Group #45688139 — subgroups

Subgroups — #45688139

Order 33344.

228 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 115 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	#30659708	C₂	1	1	yes	maximal, characteristic, complemented
2	#30659699	C₂	1	2	yes	maximal, complemented
3	#30659687	C₂	1	2	yes	maximal, complemented
4	#30659685	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₁₀₄₂	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₅₂₁	C₂	1	2	yes	maximal
7	C₂×C₄⋊Dic₅₂₁	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₅₂₁	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₁₀₄₂	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₀₄₂	C₂²	1	2	yes	—
11	C₂²×Dic₅₂₁	C₂²	1	4	yes	—
12	C₄⋊Dic₅₂₁	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₅₂₁	C₂²	1	4	yes	—
14	C₄⋊Dic₅₂₁	C₂²	1	2	yes	—
15	C₄⋊Dic₅₂₁	C₂²	1	2	yes	—
16	#22552343	C₂²	1	2	yes	—
17	#22552343	C₂²	1	4	yes	—
18	C₂²⋊Dic₅₂₁	C₂²	1	4	yes	—
19	C₄×Dic₅₂₁	C₂²	1	2	yes	—
20	C₂×D₄×C₅₂₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₂₁	C₂²	1	2	yes	—
22	C₄⋊C₄×C₅₂₁	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₅₂₁	C₂²	1	2	yes	—
24	C₄²×C₅₂₁	C₂²	1	1	yes	characteristic
25	Dic₁₀₄₂	—	2	2	no	—
26	Dic₁₀₄₂	—	4	2	no	—
27	C₂×Dic₅₂₁	—	4	4	no	—
28	C₂×Dic₅₂₁	C₂³	1	2	yes	—
29	C₂×Dic₅₂₁	C₂³	1	2	yes	—
30	C₂×Dic₅₂₁	C₂³	1	4	yes	—
31	C₂³×C₅₂₁	C₂³	1	2	yes	—
32	D₄×C₅₂₁	Q₈	1	4	yes	complemented
33	C₂×C₄×C₅₂₁	—	2	4	no	—
34	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₅₂₁	C₂³	1	2	yes	—
36	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
38	C₂²×C₅₂₁	—	2	2	no	—
39	C₂²×C₅₂₁	—	2	2	no	—
40	C₂²×C₅₂₁	C₂×Q₈	1	4	yes	—
41	C₂²×C₅₂₁	C₂⁴	1	1	yes	characteristic
42	C₄×C₅₂₁	—	2	2	no	—
43	C₄×C₅₂₁	—	2	1	no	—
44	C₄×C₅₂₁	C₂×Q₈	1	2	yes	—
45	C₄×C₅₂₁	#41	1	2	yes	—
46	Dic₅₂₁	—	2	2	no	—
47	Dic₅₂₁	—	2	2	no	—
48	Dic₅₂₁	—	2	4	no	—
49	C₂×C₅₂₁	—	2	4	no	—
50	C₂×C₅₂₁	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₅₂₁	C₂×#41	1	1	yes	characteristic
52	C₂×C₅₂₁	H(5,𝔽₂)	1	1	yes	characteristic
53	C₅₂₁	#554	1	1	yes	characteristic, complemented, Sylow
54	#554	—	521	1	no	maximal, Sylow
55	C₄⋊Q₈	—	521	1	no	—
56	#125	—	521	2	no	—
57	C₂²⋊Q₈	—	521	2	no	—
58	C₂²⋊Q₈	—	521	4	no	—
59	C₄×Q₈	—	521	1	no	—
60	C₄×D₄	—	521	2	no	—
61	C₄×D₄	D₅₂₁	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	521	2	no	—
63	C₂×Q₈	—	521	1	no	—
64	C₂×Q₈	—	521	2	no	—
65	C₂×D₄	C₂×D₅₂₁	1	1	yes	characteristic
66	C₂²×C₄	—	521	4	no	—
67	C₂²×C₄	C₂×D₅₂₁	1	2	yes	—
68	C₄⋊C₄	—	521	2	no	—
69	C₄⋊C₄	—	521	4	no	—
70	C₄⋊C₄	—	521	1	no	—
71	C₄⋊C₄	—	521	4	no	—
72	C₄⋊C₄	—	521	2	no	—
73	C₄⋊C₄	—	521	2	no	—
74	C₄⋊C₄	C₂×D₅₂₁	1	1	yes	characteristic
75	C₂²⋊C₄	—	521	4	no	—
76	C₂²⋊C₄	C₂×D₅₂₁	1	2	yes	—
77	C₄²	—	521	2	no	—
78	C₄²	C₂×D₅₂₁	1	1	yes	characteristic
79	C₂³	C₂²×D₅₂₁	1	2	yes	—
80	Q₈	—	1042	2	no	—
81	Q₈	—	2084	2	no	—
82	D₄	Dic₁₀₄₂	1	4	yes	complemented
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	521	2	no	—
85	C₂×C₄	—	521	2	no	—
86	C₂×C₄	—	521	4	no	—
87	C₂×C₄	—	2084	4	no	—
88	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₅₂₁	1	2	yes	—
90	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	—	2	2	no	—
94	C₂²	C₂×Dic₁₀₄₂	1	4	yes	—
95	C₂²	C₂³×D₅₂₁	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	1	no	—
98	C₄	—	1042	2	no	—
99	C₄	—	1042	2	no	—
100	C₄	—	1042	4	no	—
101	C₄	C₂×Dic₁₀₄₂	1	2	yes	—
102	C₄	#22552363	1	2	yes	—
103	C₂	—	2	4	no	—
104	C₂	C₂²×Dic₁₀₄₂	1	1	yes	characteristic
105	C₂	C₂×#22552363	1	1	yes	characteristic
106	C₂	#30659754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30659708

