Group #17993330 — subgroups

Subgroups — #17993330

Order 4736.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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	#14558084	C₂	1	1	yes	maximal, characteristic
2	#14557555	C₂	1	2	yes	maximal, complemented
3	#14557282	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557276	C₂	1	2	yes	maximal, complemented
5	C₃₇×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596966	C₄	1	2	yes	—
7	C₂×#10596929	—	2	2	no	—
8	#11924533	C₂²	1	1	yes	characteristic
9	#11924525	C₄	1	2	yes	—
10	#11924511	—	2	2	no	—
11	#11924498	—	2	2	no	—
12	C₄²×D₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊#64436	C₂²	1	2	yes	—
14	#11924465	—	2	2	no	—
15	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
18	#10596966	—	4	4	no	—
19	C₂×D₁₄₈	—	2	2	no	—
20	C₂×D₁₄₈	—	2	2	no	—
21	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₃₇	—	2	2	no	—
23	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₃₇	D₄	1	2	yes	—
25	C₂×C₈×C₃₇	—	2	2	no	—
26	C₄⋊C₄×C₃₇	—	2	2	no	—
27	C₄²×C₃₇	C₂³	1	1	yes	characteristic
28	#10596938	—	2	4	no	—
29	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₃₇	D₄	1	2	yes	—
31	C₂×#64436	—	2	2	no	—
32	#10596929	—	4	4	no	—
33	C₂²×D₃₇	—	2	2	no	—
34	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₃₇	—	4	2	no	—
36	C₂×C₄×C₃₇	—	2	2	no	—
37	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
39	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
40	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
41	D₁₄₈	—	4	2	no	—
42	D₁₄₈	—	4	4	no	—
43	D₁₄₈	—	4	2	no	—
44	D₁₄₈	—	4	2	no	—
45	C₄×D₃₇	—	2	2	no	—
46	C₄×D₃₇	—	2	2	no	—
47	C₄×D₃₇	—	2	2	no	—
48	C₄×D₃₇	—	4	4	no	—
49	C₈×C₃₇	—	4	2	no	—
50	#64436	—	4	2	no	—
51	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×D₃₇	—	2	1	no	—
53	C₂×D₃₇	—	2	2	no	—
54	C₂×D₃₇	—	4	2	no	—
55	C₂×D₃₇	—	4	4	no	—
56	C₄×C₃₇	—	2	1	no	—
57	C₄×C₃₇	—	4	2	no	—
58	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
59	C₄×C₃₇	#137	1	2	yes	—
60	Dic₃₇	—	2	2	no	—
61	Dic₃₇	#100	1	2	yes	—
62	#1560	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	C₂×#100	1	1	yes	characteristic
64	C₂×C₃₇	#417	1	2	yes	—
65	D₃₇	—	4	1	no	—
66	D₃₇	—	8	2	no	—
67	#532	—	37	1	no	—
68	#423	—	37	2	no	—
69	#333	—	37	1	no	—
70	#333	D₃₇	1	1	yes	characteristic, complemented
71	#331	—	37	2	no	—
72	C₃₇	#1560	1	1	yes	characteristic, complemented, Sylow
73	C₂×#41	—	37	2	no	—
74	C₂×M₁₆	—	74	2	no	—
75	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
76	#127	—	37	1	no	—
77	#124	—	37	2	no	—
78	#121	—	74	2	no	—
79	C₄×D₄	—	74	2	no	—
80	C₂×C₄²	—	37	1	no	—
81	C₄⋊C₈	—	37	2	no	—
82	C₄⋊C₈	—	74	2	no	—
83	C₄⋊C₈	C₂×D₃₇	1	2	yes	—
84	#41	—	148	4	no	—
85	C₂×Q₈	C₄×D₃₇	1	2	yes	—
86	C₂×D₄	—	37	2	no	—
87	C₂×D₄	—	74	2	no	—
88	C₂×D₄	—	74	2	no	—
89	C₂²×C₄	—	37	1	no	—
90	C₂²×C₄	—	37	2	no	—
91	C₂²×C₄	—	74	2	no	—
92	M₁₆	—	148	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	74	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂²⋊C₄	—	74	4	no	—
97	C₄²	—	37	1	no	—
98	C₄²	—	37	2	no	—
99	C₄²	C₂²×D₃₇	1	1	yes	characteristic
100	C₂³	—	37	1	no	—
101	C₂³	—	74	2	no	—
102	Q₈	—	4	2	no	—
103	D₄	—	148	2	no	—
104	D₄	—	148	4	no	—
105	D₄	—	148	2	no	—
106	D₄	—	148	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	37	1	no	—
109	C₂×C₄	—	37	2	no	—
110	C₂×C₄	—	74	2	no	—
111	C₂×C₄	—	74	2	no	—
112	C₂×C₄	—	74	2	no	—
113	C₂×C₄	—	148	4	no	—
114	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
115	C₂×C₄	D₄×D₃₇	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	148	2	no	—
118	C₂²	—	74	1	no	—
119	C₂²	—	74	2	no	—
120	C₂²	—	148	2	no	—
121	C₂²	—	148	4	no	—
122	C₂²	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	4	2	no	—
125	C₄	—	37	2	no	—
126	C₄	—	74	2	no	—
127	C₄	#11924563	1	2	yes	—
128	C₄	#11924568	1	2	yes	—
129	C₂	—	296	2	no	—
130	C₂	—	148	1	no	—
131	C₂	#100×D₃₇	1	1	yes	characteristic
132	C₂	#14557533	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558084

