Group #37527018 — subgroups

Subgroups — #37527018

Order 22976.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#19153342	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#26040013	C₂	1	1	yes	maximal, characteristic, complemented
4	#26040004	C₂	1	1	yes	maximal, characteristic, complemented
5	#26040003	C₂	1	1	yes	maximal, characteristic, complemented
6	#26040000	C₂	1	4	yes	maximal, complemented
7	#26039993	C₂	1	4	yes	maximal, complemented
8	#26039972	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#19153342	—	2	4	no	—
11	C₂×Dic₇₁₈	C₂²	1	2	yes	—
12	C₂×Dic₇₁₈	C₂²	1	1	yes	characteristic
13	Q₈×D₃₅₉	—	2	4	no	—
14	C₂×D₁₄₃₆	C₂²	1	2	yes	—
15	C₂×D₁₄₃₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₃₅₉	C₂²	1	2	yes	—
17	C₂×C₄×D₃₅₉	C₂²	1	2	yes	—
18	C₂×C₄×D₃₅₉	C₂²	1	1	yes	characteristic
19	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
20	Dic₃₅₉⋊C₄	C₂²	1	4	yes	—
21	(C₂×D₃₅₉)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₃₅₉)⋊C₄	C₂²	1	4	yes	—
23	(C₂×D₃₅₉)⋊C₄	C₂²	1	4	yes	—
24	C₄×Dic₃₅₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₃₅₉	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₃₅₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₃₅₉	C₂²	1	4	yes	—
28	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic
29	C₂²×D₃₅₉	C₂³	1	2	yes	—
30	C₂²×D₃₅₉	C₂³	1	1	yes	characteristic
31	Dic₇₁₈	—	2	2	no	—
32	Dic₇₁₈	—	2	4	no	—
33	D₁₄₃₆	—	2	4	no	—
34	D₁₄₃₆	—	2	2	no	—
35	C₂×Dic₃₅₉	C₂³	1	2	yes	—
36	C₂×Dic₃₅₉	C₂³	1	2	yes	—
37	C₂×Dic₃₅₉	C₂³	1	1	yes	characteristic
38	C₄×D₃₅₉	—	2	4	no	—
39	C₄×D₃₅₉	—	2	4	no	—
40	C₄×D₃₅₉	D₄	1	4	yes	—
41	Q₈×C₃₅₉	—	2	2	no	—
42	Q₈×C₃₅₉	—	2	2	no	—
43	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
44	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
45	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
46	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
47	C₂²×C₃₅₉	C₂⁴	1	1	yes	characteristic
48	C₂×D₃₅₉	—	2	2	no	—
49	C₂×D₃₅₉	—	2	2	no	—
50	C₂×D₃₅₉	—	2	4	no	—
51	C₂×D₃₅₉	C₂×D₄	1	2	yes	—
52	C₄×C₃₅₉	—	2	2	no	—
53	C₄×C₃₅₉	—	2	2	no	—
54	C₄×C₃₅₉	—	2	2	no	—
55	C₄×C₃₅₉	C₂×D₄	1	2	yes	—
56	Dic₃₅₉	—	2	2	no	—
57	Dic₃₅₉	—	2	2	no	—
58	Dic₃₅₉	C₂×D₄	1	2	yes	—
59	C₂×C₃₅₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₃₅₉	2₋¹⁺⁴	1	2	yes	—
61	D₃₅₉	—	2	2	no	—
62	D₃₅₉	—	4	2	no	—
63	C₃₅₉	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	359	1	no	maximal, Sylow
65	C₂×#41	—	359	1	no	—
66	C₂²×Q₈	—	359	1	no	—
67	C₄⋊Q₈	—	359	1	no	—
68	C₄⋊Q₈	D₃₅₉	1	1	yes	characteristic, complemented
69	#124	—	359	1	no	—
70	#124	—	359	1	no	—
71	#123	—	359	4	no	—
72	C₂²⋊Q₈	—	359	4	no	—
73	#117	—	359	1	no	—
74	#41	—	718	4	no	—
75	C₂×Q₈	—	359	2	no	—
76	C₂×Q₈	—	359	1	no	—
77	C₂×Q₈	—	718	4	no	—
78	C₂×Q₈	C₂×D₃₅₉	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₃₅₉	1	1	yes	characteristic
80	C₂×D₄	—	359	2	no	—
81	C₂×D₄	—	359	1	no	—
82	C₂²×C₄	—	359	1	no	—
83	C₂²×C₄	—	359	2	no	—
84	C₂²×C₄	—	359	2	no	—
85	C₄⋊C₄	—	359	2	no	—
86	C₄⋊C₄	—	359	4	no	—
87	C₄⋊C₄	C₂×D₃₅₉	1	4	yes	—
88	C₂²⋊C₄	—	359	2	no	—
89	C₂²⋊C₄	—	359	4	no	—
90	C₂²⋊C₄	—	359	4	no	—
91	C₄²	—	359	1	no	—
92	C₄²	C₂×D₃₅₉	1	1	yes	characteristic
93	C₂³	—	359	1	no	—
94	C₂³	—	359	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	718	4	no	—
98	Q₈	—	718	2	no	—
99	D₄	—	718	4	no	—
100	D₄	—	718	2	no	—
101	C₂×C₄	—	359	1	no	—
102	C₂×C₄	—	359	4	no	—
103	C₂×C₄	—	359	2	no	—
104	C₂×C₄	—	359	2	no	—
105	C₂×C₄	—	718	4	no	—
106	C₂×C₄	—	718	4	no	—
107	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
108	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
109	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
110	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
111	C₂²	—	359	2	no	—
112	C₂²	—	718	2	no	—
113	C₂²	—	718	4	no	—
114	C₂²	—	718	2	no	—
115	C₂²	C₂³×D₃₅₉	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	359	2	no	—
120	C₄	—	718	2	no	—
121	C₄	—	718	2	no	—
122	C₄	D₄×D₃₅₉	1	2	yes	—
123	C₂	—	1436	2	no	—
124	C₂	—	718	2	no	—
125	C₂	C₂×D₄×D₃₅₉	1	1	yes	characteristic
126	C₂	#26040063	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153342

