Group #17996348 — subgroups

Subgroups — C₂×#14557351 #17996348

Order 4736.

184 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×#10596925	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557351	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64436	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic
7	C₂×#10596925	—	2	2	no	—
8	C₂×#10596925	C₂²	1	2	yes	—
9	C₈×Dic₃₇	C₄	1	2	yes	—
10	C₈×Dic₃₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₃₇	—	2	2	no	—
13	C₂×C₁₆×C₃₇	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₃₇	C₈	1	1	yes	characteristic
16	C₂²×Dic₃₇	C₈	1	1	yes	characteristic
17	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₃₇	D₄	1	2	yes	—
19	C₂×C₈×C₃₇	Q₈	1	2	yes	—
20	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
21	C₄×Dic₃₇	C₈	1	2	yes	—
22	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
23	C₂×#64436	—	2	2	no	—
24	C₂×#64436	C₂×C₄	1	2	yes	—
25	C₁₆×C₃₇	—	2	2	no	—
26	#10596925	—	2	2	no	—
27	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₃₇	M₁₆	1	2	yes	—
31	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₃₇	—	2	2	no	—
33	C₂×Dic₃₇	C₁₆	1	4	yes	complemented
34	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
35	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
36	C₈×C₃₇	C₄⋊C₄	1	2	yes	—
37	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₃₇	C₂×Q₈	1	1	yes	characteristic
39	#64436	—	2	2	no	—
40	C₂²×C₃₇	C₄⋊C₈	1	2	yes	—
41	C₂²×C₃₇	C₂×C₁₆	1	2	yes	—
42	C₂²×C₃₇	M₃₂	1	2	yes	—
43	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₃₇	C₄⋊C₈	1	2	yes	—
45	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
47	Dic₃₇	—	2	2	no	—
48	Dic₃₇	C₂×C₁₆	1	4	yes	complemented
49	C₂×C₄⋊C₁₆	—	37	1	no	maximal, Sylow
50	C₂×C₃₇	C₄⋊C₁₆	1	4	yes	complemented
51	C₂×C₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₃₇	C₂²×C₁₆	1	1	yes	characteristic
53	C₂×C₃₇	C₂×M₃₂	1	1	yes	characteristic
54	C₂²×C₁₆	—	37	1	no	—
55	C₂²×C₁₆	D₃₇	1	1	yes	characteristic
56	C₂×C₄×C₈	—	37	1	no	—
57	C₄⋊C₁₆	—	37	4	no	—
58	C₃₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	37	1	no	—
60	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
61	C₂×C₄²	—	37	1	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	37	2	no	—
64	C₂×C₁₆	—	74	2	no	—
65	C₂×C₁₆	C₂×D₃₇	1	2	yes	—
66	C₄×C₈	—	37	2	no	—
67	C₄×C₈	—	37	2	no	—
68	C₂²×C₄	—	37	1	no	—
69	C₂²×C₄	—	37	1	no	—
70	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
71	C₂×C₈	—	37	2	no	—
72	C₂×C₈	—	74	2	no	—
73	C₂×C₈	Dic₇₄	1	2	yes	—
74	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
75	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
76	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
77	C₄²	—	37	2	no	—
78	C₄²	—	37	2	no	—
79	C₁₆	—	2	2	no	—
80	C₁₆	—	74	2	no	—
81	C₂³	C₈×D₃₇	1	1	yes	characteristic
82	C₂×C₄	—	37	2	no	—
83	C₂×C₄	—	37	4	no	—
84	C₂×C₄	—	37	2	no	—
85	C₂×C₄	—	74	2	no	—
86	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
87	C₂×C₄	#10596929	1	2	yes	—
88	C₂×C₄	#10596936	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
90	C₈	—	74	2	no	—
91	C₈	#10596936	1	2	yes	—
92	C₈	C₂×Dic₇₄	1	1	yes	characteristic
93	C₈	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
94	C₂²	#11924465	1	2	yes	—
95	C₂²	C₁₆×D₃₇	1	2	yes	—
96	C₂²	#11924475	1	2	yes	—
97	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
98	C₄	—	37	4	no	—
99	C₄	—	74	2	no	—
100	C₄	#11924465	1	2	yes	—
101	C₄	C₂×#10596936	1	1	yes	characteristic
102	C₄	C₂×#10596929	1	1	yes	characteristic
103	C₂	#14557351	1	4	yes	direct product factor
104	C₂	C₂×#11924465	1	1	yes	characteristic
105	C₂	C₂×C₁₆×D₃₇	1	1	yes	characteristic
106	C₂	C₂×#11924475	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10596925

