Group #38763193 — subgroups

Subgroups — #38763193

Order 25408.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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×#20263731	C₂	1	2	yes	maximal, complemented
2	#27163093	C₂	1	1	yes	maximal, characteristic, complemented
3	#27163092	C₂	1	2	yes	maximal, complemented
4	#27163070	C₂	1	2	yes	maximal, complemented
5	#27163069	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₇₉₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#27163058	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#20263731	—	2	8	no	—
10	C₂×Dic₇₉₄	C₂²	1	1	yes	characteristic
11	C₂×D₁₅₈₈	C₂²	1	2	yes	—
12	C₂×D₁₅₈₈	C₂²	1	2	yes	—
13	C₂×D₁₅₈₈	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₃₉₇	C₂²	1	4	yes	—
15	C₂×C₂²⋊D₃₉₇	C₂²	1	4	yes	—
16	C₂×C₄×D₃₉₇	C₂²	1	4	yes	—
17	C₄⋊Dic₃₉₇	C₂²	1	1	yes	characteristic
18	#20263710	C₂²	1	2	yes	—
19	#20263708	C₂²	1	4	yes	—
20	C₄×Dic₃₉₇	C₂²	1	2	yes	—
21	C₂×D₄×C₃₉₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₃₉₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₃₉₇	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₃₉₇	C₂²	1	2	yes	—
25	C₄²×C₃₉₇	C₂²	1	1	yes	characteristic
26	C₂²×D₃₉₇	C₂³	1	4	yes	—
27	Dic₇₉₄	D₄	1	4	yes	complemented
28	D₁₅₈₈	—	2	4	no	—
29	D₁₅₈₈	—	2	4	no	—
30	D₁₅₈₈	—	4	2	no	—
31	C₂²⋊D₃₉₇	—	2	8	no	—
32	C₂²⋊D₃₉₇	—	4	4	no	—
33	C₂×Dic₃₉₇	C₂³	1	2	yes	—
34	C₂×Dic₃₉₇	C₂³	1	2	yes	—
35	C₄×D₃₉₇	—	2	8	no	—
36	C₂³×C₃₉₇	C₂³	1	2	yes	—
37	D₄×C₃₉₇	—	2	2	no	—
38	C₂×C₄×C₃₉₇	—	2	4	no	—
39	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₃₉₇	C₂³	1	2	yes	—
41	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
43	C₂²×C₃₉₇	—	2	2	no	—
44	C₂²×C₃₉₇	—	2	2	no	—
45	C₂²×C₃₉₇	—	2	2	no	—
46	C₂²×C₃₉₇	C₂⁴	1	1	yes	characteristic
47	C₂×D₃₉₇	—	2	4	no	—
48	C₂×D₃₉₇	—	2	4	no	—
49	C₂×D₃₉₇	—	2	4	no	—
50	C₄×C₃₉₇	—	2	2	no	—
51	C₄×C₃₉₇	—	2	1	no	—
52	C₄×C₃₉₇	C₂×D₄	1	2	yes	—
53	C₄×C₃₉₇	#41	1	2	yes	complemented
54	Dic₃₉₇	—	2	2	no	—
55	Dic₃₉₇	C₂×D₄	1	4	yes	—
56	C₂×C₃₉₇	—	4	2	no	—
57	C₂×C₃₉₇	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₃₉₇	C₂×#41	1	1	yes	characteristic
59	C₂×C₃₉₇	H(5,𝔽₂)	1	1	yes	characteristic
60	D₃₉₇	—	4	4	no	—
61	C₃₉₇	#550	1	1	yes	characteristic, complemented, Sylow
62	#550	—	397	1	no	maximal, Sylow
63	C₂×#41	—	397	2	no	—
64	#127	—	397	2	no	—
65	#127	—	397	1	no	—
66	#121	—	397	2	no	—
67	#121	—	397	4	no	—
68	C₄×Q₈	—	397	1	no	—
69	C₄×D₄	—	397	2	no	—
70	C₄×D₄	D₃₉₇	1	1	yes	characteristic, complemented
71	#41	—	794	8	no	—
72	C₂×Q₈	—	397	1	no	—
73	C₂×D₄	—	397	4	no	—
74	C₂×D₄	—	397	4	no	—
75	C₂×D₄	—	397	2	no	—
76	C₂×D₄	—	397	2	no	—
77	C₂×D₄	—	397	2	no	—
78	C₂×D₄	C₂×D₃₉₇	1	1	yes	characteristic
79	C₂²×C₄	—	397	4	no	—
80	C₂²×C₄	C₂×D₃₉₇	1	2	yes	—
81	C₄⋊C₄	—	397	2	no	—
82	C₄⋊C₄	—	397	1	no	—
83	C₄⋊C₄	C₂×D₃₉₇	1	1	yes	characteristic
84	C₂²⋊C₄	—	397	4	no	—
85	C₂²⋊C₄	C₂×D₃₉₇	1	2	yes	—
86	C₄²	—	397	2	no	—
87	C₄²	C₂×D₃₉₇	1	1	yes	characteristic
88	C₂³	—	397	4	no	—
89	C₂³	C₂²×D₃₉₇	1	2	yes	—
90	Q₈	—	397	4	no	—
91	D₄	—	2	2	no	—
92	D₄	—	794	8	no	—
93	D₄	—	794	4	no	—
94	D₄	—	794	4	no	—
95	D₄	—	1588	2	no	—
96	D₄	—	1588	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	794	8	no	—
99	C₂×C₄	—	397	2	no	—
100	C₂×C₄	—	397	2	no	—
101	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₃₉₇	1	2	yes	—
103	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	794	4	no	—
109	C₂²	—	794	4	no	—
110	C₂²	—	794	4	no	—
111	C₂²	C₂³×D₃₉₇	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	—	794	2	no	—
115	C₄	—	397	4	no	—
116	C₄	D₄×D₃₉₇	1	2	yes	—
117	C₄	#20263731	1	2	yes	complemented
118	C₂	—	4	2	no	—
119	C₂	—	1588	4	no	—
120	C₂	C₂×D₄×D₃₉₇	1	1	yes	characteristic
121	C₂	C₂×#20263731	1	1	yes	characteristic
122	C₂	#27163142	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263731

