Group #17993736 — subgroups

Subgroups — #17993736

Order 4736.

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

normal subgroups only

This group has 206 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	#14557912	C₂	1	1	yes	maximal, characteristic
2	#14557828	C₂	1	2	yes	maximal
3	#14557282	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#64436	C₂	1	2	yes	maximal
5	Q₈²×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596942	—	2	2	no	—
7	#11924536	C₂²	1	1	yes	characteristic
8	#11924469	—	2	2	no	—
9	C₄⋊#64436	C₂²	1	2	yes	—
10	#11924460	—	2	2	no	—
11	C₄×#64436	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₃₇	—	2	2	no	—
13	C₄⋊Q₈×C₃₇	—	2	2	no	—
14	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₃₇	—	2	2	no	—
16	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
17	C₂×Q₈×C₃₇	—	2	2	no	—
18	C₂×Q₈×C₃₇	D₄	1	2	yes	—
19	C₂×Q₈×C₃₇	D₄	1	2	yes	—
20	C₂×Dic₇₄	—	2	1	no	—
21	C₄⋊C₄×C₃₇	—	2	4	no	—
22	C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₄⋊C₄×C₃₇	—	2	2	no	—
24	C₄⋊C₄×C₃₇	D₄	1	2	yes	—
25	C₄²×C₃₇	—	2	2	no	—
26	C₄²×C₃₇	—	2	2	no	—
27	C₄²×C₃₇	C₂³	1	1	yes	characteristic
28	#10596942	—	4	4	no	—
29	C₄⋊Dic₃₇	—	2	2	no	—
30	C₂×#64436	—	2	1	no	—
31	C₂×#64436	—	2	2	no	—
32	Q₈×C₃₇	—	2	2	no	—
33	Q₈×C₃₇	—	4	2	no	—
34	Q₈×C₃₇	—	8	2	no	—
35	Q₈×C₃₇	Q₁₆	1	4	yes	complemented
36	C₂×C₄×C₃₇	—	2	2	no	—
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
41	C₂×Dic₃₇	—	4	1	no	—
42	Dic₇₄	—	4	2	no	—
43	#64436	—	2	2	no	—
44	#64436	—	4	2	no	—
45	C₂²×C₃₇	#120	1	1	yes	characteristic
46	C₄×C₃₇	—	2	4	no	—
47	C₄×C₃₇	—	2	1	no	—
48	C₄×C₃₇	—	4	2	no	—
49	C₄×C₃₇	—	4	2	no	—
50	C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
51	C₄×C₃₇	#137	1	2	yes	—
52	Dic₃₇	—	8	1	no	—
53	#1633	—	37	1	no	maximal, Sylow
54	C₂×C₃₇	#451	1	2	yes	—
55	C₂×C₃₇	#453	1	1	yes	characteristic
56	Q₈²	D₃₇	1	1	yes	characteristic
57	#494	—	37	1	no	—
58	#477	—	37	2	no	—
59	#333	—	37	1	no	—
60	Q₈⋊C₈	—	37	2	no	—
61	C₃₇	#1633	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₁₆	—	74	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	37	1	no	—
66	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₃₇	1	2	yes	—
69	#107	—	74	2	no	—
70	C₄⋊C₈	—	37	2	no	—
71	#103	—	74	2	no	—
72	C₄×C₈	—	37	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	74	1	no	—
75	C₂×Q₈	C₂²⋊D₃₇	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₃₇	1	2	yes	—
77	Q₁₆	—	148	4	no	—
78	C₂×C₈	—	74	1	no	—
79	C₂×C₈	—	74	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	74	2	no	—
84	C₄⋊C₄	C₂²⋊D₃₇	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₃₇	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	148	2	no	—
92	Q₈	#10596942	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	148	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
99	C₈	—	74	2	no	—
100	C₈	—	148	2	no	—
101	C₂²	#11924505	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	296	1	no	—
107	C₄	C₂×#10596942	1	2	yes	—
108	C₄	#11924567	1	2	yes	—
109	C₂	#14557689	1	2	yes	—
110	C₂	#14557701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557912

C₂

yes

maximal, characteristic

#14557828

C₂

yes

maximal

#14557282

C₂

yes

maximal, characteristic

Q₈⋊#64436

C₂

yes

maximal

Q₈²×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596942

—

#11924536

C₂²

yes

characteristic

#11924469

—

C₄⋊#64436

C₂²

yes

—

#11924460

—

C₄×#64436

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

—

C₄⋊Q₈×C₃₇

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄×Q₈×C₃₇

—

C₄×Q₈×C₃₇

C₂²

yes

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂×Dic₇₄

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

—

C₄²×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596942

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

Q₁₆

yes

complemented

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

—

Dic₇₄

—

#64436

—

#64436

—

C₂²×C₃₇

#120

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×Q₁₆

yes

—

C₄×C₃₇

#137

yes

—

Dic₃₇

—

#1633

—

maximal, Sylow

C₂×C₃₇

#451

yes

—

C₂×C₃₇

#453

yes

characteristic

Q₈²

D₃₇

yes

—

—

—

Q₈⋊C₈

—

C₃₇

#1633

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₃₇

yes

—

#107

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₇

yes

—

C₂×Q₈

C₂²⋊D₃₇

yes

—

Q₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₃₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

148

—

Q₈

#10596942

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

C₈

—

100

C₈

—

148

—

101

C₂²

#11924505

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

296

—

107

C₄

C₂×#10596942

yes

—

108

C₄

#11924567

yes

—

109

C₂

#14557689

yes

—

110

C₂

#14557701

yes

characteristic

	subgroup	G/H
center	C₂²	#11924505
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	Q₈²×C₃₇	C₂
socle, prime radical	C₂²×C₃₇	#120
hypercenter	Q₈²	D₃₇
hypocenter	C₃₇	#1633
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#11924505

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

Q₈²×C₃₇

C₂

socle, prime radical

C₂²×C₃₇

#120

hypercenter

Q₈²

D₃₇

hypocenter

C₃₇

#1633

LCS

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
35. Q₈×C₃₇	Q₁₆	semidirect
61. C₃₇	#1633	semidirect
92. Q₈	#10596942	semidirect

H ≅ G/N

type

35. Q₈×C₃₇

Q₁₆

semidirect

61. C₃₇

#1633

semidirect

92. Q₈

#10596942

semidirect

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

subgroup	index
#14557912	2
#14557828	2
#14557282	2
Q₈⋊#64436	2
Q₈²×C₃₇	2
#1633	37

subgroup

index

Q₈²×C₃₇

Subgroups — #17993736

Conjugacy classes of subgroups

Complements

Maximal subgroups