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

#14557351

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₄

yes

characteristic

C₂×C₄×Dic₃₇

C₄

yes

characteristic

C₂×#10596925

—

C₂×#10596925

C₂²

yes

—

C₈×Dic₃₇

C₄

yes

—

C₈×Dic₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₃₇

—

C₂×C₁₆×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₈

yes

characteristic

C₂²×Dic₃₇

C₈

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄×Dic₃₇

C₈

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₂×#64436

—

C₂×#64436

C₂×C₄

yes

—

C₁₆×C₃₇

—

#10596925

—

C₂³×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

M₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₁₆

yes

complemented

C₂×Dic₃₇

C₂×C₈

yes

—

C₂×Dic₃₇

C₂×C₈

yes

—

C₈×C₃₇

C₄⋊C₄

yes

—

C₈×C₃₇

C₂×D₄

yes

characteristic

C₈×C₃₇

C₂×Q₈

yes

characteristic

#64436

—

C₂²×C₃₇

C₄⋊C₈

yes

—

C₂²×C₃₇

C₂×C₁₆

yes

—

C₂²×C₃₇

M₃₂

yes

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

C₄⋊C₈

yes

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₂×M₁₆

yes

characteristic

Dic₃₇

—

Dic₃₇

C₂×C₁₆

yes

complemented

C₂×C₄⋊C₁₆

—

maximal, Sylow

C₂×C₃₇

C₄⋊C₁₆

yes

complemented

C₂×C₃₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₃₇

C₂²×C₁₆

yes

characteristic

C₂×C₃₇

C₂×M₃₂

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

D₃₇

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₃₇

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

—

C₂×C₈

C₄×D₃₇

yes

characteristic

C₂×C₈

C₂²⋊D₃₇

yes

—

C₂×C₈

C₂²×D₃₇

yes

characteristic

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

C₈×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₃₇

yes

characteristic

C₂×C₄

#10596929

yes

—

C₂×C₄

#10596936

yes

—

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

C₈

—

C₈

#10596936

yes

—

C₈

C₂×Dic₇₄

yes

characteristic

C₈

C₂×C₂²⋊D₃₇

yes

characteristic

C₂²

#11924465

yes

—

C₂²

C₁₆×D₃₇

yes

—

C₂²

#11924475

yes

—

C₂²

C₂×C₈×D₃₇

yes

characteristic

C₄

—

C₄

—

100

C₄

#11924465

yes

—

101

C₄

C₂×#10596936

yes

characteristic

102

C₄

C₂×#10596929

yes

characteristic

103

C₂

#14557351

yes

direct product factor

104

C₂

C₂×#11924465

yes

characteristic

105

C₂

C₂×C₁₆×D₃₇

yes

characteristic

106

C₂

C₂×#11924475

yes

characteristic

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

subgroup

G/H

center

C₂²×C₈

C₂×D₃₇

commutator

C₂×C₃₇

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₃₇

Fitting

C₂²×C₁₆×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×C₈

cosocle

C₂×C₈×C₃₇

C₂³

hypercenter

C₂²×C₁₆

D₃₇

hypocenter

C₃₇

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. #14557351	C₂	direct (N × H)
33. C₂×Dic₃₇	C₁₆	semidirect
48. Dic₃₇	C₂×C₁₆	semidirect
50. C₂×C₃₇	C₄⋊C₁₆	semidirect
58. C₃₇	C₂×C₄⋊C₁₆	semidirect
103. C₂	#14557351	direct (N × H)

H ≅ G/N

type

3. #14557351

C₂

direct (N × H)

33. C₂×Dic₃₇

C₁₆

semidirect

48. Dic₃₇

C₂×C₁₆

semidirect

50. C₂×C₃₇

C₄⋊C₁₆

semidirect

58. C₃₇

C₂×C₄⋊C₁₆

semidirect

103. C₂

#14557351

direct (N × H)

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

subgroup	index
C₂²×#10596925	2
C₂×C₈×Dic₃₇	2
#14557351	2
C₂²×C₁₆×C₃₇	2
C₂×C₄⋊C₁₆	37

subgroup

index

C₂²×#10596925

C₂×C₈×Dic₃₇

#14557351

C₂²×C₁₆×C₃₇

C₂×C₄⋊C₁₆

Subgroups — C₂×#14557351 #17996348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557351 #17996348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557351 #17996348