Group #17996647 — subgroups

Subgroups — #17996647

Order 4736.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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×Dic₂₉₆	C₂	1	2	yes	maximal
2	#14557913	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₁₄₈	C₂	1	1	yes	maximal, characteristic
4	#14557338	C₂	1	2	yes	maximal
5	C₃₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₄₈	—	2	1	no	—
7	C₂×Dic₁₄₈	C₂²	1	2	yes	—
8	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic
9	#11924536	—	2	1	no	—
10	C₄×Dic₇₄	—	2	1	no	—
11	Dic₂₉₆	—	2	4	no	—
12	#11924470	C₂²	1	1	yes	characteristic
13	#11924461	—	2	1	no	—
14	C₂×C₁₆×C₃₇	C₂²	1	2	yes	—
15	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₇₄	—	2	1	no	—
17	C₂×Dic₇₄	—	2	2	no	—
18	C₂×C₈×C₃₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
20	C₄²×C₃₇	D₄	1	1	yes	characteristic
21	C₄⋊Dic₃₇	—	2	1	no	—
22	C₄⋊Dic₃₇	—	4	1	no	—
23	C₄⋊Dic₃₇	—	4	1	no	—
24	#10596936	—	4	1	no	—
25	C₄×Dic₃₇	—	4	1	no	—
26	Dic₁₄₈	—	2	2	no	—
27	Dic₁₄₈	—	2	1	no	—
28	Dic₁₄₈	—	2	2	no	—
29	Dic₁₄₈	—	4	2	no	—
30	Dic₁₄₈	D₄	1	2	yes	—
31	C₁₆×C₃₇	—	2	2	no	—
32	C₂×C₄×C₃₇	D₈	1	1	yes	characteristic
33	C₂×C₄×C₃₇	D₈	1	1	yes	characteristic
34	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₃₇	—	4	1	no	—
36	C₂×Dic₃₇	—	4	1	no	—
37	C₂×Dic₃₇	—	4	2	no	—
38	Dic₇₄	—	2	2	no	—
39	Dic₇₄	—	4	1	no	—
40	Dic₇₄	—	4	2	no	—
41	Dic₇₄	—	4	2	no	—
42	C₈×C₃₇	—	2	1	no	—
43	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
44	C₈×C₃₇	#41	1	1	yes	characteristic
45	C₂²×C₃₇	C₂×D₈	1	1	yes	characteristic
46	C₄×C₃₇	—	2	1	no	—
47	C₄×C₃₇	Q₃₂	1	2	yes	complemented
48	C₄×C₃₇	#121	1	1	yes	characteristic
49	C₄×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₃₇	—	4	2	no	—
51	Dic₃₇	—	8	2	no	—
52	Dic₃₇	—	8	1	no	—
53	#2209	—	37	1	no	maximal, Sylow
54	C₂×C₃₇	#459	1	1	yes	characteristic
55	C₂×C₃₇	C₂×Q₃₂	1	1	yes	characteristic
56	C₂×C₃₇	#510	1	1	yes	characteristic
57	C₂×Q₃₂	—	37	2	no	—
58	#494	—	37	1	no	—
59	C₄×Q₁₆	—	37	1	no	—
60	C₄⋊C₁₆	D₃₇	1	1	yes	characteristic
61	#358	—	37	2	no	—
62	C₃₇	#2209	1	1	yes	characteristic, complemented, Sylow
63	C₂×Q₁₆	—	37	1	no	—
64	C₂×Q₁₆	—	37	2	no	—
65	C₂×Q₁₆	—	74	1	no	—
66	C₄⋊Q₈	—	74	1	no	—
67	C₄×Q₈	—	74	1	no	—
68	Q₃₂	—	74	4	no	—
69	C₂×C₁₆	C₂×D₃₇	1	2	yes	—
70	#107	—	37	1	no	—
71	#103	—	74	1	no	—
72	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
73	C₂×Q₈	—	74	1	no	—
74	C₂×Q₈	—	74	2	no	—
75	Q₁₆	—	37	2	no	—
76	Q₁₆	—	74	2	no	—
77	Q₁₆	—	74	1	no	—
78	Q₁₆	—	74	2	no	—
79	Q₁₆	—	148	2	no	—
80	C₂×C₈	D₁₄₈	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
82	C₄⋊C₄	—	74	1	no	—
83	C₄⋊C₄	—	148	1	no	—
84	C₄⋊C₄	—	148	1	no	—
85	C₄⋊C₄	—	148	1	no	—
86	C₄²	—	148	1	no	—
87	C₄²	D₁₄₈	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	74	2	no	—
90	Q₈	—	148	2	no	—
91	Q₈	—	148	2	no	—
92	Q₈	—	148	1	no	—
93	C₂×C₄	—	148	1	no	—
94	C₂×C₄	—	148	2	no	—
95	C₂×C₄	—	148	1	no	—
96	C₂×C₄	D₂₉₆	1	1	yes	characteristic
97	C₂×C₄	D₂₉₆	1	1	yes	characteristic
98	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₃₇	1	1	yes	characteristic
101	C₈	#10596966	1	1	yes	characteristic
102	C₂²	C₂×D₂₉₆	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	296	1	no	—
105	C₄	—	296	2	no	—
106	C₄	—	148	2	no	—
107	C₄	Dic₂₉₆	1	2	yes	complemented
108	C₄	#11924511	1	1	yes	characteristic
109	C₄	#11924564	1	1	yes	characteristic
110	C₂	#14557729	1	1	yes	characteristic
111	C₂	C₂×Dic₂₉₆	1	1	yes	characteristic
112	C₂	#14557976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₉₆