C₂

yes

maximal, characteristic

#14557555

C₂

yes

maximal, complemented

#14557282

C₂

yes

maximal, characteristic, complemented

#14557276

C₂

yes

maximal, complemented

C₃₇×#333

C₂

yes

maximal, characteristic, complemented

C₂×#10596966

C₄

yes

—

C₂×#10596929

—

#11924533

C₂²

yes

characteristic

#11924525

C₄

yes

—

#11924511

—

#11924498

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊#64436

C₂²

yes

—

#11924465

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

#10596966

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596938

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₂×#64436

—

#10596929

—

C₂²×D₃₇

—

C₂²×D₃₇

C₂²⋊C₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

C₄×C₃₇

#137

yes

—

Dic₃₇

—

Dic₃₇

#100

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

D₃₇

—

—

—

—

D₃₇

yes

characteristic, complemented

#331

—

C₃₇

#1560

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₃₇

yes

—

—

—

C₄×D₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₇

yes

—

#41

—

148

—

C₂×Q₈

C₄×D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

100

C₂³

—

101

C₂³

—

102

Q₈

—

103

D₄

—

148

—

104

D₄

—

148

—

105

D₄

—

148

—

106

D₄

—

148

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

148

—

114

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

115

C₂×C₄

D₄×D₃₇

yes

—

116

C₈

—

117

C₈

—

148

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

148

—

121

C₂²

—

148

—

122

C₂²

C₂²⋊C₄×D₃₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#11924563

yes

—

128

C₄

#11924568

yes

—

129

C₂

—

296

—

130

C₂

—

148

—

131

C₂

#100×D₃₇

yes

characteristic

132

C₂

#14557533

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#333	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	#14558084	C₂
cosocle	C₄²×C₃₇	C₂³
hypercenter	#333	D₃₇
hypocenter	C₃₇	#1560
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#333

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

#14558084

C₂

cosocle

C₄²×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #14557555	C₂	semidirect
3. #14557282	C₂	semidirect
4. #14557276	C₂	semidirect
5. C₃₇×#333	C₂	semidirect
70. #333	D₃₇	semidirect
72. C₃₇	#1560	semidirect

H ≅ G/N

type

2. #14557555

C₂

semidirect

3. #14557282

C₂

semidirect

4. #14557276

C₂

semidirect

5. C₃₇×#333

C₂

semidirect

70. #333

D₃₇

semidirect

72. C₃₇

#1560

semidirect

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

subgroup	index
#14558084	2
#14557555	2
#14557282	2
#14557276	2
C₃₇×#333	2
#1560	37

subgroup

index

C₃₇×#333

Subgroups — #17993330

Conjugacy classes of subgroups

Complements

Maximal subgroups