Group #17994449 — subgroups

Subgroups — #17994449

Order 4736.

292 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄⋊#64436	C₂	1	2	yes	maximal
2	C₂×#11924447	C₂	1	2	yes	maximal
3	#14557288	C₂	1	2	yes	maximal
4	C₄³×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64436	C₂²	1	4	yes	—
6	C₄⋊#64436	C₄	1	8	yes	complemented
7	#11924447	—	2	4	no	—
8	C₂×C₄²×C₃₇	C₄	1	2	yes	—
9	C₂×C₄²×C₃₇	C₄	1	2	yes	—
10	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₄²×C₃₇	—	2	4	no	—
19	C₄²×C₃₇	—	2	4	no	—
20	C₄²×C₃₇	—	2	2	no	—
21	C₄²×C₃₇	C₂×C₄	1	4	yes	—
22	C₄²×C₃₇	D₄	1	2	yes	—
23	C₄²×C₃₇	Q₈	1	2	yes	—
24	C₂×#64436	—	2	4	no	—
25	C₂×#64436	—	2	4	no	—
26	C₂×#64436	C₂×C₄	1	8	yes	—
27	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₃₇	—	2	2	no	—
29	C₂×C₄×C₃₇	—	2	2	no	—
30	C₂×C₄×C₃₇	—	2	2	no	—
31	C₂×C₄×C₃₇	—	2	2	no	—
32	C₂×C₄×C₃₇	—	2	2	no	—
33	C₂×C₄×C₃₇	C₄²	1	2	yes	—
34	C₂×C₄×C₃₇	C₄²	1	2	yes	—
35	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₃₇	M₁₆	1	4	yes	—
37	C₂×C₄×C₃₇	M₁₆	1	4	yes	—
38	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₃₇	#41	1	2	yes	—
42	#64436	—	2	8	no	—
43	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₃₇	#117	1	1	yes	characteristic
46	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
47	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
48	C₄×C₃₇	—	2	2	no	—
49	C₄×C₃₇	—	2	4	no	—
50	C₄×C₃₇	—	2	4	no	—
51	C₄×C₃₇	#97	1	4	yes	complemented
52	C₄×C₃₇	C₄×D₄	1	2	yes	—
53	C₄×C₃₇	C₄×Q₈	1	2	yes	—
54	#1767	—	37	1	no	maximal, Sylow
55	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₃₇	C₂×#97	1	1	yes	characteristic
57	C₂×C₃₇	C₄×M₁₆	1	1	yes	characteristic
58	C₂×C₃₇	#423	1	2	yes	—
59	C₂×C₃₇	#432	1	2	yes	—
60	C₂×C₄⋊C₈	—	37	2	no	—
61	C₂×#97	—	37	2	no	—
62	C₄³	D₃₇	1	1	yes	characteristic
63	#336	—	37	2	no	—
64	C₃₇	#1767	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	—	37	4	no	—
66	C₂×C₄²	Dic₃₇	1	2	yes	—
67	C₂×C₄²	Dic₃₇	1	2	yes	—
68	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
71	C₄⋊C₈	—	37	8	no	—
72	#97	—	74	4	no	—
73	C₂²×C₄	C₂×Dic₃₇	1	2	yes	—
74	C₂²×C₄	C₂×Dic₃₇	1	1	yes	characteristic
75	C₂²×C₄	C₂×Dic₃₇	1	2	yes	—
76	C₂²×C₄	C₂×Dic₃₇	1	1	yes	characteristic
77	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
78	C₂×C₈	—	37	8	no	—
79	C₂×C₈	—	74	4	no	—
80	C₂×C₈	—	74	4	no	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	4	no	—
83	C₄²	—	2	2	no	—
84	C₄²	Dic₇₄	1	2	yes	—
85	C₄²	C₄×D₃₇	1	4	yes	—
86	C₄²	D₁₄₈	1	2	yes	—
87	C₂³	C₂²×Dic₃₇	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	#10596934	1	4	yes	—
94	C₂×C₄	#10596934	1	4	yes	—
95	C₂×C₄	C₄×Dic₃₇	1	2	yes	—
96	C₂×C₄	C₄×Dic₃₇	1	2	yes	—
97	C₂×C₄	C₄⋊Dic₃₇	1	4	yes	—
98	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
99	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
100	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
101	C₂×C₄	#10596962	1	2	yes	—
102	C₈	—	74	8	no	—
103	C₂²	C₂×C₄×Dic₃₇	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊Dic₃₇	1	1	yes	characteristic
105	C₂²	#11924495	1	1	yes	characteristic
106	C₂²	C₂×#10596934	1	2	yes	—
107	C₂²	C₂×#10596934	1	2	yes	—
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	4	no	—
111	C₄	#11924447	1	4	yes	complemented
112	C₄	C₄×D₁₄₈	1	2	yes	—
113	C₄	C₄×Dic₇₄	1	2	yes	—
114	C₂	C₄×C₄⋊Dic₃₇	1	1	yes	characteristic
115	C₂	C₂×#11924447	1	1	yes	characteristic
116	C₂	C₄×#10596934	1	1	yes	characteristic
117	C₂	#14557557	1	2	yes	—
118	C₂	#14557590	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#64436