C₂

yes

maximal

#14557913

C₂

yes

maximal, characteristic

C₄×Dic₁₄₈

C₂

yes

maximal, characteristic

#14557338

C₂

yes

maximal

C₃₇×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₁₄₈

—

C₂×Dic₁₄₈

C₂²

yes

—

C₂×Dic₁₄₈

C₂²

yes

characteristic

#11924536

—

C₄×Dic₇₄

—

Dic₂₉₆

—

#11924470

C₂²

yes

characteristic

#11924461

—

C₂×C₁₆×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×C₈×C₃₇

D₄

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

D₄

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

—

Dic₁₄₈

—

Dic₁₄₈

—

Dic₁₄₈

—

Dic₁₄₈

—

Dic₁₄₈

D₄

yes

—

C₁₆×C₃₇

—

C₂×C₄×C₃₇

D₈

yes

characteristic

C₂×C₄×C₃₇

D₈

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

C₈×C₃₇

C₂×D₄

yes

characteristic

C₈×C₃₇

#41

yes

characteristic

C₂²×C₃₇

C₂×D₈

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

Q₃₂

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₂×Q₃₂

yes

characteristic

C₂×C₃₇

#510

yes

characteristic

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

D₃₇

yes

characteristic

#358

—

C₃₇

#2209

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

C₂×D₃₇

yes

—

#107

—

#103

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

148

—

C₂×C₈

D₁₄₈

yes

characteristic

C₂×C₈

C₂²×D₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄⋊C₄

—

148

—

C₄⋊C₄

—

148

—

C₄²

—

148

—

C₄²

D₁₄₈

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

148

—

Q₈

—

148

—

Q₈

—

148

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

D₂₉₆

yes

characteristic

C₂×C₄

D₂₉₆

yes

characteristic

C₂×C₄

C₂×D₁₄₈

yes

characteristic

C₈

—

100

C₈

D₄×D₃₇

yes

characteristic

101

C₈

#10596966

yes

characteristic

102

C₂²

C₂×D₂₉₆

yes

characteristic

103

C₄

—

104

C₄

—

296

—

105

C₄

—

296

—

106

C₄

—

148

—

107

C₄

Dic₂₉₆

yes

complemented

108

C₄

#11924511

yes

characteristic

109

C₄

#11924564

yes

characteristic

110

C₂

#14557729

yes

characteristic

111

C₂

C₂×Dic₂₉₆

yes

characteristic

112

C₂

#14557976

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₉₆
commutator, cosocle	C₂×C₈×C₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₃₇
Fitting	C₃₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃₇	C₂×D₈
hypercenter	C₄⋊C₁₆	D₃₇
hypocenter	C₃₇	#2209
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₂₉₆

commutator, cosocle

C₂×C₈×C₃₇

C₂³

Frattini

C₂×C₈

C₂²×D₃₇

Fitting

C₃₇×C₄⋊C₁₆

C₂

socle, prime radical

C₂²×C₃₇

C₂×D₈

hypercenter

C₄⋊C₁₆

D₃₇

hypocenter

C₃₇

#2209

LCS

G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₃₇	Q₃₂	semidirect
62. C₃₇	#2209	semidirect
107. C₄	Dic₂₉₆	semidirect

H ≅ G/N

type

47. C₄×C₃₇

Q₃₂

semidirect

62. C₃₇

#2209

semidirect

107. C₄

Dic₂₉₆

semidirect

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

subgroup	index
C₂×Dic₂₉₆	2
#14557913	2
C₄×Dic₁₄₈	2
#14557338	2
C₃₇×C₄⋊C₁₆	2
#2209	37

subgroup

index

C₂×Dic₂₉₆

#14557913

C₄×Dic₁₄₈

#14557338

C₃₇×C₄⋊C₁₆

#2209

Subgroups — #17996647

Conjugacy classes of subgroups

Complements

Maximal subgroups