Group #38763033 — subgroups

Subgroups — #38763033

Order 25408.

330 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 119 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	1	yes	maximal, characteristic, complemented
2	#27163079	C₂	1	2	yes	maximal, complemented
3	#27163075	C₂	1	2	yes	maximal, complemented
4	#27163070	C₂	1	2	yes	maximal, complemented
5	#27163065	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊D₃₉₇	C₂	1	4	yes	maximal, complemented
7	#27163055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#20263731	—	2	4	no	—
10	C₂×Dic₇₉₄	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₁₅₈₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊D₃₉₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₃₉₇	C₂²	1	4	yes	—
14	C₂×C₄×D₃₉₇	C₂²	1	2	yes	complemented
15	C₄⋊Dic₃₉₇	C₂²	1	2	yes	complemented
16	#20263710	C₂²	1	4	yes	complemented
17	C₂²⋊Dic₃₉₇	C₂²	1	2	yes	—
18	C₂²⋊Dic₃₉₇	C₂²	1	4	yes	—
19	#20263708	C₂²	1	4	yes	complemented
20	C₄×Dic₃₉₇	C₂²	1	2	yes	complemented
21	C₂⁴×C₃₉₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₃₉₇	—	2	4	no	—
23	C₂²×C₄×C₃₉₇	C₂²	1	2	yes	—
24	C₂²×C₄×C₃₉₇	C₂²	1	2	yes	—
25	C₂²×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
27	C₂²×D₃₉₇	C₂³	1	2	yes	—
28	Dic₇₉₄	—	2	2	no	—
29	D₁₅₈₈	—	2	2	no	—
30	C₂²⋊D₃₉₇	—	2	4	no	—
31	C₂²⋊D₃₉₇	—	2	8	no	—
32	C₂×Dic₃₉₇	C₂³	1	4	yes	—
33	C₂×Dic₃₉₇	C₂³	1	2	yes	—
34	C₄×D₃₉₇	—	2	4	no	—
35	C₂³×C₃₉₇	—	2	2	no	—
36	C₂³×C₃₉₇	—	2	4	no	—
37	C₂³×C₃₉₇	C₂³	1	1	yes	characteristic
38	C₂³×C₃₉₇	C₂³	1	2	yes	—
39	C₂×C₄×C₃₉₇	—	2	2	no	—
40	C₂×C₄×C₃₉₇	—	2	4	no	—
41	C₂×C₄×C₃₉₇	—	2	4	no	—
42	C₂×C₄×C₃₉₇	D₄	1	4	yes	complemented
43	C₂×C₄×C₃₉₇	C₂³	1	2	yes	—
44	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
46	C₂²×C₃₉₇	—	2	8	no	—
47	C₂²×C₃₉₇	—	2	2	no	—
48	C₂²×C₃₉₇	—	2	2	no	—
49	C₂²×C₃₉₇	—	2	2	no	—
50	C₂²×C₃₉₇	C₂×D₄	1	2	yes	—
51	C₂²×C₃₉₇	#41	1	4	yes	complemented
52	C₂²×C₃₉₇	C₂⁴	1	1	yes	characteristic
53	C₂×D₃₉₇	—	2	2	no	—
54	C₂×D₃₉₇	—	2	4	no	—
55	C₄×C₃₉₇	—	2	2	no	—
