Group #17993655 — subgroups

Subgroups — #17993655

Order 4736.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14557909	C₂	1	1	yes	maximal, characteristic
2	#14557733	C₂	1	2	yes	maximal, complemented
3	#14557277	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#64436	C₂	1	2	yes	maximal, complemented
5	C₃₇×#559	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596941	—	2	2	no	—
7	C₂×#10596939	—	2	2	no	—
8	#11924533	C₂²	1	1	yes	characteristic
9	C₄⋊#64436	C₂²	1	2	yes	—
10	#11924456	—	2	2	no	—
11	C₄×#64436	C₂²	1	1	yes	characteristic
12	#127×C₃₇	C₂²	1	1	yes	characteristic
13	#124×C₃₇	—	2	2	no	—
14	#123×C₃₇	—	2	2	no	—
15	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
16	C₄×D₄×C₃₇	—	2	2	no	—
17	C₂×Q₈×C₃₇	D₄	1	2	yes	—
18	C₂×D₄×C₃₇	—	2	2	no	—
19	C₂×D₄×C₃₇	D₄	1	2	yes	—
20	C₂²×C₄×C₃₇	—	2	2	no	—
21	C₂×D₁₄₈	—	2	1	no	—
22	C₂×D₁₄₈	—	2	2	no	—
23	C₄⋊C₄×C₃₇	—	2	2	no	—
24	C₄⋊C₄×C₃₇	D₄	1	2	yes	—
25	C₂²⋊C₄×C₃₇	—	2	4	no	—
26	C₂²⋊C₄×C₃₇	—	2	2	no	—
27	C₄²×C₃₇	—	2	2	no	—
28	C₄²×C₃₇	C₂³	1	1	yes	characteristic
29	#10596941	—	4	4	no	—
30	#10596939	—	4	4	no	—
31	C₂×#64436	—	2	1	no	—
32	C₂×#64436	—	2	2	no	—
33	C₂³×C₃₇	—	2	2	no	—
34	C₂²×D₃₇	—	4	1	no	—
35	Q₈×C₃₇	—	2	2	no	—
36	Q₈×C₃₇	—	2	2	no	—
37	D₄×C₃₇	—	4	2	no	—
38	D₄×C₃₇	—	4	2	no	—
39	D₄×C₃₇	—	4	2	no	—
40	C₂×C₄×C₃₇	—	2	2	no	—
41	C₂×C₄×C₃₇	—	2	2	no	—
42	C₂×C₄×C₃₇	—	8	2	no	—
43	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
45	D₁₄₈	—	4	2	no	—
46	D₁₄₈	—	4	2	no	—
47	D₁₄₈	—	4	2	no	—
48	#64436	—	2	2	no	—
49	#64436	—	4	2	no	—
50	C₂²×C₃₇	—	4	4	no	—
51	C₂²×C₃₇	—	4	2	no	—
52	C₂²×C₃₇	#120	1	1	yes	characteristic
53	C₂×D₃₇	—	8	2	no	—
54	C₂×D₃₇	—	8	1	no	—
55	C₄×C₃₇	—	2	1	no	—
56	C₄×C₃₇	—	4	2	no	—
57	C₄×C₃₇	—	4	2	no	—
58	C₄×C₃₇	#135	1	2	yes	—
59	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
60	#1620	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	—	8	2	no	—
62	C₂×C₃₇	#449	1	2	yes	—
63	C₂×C₃₇	#453	1	1	yes	characteristic
64	D₃₇	—	16	1	no	—
65	#559	D₃₇	1	1	yes	characteristic, complemented
66	#493	—	37	1	no	—
67	#460	—	37	2	no	—
68	#331	—	37	1	no	—
69	Q₈⋊C₈	—	37	2	no	—
70	C₃₇	#1620	1	1	yes	characteristic, complemented, Sylow
71	C₂×SD₁₆	—	74	2	no	—
72	C₂×D₈	—	74	2	no	—
73	#127	—	37	1	no	—
74	#127	C₂×D₃₇	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₃₇	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	37	2	no	—
80	#102	—	74	2	no	—
81	C₄×C₈	—	37	1	no	—
82	C₂×Q₈	C₂²⋊D₃₇	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	74	2	no	—
85	C₂×D₄	—	74	1	no	—
86	C₂×D₄	C₂²⋊D₃₇	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	148	4	no	—
89	D₈	—	148	4	no	—
90	C₂×C₈	—	74	1	no	—
91	C₂×C₈	—	74	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₃₇	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₃₇	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	148	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	148	2	no	—
106	D₄	—	148	2	no	—
107	D₄	—	148	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
113	C₈	—	74	2	no	—
114	C₈	—	148	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	4	no	—
117	C₂²	—	296	1	no	—
118	C₂²	—	296	2	no	—
119	C₂²	#11924505	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#11924556	1	2	yes	—
124	C₄	#11924561	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	592	1	no	—
127	C₂	#14557678	1	2	yes	—
128	C₂	#14557701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557909

