Group #17995417 — subgroups

Subgroups — #17995417

Order 4736.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11924460	C₂	1	4	yes	maximal
2	C₂×#11924447	C₂	1	1	yes	maximal, characteristic
3	#14557399	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64436	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
7	#11924460	—	2	8	no	—
8	#11924447	—	2	2	no	—
9	#11924443	—	2	2	no	—
10	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₃₇	—	2	2	no	—
12	C₄⋊Q₈×C₃₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₃₇	—	2	2	no	—
16	C₂×Q₈×C₃₇	—	2	4	no	—
17	C₂×Q₈×C₃₇	—	2	4	no	—
18	C₂×Q₈×C₃₇	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₃₇	—	2	2	no	—
20	C₂²×C₄×C₃₇	D₄	1	2	yes	—
21	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₇	—	2	2	no	—
23	C₄⋊C₄×C₃₇	—	2	8	no	—
24	C₄²×C₃₇	C₂×C₄	1	2	yes	—
25	C₄²×C₃₇	D₄	1	2	yes	—
26	C₄⋊Dic₃₇	—	2	4	no	—
27	C₂×#64436	—	2	2	no	—
28	C₂×#64436	—	2	2	no	—
29	C₂×#64436	D₄	1	4	yes	—
30	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₃₇	—	2	8	no	—
32	Q₈×C₃₇	—	4	4	no	—
33	C₂×C₄×C₃₇	—	2	4	no	—
34	C₂×C₄×C₃₇	—	2	4	no	—
35	C₂×C₄×C₃₇	—	4	2	no	—
36	C₂×C₄×C₃₇	—	4	2	no	—
37	C₂×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	#41	1	2	yes	—
42	C₂×Dic₃₇	—	4	2	no	—
43	C₂×Dic₃₇	—	4	4	no	—
44	#64436	—	2	4	no	—
45	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₃₇	#124	1	1	yes	characteristic
47	C₂²×C₃₇	#127	1	1	yes	characteristic
48	C₂²×C₃₇	#137	1	4	yes	—
49	C₄×C₃₇	—	2	4	no	—
50	C₄×C₃₇	—	4	4	no	—
51	C₄×C₃₇	C₄×D₄	1	2	yes	—
52	C₄×C₃₇	#121	1	2	yes	—
53	Dic₃₇	—	8	2	no	—
54	#1960	—	37	1	no	maximal, Sylow
55	C₂×C₃₇	#390	1	1	yes	characteristic
56	C₂×C₃₇	#419	1	2	yes	—
57	C₂×C₃₇	#491	1	2	yes	—
58	C₂×C₃₇	#497	1	2	yes	—
59	C₂×C₄⋊Q₈	D₃₇	1	1	yes	characteristic
60	C₂×#103	—	37	4	no	—
61	C₂×#97	—	37	1	no	—
62	#382	—	37	1	no	—
63	C₃₇	#1960	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	C₂×D₃₇	1	2	yes	—
65	C₂²×C₈	—	37	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₃₇	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	37	2	no	—
70	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
71	#103	—	74	8	no	—
72	#97	—	74	2	no	—
73	#95	—	74	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₃₇	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	74	2	no	—
80	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
82	C₂×C₈	—	37	4	no	—
83	C₂×C₈	—	74	2	no	—
84	C₂×C₈	—	74	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	74	4	no	—
87	C₄²	C₂×Dic₃₇	1	2	yes	—
88	C₄²	C₂²⋊D₃₇	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	148	2	no	—
97	C₂×C₄	—	148	4	no	—
98	C₂×C₄	C₂²⋊Dic₃₇	1	4	yes	—
99	C₂×C₄	D₄×D₃₇	1	2	yes	—
100	C₂×C₄	#10596964	1	2	yes	—
101	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
103	C₈	—	74	4	no	—
104	C₂²	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
105	C₂²	#11924523	1	1	yes	characteristic
106	C₂²	#11924532	1	1	yes	characteristic
107	C₂²	#11924567	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	296	2	no	—
111	C₄	D₄×Dic₃₇	1	2	yes	—
112	C₄	#11924507	1	2	yes	—
113	C₂	#14557436	1	1	yes	characteristic
114	C₂	#14557542	1	2	yes	—
115	C₂	#14557901	1	2	yes	—
116	C₂	#14557925	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924460

C₂

yes

maximal

C₂×#11924447

C₂

yes

maximal, characteristic

#14557399

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924460

—

#11924447

—

#11924443

—

C₂²×Q₈×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

—

C₄⋊Q₈×C₃₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

D₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

#121

yes

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#491

yes

—

C₂×C₃₇

#497

yes

—

C₂×C₄⋊Q₈

D₃₇

yes

characteristic

C₂×#103

—

C₂×#97

—

#382

—

C₃₇

#1960

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₃₇

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₃₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₃₇

yes

—

C₄²

C₂²⋊D₃₇

yes

—

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

C₂²⋊Dic₃₇

yes

—

C₂×C₄

D₄×D₃₇

yes

—

100

C₂×C₄

#10596964

yes

—

101

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

103

C₈

—

104

C₂²

C₂×C₂²⋊Dic₃₇

yes

characteristic

105

C₂²

#11924523

yes

characteristic

106

C₂²

#11924532

yes

characteristic

107

C₂²

#11924567

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

296

—

111

C₄

D₄×Dic₃₇

yes

—

112

C₄

#11924507

yes

—

113

C₂

#14557436

yes

characteristic

114

C₂

#14557542

yes

—

115

C₂

#14557901

yes

—

116

C₂

#14557925

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×C₄⋊Q₈×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃₇
hypocenter	C₃₇	#1960
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×C₄⋊Q₈×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₃₇

hypocenter

C₃₇

#1960

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₃₇	C₄	semidirect
63. C₃₇	#1960	semidirect
67. C₄⋊Q₈	Dic₃₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₃₇

C₄

semidirect

63. C₃₇

#1960

semidirect

67. C₄⋊Q₈

Dic₃₇

semidirect

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

subgroup	index
C₂×#11924460	2
C₂×#11924447	2
#14557399	2
C₂×C₄⋊Q₈×C₃₇	2
#1960	37

subgroup

index

C₂×C₄⋊Q₈×C₃₇

Subgroups — #17995417

Conjugacy classes of subgroups

Complements

Maximal subgroups