Group #17994260 — subgroups

Subgroups — #17994260

Order 4736.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#14557910	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557909	C₂	1	2	yes	maximal, complemented
3	#14557881	C₂	1	2	yes	maximal, complemented
4	#14557254	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₉₆	—	2	2	no	—
7	C₂×#10596939	—	2	4	no	—
8	#11924533	C₂²	1	2	yes	—
9	#11924456	—	2	4	no	—
10	C₄×#64436	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₃₇	—	2	2	no	—
12	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	#127×C₃₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₃₇	—	2	1	no	—
16	C₂×D₄×C₃₇	—	2	2	no	—
17	C₂×D₄×C₃₇	—	2	1	no	—
18	C₂×D₁₄₈	—	2	2	no	—
19	C₂×D₁₄₈	—	2	2	no	—
20	C₂×D₁₄₈	—	2	2	no	—
21	SD₁₆×C₃₇	—	4	4	no	—
22	C₂×C₈×C₃₇	D₄	1	2	yes	—
23	C₄⋊C₄×C₃₇	—	2	2	no	—
24	C₄²×C₃₇	C₂³	1	1	yes	characteristic
25	#10596939	—	4	8	no	—
26	C₂×#64436	D₄	1	4	yes	—
27	D₂₉₆	—	4	4	no	—
28	C₂³×C₃₇	—	4	1	no	—
29	C₂²×D₃₇	—	4	2	no	—
30	Q₈×C₃₇	—	4	2	no	—
31	D₄×C₃₇	—	4	2	no	—
32	D₄×C₃₇	—	4	4	no	—
33	C₂×C₄×C₃₇	—	4	1	no	—
34	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
36	D₁₄₈	—	4	4	no	—
37	D₁₄₈	—	4	4	no	—
38	D₁₄₈	—	4	4	no	—
39	C₈×C₃₇	D₈	1	4	yes	complemented
40	#64436	—	2	4	no	—
41	C₂²×C₃₇	—	8	1	no	—
42	C₂²×C₃₇	—	8	2	no	—
43	C₂²×C₃₇	#127	1	1	yes	characteristic
44	C₂×D₃₇	—	8	2	no	—
45	C₂×D₃₇	—	8	2	no	—
46	C₂×D₃₇	—	8	2	no	—
47	C₄×C₃₇	—	8	1	no	—
48	C₄×C₃₇	C₂×D₈	1	2	yes	—
49	C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
50	#1721	—	37	1	no	maximal, Sylow
51	C₂×C₃₇	—	16	1	no	—
52	C₂×C₃₇	#493	1	1	yes	characteristic
53	C₂×C₃₇	#496	1	2	yes	—
54	D₃₇	—	16	2	no	—
55	#493	—	37	2	no	—
56	#493	—	37	1	no	—
57	#492	D₃₇	1	1	yes	characteristic, complemented
58	#486	—	37	2	no	—
59	#322	—	37	1	no	—
60	C₃₇	#1721	1	1	yes	characteristic, complemented, Sylow
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	74	4	no	—
63	C₂×D₈	—	74	2	no	—
64	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
65	#127	—	37	2	no	—
66	#127	C₂×D₃₇	1	1	yes	characteristic
67	#102	—	74	4	no	—
68	C₄×C₈	—	37	2	no	—
69	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	74	2	no	—
74	C₂×D₄	—	74	2	no	—
75	C₂×D₄	—	74	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	148	8	no	—
78	D₈	—	148	4	no	—
79	C₂×C₈	—	37	4	no	—
80	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₃₇	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	148	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	2	no	—
87	D₄	—	4	4	no	—
88	D₄	—	148	4	no	—
89	D₄	—	148	4	no	—
90	D₄	—	148	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₃₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
94	C₈	—	74	4	no	—
95	C₈	#10596939	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	296	2	no	—
99	C₂²	—	296	2	no	—
100	C₂²	—	296	2	no	—
101	C₂²	#11924532	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	C₂×#10596939	1	2	yes	—
104	C₄	#11924563	1	4	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	592	2	no	—
107	C₂	#14557909	1	1	yes	characteristic
108	C₂	#14557919	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557910

C₂

yes

maximal, characteristic, complemented

#14557909

C₂

yes

maximal, complemented

#14557881

C₂

yes

maximal, complemented

#14557254

C₂

yes

maximal, characteristic, complemented

C₃₇×#492

C₂

yes

maximal, characteristic, complemented

C₂×D₂₉₆

—

C₂×#10596939

—

#11924533

C₂²

yes

—

#11924456

—

C₄×#64436

C₂²

yes

—

C₂×SD₁₆×C₃₇

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

#127×C₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

SD₁₆×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596939

—

C₂×#64436

D₄

yes

—

D₂₉₆

—

C₂³×C₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₈×C₃₇

D₈

yes

complemented

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#127

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₈

yes

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

#1721

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#493

yes

characteristic

C₂×C₃₇

#496

yes

—

D₃₇

—

#493

—

#493

—

#492

D₃₇

yes

characteristic, complemented

#486

—

#322

—

C₃₇

#1721

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

#127

—

#127

C₂×D₃₇

yes

characteristic

#102

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

148

—

D₈

—

148

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

—

C₂³

—

148

—

Q₈

—

D₄

—

D₄

—

D₄

—

148

—

D₄

—

148

—

D₄

—

148

—

C₂×C₄

—

C₂×C₄

D₄×D₃₇

yes

—

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

C₈

—

C₈

#10596939

yes

complemented

C₂²

—

C₂²

—

C₂²

—

296

—

C₂²

—

296

—

100

C₂²

—

296

—

101

C₂²

#11924532

yes

characteristic

102

C₄

—

103

C₄

C₂×#10596939

yes

—

104

C₄

#11924563

yes

—

105

C₂

—

106

C₂

—

592

—

107

C₂

#14557909

yes

characteristic

108

C₂

#14557919

yes

—

	subgroup	G/H
center	C₂²	#11924532
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#492	C₂
socle	C₂²×C₃₇	#127
hypercenter	#492	D₃₇
hypocenter	C₃₇	#1721
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#11924532

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#492

C₂

C₂²×C₃₇

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #14557910	C₂	semidirect
2. #14557909	C₂	semidirect
3. #14557881	C₂	semidirect
4. #14557254	C₂	semidirect
5. C₃₇×#492	C₂	semidirect
39. C₈×C₃₇	D₈	semidirect
57. #492	D₃₇	semidirect
60. C₃₇	#1721	semidirect
95. C₈	#10596939	semidirect

H ≅ G/N

type

1. #14557910

C₂

semidirect

2. #14557909

C₂

semidirect

3. #14557881

C₂

semidirect

4. #14557254

C₂

semidirect

5. C₃₇×#492

C₂

semidirect

39. C₈×C₃₇

D₈

semidirect

57. #492

D₃₇

semidirect

60. C₃₇

#1721

semidirect

95. C₈

#10596939

semidirect

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

subgroup	index
#14557910	2
#14557909	2
#14557881	2
#14557254	2
C₃₇×#492	2
#1721	37

subgroup

index

C₃₇×#492

Subgroups — #17994260

Conjugacy classes of subgroups

Complements

Maximal subgroups