Group #39909996 — subgroups

Subgroups — #39909996

Order 26176.

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	#27189715	C₂	1	1	yes	maximal, characteristic, complemented
2	#27189712	C₂	1	2	yes	maximal, complemented
3	#27189711	C₂	1	2	yes	maximal, complemented
4	#27189707	C₂	1	2	yes	maximal, complemented
5	#27189706	C₂	1	1	yes	maximal, characteristic
6	#27189702	C₂	1	2	yes	maximal
7	#27189695	C₂	1	2	yes	maximal, complemented
8	#27189693	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₄₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₈₁₈	C₂²	1	1	yes	characteristic
11	C₂×D₁₆₃₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄₀₉	C₂²	1	2	yes	—
13	C₄⋊Dic₄₀₉	C₂²	1	2	yes	—
14	#20266967	C₂²	1	2	yes	—
15	#20266967	C₂²	1	2	yes	—
16	#20266967	C₂²	1	2	yes	—
17	#20266967	C₂²	1	2	yes	—
18	#20266967	C₂²	1	2	yes	—
19	#20266965	C₂²	1	2	yes	—
20	#20266965	C₂²	1	2	yes	—
21	#20266965	C₂²	1	2	yes	—
22	#20266965	C₂²	1	2	yes	—
23	#20266965	C₂²	1	2	yes	—
24	C₄×Dic₄₀₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₄₀₉	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₄₀₉	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₄₀₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₄₀₉	C₂²	1	2	yes	—
29	C₄⋊C₄×C₄₀₉	C₂²	1	2	yes	—
30	C₄²×C₄₀₉	C₂²	1	1	yes	characteristic
31	C₂²×D₄₀₉	C₂³	1	2	yes	—
32	Dic₈₁₈	—	4	1	no	—
33	D₁₆₃₆	—	4	1	no	—
34	C₂×Dic₄₀₉	C₂³	1	2	yes	—
35	C₂×Dic₄₀₉	C₂³	1	2	yes	—
36	C₂×Dic₄₀₉	C₂³	1	2	yes	—
37	C₄×D₄₀₉	—	4	2	no	—
38	Q₈×C₄₀₉	—	4	1	no	—
39	Q₈×C₄₀₉	—	4	1	no	—
40	C₂×C₄×C₄₀₉	C₂³	1	2	yes	—
41	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₄₀₉	C₂³	1	2	yes	—
45	C₂²×C₄₀₉	C₂⁴	1	1	yes	characteristic
46	C₂×D₄₀₉	—	2	2	no	—
47	C₂×D₄₀₉	—	2	4	no	—
48	C₄×C₄₀₉	—	2	2	no	—
49	C₄×C₄₀₉	—	2	1	no	—
50	C₄×C₄₀₉	—	2	1	no	—
51	C₄×C₄₀₉	—	2	1	no	—
52	C₄×C₄₀₉	—	2	2	no	—
53	Dic₄₀₉	—	2	2	no	—
54	Dic₄₀₉	—	2	2	no	—
55	Dic₄₀₉	—	2	2	no	—
56	C₂×C₄₀₉	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₄₀₉	2₋¹⁺⁴	1	2	yes	—
58	D₄₀₉	—	4	2	no	—
59	C₄₀₉	#563	1	1	yes	characteristic, complemented, Sylow
60	#563	—	409	1	no	maximal, Sylow
61	C₄⋊Q₈	—	409	1	no	—
62	C₄⋊Q₈	D₄₀₉	1	1	yes	characteristic, complemented
63	#126	—	409	2	no	—
64	#126	—	409	2	no	—
65	#125	—	409	2	no	—
66	#124	—	409	1	no	—
67	#123	—	409	2	no	—
68	C₂²⋊Q₈	—	409	2	no	—
69	C₂²⋊Q₈	—	409	2	no	—
70	C₂×Q₈	—	409	1	no	—
71	C₂×Q₈	C₂×D₄₀₉	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₄₀₉	1	1	yes	characteristic
73	C₂×D₄	—	409	1	no	—
74	C₂²×C₄	—	409	2	no	—
75	C₄⋊C₄	—	409	2	no	—
76	C₄⋊C₄	—	409	2	no	—
77	C₄⋊C₄	—	409	2	no	—
78	C₄⋊C₄	—	409	2	no	—
79	C₄⋊C₄	—	409	2	no	—
80	C₄⋊C₄	—	409	2	no	—
81	C₄⋊C₄	C₂×D₄₀₉	1	2	yes	—
82	C₄⋊C₄	C₂×D₄₀₉	1	2	yes	—
83	C₂²⋊C₄	—	409	2	no	—
84	C₂²⋊C₄	—	409	2	no	—
85	C₂²⋊C₄	—	409	2	no	—
86	C₂²⋊C₄	—	409	2	no	—
87	C₂²⋊C₄	—	409	2	no	—
88	C₄²	—	409	1	no	—
89	C₄²	—	409	1	no	—
90	C₄²	C₂×D₄₀₉	1	1	yes	characteristic
91	C₂³	—	409	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	1636	1	no	—
95	D₄	—	1636	1	no	—
96	C₂×C₄	—	1636	2	no	—
97	C₂×C₄	—	409	2	no	—
98	C₂×C₄	—	409	2	no	—
99	C₂×C₄	—	409	2	no	—
100	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₄₀₉	1	2	yes	—
103	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₄₀₉	1	2	yes	—
105	C₂²	—	818	2	no	—
106	C₂²	—	818	4	no	—
107	C₂²	C₂³×D₄₀₉	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	—	818	2	no	—
114	C₄	—	818	2	no	—
115	C₄	—	818	2	no	—
116	C₂	—	1636	2	no	—
117	C₂	#27189762	1	1	yes	characteristic
118	C₂	#27189765	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27189715

