Group #14581085 — subgroups

Subgroups — #14581085

Order 2624.

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

normal subgroups only

This group has 196 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	#13050162	C₂	1	1	yes	maximal, characteristic, complemented
2	#13050159	C₂	1	2	yes	maximal, complemented
3	#13050158	C₂	1	2	yes	maximal, complemented
4	#13050154	C₂	1	2	yes	maximal, complemented
5	#13050153	C₂	1	1	yes	maximal, characteristic
6	#13050149	C₂	1	2	yes	maximal
7	#13050142	C₂	1	2	yes	maximal, complemented
8	#13050140	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄₁	C₂²	1	2	yes	—
14	C₂×Dic₈₂	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
16	C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
17	C₄²×C₄₁	C₂²	1	1	yes	characteristic
18	#10620281	C₂²	1	2	yes	—
19	#10620281	C₂²	1	2	yes	—
20	#10620281	C₂²	1	2	yes	—
21	#10620281	C₂²	1	2	yes	—
22	#10620281	C₂²	1	2	yes	—
23	C₄⋊Dic₄₁	C₂²	1	2	yes	—
24	#10620279	C₂²	1	2	yes	—
25	#10620279	C₂²	1	2	yes	—
26	#10620279	C₂²	1	2	yes	—
27	#10620279	C₂²	1	2	yes	—
28	#10620279	C₂²	1	2	yes	—
29	C₄×Dic₄₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₄₁	C₂²	1	1	yes	characteristic
31	C₂²×D₄₁	C₂³	1	2	yes	—
32	Q₈×C₄₁	—	4	1	no	—
33	Q₈×C₄₁	—	4	1	no	—
34	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₄₁	C₂³	1	2	yes	—
37	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₄₁	C₂³	1	2	yes	—
39	C₂×Dic₄₁	C₂³	1	2	yes	—
40	C₂×Dic₄₁	C₂³	1	2	yes	—
41	C₂×Dic₄₁	C₂³	1	2	yes	—
42	D₁₆₄	—	4	1	no	—
43	C₄×D₄₁	—	4	2	no	—
44	Dic₈₂	—	4	1	no	—
45	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
46	C₂×D₄₁	—	2	2	no	—
47	C₂×D₄₁	—	2	4	no	—
48	C₄×C₄₁	—	2	1	no	—
49	C₄×C₄₁	—	2	2	no	—
50	C₄×C₄₁	—	2	1	no	—
51	C₄×C₄₁	—	2	1	no	—
52	C₄×C₄₁	—	2	2	no	—
53	Dic₄₁	—	2	2	no	—
54	Dic₄₁	—	2	2	no	—
55	Dic₄₁	—	2	2	no	—
56	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₄₁	2₋¹⁺⁴	1	2	yes	—
58	D₄₁	—	4	2	no	—
59	#563	—	41	1	no	maximal, Sylow
60	C₄₁	#563	1	1	yes	characteristic, complemented, Sylow
61	C₄⋊Q₈	—	41	1	no	—
62	C₄⋊Q₈	D₄₁	1	1	yes	characteristic, complemented
63	#126	—	41	2	no	—
64	#126	—	41	2	no	—
65	#125	—	41	2	no	—
66	#124	—	41	1	no	—
67	#123	—	41	2	no	—
68	C₂²⋊Q₈	—	41	2	no	—
69	C₂²⋊Q₈	—	41	2	no	—
70	C₂×Q₈	—	41	1	no	—
71	C₂×Q₈	C₂×D₄₁	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₄₁	1	1	yes	characteristic
73	C₂×D₄	—	41	1	no	—
74	C₂²×C₄	—	41	2	no	—
75	C₄⋊C₄	—	41	2	no	—
76	C₄⋊C₄	—	41	2	no	—
77	C₄⋊C₄	—	41	2	no	—
78	C₄⋊C₄	—	41	2	no	—
79	C₄⋊C₄	—	41	2	no	—
80	C₄⋊C₄	—	41	2	no	—
81	C₄⋊C₄	C₂×D₄₁	1	2	yes	—
82	C₄⋊C₄	C₂×D₄₁	1	2	yes	—
83	C₂²⋊C₄	—	41	2	no	—
84	C₂²⋊C₄	—	41	2	no	—
85	C₂²⋊C₄	—	41	2	no	—
86	C₂²⋊C₄	—	41	2	no	—
87	C₂²⋊C₄	—	41	2	no	—
88	C₄²	—	41	1	no	—
89	C₄²	—	41	1	no	—
90	C₄²	C₂×D₄₁	1	1	yes	characteristic
91	C₂³	—	41	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	164	1	no	—
95	D₄	—	164	1	no	—
96	C₂×C₄	—	41	2	no	—
97	C₂×C₄	—	41	2	no	—
98	C₂×C₄	—	41	2	no	—
99	C₂×C₄	—	164	2	no	—
100	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₄₁	1	2	yes	—
103	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₄₁	1	2	yes	—
105	C₂²	—	82	2	no	—
106	C₂²	—	82	4	no	—
107	C₂²	C₂³×D₄₁	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	82	2	no	—
114	C₄	—	82	2	no	—
115	C₄	—	82	2	no	—
116	C₂	—	164	2	no	—
117	C₂	#13050209	1	1	yes	characteristic
118	C₂	#13050212	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13050162

