Group #37589105 — subgroups

Subgroups — #37589105

Order 24512.

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	#27133840	C₂	1	2	yes	maximal, complemented
2	#27133836	C₂	1	2	yes	maximal, complemented
3	#27133836	C₂	1	2	yes	maximal, complemented
4	#27133833	C₂	1	1	yes	maximal, characteristic, complemented
5	#27133829	C₂	1	1	yes	maximal, characteristic
6	#27133826	C₂	1	2	yes	maximal
7	#27133821	C₂	1	2	yes	maximal, complemented
8	#27133819	C₂	1	2	yes	maximal, complemented
9	#125×C₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₇₆₆	C₂²	1	2	yes	—
11	C₂×Dic₇₆₆	C₂²	1	1	yes	characteristic
12	C₂×D₁₅₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₈₃	C₂²	1	2	yes	—
14	C₄⋊Dic₃₈₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃₈₃	C₂²	1	2	yes	—
16	C₄⋊Dic₃₈₃	C₂²	1	1	yes	characteristic
17	Dic₃₈₃⋊C₄	C₂²	1	2	yes	—
18	Dic₃₈₃⋊C₄	C₂²	1	2	yes	—
19	Dic₃₈₃⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₃₈₃)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₃₈₃)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₃₈₃)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₃₈₃)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₈₃)⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₃₈₃	C₂²	1	2	yes	—
26	C₄⋊C₄×C₃₈₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₃₈₃	C₂²	1	2	yes	—
28	C₄⋊C₄×C₃₈₃	C₂²	1	2	yes	—
29	C₄⋊C₄×C₃₈₃	C₂²	1	1	yes	characteristic
30	C₄²×C₃₈₃	C₂²	1	1	yes	characteristic
31	C₂²×D₃₈₃	C₂³	1	2	yes	—
32	Dic₇₆₆	—	4	2	no	—
33	Dic₇₆₆	—	4	1	no	—
34	D₁₅₃₂	—	4	1	no	—
35	C₂×Dic₃₈₃	C₂³	1	2	yes	—
36	C₂×Dic₃₈₃	C₂³	1	2	yes	—
37	C₂×Dic₃₈₃	C₂³	1	2	yes	—
38	C₄×D₃₈₃	—	4	2	no	—
39	C₂×C₄×C₃₈₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₃₈₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₃₈₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₃₈₃	C₂³	1	2	yes	—
43	C₂×C₄×C₃₈₃	C₂³	1	2	yes	—
44	C₂²×C₃₈₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₃₈₃	—	2	2	no	—
46	C₂×D₃₈₃	—	2	2	no	—
47	C₂×D₃₈₃	—	2	2	no	—
48	C₄×C₃₈₃	—	2	2	no	—
49	C₄×C₃₈₃	—	2	1	no	—
50	C₄×C₃₈₃	—	2	1	no	—
51	C₄×C₃₈₃	—	2	2	no	—
52	C₄×C₃₈₃	—	2	1	no	—
53	Dic₃₈₃	—	2	2	no	—
54	Dic₃₈₃	—	2	2	no	—
55	Dic₃₈₃	—	2	2	no	—
56	C₂×C₃₈₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₃₈₃	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₃₈₃	2₋¹⁺⁴	1	1	yes	characteristic
59	D₃₈₃	—	4	2	no	—
60	C₃₈₃	#563	1	1	yes	characteristic, complemented, Sylow
61	#563	—	383	1	no	maximal, Sylow
62	C₄⋊Q₈	—	383	2	no	—
63	#126	—	383	2	no	—
64	#126	—	383	2	no	—
65	#125	—	383	1	no	—
66	#125	D₃₈₃	1	1	yes	characteristic, complemented
67	#124	—	383	1	no	—
68	#123	—	383	2	no	—
69	C₂²⋊Q₈	—	383	2	no	—
70	C₂²⋊Q₈	—	383	2	no	—
71	C₂×Q₈	—	383	2	no	—
72	C₂×Q₈	—	383	1	no	—
73	C₂×D₄	—	383	1	no	—
74	C₂²×C₄	—	383	2	no	—
75	C₄⋊C₄	—	383	2	no	—
76	C₄⋊C₄	—	383	2	no	—
77	C₄⋊C₄	—	383	2	no	—
78	C₄⋊C₄	—	383	1	no	—
79	C₄⋊C₄	—	383	2	no	—
80	C₄⋊C₄	—	383	1	no	—
81	C₄⋊C₄	C₂×D₃₈₃	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₃₈₃	1	2	yes	—
83	C₄⋊C₄	C₂×D₃₈₃	1	2	yes	—
84	C₄⋊C₄	C₂×D₃₈₃	1	1	yes	characteristic
85	C₂²⋊C₄	—	383	2	no	—
86	C₂²⋊C₄	—	383	2	no	—
87	C₂²⋊C₄	—	383	2	no	—
88	C₂²⋊C₄	—	383	2	no	—
89	C₂²⋊C₄	—	383	2	no	—
90	C₄²	—	383	2	no	—
91	C₄²	C₂×D₃₈₃	1	1	yes	characteristic
92	C₂³	—	383	2	no	—
93	Q₈	—	1532	2	no	—
94	Q₈	—	1532	1	no	—
95	D₄	—	1532	1	no	—
96	C₂×C₄	—	383	2	no	—
97	C₂×C₄	—	383	2	no	—
98	C₂×C₄	—	383	2	no	—
99	C₂×C₄	—	1532	2	no	—
100	C₂×C₄	C₂²×D₃₈₃	1	2	yes	—
101	C₂×C₄	C₂²×D₃₈₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₃₈₃	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₃₈₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₃₈₃	1	2	yes	—
105	C₂²	—	766	2	no	—
106	C₂²	—	766	2	no	—
107	C₂²	—	766	2	no	—
108	C₂²	C₂³×D₃₈₃	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	766	2	no	—
115	C₄	—	766	2	no	—
116	C₄	—	766	2	no	—
117	C₂	—	1532	2	no	—
118	C₂	#27133887	1	1	yes	characteristic
119	C₂	#27133888	1	1	yes	characteristic
120	C₂	#27133889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133840

