Group #14580903 — subgroups

Subgroups — #14580903

Order 2624.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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×#10620306	C₂	1	1	yes	maximal, characteristic, complemented
2	#13050160	C₂	1	2	yes	maximal, complemented
3	#13050151	C₂	1	2	yes	maximal, complemented
4	#13050135	C₂	1	4	yes	maximal, complemented
5	#13050133	C₂	1	4	yes	maximal, complemented
6	#13050123	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×C₄₁	C₂²	1	2	yes	—
9	C₂×D₄×C₄₁	—	2	4	no	—
10	C₂×D₄×C₄₁	C₂²	1	2	yes	—
11	C₂×D₄×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₄₁	C₂²	1	8	yes	complemented
15	#10620306	—	2	4	no	—
16	C₂×D₁₆₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄×D₄₁	C₂²	1	2	yes	complemented
18	C₂×Dic₈₂	C₂²	1	1	yes	characteristic, complemented
19	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
20	C₂²⋊Dic₄₁	C₂²	1	8	yes	complemented
21	C₄⋊Dic₄₁	C₂²	1	2	yes	complemented
22	C₄×Dic₄₁	C₂²	1	2	yes	complemented
23	C₂³×C₄₁	—	2	4	no	—
24	C₂³×C₄₁	—	4	4	no	—
25	C₂³×C₄₁	C₂³	1	4	yes	—
26	C₂³×C₄₁	C₂³	1	1	yes	characteristic
27	C₂²×D₄₁	C₂³	1	2	yes	—
28	D₄×C₄₁	—	2	4	no	—
29	D₄×C₄₁	—	2	4	no	—
30	C₂×C₄×C₄₁	D₄	1	4	yes	complemented
31	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
33	C₂²⋊D₄₁	—	2	4	no	—
34	C₂²⋊D₄₁	—	4	8	no	—
35	C₂×Dic₄₁	C₂³	1	4	yes	—
36	C₂×Dic₄₁	C₂³	1	2	yes	—
37	D₁₆₄	—	2	2	no	—
38	C₄×D₄₁	—	2	4	no	—
39	Dic₈₂	—	2	2	no	—
40	C₂²×C₄₁	—	2	2	no	—
41	C₂²×C₄₁	—	2	4	no	—
42	C₂²×C₄₁	—	2	8	no	—
43	C₂²×C₄₁	—	4	8	no	—
44	C₂²×C₄₁	C₂×D₄	1	2	yes	—
45	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
46	C₂×D₄₁	—	2	2	no	—
47	C₂×D₄₁	—	2	4	no	—
48	C₄×C₄₁	C₂×D₄	1	2	yes	—
49	C₄×C₄₁	C₂×D₄	1	2	yes	—
50	Dic₄₁	—	2	2	no	—
51	Dic₄₁	—	2	4	no	—
52	C₂×C₄₁	—	2	2	no	—
53	C₂×C₄₁	—	4	4	no	—
54	C₂×C₄₁	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄₁	H(5,𝔽₂)	1	2	yes	—
56	D₄₁	—	4	2	no	—
57	#535	—	41	1	no	maximal, Sylow
58	C₄₁	#535	1	1	yes	characteristic, complemented, Sylow
59	C₂×#41	—	41	1	no	—
60	C₂²×D₄	D₄₁	1	1	yes	characteristic, complemented
61	#127	—	41	2	no	—
62	#124	—	41	2	no	—
63	#121	—	41	4	no	—
64	#120	—	41	4	no	—
65	#117	—	41	1	no	—
66	C₂⁴	C₂×D₄₁	1	2	yes	—
67	#41	—	82	4	no	—
68	C₂×Q₈	—	41	1	no	—
69	C₂×D₄	—	2	4	no	—
70	C₂×D₄	—	41	2	no	—
71	C₂×D₄	—	41	8	no	—
72	C₂×D₄	—	41	1	no	—
73	C₂×D₄	C₂×D₄₁	1	2	yes	—
74	C₂×D₄	C₂×D₄₁	1	2	yes	—
75	C₂²×C₄	—	41	2	no	—
76	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
77	C₄⋊C₄	—	41	2	no	—
78	C₂²⋊C₄	—	41	2	no	—
79	C₂²⋊C₄	—	41	8	no	—
80	C₄²	—	41	2	no	—
81	C₂³	—	2	4	no	—
82	C₂³	—	4	4	no	—
83	C₂³	—	41	2	no	—
84	C₂³	C₂²×D₄₁	1	4	yes	—
85	C₂³	C₂²×D₄₁	1	1	yes	characteristic
86	Q₈	—	82	2	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	4	no	—
89	D₄	—	82	4	no	—
90	D₄	—	82	2	no	—
91	D₄	—	164	8	no	—
92	C₂×C₄	—	41	4	no	—
93	C₂×C₄	—	41	2	no	—
94	C₂×C₄	—	82	4	no	—
95	C₂×C₄	C₂²⋊D₄₁	1	4	yes	complemented
96	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	2	4	no	—
100	C₂²	—	2	8	no	—
101	C₂²	—	4	8	no	—
102	C₂²	—	82	2	no	—
103	C₂²	—	82	4	no	—
104	C₂²	C₂×C₂²⋊D₄₁	1	2	yes	—
105	C₂²	C₂³×D₄₁	1	1	yes	characteristic
106	C₄	—	82	2	no	—
107	C₄	—	82	4	no	—
108	C₄	C₂×C₂²⋊D₄₁	1	2	yes	—
109	C₄	C₂×C₂²⋊D₄₁	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	4	no	—
112	C₂	—	164	2	no	—
113	C₂	C₂²×C₂²⋊D₄₁	1	1	yes	characteristic
114	C₂	#13050209	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10620306