C₂

yes

maximal

C₂×#11924447

C₂

yes

maximal

#14557288

C₂

yes

maximal

C₄³×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

C₄⋊#64436

C₄

yes

complemented

#11924447

—

C₂×C₄²×C₃₇

C₄

yes

—

C₂×C₄²×C₃₇

C₄

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

Q₈

yes

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

M₁₆

yes

—

C₂×C₄×C₃₇

M₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

#64436

—

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#117

yes

characteristic

C₂²×C₃₇

C₂×M₁₆

yes

—

C₂²×C₃₇

C₂×M₁₆

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#97

yes

complemented

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

C₄×Q₈

yes

—

#1767

—

maximal, Sylow

C₂×C₃₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₇

C₂×#97

yes

characteristic

C₂×C₃₇

C₄×M₁₆

yes

characteristic

C₂×C₃₇

#423

yes

—

C₂×C₃₇

#432

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

D₃₇

yes

characteristic

#336

—

C₃₇

#1767

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

Dic₃₇

yes

—

C₂×C₄²

Dic₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

#97

—

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂×Dic₃₇

yes

characteristic

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂×Dic₃₇

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₇₄

yes

—

C₄²

C₄×D₃₇

yes

—

C₄²

D₁₄₈

yes

—

C₂³

C₂²×Dic₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#10596934

yes

—

C₂×C₄

#10596934

yes

—

C₂×C₄

C₄×Dic₃₇

yes

—

C₂×C₄

C₄×Dic₃₇

yes

—

C₂×C₄

C₄⋊Dic₃₇

yes

—

C₂×C₄

C₂×Dic₇₄

yes

characteristic

C₂×C₄

C₂×C₄×D₃₇

yes

—

100

C₂×C₄

C₂×D₁₄₈

yes

characteristic

101

C₂×C₄

#10596962

yes

—

102

C₈

—

103

C₂²

C₂×C₄×Dic₃₇

yes

characteristic

104

C₂²

C₂×C₄⋊Dic₃₇

yes

characteristic

105

C₂²

#11924495

yes

characteristic

106

C₂²

C₂×#10596934

yes

—

107

C₂²

C₂×#10596934

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#11924447

yes

complemented

112

C₄

C₄×D₁₄₈

yes

—

113

C₄

C₄×Dic₇₄

yes

—

114

C₂

C₄×C₄⋊Dic₃₇

yes

characteristic

115

C₂

C₂×#11924447

yes

characteristic

116

C₂

C₄×#10596934

yes

characteristic

117

C₂

#14557557

yes

—

118

C₂

#14557590

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₇
commutator	C₂²×C₃₇	C₂×C₄²
Fitting	C₄³×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₄³	D₃₇
hypocenter	C₃₇	#1767
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₇

commutator

C₂²×C₃₇

C₂×C₄²

Fitting

C₄³×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂²×C₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₄³

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
6. C₄⋊#64436	C₄	semidirect
51. C₄×C₃₇	#97	semidirect
64. C₃₇	#1767	semidirect
111. C₄	#11924447	semidirect

H ≅ G/N

type

6. C₄⋊#64436

C₄

semidirect

51. C₄×C₃₇

#97

semidirect

64. C₃₇

#1767

semidirect

111. C₄

#11924447

semidirect

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

subgroup	index
C₂×C₄⋊#64436	2
C₂×#11924447	2
#14557288	2
C₄³×C₃₇	2
#1767	37

subgroup

index

C₂×C₄⋊#64436

C₂×#11924447

#14557288

C₄³×C₃₇

#1767

Subgroups — #17994449

Conjugacy classes of subgroups

Complements

Maximal subgroups