C₂

yes

maximal, complemented

#27133836

C₂

yes

maximal, complemented

#27133836

C₂

yes

maximal, complemented

#27133833

C₂

yes

maximal, characteristic, complemented

#27133829

C₂

yes

maximal, characteristic

#27133826

C₂

yes

maximal

#27133821

C₂

yes

maximal, complemented

#27133819

C₂

yes

maximal, complemented

#125×C₃₈₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₆₆

C₂²

yes

—

C₂×Dic₇₆₆

C₂²

yes

characteristic

C₂×D₁₅₃₂

C₂²

yes

characteristic

C₂×C₄×D₃₈₃

C₂²

yes

—

C₄⋊Dic₃₈₃

C₂²

yes

characteristic

C₄⋊Dic₃₈₃

C₂²

yes

—

C₄⋊Dic₃₈₃

C₂²

yes

characteristic

Dic₃₈₃⋊C₄

C₂²

yes

—

Dic₃₈₃⋊C₄

C₂²

yes

—

Dic₃₈₃⋊C₄

C₂²

yes

—

(C₂×D₃₈₃)⋊C₄

C₂²

yes

—

(C₂×D₃₈₃)⋊C₄

C₂²

yes

—

(C₂×D₃₈₃)⋊C₄

C₂²

yes

—

(C₂×D₃₈₃)⋊C₄

C₂²

yes

—

(C₂×D₃₈₃)⋊C₄

C₂²

yes

—

C₄×Dic₃₈₃

C₂²

yes

—

C₄⋊C₄×C₃₈₃

C₂²

yes

characteristic

C₄⋊C₄×C₃₈₃

C₂²

yes

—

C₄⋊C₄×C₃₈₃

C₂²

yes

—

C₄⋊C₄×C₃₈₃

C₂²

yes

characteristic

C₄²×C₃₈₃

C₂²

yes

characteristic

C₂²×D₃₈₃

C₂³

yes

—

Dic₇₆₆

—

Dic₇₆₆

—

D₁₅₃₂

—

C₂×Dic₃₈₃

C₂³

yes

—

C₂×Dic₃₈₃

C₂³

yes

—

C₂×Dic₃₈₃

C₂³

yes

—

C₄×D₃₈₃

—

C₂×C₄×C₃₈₃

C₂³

yes

characteristic

C₂×C₄×C₃₈₃

C₂³

yes

characteristic

C₂×C₄×C₃₈₃

C₂³

yes

characteristic

C₂×C₄×C₃₈₃

C₂³

yes

—

C₂×C₄×C₃₈₃

C₂³

yes

—

C₂²×C₃₈₃

C₂⁴

yes

characteristic

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₄×C₃₈₃

—

C₄×C₃₈₃

—

C₄×C₃₈₃

—

C₄×C₃₈₃

—

C₄×C₃₈₃

—

Dic₃₈₃

—

Dic₃₈₃

—

Dic₃₈₃

—

C₂×C₃₈₃

H(5,𝔽₂)