56	C₄×C₃₉₇	C₂×D₄	1	2	yes	—
57	C₄×C₃₉₇	C₂×D₄	1	2	yes	—
58	Dic₃₉₇	—	2	2	no	—
59	Dic₃₉₇	—	2	4	no	—
60	C₂×C₃₉₇	—	2	2	no	—
61	C₂×C₃₉₇	—	2	4	no	—
62	C₂×C₃₉₇	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₃₉₇	C₂×#41	1	2	yes	—
64	D₃₉₇	—	4	2	no	—
65	C₃₉₇	#525	1	1	yes	characteristic, complemented, Sylow
66	#525	—	397	1	no	maximal, Sylow
67	C₂×#41	—	397	1	no	—
68	C₂³×C₄	D₃₉₇	1	1	yes	characteristic, complemented
69	#123	—	397	2	no	—
70	C₂²⋊Q₈	—	397	2	no	—
71	#121	—	397	2	no	—
72	#120	—	397	2	no	—
73	C₄×D₄	—	397	4	no	—
74	#117	—	397	1	no	—
75	C₂⁴	C₂×D₃₉₇	1	1	yes	characteristic
76	#41	—	794	4	no	—
77	C₂×Q₈	—	397	1	no	—
78	C₂×D₄	—	397	4	no	—
79	C₂×D₄	—	397	2	no	—
80	C₂×D₄	—	397	1	no	—
81	C₂²×C₄	—	2	4	no	—
82	C₂²×C₄	—	397	2	no	—
83	C₂²×C₄	C₂×D₃₉₇	1	2	yes	—
84	C₂²×C₄	C₂×D₃₉₇	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₃₉₇	1	2	yes	—
86	C₂²×C₄	C₂×D₃₉₇	1	1	yes	characteristic
87	C₄⋊C₄	—	397	4	no	—
88	C₄⋊C₄	—	397	2	no	—
89	C₂²⋊C₄	—	397	2	no	—
90	C₂²⋊C₄	—	397	4	no	—
91	C₂²⋊C₄	—	397	4	no	—
92	C₄²	—	397	2	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	397	2	no	—
96	C₂³	C₂²×D₃₉₇	1	1	yes	characteristic
97	C₂³	C₂²×D₃₉₇	1	2	yes	—
98	Q₈	—	794	2	no	—
99	D₄	—	794	4	no	—
100	D₄	—	794	8	no	—
101	D₄	—	794	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	794	4	no	—
106	C₂×C₄	—	397	4	no	—
107	C₂×C₄	—	397	2	no	—
108	C₂×C₄	C₂²⋊D₃₉₇	1	4	yes	complemented
109	C₂×C₄	C₂²×D₃₉₇	1	2	yes	—
110	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
112	C₂²	—	2	8	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	794	4	no	—
117	C₂²	—	794	2	no	—
118	C₂²	C₂×C₂²⋊D₃₉₇	1	2	yes	—
119	C₂²	#20263731	1	4	yes	complemented
120	C₂²	C₂³×D₃₉₇	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	794	4	no	—
123	C₄	—	794	2	no	—
124	C₄	C₂×C₂²⋊D₃₉₇	1	2	yes	—
125	C₄	C₂×C₂²⋊D₃₉₇	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	2	4	no	—
128	C₂	—	1588	2	no	—
129	C₂	C₂²×C₂²⋊D₃₉₇	1	1	yes	characteristic
130	C₂	C₂×#20263731	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263731

