541

—

100

C₂³

C₂²×D₅₄₁

yes

—

101

Q₈

—

102

Q₈

—

2164

—

103

D₄

—

104

D₄

—

2164

—

105

D₄

—

2164

—

106

D₄

—

2164

—

107

C₂×C₄

—

541

—

108

C₂×C₄

—

541

—

109

C₂×C₄

—

541

—

110

C₂×C₄

—

2164

—

111

C₂×C₄

—

2164

—

112

C₂×C₄

C₂²×D₅₄₁

yes

characteristic

113

C₂×C₄

C₂²×D₅₄₁

yes

characteristic

114

C₂×C₄

C₂²×D₅₄₁

yes

characteristic

115

C₂×C₄

C₂²×D₅₄₁

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

1082

—

120

C₂²

—

1082

—

121

C₂²

—

1082

—

122

C₂²

C₂³×D₅₄₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

1082

—

128

C₄

—

1082

—

129

C₄

—

1082

—

130

C₂

—

131

C₂

—

2164

—

132

C₂

#31772673

yes

characteristic

133

C₂

#31772674

yes

characteristic

134

C₂

#31772675

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₄₁
commutator, socle, cosocle	C₂²×C₅₄₁	C₂⁴
Fitting	#124×C₅₄₁	C₂
hypercenter	#124	D₅₄₁
hypocenter	C₅₄₁	#562
LCS	G ▷ C₂²×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₄₁

commutator, socle, cosocle

C₂²×C₅₄₁

C₂⁴

Fitting

#124×C₅₄₁

C₂

D₅₄₁

C₅₄₁

G ▷ C₂²×C₅₄₁ ▷ C₅₄₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #31772620	C₂	semidirect
2. #31772616	C₂	semidirect
3. #31772612	C₂	semidirect
4. #31772610	C₂	semidirect
5. #31772605	C₂	semidirect
6. #31772604	C₂	semidirect
7. #31772601	C₂	semidirect
8. #31772600	C₂	semidirect
9. #124×C₅₄₁	C₂	semidirect
67. C₅₄₁	#562	semidirect
71. #124	D₅₄₁	semidirect

H ≅ G/N

type

1. #31772620

C₂

semidirect

2. #31772616

C₂

semidirect

3. #31772612

C₂

semidirect

4. #31772610

C₂

semidirect

5. #31772605

C₂

semidirect

6. #31772604

C₂

semidirect

7. #31772601

C₂

semidirect

8. #31772600

C₂

semidirect

9. #124×C₅₄₁

C₂

semidirect

67. C₅₄₁

#562

semidirect

71. #124

D₅₄₁

semidirect

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

subgroup	index
#31772620	2
#31772616	2
#31772612	2
#31772610	2
#31772605	2
#31772604	2
#31772601	2
#31772600	2
#124×C₅₄₁	2
#562	541

subgroup

index

#124×C₅₄₁

541

Subgroups — #46830919

Conjugacy classes of subgroups

Complements

Maximal subgroups