C₂

yes

maximal, complemented

#27163093

C₂

yes

maximal, characteristic, complemented

#27163092

C₂

yes

maximal, complemented

#27163070

C₂

yes

maximal, complemented

#27163069

C₂

yes

maximal, complemented

C₄×Dic₇₉₄

C₂

yes

maximal, characteristic, complemented

#27163058

C₂

yes

maximal, complemented

C₄×D₄×C₃₉₇

C₂

yes

maximal, characteristic, complemented

#20263731

—

C₂×Dic₇₉₄

C₂²

yes

characteristic

C₂×D₁₅₈₈

C₂²

yes

—

C₂×D₁₅₈₈

C₂²

yes

—

C₂×D₁₅₈₈

C₂²

yes

—

C₂×C₂²⋊D₃₉₇

C₂²

yes

—

C₂×C₂²⋊D₃₉₇

C₂²

yes

—

C₂×C₄×D₃₉₇

C₂²

yes

—

C₄⋊Dic₃₉₇

C₂²

yes

characteristic

#20263710

C₂²

yes

—

#20263708

C₂²

yes

—

C₄×Dic₃₉₇

C₂²

yes

—

C₂×D₄×C₃₉₇

C₂²

yes

characteristic

C₂²×C₄×C₃₉₇

C₂²

yes

—

C₄⋊C₄×C₃₉₇

C₂²

yes

characteristic

C₂²⋊C₄×C₃₉₇

C₂²

yes

—

C₄²×C₃₉₇

C₂²

yes

characteristic

C₂²×D₃₉₇

C₂³

yes

—

Dic₇₉₄

D₄

yes

complemented

D₁₅₈₈

—

D₁₅₈₈

—

D₁₅₈₈

—

C₂²⋊D₃₉₇

—

C₂²⋊D₃₉₇

—

C₂×Dic₃₉₇

C₂³

yes

—

C₂×Dic₃₉₇

C₂³

yes

—

C₄×D₃₉₇

—

C₂³×C₃₉₇

C₂³

yes

—

D₄×C₃₉₇

—

C₂×C₄×C₃₉₇

—

C₂×C₄×C₃₉₇

C₂³

yes

characteristic

C₂×C₄×C₃₉₇

C₂³

yes

—

C₂×C₄×C₃₉₇

C₂³

yes

characteristic

C₂×C₄×C₃₉₇

C₂³

yes

characteristic

C₂²×C₃₉₇

—

C₂²×C₃₉₇

—

C₂²×C₃₉₇

—

C₂²×C₃₉₇

C₂⁴

yes

characteristic

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₄×C₃₉₇

—

C₄×C₃₉₇

—

C₄×C₃₉₇

C₂×D₄

yes

—

C₄×C₃₉₇

#41

yes

complemented

Dic₃₉₇

—

Dic₃₉₇

C₂×D₄

yes

—

C₂×C₃₉₇

—

C₂×C₃₉₇

C₂²×D₄

yes

characteristic

C₂×C₃₉₇

C₂×#41

yes

characteristic

C₂×C₃₉₇

H(5,𝔽₂)