C₂

yes

maximal, characteristic

#14557733

C₂

yes

maximal, complemented

#14557277

C₂

yes

maximal, characteristic, complemented

Q₈⋊#64436

C₂

yes

maximal, complemented

C₃₇×#559

C₂

yes

maximal, characteristic, complemented

C₂×#10596941

—

C₂×#10596939

—

#11924533

C₂²

yes

characteristic

C₄⋊#64436

C₂²

yes

—

#11924456

—

C₄×#64436

C₂²

yes

characteristic

#127×C₃₇

C₂²

yes

characteristic

#124×C₃₇

—

#123×C₃₇

—

C₄×Q₈×C₃₇

C₂²

yes

—

C₄×D₄×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

—

C₂²⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

C₂³

yes

—

—

—

—

C₂³×C₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#120

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#135

yes

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

#1620

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#449

yes

—

C₂×C₃₇

#453

yes

characteristic

D₃₇

—

#559

D₃₇

yes

characteristic, complemented

#493

—

#460

—

#331

—

Q₈⋊C₈

—

C₃₇

#1620

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

#127

—

#127

C₂×D₃₇

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₃₇

yes

—

C₄×D₄

—

C₄⋊C₈

—

#102

—

C₄×C₈

—

C₂×Q₈

C₂²⋊D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

—

SD₁₆

—

148

—

D₈

—

148

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₃₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

—

C₂³

—

148

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

148

—

106

D₄

—

148

—

107

D₄

—

148

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

113

C₈

—

114

C₈

—

148

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

296

—

118

C₂²

—

296

—

119

C₂²

#11924505

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#11924556

yes

—

124

C₄

#11924561

yes

—

125

C₂

—

126

C₂

—

592

—

127

C₂

#14557678

yes

—

128

C₂

#14557701

yes

characteristic

	subgroup	G/H
center	C₂²	#11924505
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#559	C₂
socle	C₂²×C₃₇	#120
prime radical	#14557909	C₂
hypercenter	#559	D₃₇
hypocenter	C₃₇	#1620
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#11924505

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#559

C₂

C₂²×C₃₇

C₂

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #14557733	C₂	semidirect
3. #14557277	C₂	semidirect
4. Q₈⋊#64436	C₂	semidirect
5. C₃₇×#559	C₂	semidirect
65. #559	D₃₇	semidirect
70. C₃₇	#1620	semidirect

H ≅ G/N

type

2. #14557733

C₂

semidirect

3. #14557277

C₂

semidirect

4. Q₈⋊#64436

C₂

semidirect

5. C₃₇×#559

C₂

semidirect

65. #559

D₃₇

semidirect

70. C₃₇

#1620

semidirect

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

subgroup	index
#14557909	2
#14557733	2
#14557277	2
Q₈⋊#64436	2
C₃₇×#559	2
#1620	37

subgroup

index

C₃₇×#559

Subgroups — #17993655

Conjugacy classes of subgroups

Complements

Maximal subgroups