Group #46830890 — subgroups

Subgroups — #46830890

Order 34624.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#31772628	C₂	1	1	yes	maximal, characteristic, complemented
2	#31772614	C₂	1	4	yes	maximal, complemented
3	#31772602	C₂	1	2	yes	maximal, complemented
4	#31772599	C₂	1	2	yes	maximal, complemented
5	C₄×D₂₁₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#31772587	C₂	1	2	yes	maximal, complemented
7	#31772586	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₁₀₈₂	C₂²	1	2	yes	—
10	C₂×D₂₁₆₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₅₄₁	C₂²	1	4	yes	—
12	C₂×C₄×D₅₄₁	C₂²	1	2	yes	—
13	C₄⋊Dic₅₄₁	C₂²	1	2	yes	—
14	C₄⋊Dic₅₄₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₅₄₁	C₂²	1	2	yes	—
16	C₂²⋊Dic₅₄₁	C₂²	1	4	yes	—
17	#22575813	C₂²	1	4	yes	—
18	#22575813	C₂²	1	2	yes	—
19	C₄×Dic₅₄₁	C₂²	1	4	yes	—
20	C₂×D₄×C₅₄₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₄₁	C₂²	1	2	yes	—
22	C₄⋊C₄×C₅₄₁	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₅₄₁	C₂²	1	2	yes	—
24	C₄²×C₅₄₁	C₂²	1	1	yes	characteristic
25	C₂²×D₅₄₁	C₂³	1	2	yes	—
26	Dic₁₀₈₂	—	4	2	no	—
27	D₂₁₆₄	—	2	2	no	—
28	C₂²⋊D₅₄₁	—	4	4	no	—
29	C₂×Dic₅₄₁	C₂³	1	2	yes	—
30	C₂×Dic₅₄₁	C₂³	1	4	yes	—
31	C₄×D₅₄₁	—	2	4	no	—
32	C₂³×C₅₄₁	C₂³	1	2	yes	—
33	D₄×C₅₄₁	—	2	2	no	—
34	C₂×C₄×C₅₄₁	—	2	4	no	—
35	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₅₄₁	C₂³	1	2	yes	—
37	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
39	C₂²×C₅₄₁	—	2	2	no	—
40	C₂²×C₅₄₁	—	2	2	no	—
41	C₂²×C₅₄₁	—	2	2	no	—
42	C₂²×C₅₄₁	C₂⁴	1	1	yes	characteristic
43	C₂×D₅₄₁	—	2	2	no	—
44	C₂×D₅₄₁	—	2	2	no	—
45	C₂×D₅₄₁	—	2	2	no	—
46	C₄×C₅₄₁	—	2	2	no	—
47	C₄×C₅₄₁	—	2	1	no	—
48	C₄×C₅₄₁	#41	1	2	yes	—
49	C₄×C₅₄₁	#41	1	2	yes	—
50	Dic₅₄₁	—	2	2	no	—
51	Dic₅₄₁	—	2	4	no	—
52	C₂×C₅₄₁	—	4	2	no	—
53	C₂×C₅₄₁	C₂×#41	1	1	yes	characteristic
54	C₂×C₅₄₁	C₂×#41	1	1	yes	characteristic
55	C₂×C₅₄₁	H(5,𝔽₂)	1	1	yes	characteristic
56	D₅₄₁	—	4	2	no	—
57	C₅₄₁	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	541	1	no	maximal, Sylow
59	C₄⋊Q₈	—	541	1	no	—
60	#124	—	541	4	no	—
61	#121	—	541	2	no	—
62	#121	—	541	2	no	—
63	C₄×D₄	—	541	1	no	—
64	C₄×D₄	D₅₄₁	1	1	yes	characteristic, complemented
65	#117	—	541	2	no	—
66	#117	—	541	2	no	—
67	C₂×Q₈	—	541	2	no	—
68	C₂×D₄	—	541	4	no	—
69	C₂×D₄	—	541	1	no	—
70	C₂×D₄	C₂×D₅₄₁	1	1	yes	characteristic
71	C₂²×C₄	—	541	2	no	—
72	C₂²×C₄	C₂×D₅₄₁	1	2	yes	—
73	C₄⋊C₄	—	541	1	no	—
74	C₄⋊C₄	—	541	2	no	—
75	C₄⋊C₄	—	541	2	no	—
76	C₄⋊C₄	C₂×D₅₄₁	1	1	yes	characteristic
77	C₂²⋊C₄	—	541	4	no	—
78	C₂²⋊C₄	—	541	4	no	—
79	C₂²⋊C₄	—	541	2	no	—
80	C₂²⋊C₄	C₂×D₅₄₁	1	2	yes	—
81	C₄²	—	541	4	no	—
82	C₄²	C₂×D₅₄₁	1	1	yes	characteristic
83	C₂³	—	541	2	no	—
84	C₂³	C₂²×D₅₄₁	1	2	yes	—
85	Q₈	—	2164	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	1082	2	no	—
88	D₄	—	2164	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	541	2	no	—
91	C₂×C₄	—	541	4	no	—
92	C₂×C₄	—	1082	4	no	—
93	C₂×C₄	C₂²×D₅₄₁	1	2	yes	—
94	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	1082	2	no	—
101	C₂²	—	1082	2	no	—
102	C₂²	—	1082	2	no	—
103	C₂²	C₂³×D₅₄₁	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	—	1082	4	no	—
107	C₄	—	1082	2	no	—
108	C₄	#22575835	1	2	yes	—
109	C₄	#22575836	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	2164	2	no	—
112	C₂	C₂×#22575835	1	1	yes	characteristic
113	C₂	C₂×#22575836	1	1	yes	characteristic
114	C₂	#31772674	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31772628