C₂

yes

maximal, characteristic, complemented

#27163079

C₂

yes

maximal, complemented

#27163075

C₂

yes

maximal, complemented

#27163070

C₂

yes

maximal, complemented

#27163065

C₂

yes

maximal, complemented

C₄×C₂²⋊D₃₉₇

C₂

yes

maximal, complemented

#27163055

C₂

yes

maximal, characteristic, complemented

C₂³×C₄×C₃₉₇

C₂

yes

maximal, characteristic, complemented

#20263731

—

C₂×Dic₇₉₄

C₂²

yes

characteristic, complemented

C₂×D₁₅₈₈

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₃₉₇

C₂²

yes

—

C₂×C₂²⋊D₃₉₇

C₂²

yes

—

C₂×C₄×D₃₉₇

C₂²

yes

complemented

C₄⋊Dic₃₉₇

C₂²

yes

complemented

#20263710

C₂²

yes

complemented

C₂²⋊Dic₃₉₇

C₂²

yes

—

C₂²⋊Dic₃₉₇

C₂²

yes

—

#20263708

C₂²

yes

complemented

C₄×Dic₃₉₇

C₂²

yes

complemented

C₂⁴×C₃₉₇

C₂²

yes

characteristic

C₂²×C₄×C₃₉₇

—

C₂²×C₄×C₃₉₇

C₂²

yes

—

C₂²×C₄×C₃₉₇

C₂²

yes

—

C₂²×C₄×C₃₉₇

C₂²

yes

characteristic

C₂²×C₄×C₃₉₇

C₂²

yes

characteristic

C₂²×D₃₉₇

C₂³

yes

—

Dic₇₉₄

—

D₁₅₈₈

—

C₂²⋊D₃₉₇

—

C₂²⋊D₃₉₇

—

C₂×Dic₃₉₇

C₂³

yes

—

C₂×Dic₃₉₇

C₂³

yes

—

C₄×D₃₉₇

—

C₂³×C₃₉₇

—

C₂³×C₃₉₇

—

C₂³×C₃₉₇

C₂³

yes

characteristic

C₂³×C₃₉₇

C₂³

yes

—

C₂×C₄×C₃₉₇

—

C₂×C₄×C₃₉₇

—

C₂×C₄×C₃₉₇

—

C₂×C₄×C₃₉₇

D₄

yes

complemented

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

—

C₂²×C₃₉₇

C₂×D₄

yes

—

C₂²×C₃₉₇

#41

yes

complemented

C₂²×C₃₉₇

C₂⁴

yes

characteristic

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₄×C₃₉₇

—

C₄×C₃₉₇

C₂×D₄

yes

—

C₄×C₃₉₇

C₂×D₄

yes

—

Dic₃₉₇

—

Dic₃₉₇

—

C₂×C₃₉₇

—

C₂×C₃₉₇

—

C₂×C₃₉₇

C₂²×D₄

yes

characteristic

C₂×C₃₉₇

C₂×#41

yes

—

D₃₉₇

—

C₃₉₇

#525

yes

characteristic, complemented, Sylow

#525

—

397

maximal, Sylow

C₂×#41

—

397

—

C₂³×C₄

D₃₉₇

yes

characteristic, complemented

#123

—

397

—

C₂²⋊Q₈

—

397

—

#121

—

397

—

#120

—

397

—

C₄×D₄

—

397

—

#117

—

397

—

C₂⁴

C₂×D₃₉₇

yes

characteristic

#41

—

794

—

C₂×Q₈

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂²×C₄

—

C₂²×C₄

—

397

—

C₂²×C₄

C₂×D₃₉₇

yes

—

C₂²×C₄

C₂×D₃₉₇

yes

characteristic

C₂²×C₄

C₂×D₃₉₇

yes

—

C₂²×C₄

C₂×D₃₉₇

yes

characteristic

C₄⋊C₄

—

397

—

C₄⋊C₄

—

397

—

C₂²⋊C₄

—

397

—

C₂²⋊C₄

—

397

—

C₂²⋊C₄

—

397

—

C₄²

—

397

—

C₂³

—

C₂³

—

C₂³

—

397

—

C₂³

C₂²×D₃₉₇

yes

characteristic

C₂³

C₂²×D₃₉₇

yes

—

Q₈

—

794

—

D₄

—

794

—

100

D₄

—

794

—

101

D₄

—

794

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

794

—

106

C₂×C₄

—

397

—

107

C₂×C₄

—

397

—

108

C₂×C₄

C₂²⋊D₃₉₇

yes

complemented

109

C₂×C₄

C₂²×D₃₉₇

yes

—

110

C₂×C₄

C₂²×D₃₉₇

yes

characteristic

111

C₂×C₄

C₂²×D₃₉₇

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

794

—

117

C₂²

—

794

—

118

C₂²

C₂×C₂²⋊D₃₉₇

yes

—

119

C₂²

#20263731

yes

complemented

120

C₂²

C₂³×D₃₉₇

yes

characteristic

121

C₄

—

122

C₄

—

794

—

123

C₄

—

794

—

124

C₄

C₂×C₂²⋊D₃₉₇

yes

—

125

C₄

C₂×C₂²⋊D₃₉₇

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

1588

—

129

C₂

C₂²×C₂²⋊D₃₉₇

yes

characteristic

130

C₂

C₂×#20263731

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₉₇
commutator, socle, cosocle	C₂²×C₃₉₇	C₂⁴
Frattini	C₂²	C₂³×D₃₉₇
Fitting	C₂³×C₄×C₃₉₇	C₂
hypercenter	C₂³×C₄	D₃₉₇
hypocenter	C₃₉₇	#525
LCS	G ▷ C₂²×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