yes

characteristic

C₂×C₃₈₃

2₋¹⁺⁴

yes

characteristic

C₂×C₃₈₃

2₋¹⁺⁴

yes

characteristic

D₃₈₃

—

C₃₈₃

#563

yes

characteristic, complemented, Sylow

#563

—

383

maximal, Sylow

C₄⋊Q₈

—

383

—

#126

—

383

—

#126

—

383

—

#125

—

383

—

#125

D₃₈₃

yes

characteristic, complemented

#124

—

383

—

#123

—

383

—

C₂²⋊Q₈

—

383

—

C₂²⋊Q₈

—

383

—

C₂×Q₈

—

383

—

C₂×Q₈

—

383

—

C₂×D₄

—

383

—

C₂²×C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

C₂×D₃₈₃

yes

characteristic

C₄⋊C₄

C₂×D₃₈₃

yes

—

C₄⋊C₄

C₂×D₃₈₃

yes

—

C₄⋊C₄

C₂×D₃₈₃

yes

characteristic

C₂²⋊C₄

—

383

—

C₂²⋊C₄

—

383

—

C₂²⋊C₄

—

383

—

C₂²⋊C₄

—

383

—

C₂²⋊C₄

—

383

—

C₄²

—

383

—

C₄²

C₂×D₃₈₃

yes

characteristic

C₂³

—

383

—

Q₈

—

1532

—

Q₈

—

1532

—

D₄

—

1532

—

C₂×C₄

—

383

—

C₂×C₄

—

383

—

C₂×C₄

—

383

—

C₂×C₄

—

1532

—

100

C₂×C₄

C₂²×D₃₈₃

yes

—

101

C₂×C₄

C₂²×D₃₈₃

yes

characteristic

102

C₂×C₄

C₂²×D₃₈₃

yes

characteristic

103

C₂×C₄

C₂²×D₃₈₃

yes

characteristic

104

C₂×C₄

C₂²×D₃₈₃

yes

—

105

C₂²

—

766

—

106

C₂²

—

766

—

107

C₂²

—

766

—

108

C₂²

C₂³×D₃₈₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

766

—

115

C₄

—

766

—

116

C₄

—

766

—

117

C₂

—

1532

—

118

C₂

#27133887

yes

characteristic

119

C₂

#27133888

yes

characteristic

120

C₂

#27133889

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₈₃
commutator, socle, cosocle	C₂²×C₃₈₃	C₂⁴
Fitting	#125×C₃₈₃	C₂
prime radical	C₂×D₁₅₃₂	C₂²
hypercenter	#125	D₃₈₃
hypocenter	C₃₈₃	#563
LCS	G ▷ C₂²×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₈₃

commutator, socle, cosocle

C₂²×C₃₈₃

C₂⁴

Fitting

#125×C₃₈₃

C₂

prime radical

C₂×D₁₅₃₂

C₂²

D₃₈₃

C₃₈₃

G ▷ C₂²×C₃₈₃ ▷ C₃₈₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #27133840	C₂	semidirect
2. #27133836	C₂	semidirect
3. #27133836	C₂	semidirect
4. #27133833	C₂	semidirect
7. #27133821	C₂	semidirect
8. #27133819	C₂	semidirect
9. #125×C₃₈₃	C₂	semidirect
60. C₃₈₃	#563	semidirect
66. #125	D₃₈₃	semidirect

H ≅ G/N

type

1. #27133840

C₂

semidirect

2. #27133836

C₂

semidirect

3. #27133836

C₂

semidirect

4. #27133833

C₂

semidirect

7. #27133821

C₂

semidirect

8. #27133819

C₂

semidirect

9. #125×C₃₈₃

C₂

semidirect

60. C₃₈₃

#563

semidirect

66. #125

D₃₈₃

semidirect

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

subgroup	index
#27133840	2
#27133836	2
#27133833	2
#27133829	2
#27133826	2
#27133821	2
#27133819	2
#125×C₃₈₃	2
#563	383

subgroup

index

#125×C₃₈₃

383

Subgroups — #37589105

Conjugacy classes of subgroups

Complements

Maximal subgroups