C₂

yes

maximal, characteristic, complemented

#27189712

C₂

yes

maximal, complemented

#27189711

C₂

yes

maximal, complemented

#27189707

C₂

yes

maximal, complemented

#27189706

C₂

yes

maximal, characteristic

#27189702

C₂

yes

maximal

#27189695

C₂

yes

maximal, complemented

#27189693

C₂

yes

maximal, complemented

C₄⋊Q₈×C₄₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₈₁₈

C₂²

yes

characteristic

C₂×D₁₆₃₆

C₂²

yes

characteristic

C₂×C₄×D₄₀₉

C₂²

yes

—

C₄⋊Dic₄₀₉

C₂²

yes

—

#20266967

C₂²

yes

—

#20266967

C₂²

yes

—

#20266967

C₂²

yes

—

#20266967

C₂²

yes

—

#20266967

C₂²

yes

—

#20266965

C₂²

yes

—

#20266965

C₂²

yes

—

#20266965

C₂²

yes

—

#20266965

C₂²

yes

—

#20266965

C₂²

yes

—

C₄×Dic₄₀₉

C₂²

yes

characteristic

C₄×Dic₄₀₉

C₂²

yes

characteristic

C₂×Q₈×C₄₀₉

C₂²

yes

characteristic

C₂×Q₈×C₄₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₄₀₉

C₂²

yes

—

C₄⋊C₄×C₄₀₉

C₂²

yes

—

C₄²×C₄₀₉

C₂²

yes

characteristic

C₂²×D₄₀₉

C₂³

yes

—

Dic₈₁₈

—

D₁₆₃₆

—

C₂×Dic₄₀₉

C₂³

yes

—

C₂×Dic₄₀₉

C₂³

yes

—

C₂×Dic₄₀₉

C₂³

yes

—

C₄×D₄₀₉

—

Q₈×C₄₀₉

—

Q₈×C₄₀₉

—

C₂×C₄×C₄₀₉

C₂³

yes

—

C₂×C₄×C₄₀₉

C₂³

yes

characteristic

C₂×C₄×C₄₀₉

C₂³

yes

characteristic

C₂×C₄×C₄₀₉

C₂³

yes

characteristic

C₂×C₄×C₄₀₉

C₂³

yes

—

C₂²×C₄₀₉

C₂⁴

yes

characteristic

C₂×D₄₀₉

—

C₂×D₄₀₉

—

C₄×C₄₀₉

—

C₄×C₄₀₉

—

C₄×C₄₀₉

—

C₄×C₄₀₉

—

C₄×C₄₀₉

—

Dic₄₀₉

—

Dic₄₀₉

—

Dic₄₀₉

—

C₂×C₄₀₉

H(5,𝔽₂)