C₂

yes

maximal, characteristic, complemented

#30659699

C₂

yes

maximal, complemented

#30659687

C₂

yes

maximal, complemented

#30659685

C₂

yes

maximal, complemented

C₄×Dic₁₀₄₂

C₂

yes

maximal, characteristic, complemented

D₄×Dic₅₂₁

C₂

yes

maximal

C₂×C₄⋊Dic₅₂₁

C₂

yes

maximal, complemented

C₄×D₄×C₅₂₁

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄₂

C₂²

yes

characteristic

C₂×Dic₁₀₄₂

C₂²

yes

—

C₂²×Dic₅₂₁

C₂²

yes

—

C₄⋊Dic₅₂₁

C₂²

yes

characteristic

C₄⋊Dic₅₂₁

C₂²

yes

—

C₄⋊Dic₅₂₁

C₂²

yes

—

C₄⋊Dic₅₂₁

C₂²

yes

—

#22552343

C₂²

yes

—

#22552343

C₂²

yes

—

C₂²⋊Dic₅₂₁

C₂²

yes

—

C₄×Dic₅₂₁

C₂²

yes

—

C₂×D₄×C₅₂₁

C₂²

yes

characteristic

C₂²×C₄×C₅₂₁

C₂²

yes

—

C₄⋊C₄×C₅₂₁

C₂²

yes

characteristic

C₂²⋊C₄×C₅₂₁

C₂²

yes

—

C₄²×C₅₂₁

C₂²

yes

characteristic

Dic₁₀₄₂

—

Dic₁₀₄₂

—

C₂×Dic₅₂₁

—

C₂×Dic₅₂₁

C₂³

yes

—

C₂×Dic₅₂₁

C₂³

yes

—

C₂×Dic₅₂₁

C₂³

yes

—

C₂³×C₅₂₁

C₂³

yes

—

D₄×C₅₂₁

Q₈

yes

complemented

C₂×C₄×C₅₂₁

—

C₂×C₄×C₅₂₁

C₂³

yes

characteristic

C₂×C₄×C₅₂₁

C₂³

yes

—

C₂×C₄×C₅₂₁

C₂³

yes

characteristic

C₂×C₄×C₅₂₁

C₂³

yes

characteristic

C₂²×C₅₂₁

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

C₂×Q₈

yes

—

C₂²×C₅₂₁

C₂⁴

yes

characteristic

C₄×C₅₂₁

—

C₄×C₅₂₁

—

C₄×C₅₂₁

C₂×Q₈

yes

—

C₄×C₅₂₁

#41

yes

—

Dic₅₂₁

—

Dic₅₂₁

—

Dic₅₂₁

—

C₂×C₅₂₁

—

C₂×C₅₂₁

C₂²×Q₈

yes

characteristic

C₂×C₅₂₁

C₂×#41

yes

C₂×C₅₂₁

yes

C₅₂₁

yes

characteristic, complemented, Sylow

#554

—

521

maximal, Sylow

C₄⋊Q₈

—

521

—

#125

—

521

—

C₂²⋊Q₈

—

521

—

C₂²⋊Q₈

—

521

—

C₄×Q₈

—

521

—

C₄×D₄

—

521

—

C₄×D₄

D₅₂₁

yes

characteristic

C₂×C₄⋊C₄

—

521

—

C₂×Q₈

—

521

—

C₂×Q₈

—

521

—

C₂×D₄

C₂×D₅₂₁

yes

characteristic

C₂²×C₄

—

521

—

C₂²×C₄

C₂×D₅₂₁

yes

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

C₂×D₅₂₁

yes

characteristic

C₂²⋊C₄

—

521

—

C₂²⋊C₄

C₂×D₅₂₁

yes

—

C₄²

—

521

—

C₄²

C₂×D₅₂₁

yes