subgroup

G/H

center

C₂×C₄

C₂²×D₃₉₇

commutator, socle, cosocle

C₂²×C₃₉₇

C₂⁴

Frattini

C₂²

C₂³×D₃₉₇

Fitting

C₂³×C₄×C₃₉₇

C₂

hypercenter

C₂³×C₄

D₃₉₇

C₃₉₇

G ▷ C₂²×C₃₉₇ ▷ C₃₉₇

C₁ ◁ C₂×C₄ ◁ C₂³×C₄

N	H ≅ G/N	type
1. C₂×#20263731	C₂	semidirect
2. #27163079	C₂	semidirect
3. #27163075	C₂	semidirect
4. #27163070	C₂	semidirect
5. #27163065	C₂	semidirect
6. C₄×C₂²⋊D₃₉₇	C₂	semidirect
7. #27163055	C₂	semidirect
8. C₂³×C₄×C₃₉₇	C₂	semidirect
10. C₂×Dic₇₉₄	C₂²	semidirect
11. C₂×D₁₅₈₈	C₂²	semidirect
14. C₂×C₄×D₃₉₇	C₂²	semidirect
15. C₄⋊Dic₃₉₇	C₂²	semidirect
16. #20263710	C₂²	semidirect
19. #20263708	C₂²	semidirect
20. C₄×Dic₃₉₇	C₂²	semidirect
42. C₂×C₄×C₃₉₇	D₄	semidirect
51. C₂²×C₃₉₇	#41	semidirect
65. C₃₉₇	#525	semidirect
68. C₂³×C₄	D₃₉₇	semidirect
108. C₂×C₄	C₂²⋊D₃₉₇	semidirect
119. C₂²	#20263731	semidirect

H ≅ G/N

type

1. C₂×#20263731

C₂

semidirect

2. #27163079

C₂

semidirect

3. #27163075

C₂

semidirect

4. #27163070

C₂

semidirect

5. #27163065

C₂

semidirect

6. C₄×C₂²⋊D₃₉₇

C₂

semidirect

7. #27163055

C₂

semidirect

8. C₂³×C₄×C₃₉₇

C₂

semidirect

10. C₂×Dic₇₉₄

C₂²

semidirect

11. C₂×D₁₅₈₈

C₂²

semidirect

14. C₂×C₄×D₃₉₇

C₂²

semidirect

15. C₄⋊Dic₃₉₇

C₂²

semidirect

16. #20263710

C₂²

semidirect

19. #20263708

C₂²

semidirect

20. C₄×Dic₃₉₇

C₂²

semidirect

42. C₂×C₄×C₃₉₇

D₄

semidirect

51. C₂²×C₃₉₇

#41

semidirect

65. C₃₉₇

#525

semidirect

68. C₂³×C₄

D₃₉₇

semidirect

108. C₂×C₄

C₂²⋊D₃₉₇

semidirect

119. C₂²

#20263731

semidirect

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

subgroup	index
C₂×#20263731	2
#27163079	2
#27163075	2
#27163070	2
#27163065	2
C₄×C₂²⋊D₃₉₇	2
#27163055	2
C₂³×C₄×C₃₉₇	2
#525	397

subgroup

index

C₄×C₂²⋊D₃₉₇

C₂³×C₄×C₃₉₇

397

Subgroups — #38763033

Conjugacy classes of subgroups

Complements

Maximal subgroups