C₂

yes

maximal, characteristic

C₂×Q₈×D₃₅₉

C₂

yes

maximal, characteristic, complemented

#26040013

C₂

yes

maximal, characteristic, complemented

#26040004

C₂

yes

maximal, characteristic, complemented

#26040003

C₂

yes

maximal, characteristic, complemented

#26040000

C₂

yes

maximal, complemented

#26039993

C₂

yes

maximal, complemented

#26039972

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₃₅₉

C₂

yes

maximal, characteristic, complemented

#19153342

—

C₂×Dic₇₁₈

C₂²

yes

—

C₂×Dic₇₁₈

C₂²

yes

characteristic

Q₈×D₃₅₉

—

C₂×D₁₄₃₆

C₂²

yes

—

C₂×D₁₄₃₆

C₂²

yes

characteristic

C₂×C₄×D₃₅₉

C₂²

yes

—

C₂×C₄×D₃₅₉

C₂²

yes

—

C₂×C₄×D₃₅₉

C₂²

yes

characteristic

Dic₃₅₉⋊C₄

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

(C₂×D₃₅₉)⋊C₄

C₂²

yes

—

(C₂×D₃₅₉)⋊C₄

C₂²

yes

—

(C₂×D₃₅₉)⋊C₄

C₂²

yes

—

C₄×Dic₃₅₉

C₂²

yes

characteristic

C₂×Q₈×C₃₅₉

C₂²

yes

characteristic

C₂×Q₈×C₃₅₉

C₂²

yes

characteristic

C₄⋊C₄×C₃₅₉

C₂²

yes

—

C₄²×C₃₅₉

C₂²

yes

characteristic

C₂²×D₃₅₉

C₂³

yes

—

C₂²×D₃₅₉

C₂³

yes

characteristic

Dic₇₁₈

—

Dic₇₁₈

—

D₁₄₃₆

—

D₁₄₃₆

—

C₂×Dic₃₅₉

C₂³

yes

—

C₂×Dic₃₅₉

C₂³

yes

—

C₂×Dic₃₅₉

C₂³

yes

characteristic

C₄×D₃₅₉

—

C₄×D₃₅₉

—

C₄×D₃₅₉

D₄

yes

—

Q₈×C₃₅₉

—

Q₈×C₃₅₉

—

C₂×C₄×C₃₅₉

C₂³

yes

—

C₂×C₄×C₃₅₉

C₂³

yes

—

C₂×C₄×C₃₅₉

C₂³

yes

—

C₂×C₄×C₃₅₉

C₂³

yes

characteristic

C₂²×C₃₅₉

C₂⁴

yes

characteristic

C₂×D₃₅₉

—

C₂×D₃₅₉

—

C₂×D₃₅₉

—

C₂×D₃₅₉

C₂×D₄

yes

—

C₄×C₃₅₉

—

C₄×C₃₅₉

—

C₄×C₃₅₉

—

C₄×C₃₅₉

C₂×D₄

yes

—

Dic₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

C₂×D₄

yes

—

C₂×C₃₅₉

C₂²×D₄

yes

characteristic

C₂×C₃₅₉

2₋¹⁺⁴

yes

—

D₃₅₉

—

D₃₅₉

—

C₃₅₉

#536

yes

characteristic, complemented, Sylow

#536

—

359

maximal, Sylow

C₂×#41

—

359

—

C₂²×Q₈

—

359

—

C₄⋊Q₈

—

359

—

C₄⋊Q₈

D₃₅₉

yes

characteristic, complemented

#124

—

359

—

#124

—

359

—

#123

—

359

—

C₂²⋊Q₈

—

359

—

#117

—

359

—

#41

—

718

—

C₂×Q₈

—

359

—

C₂×Q₈

—

359

—

C₂×Q₈

—

718

—

C₂×Q₈

C₂×D₃₅₉

yes

characteristic

C₂×Q₈

C₂×D₃₅₉

yes

characteristic

C₂×D₄

—

359

—

C₂×D₄

—

359

—

C₂²×C₄

—

359

—

C₂²×C₄

—

359

—

C₂²×C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