characteristic

C₂³

C₂²×D₅₂₁

yes

—

Q₈

—

1042

—

Q₈

—

2084

—

D₄

Dic₁₀₄₂

yes

complemented

C₂×C₄

—

C₂×C₄

—

521

—

C₂×C₄

—

521

—

C₂×C₄

—

521

—

C₂×C₄

—

2084

—

C₂×C₄

C₂²×D₅₂₁

yes

characteristic

C₂×C₄

C₂²×D₅₂₁

yes

—

C₂×C₄

C₂²×D₅₂₁

yes

characteristic

C₂×C₄

C₂²×D₅₂₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₁₀₄₂

yes

—

C₂²

C₂³×D₅₂₁

yes

characteristic

C₄

—

C₄

—

C₄

—

1042

—

C₄

—

1042

—

100

C₄

—

1042

—

101

C₄

C₂×Dic₁₀₄₂

yes

—

102

C₄

#22552363

yes

—

103

C₂

—

104

C₂

C₂²×Dic₁₀₄₂

yes

characteristic

105

C₂

C₂×#22552363

yes

characteristic

106

C₂

#30659754

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₂₁
commutator, socle, cosocle	C₂²×C₅₂₁	C₂⁴
Fitting	C₄×D₄×C₅₂₁	C₂
prime radical	C₂×D₄×C₅₂₁	C₂²
hypercenter	C₄×D₄	D₅₂₁
hypocenter	C₅₂₁	#554
LCS	G ▷ C₂²×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₂₁

commutator, socle, cosocle

C₂²×C₅₂₁

C₂⁴

Fitting

C₄×D₄×C₅₂₁

C₂

prime radical

C₂×D₄×C₅₂₁

C₂²

hypercenter

C₄×D₄

D₅₂₁

C₅₂₁

G ▷ C₂²×C₅₂₁ ▷ C₅₂₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #30659708	C₂	semidirect
2. #30659699	C₂	semidirect
3. #30659687	C₂	semidirect
4. #30659685	C₂	semidirect
5. C₄×Dic₁₀₄₂	C₂	semidirect
7. C₂×C₄⋊Dic₅₂₁	C₂	semidirect
32. D₄×C₅₂₁	Q₈	semidirect
53. C₅₂₁	#554	semidirect
82. D₄	Dic₁₀₄₂	semidirect

H ≅ G/N

type

1. #30659708

C₂

semidirect

2. #30659699

C₂

semidirect

3. #30659687

C₂

semidirect

4. #30659685

C₂

semidirect

5. C₄×Dic₁₀₄₂

C₂

semidirect

7. C₂×C₄⋊Dic₅₂₁

C₂

semidirect

32. D₄×C₅₂₁

Q₈

semidirect

53. C₅₂₁

#554

semidirect

82. D₄

Dic₁₀₄₂

semidirect

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

subgroup	index
#30659708	2
#30659699	2
#30659687	2
#30659685	2
C₄×Dic₁₀₄₂	2
D₄×Dic₅₂₁	2
C₂×C₄⋊Dic₅₂₁	2
C₄×D₄×C₅₂₁	2
#554	521

subgroup

index

C₄×Dic₁₀₄₂

D₄×Dic₅₂₁

C₂×C₄⋊Dic₅₂₁

C₄×D₄×C₅₂₁

#554

521

Subgroups — #45688139

Conjugacy classes of subgroups

Complements

Maximal subgroups