yes

characteristic

C₂×C₄₀₉

2₋¹⁺⁴

yes

—

D₄₀₉

—

C₄₀₉

#563

yes

characteristic, complemented, Sylow

#563

—

409

maximal, Sylow

C₄⋊Q₈

—

409

—

C₄⋊Q₈

D₄₀₉

yes

characteristic, complemented

#126

—

409

—

#126

—

409

—

#125

—

409

—

#124

—

409

—

#123

—

409

—

C₂²⋊Q₈

—

409

—

C₂²⋊Q₈

—

409

—

C₂×Q₈

—

409

—

C₂×Q₈

C₂×D₄₀₉

yes

characteristic

C₂×Q₈

C₂×D₄₀₉

yes

characteristic

C₂×D₄

—

409

—

C₂²×C₄

—

409

—

C₄⋊C₄

—

409

—

C₄⋊C₄

—

409

—

C₄⋊C₄

—

409

—

C₄⋊C₄

—

409

—

C₄⋊C₄

—

409

—

C₄⋊C₄

—

409

—

C₄⋊C₄

C₂×D₄₀₉

yes

—

C₄⋊C₄

C₂×D₄₀₉

yes

—

C₂²⋊C₄

—

409

—

C₂²⋊C₄

—

409

—

C₂²⋊C₄

—

409

—

C₂²⋊C₄

—

409

—

C₂²⋊C₄

—

409

—

C₄²

—

409

—

C₄²

—

409

—

C₄²

C₂×D₄₀₉

yes

characteristic

C₂³

—

409

—

Q₈

—

Q₈

—

Q₈

—

1636

—

D₄

—

1636

—

C₂×C₄

—

1636

—

C₂×C₄

—

409

—

C₂×C₄

—

409

—

C₂×C₄

—

409

—

100

C₂×C₄

C₂²×D₄₀₉

yes

characteristic

101

C₂×C₄

C₂²×D₄₀₉

yes

characteristic

102

C₂×C₄

C₂²×D₄₀₉

yes

—

103

C₂×C₄

C₂²×D₄₀₉

yes

characteristic

104

C₂×C₄

C₂²×D₄₀₉

yes

—

105

C₂²

—

818

—

106

C₂²

—

818

—

107

C₂²

C₂³×D₄₀₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

818

—

114

C₄

—

818

—

115

C₄

—

818

—

116

C₂

—

1636

—

117

C₂

#27189762

yes

characteristic

118

C₂

#27189765

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₀₉
commutator, socle, cosocle	C₂²×C₄₀₉	C₂⁴
Fitting	C₄⋊Q₈×C₄₀₉	C₂
prime radical	C₂×D₁₆₃₆	C₂²
hypercenter	C₄⋊Q₈	D₄₀₉
hypocenter	C₄₀₉	#563
LCS	G ▷ C₂²×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₀₉

commutator, socle, cosocle

C₂²×C₄₀₉

C₂⁴

Fitting

C₄⋊Q₈×C₄₀₉

C₂

prime radical

C₂×D₁₆₃₆

C₂²

hypercenter

C₄⋊Q₈

D₄₀₉

C₄₀₉

G ▷ C₂²×C₄₀₉ ▷ C₄₀₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #27189715	C₂	semidirect
2. #27189712	C₂	semidirect
3. #27189711	C₂	semidirect
4. #27189707	C₂	semidirect
7. #27189695	C₂	semidirect
8. #27189693	C₂	semidirect
9. C₄⋊Q₈×C₄₀₉	C₂	semidirect
59. C₄₀₉	#563	semidirect
62. C₄⋊Q₈	D₄₀₉	semidirect

H ≅ G/N

type

1. #27189715

C₂

semidirect

2. #27189712

C₂

semidirect

3. #27189711

C₂

semidirect

4. #27189707

C₂

semidirect

7. #27189695

C₂

semidirect

8. #27189693

C₂

semidirect

9. C₄⋊Q₈×C₄₀₉

C₂

semidirect

59. C₄₀₉

#563

semidirect

62. C₄⋊Q₈

D₄₀₉

semidirect

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

subgroup	index
#27189715	2
#27189712	2
#27189711	2
#27189707	2
#27189706	2
#27189702	2
#27189695	2
#27189693	2
C₄⋊Q₈×C₄₀₉	2
#563	409

subgroup

index

C₄⋊Q₈×C₄₀₉

409

Subgroups — #39909996

Conjugacy classes of subgroups

Complements

Maximal subgroups