C₂

yes

maximal, characteristic, complemented

#13050159

C₂

yes

maximal, complemented

#13050158

C₂

yes

maximal, complemented

#13050154

C₂

yes

maximal, complemented

#13050153

C₂

yes

maximal, characteristic

#13050149

C₂

yes

maximal

#13050142

C₂

yes

maximal, complemented

#13050140

C₂

yes

maximal, complemented

C₄⋊Q₈×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₁

C₂²

yes

characteristic

C₂×Q₈×C₄₁

C₂²

yes

characteristic

C₂×D₁₆₄

C₂²

yes

characteristic

C₂×C₄×D₄₁

C₂²

yes

—

C₂×Dic₈₂

C₂²

yes

characteristic

C₄⋊C₄×C₄₁

C₂²

yes

—

C₄⋊C₄×C₄₁

C₂²

yes

—

C₄²×C₄₁

C₂²

yes

characteristic

#10620281

C₂²

yes

—

#10620281

C₂²

yes

—

#10620281

C₂²

yes

—

#10620281

C₂²

yes

—

#10620281

C₂²

yes

—

C₄⋊Dic₄₁

C₂²

yes

—

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

C₄×Dic₄₁

C₂²

yes

characteristic

C₄×Dic₄₁

C₂²

yes

characteristic

C₂²×D₄₁

C₂³

yes

—

Q₈×C₄₁

—

Q₈×C₄₁

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

D₁₆₄

—

C₄×D₄₁

—

Dic₈₂

—

C₂²×C₄₁

C₂⁴

yes

characteristic

C₂×D₄₁

—

C₂×D₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

Dic₄₁

—

Dic₄₁

—

Dic₄₁

—

C₂×C₄₁

H(5,𝔽₂)

yes

characteristic

C₂×C₄₁

2₋¹⁺⁴

yes

—

D₄₁

—

#563

—

maximal, Sylow

C₄₁

#563

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

D₄₁

yes

characteristic, complemented

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₁

yes

characteristic

C₂×Q₈

C₂×D₄₁

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₁

yes

—

C₄⋊C₄

C₂×D₄₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

164

—

D₄

—

164

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

164

—

100

C₂×C₄

C₂²×D₄₁

yes

characteristic

101

C₂×C₄

C₂²×D₄₁

yes

characteristic

102

C₂×C₄

C₂²×D₄₁

yes

—

103

C₂×C₄

C₂²×D₄₁

yes

characteristic

104

C₂×C₄

C₂²×D₄₁

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂³×D₄₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₂

—

164

—

117

C₂

#13050209

yes

characteristic

118

C₂

#13050212

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	C₄⋊Q₈×C₄₁	C₂
prime radical	C₂×D₁₆₄	C₂²
hypercenter	C₄⋊Q₈	D₄₁
hypocenter	C₄₁	#563
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₁

commutator, socle, cosocle

C₂²×C₄₁

C₂⁴

Fitting

C₄⋊Q₈×C₄₁

C₂

prime radical

C₂×D₁₆₄

C₂²

hypercenter

C₄⋊Q₈

D₄₁

C₄₁

G ▷ C₂²×C₄₁ ▷ C₄₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #13050162	C₂	semidirect
2. #13050159	C₂	semidirect
3. #13050158	C₂	semidirect
4. #13050154	C₂	semidirect
7. #13050142	C₂	semidirect
8. #13050140	C₂	semidirect
9. C₄⋊Q₈×C₄₁	C₂	semidirect
60. C₄₁	#563	semidirect
62. C₄⋊Q₈	D₄₁	semidirect

H ≅ G/N

type

1. #13050162

C₂

semidirect

2. #13050159

C₂

semidirect

3. #13050158

C₂

semidirect

4. #13050154

C₂

semidirect

7. #13050142

C₂

semidirect

8. #13050140

C₂

semidirect

9. C₄⋊Q₈×C₄₁

C₂

semidirect

60. C₄₁

#563

semidirect

62. C₄⋊Q₈

D₄₁

semidirect

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

subgroup	index
#13050162	2
#13050159	2
#13050158	2
#13050154	2
#13050153	2
#13050149	2
#13050142	2
#13050140	2
C₄⋊Q₈×C₄₁	2
#563	41

subgroup

index

C₄⋊Q₈×C₄₁

Subgroups — #14581085

Conjugacy classes of subgroups

Complements

Maximal subgroups