yes

characteristic

D₃₉₇

—

C₃₉₇

#550

yes

characteristic, complemented, Sylow

#550

—

397

maximal, Sylow

C₂×#41

—

397

—

#127

—

397

—

#127

—

397

—

#121

—

397

—

#121

—

397

—

C₄×Q₈

—

397

—

C₄×D₄

—

397

—

C₄×D₄

D₃₉₇

yes

characteristic, complemented

#41

—

794

—

C₂×Q₈

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂×D₄

C₂×D₃₉₇

yes

characteristic

C₂²×C₄

—

397

—

C₂²×C₄

C₂×D₃₉₇

yes

—

C₄⋊C₄

—

397

—

C₄⋊C₄

—

397

—

C₄⋊C₄

C₂×D₃₉₇

yes

characteristic

C₂²⋊C₄

—

397

—

C₂²⋊C₄

C₂×D₃₉₇

yes

—

C₄²

—

397

—

C₄²

C₂×D₃₉₇

yes

characteristic

C₂³

—

397

—

C₂³

C₂²×D₃₉₇

yes

—

Q₈

—

397

—

D₄

—

D₄

—

794

—

D₄

—

794

—

D₄

—

794

—

D₄

—

1588

—

D₄

—

1588

—

C₂×C₄

—

C₂×C₄

—

794

—

C₂×C₄

—

397

—

100

C₂×C₄

—

397

—

101

C₂×C₄

C₂²×D₃₉₇

yes

characteristic

102

C₂×C₄

C₂²×D₃₉₇

yes

—

103

C₂×C₄

C₂²×D₃₉₇

yes

characteristic

104

C₂×C₄

C₂²×D₃₉₇

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

794

—

109

C₂²

—

794

—

110

C₂²

—

794

—

111

C₂²

C₂³×D₃₉₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

794

—

115

C₄

—

397

—

116

C₄

D₄×D₃₉₇

yes

—

117

C₄

#20263731

yes

complemented

118

C₂

—

119

C₂

—

1588

—

120

C₂

C₂×D₄×D₃₉₇

yes

characteristic

121

C₂

C₂×#20263731

yes

characteristic

122

C₂

#27163142

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₉₇
commutator, socle, cosocle	C₂²×C₃₉₇	C₂⁴
Fitting	C₄×D₄×C₃₉₇	C₂
hypercenter	C₄×D₄	D₃₉₇
hypocenter	C₃₉₇	#550
LCS	G ▷ C₂²×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₉₇

commutator, socle, cosocle

C₂²×C₃₉₇

C₂⁴

Fitting

C₄×D₄×C₃₉₇

C₂

hypercenter

C₄×D₄

D₃₉₇

C₃₉₇

G ▷ C₂²×C₃₉₇ ▷ C₃₉₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#20263731	C₂	semidirect
2. #27163093	C₂	semidirect
3. #27163092	C₂	semidirect
4. #27163070	C₂	semidirect
5. #27163069	C₂	semidirect
6. C₄×Dic₇₉₄	C₂	semidirect
7. #27163058	C₂	semidirect
8. C₄×D₄×C₃₉₇	C₂	semidirect
27. Dic₇₉₄	D₄	semidirect
53. C₄×C₃₉₇	#41	semidirect
61. C₃₉₇	#550	semidirect
70. C₄×D₄	D₃₉₇	semidirect
117. C₄	#20263731	semidirect

H ≅ G/N

type

1. C₂×#20263731

C₂

semidirect

2. #27163093

C₂

semidirect

3. #27163092

C₂

semidirect

4. #27163070

C₂

semidirect

5. #27163069

C₂

semidirect

6. C₄×Dic₇₉₄

C₂

semidirect

7. #27163058

C₂

semidirect

8. C₄×D₄×C₃₉₇

C₂

semidirect

27. Dic₇₉₄

D₄

semidirect

53. C₄×C₃₉₇

#41

semidirect

61. C₃₉₇

#550

semidirect

70. C₄×D₄

D₃₉₇

semidirect

117. C₄

#20263731

semidirect

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

subgroup	index
C₂×#20263731	2
#27163093	2
#27163092	2
#27163070	2
#27163069	2
C₄×Dic₇₉₄	2
#27163058	2
C₄×D₄×C₃₉₇	2
#550	397

subgroup

index

C₄×Dic₇₉₄

C₄×D₄×C₃₉₇

397

Subgroups — #38763193

Conjugacy classes of subgroups

Complements

Maximal subgroups