C₂

yes

maximal, characteristic, complemented

#13050160

C₂

yes

maximal, complemented

#13050151

C₂

yes

maximal, complemented

#13050135

C₂

yes

maximal, complemented

#13050133

C₂

yes

maximal, complemented

#13050123

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₄₁

C₂²

yes

—

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

C₂²

yes

—

C₂×D₄×C₄₁

C₂²

yes

—

C₂²×C₄×C₄₁

C₂²

yes

characteristic

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂×C₂²⋊D₄₁

C₂²

yes

complemented

#10620306

—

C₂×D₁₆₄

C₂²

yes

characteristic, complemented

C₂×C₄×D₄₁

C₂²

yes

complemented

C₂×Dic₈₂

C₂²

yes

characteristic, complemented

C₂²⋊Dic₄₁

C₂²

yes

—

C₂²⋊Dic₄₁

C₂²

yes

complemented

C₄⋊Dic₄₁

C₂²

yes

complemented

C₄×Dic₄₁

C₂²

yes

complemented

C₂³×C₄₁

—

C₂³×C₄₁

—

C₂³×C₄₁

C₂³

yes

—

C₂³×C₄₁

C₂³

yes

characteristic

C₂²×D₄₁

C₂³

yes

—

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

D₄

yes

complemented

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂²⋊D₄₁

—

C₂²⋊D₄₁

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

D₁₆₄

—

C₄×D₄₁

—

Dic₈₂

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

C₂×D₄

yes

—

C₂²×C₄₁

C₂⁴

yes

characteristic

C₂×D₄₁

—

C₂×D₄₁

—

C₄×C₄₁

C₂×D₄

yes

—

C₄×C₄₁

C₂×D₄

yes

—

Dic₄₁

—

Dic₄₁

—

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

C₂²×D₄

yes

characteristic

C₂×C₄₁

H(5,𝔽₂)

yes

—

D₄₁

—

#535

—

maximal, Sylow

C₄₁

#535

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×D₄

D₄₁

yes

characteristic, complemented

—

—

—

—

—

C₂⁴

C₂×D₄₁

yes

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₁

yes

—

C₂×D₄

C₂×D₄₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₄₁

yes

—

C₂³

C₂²×D₄₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

164

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₄₁

yes

complemented

C₂×C₄

C₂²×D₄₁

yes

characteristic

C₂×C₄

C₂²×D₄₁

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₂²⋊D₄₁

yes

—

105

C₂²

C₂³×D₄₁

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×C₂²⋊D₄₁

yes

—

109

C₄

C₂×C₂²⋊D₄₁

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

164

—

113

C₂

C₂²×C₂²⋊D₄₁

yes

characteristic

114

C₂

#13050209

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	C₂²×D₄×C₄₁	C₂
hypercenter	C₂²×D₄	D₄₁
hypocenter	C₄₁	#535
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₁

commutator, socle, cosocle

C₂²×C₄₁

C₂⁴

Fitting

C₂²×D₄×C₄₁

C₂

hypercenter

C₂²×D₄

D₄₁

C₄₁

G ▷ C₂²×C₄₁ ▷ C₄₁

C₁ ◁ C₂² ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×#10620306	C₂	semidirect
2. #13050160	C₂	semidirect
3. #13050151	C₂	semidirect
4. #13050135	C₂	semidirect
5. #13050133	C₂	semidirect
6. #13050123	C₂	semidirect
7. C₂²×D₄×C₄₁	C₂	semidirect
14. C₂×C₂²⋊D₄₁	C₂²	semidirect
16. C₂×D₁₆₄	C₂²	semidirect
17. C₂×C₄×D₄₁	C₂²	semidirect
18. C₂×Dic₈₂	C₂²	semidirect
20. C₂²⋊Dic₄₁	C₂²	semidirect
21. C₄⋊Dic₄₁	C₂²	semidirect
22. C₄×Dic₄₁	C₂²	semidirect
30. C₂×C₄×C₄₁	D₄	semidirect
58. C₄₁	#535	semidirect
60. C₂²×D₄	D₄₁	semidirect
95. C₂×C₄	C₂²⋊D₄₁	semidirect

H ≅ G/N

type

1. C₂×#10620306

C₂

semidirect

2. #13050160

C₂

semidirect

3. #13050151

C₂

semidirect

4. #13050135

C₂

semidirect

5. #13050133

C₂

semidirect

6. #13050123

C₂

semidirect

7. C₂²×D₄×C₄₁

C₂

semidirect

14. C₂×C₂²⋊D₄₁

C₂²

semidirect

16. C₂×D₁₆₄

C₂²

semidirect

17. C₂×C₄×D₄₁

C₂²

semidirect

18. C₂×Dic₈₂

C₂²

semidirect

20. C₂²⋊Dic₄₁

C₂²

semidirect

21. C₄⋊Dic₄₁

C₂²

semidirect

22. C₄×Dic₄₁

C₂²

semidirect

30. C₂×C₄×C₄₁

D₄

semidirect

58. C₄₁

#535

semidirect

60. C₂²×D₄

D₄₁

semidirect

95. C₂×C₄

C₂²⋊D₄₁

semidirect

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

subgroup	index
C₂×#10620306	2
#13050160	2
#13050151	2
#13050135	2
#13050133	2
#13050123	2
C₂²×D₄×C₄₁	2
#535	41

subgroup

index

C₂²×D₄×C₄₁

Subgroups — #14580903

Conjugacy classes of subgroups

Complements

Maximal subgroups