C₂×D₃₅₉

yes

—

C₂²⋊C₄

—

359

—

C₂²⋊C₄

—

359

—

C₂²⋊C₄

—

359

—

C₄²

—

359

—

C₄²

C₂×D₃₅₉

yes

characteristic

C₂³

—

359

—

C₂³

—

359

—

Q₈

—

Q₈

—

Q₈

—

718

—

Q₈

—

718

—

D₄

—

718

—

100

D₄

—

718

—

101

C₂×C₄

—

359

—

102

C₂×C₄

—

359

—

103

C₂×C₄

—

359

—

104

C₂×C₄

—

359

—

105

C₂×C₄

—

718

—

106

C₂×C₄

—

718

—

107

C₂×C₄

C₂²×D₃₅₉

yes

—

108

C₂×C₄

C₂²×D₃₅₉

yes

—

109

C₂×C₄

C₂²×D₃₅₉

yes

—

110

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

111

C₂²

—

359

—

112

C₂²

—

718

—

113

C₂²

—

718

—

114

C₂²

—

718

—

115

C₂²

C₂³×D₃₅₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

359

—

120

C₄

—

718

—

121

C₄

—

718

—

122

C₄

D₄×D₃₅₉

yes

—

123

C₂

—

1436

—

124

C₂

—

718

—

125

C₂

C₂×D₄×D₃₅₉

yes

characteristic

126

C₂

#26040063

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₉
commutator, socle, cosocle	C₂²×C₃₅₉	C₂⁴
Fitting	C₄⋊Q₈×C₃₅₉	C₂
prime radical	C₂×#19153342	C₂
hypercenter	C₄⋊Q₈	D₃₅₉
hypocenter	C₃₅₉	#536
LCS	G ▷ C₂²×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₅₉

commutator, socle, cosocle

C₂²×C₃₅₉

C₂⁴

Fitting

C₄⋊Q₈×C₃₅₉

C₂

prime radical

C₂×#19153342

C₂

hypercenter

C₄⋊Q₈

D₃₅₉

C₃₅₉

G ▷ C₂²×C₃₅₉ ▷ C₃₅₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₃₅₉	C₂	semidirect
3. #26040013	C₂	semidirect
4. #26040004	C₂	semidirect
5. #26040003	C₂	semidirect
6. #26040000	C₂	semidirect
7. #26039993	C₂	semidirect
8. #26039972	C₂	semidirect
9. C₄⋊Q₈×C₃₅₉	C₂	semidirect
63. C₃₅₉	#536	semidirect
68. C₄⋊Q₈	D₃₅₉	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₃₅₉

C₂

semidirect

3. #26040013

C₂

semidirect

4. #26040004

C₂

semidirect

5. #26040003

C₂

semidirect

6. #26040000

C₂

semidirect

7. #26039993

C₂

semidirect

8. #26039972

C₂

semidirect

9. C₄⋊Q₈×C₃₅₉

C₂

semidirect

63. C₃₅₉

#536

semidirect

68. C₄⋊Q₈

D₃₅₉

semidirect

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

subgroup	index
C₂×#19153342	2
C₂×Q₈×D₃₅₉	2
#26040013	2
#26040004	2
#26040003	2
#26040000	2
#26039993	2
#26039972	2
C₄⋊Q₈×C₃₅₉	2
#536	359

subgroup

index

C₂×Q₈×D₃₅₉

C₄⋊Q₈×C₃₅₉

359

Subgroups — #37527018

Conjugacy classes of subgroups

Complements

Maximal subgroups