C₂

yes

maximal, characteristic, complemented

#31772614

C₂

yes

maximal, complemented

#31772602

C₂

yes

maximal, complemented

#31772599

C₂

yes

maximal, complemented

C₄×D₂₁₆₄

C₂

yes

maximal, characteristic, complemented

#31772587

C₂

yes

maximal, complemented

#31772586

C₂

yes

maximal, complemented

C₄×D₄×C₅₄₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₀₈₂

C₂²

yes

—

C₂×D₂₁₆₄

C₂²

yes

characteristic

C₂×C₂²⋊D₅₄₁

C₂²

yes

—

C₂×C₄×D₅₄₁

C₂²

yes

—

C₄⋊Dic₅₄₁

C₂²

yes

—

C₄⋊Dic₅₄₁

C₂²

yes

characteristic

C₄⋊Dic₅₄₁

C₂²

yes

—

C₂²⋊Dic₅₄₁

C₂²

yes

—

#22575813

C₂²

yes

—

#22575813

C₂²

yes

—

C₄×Dic₅₄₁

C₂²

yes

—

C₂×D₄×C₅₄₁

C₂²

yes

characteristic

C₂²×C₄×C₅₄₁

C₂²

yes

—

C₄⋊C₄×C₅₄₁

C₂²

yes

characteristic

C₂²⋊C₄×C₅₄₁

C₂²

yes

—

C₄²×C₅₄₁

C₂²

yes

characteristic

C₂²×D₅₄₁

C₂³

yes

—

Dic₁₀₈₂

—

D₂₁₆₄

—

C₂²⋊D₅₄₁

—

C₂×Dic₅₄₁

C₂³

yes

—

C₂×Dic₅₄₁

C₂³

yes

—

C₄×D₅₄₁

—

C₂³×C₅₄₁

C₂³

yes

—

D₄×C₅₄₁

—

C₂×C₄×C₅₄₁

—

C₂×C₄×C₅₄₁

C₂³

yes

characteristic

C₂×C₄×C₅₄₁

C₂³

yes

—

C₂×C₄×C₅₄₁

C₂³

yes

characteristic

C₂×C₄×C₅₄₁

C₂³

yes

characteristic

C₂²×C₅₄₁

—

C₂²×C₅₄₁

—

C₂²×C₅₄₁

—

C₂²×C₅₄₁

C₂⁴

yes

characteristic

C₂×D₅₄₁

—

C₂×D₅₄₁

—

C₂×D₅₄₁

—

C₄×C₅₄₁

—

C₄×C₅₄₁

—

C₄×C₅₄₁

#41

yes

—

C₄×C₅₄₁

#41

yes

—

Dic₅₄₁

—

Dic₅₄₁

—

C₂×C₅₄₁

—

C₂×C₅₄₁

C₂×#41

yes

characteristic

C₂×C₅₄₁

C₂×#41

yes

characteristic

C₂×C₅₄₁

H(5,𝔽₂)

