Group #17994271 — subgroups

Subgroups — #17994271

Order 4736.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14557936	C₂	1	1	yes	maximal, characteristic
2	#14557936	C₂	1	1	yes	maximal, characteristic
3	#14557929	C₂	1	1	yes	maximal, characteristic
4	#14557885	C₂	1	1	yes	maximal, characteristic
5	#14557884	C₂	1	1	yes	maximal, characteristic
6	#14557255	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#494	C₂	1	1	yes	maximal, characteristic
8	#11924536	C₂²	1	1	yes	characteristic
9	#11924536	C₂²	1	1	yes	characteristic
10	#11924469	—	2	2	no	—
11	#11924469	—	2	2	no	—
12	#11924467	—	2	2	no	—
13	#11924461	—	2	2	no	—
14	#11924459	—	2	2	no	—
15	C₄×#64436	C₂²	1	1	yes	characteristic
16	C₄×#64436	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₃₇	—	2	2	no	—
18	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₇	—	2	1	no	—
22	C₂×Q₈×C₃₇	—	2	1	no	—
23	C₂×Dic₇₄	—	2	1	no	—
24	C₂×Dic₇₄	—	2	1	no	—
25	Q₁₆×C₃₇	—	4	4	no	—
26	C₂×C₈×C₃₇	D₄	1	2	yes	—
27	C₄⋊C₄×C₃₇	—	2	1	no	—
28	C₄⋊C₄×C₃₇	—	2	1	no	—
29	C₄⋊C₄×C₃₇	—	2	1	no	—
30	C₄⋊C₄×C₃₇	—	2	1	no	—
31	C₄²×C₃₇	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₃₇	—	2	1	no	—
33	C₄⋊Dic₃₇	—	2	1	no	—
34	C₄⋊Dic₃₇	—	2	1	no	—
35	C₄⋊Dic₃₇	—	2	1	no	—
36	C₂×#64436	D₄	1	2	yes	—
37	C₂×#64436	D₄	1	2	yes	—
38	Q₈×C₃₇	—	4	2	no	—
39	Q₈×C₃₇	—	4	2	no	—
40	C₂×C₄×C₃₇	—	4	1	no	—
41	C₂×C₄×C₃₇	—	4	1	no	—
42	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₃₇	—	4	1	no	—
46	C₂×Dic₃₇	—	4	1	no	—
47	Dic₇₄	—	4	2	no	—
48	Dic₇₄	—	4	2	no	—
49	C₈×C₃₇	—	2	2	no	—
50	#64436	—	2	2	no	—
51	#64436	Q₁₆	1	4	yes	complemented
52	C₂²×C₃₇	#127	1	1	yes	characteristic
53	C₄×C₃₇	—	8	1	no	—
54	C₄×C₃₇	—	8	1	no	—
55	C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
56	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₃₇	#137	1	2	yes	—
58	Dic₃₇	—	8	1	no	—
59	Dic₃₇	—	8	1	no	—
60	#1723	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	#494	1	1	yes	characteristic
62	C₂×C₃₇	#496	1	1	yes	characteristic
63	C₂×C₃₇	#497	1	1	yes	characteristic
64	#500	—	37	1	no	—
65	#500	—	37	1	no	—
66	#498	—	37	1	no	—
67	#494	D₃₇	1	1	yes	characteristic
68	#487	—	37	1	no	—
69	#487	—	37	1	no	—
70	#322	—	37	1	no	—
71	C₃₇	#1723	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₁₆	—	2	2	no	—
73	C₄⋊Q₈	—	37	1	no	—
74	C₄⋊Q₈	—	37	1	no	—
75	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
77	#107	—	74	2	no	—
78	#107	—	74	2	no	—
79	#106	—	74	2	no	—
80	#103	—	74	2	no	—
81	#103	—	74	2	no	—
82	C₄×C₈	—	37	1	no	—
83	C₄×C₈	—	37	1	no	—
84	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	74	1	no	—
88	C₂×Q₈	—	74	1	no	—
89	Q₁₆	—	4	4	no	—
90	C₂×C₈	—	37	2	no	—
91	C₂×C₈	—	37	2	no	—
92	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	74	1	no	—
98	C₄⋊C₄	—	74	1	no	—
99	C₄⋊C₄	—	74	1	no	—
100	C₄⋊C₄	—	74	1	no	—
101	C₄²	C₂²×D₃₇	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	148	2	no	—
105	Q₈	—	148	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	148	1	no	—
109	C₂×C₄	—	148	1	no	—
110	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	37	4	no	—
115	C₈	—	74	2	no	—
116	C₂²	#11924532	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	296	1	no	—
120	C₄	—	296	1	no	—
121	C₄	Q₁₆×D₃₇	1	2	yes	—
122	C₄	#11924562	1	2	yes	—
123	C₄	#11924568	1	2	yes	—
124	C₂	#14557911	1	1	yes	characteristic
125	C₂	#14557920	1	1	yes	characteristic
126	C₂	#14557926	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557936

C₂

yes

maximal, characteristic

#14557936

C₂

yes

maximal, characteristic

#14557929

C₂

yes

maximal, characteristic

#14557885

C₂

yes

maximal, characteristic

#14557884

C₂

yes

maximal, characteristic

#14557255

C₂

yes

maximal, characteristic

C₃₇×#494

C₂

yes

maximal, characteristic

#11924536

C₂²

yes

characteristic

#11924536

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#64436

C₂²

yes

characteristic

C₂×Q₁₆×C₃₇

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂×#64436

D₄

yes

—

C₂×#64436

D₄

yes

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

#64436

Q₁₆

yes

complemented

C₂²×C₃₇

#127

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×Q₁₆

yes

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

C₄×C₃₇

#137

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

—

—

D₃₇

yes

—

—

—

C₃₇

#1723

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

C₄⋊Q₈

C₂×D₃₇

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

C₂²×D₃₇

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

148

—

105

Q₈

—

148

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

148

—

109

C₂×C₄

—

148

—

110

C₂×C₄

D₄×D₃₇

yes

characteristic

111

C₂×C₄

D₄×D₃₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂²

#11924532

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

296

—

120

C₄

—

296

—

121

C₄

Q₁₆×D₃₇

yes

—

122

C₄

#11924562

yes

—

123

C₄

#11924568

yes

—

124

C₂

#14557911

yes

characteristic

125

C₂

#14557920

yes

characteristic

126

C₂

#14557926

yes

characteristic

	subgroup	G/H
center	C₂²	#11924532
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#494	C₂
socle, prime radical	C₂²×C₃₇	#127
hypercenter	#494	D₃₇
hypocenter	C₃₇	#1723
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#11924532

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#494

C₂

C₂²×C₃₇

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
51. #64436	Q₁₆	semidirect
71. C₃₇	#1723	semidirect

H ≅ G/N

type

51. #64436

Q₁₆

semidirect

71. C₃₇

#1723

semidirect

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

subgroup	index
#14557936	2
#14557929	2
#14557885	2
#14557884	2
#14557255	2
C₃₇×#494	2
#1723	37

subgroup

index

C₃₇×#494

Subgroups — #17994271

Conjugacy classes of subgroups

Complements

Maximal subgroups