yes

characteristic

D₅₄₁

—

C₅₄₁

#555

yes

characteristic, complemented, Sylow

#555

—

541

maximal, Sylow

C₄⋊Q₈

—

541

—

#124

—

541

—

#121

—

541

—

#121

—

541

—

C₄×D₄

—

541

—

C₄×D₄

D₅₄₁

yes

characteristic, complemented

#117

—

541

—

#117

—

541

—

C₂×Q₈

—

541

—

C₂×D₄

—

541

—

C₂×D₄

—

541

—

C₂×D₄

C₂×D₅₄₁

yes

characteristic

C₂²×C₄

—

541

—

C₂²×C₄

C₂×D₅₄₁

yes

—

C₄⋊C₄

—

541

—

C₄⋊C₄

—

541

—

C₄⋊C₄

—

541

—

C₄⋊C₄

C₂×D₅₄₁

yes

characteristic

C₂²⋊C₄

—

541

—

C₂²⋊C₄

—

541

—

C₂²⋊C₄

—

541

—

C₂²⋊C₄

C₂×D₅₄₁

yes

—

C₄²

—

541

—

C₄²

C₂×D₅₄₁

yes

characteristic

C₂³

—

541

—

C₂³

C₂²×D₅₄₁

yes

—

Q₈

—

2164

—

D₄

—

D₄

—

1082

—

D₄

—

2164

—

C₂×C₄

—

C₂×C₄

—

541

—

C₂×C₄

—

541

—

C₂×C₄

—

1082

—

C₂×C₄

C₂²×D₅₄₁

yes

—

C₂×C₄

C₂²×D₅₄₁

yes

characteristic

C₂×C₄

C₂²×D₅₄₁

yes

characteristic

C₂×C₄

C₂²×D₅₄₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

1082

—

101

C₂²

—

1082

—

102

C₂²

—

1082

—

103

C₂²

C₂³×D₅₄₁

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

1082

—

107

C₄

—

1082

—

108

C₄

#22575835

yes

—

109

C₄

#22575836

yes

—

110

C₂

—

111

C₂

—

2164

—

112

C₂

C₂×#22575835

yes

characteristic

113

C₂

C₂×#22575836

yes

characteristic

114

C₂

#31772674

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₄₁
commutator, socle, cosocle	C₂²×C₅₄₁	C₂⁴
Fitting	C₄×D₄×C₅₄₁	C₂
hypercenter	C₄×D₄	D₅₄₁
hypocenter	C₅₄₁	#555
LCS	G ▷ C₂²×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₄₁

commutator, socle, cosocle

C₂²×C₅₄₁

C₂⁴

Fitting

C₄×D₄×C₅₄₁

C₂

hypercenter

C₄×D₄

D₅₄₁

C₅₄₁

G ▷ C₂²×C₅₄₁ ▷ C₅₄₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #31772628	C₂	semidirect
2. #31772614	C₂	semidirect
3. #31772602	C₂	semidirect
4. #31772599	C₂	semidirect
5. C₄×D₂₁₆₄	C₂	semidirect
6. #31772587	C₂	semidirect
7. #31772586	C₂	semidirect
8. C₄×D₄×C₅₄₁	C₂	semidirect
57. C₅₄₁	#555	semidirect
64. C₄×D₄	D₅₄₁	semidirect

H ≅ G/N

type

1. #31772628

C₂

semidirect

2. #31772614

C₂

semidirect

3. #31772602

C₂

semidirect

4. #31772599

C₂

semidirect

5. C₄×D₂₁₆₄

C₂

semidirect

6. #31772587

C₂

semidirect

7. #31772586

C₂

semidirect

8. C₄×D₄×C₅₄₁

C₂

semidirect

57. C₅₄₁

#555

semidirect

64. C₄×D₄

D₅₄₁

semidirect

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

subgroup	index
#31772628	2
#31772614	2
#31772602	2
#31772599	2
C₄×D₂₁₆₄	2
#31772587	2
#31772586	2
C₄×D₄×C₅₄₁	2
#555	541

subgroup

index

C₄×D₂₁₆₄

C₄×D₄×C₅₄₁

541

Subgroups — #46830890

Conjugacy classes of subgroups

Complements

Maximal subgroups