Group #17994209 — subgroups

Subgroups — #17994209

Order 4736.

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

normal subgroups only

This group has 178 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	#14557885	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557883	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557881	C₂	1	1	yes	maximal, characteristic
4	#14557881	C₂	1	1	yes	maximal, characteristic
5	#14557880	C₂	1	1	yes	maximal, characteristic
6	C₈×#64436	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	#11924536	C₂²	1	1	yes	characteristic
9	#11924533	C₂²	1	1	yes	characteristic
10	#11924460	—	2	2	no	—
11	#11924459	—	2	2	no	—
12	#11924458	—	2	2	no	—
13	#11924456	—	2	2	no	—
14	#11924456	—	2	2	no	—
15	C₄×#64436	C₂²	1	1	yes	characteristic
16	C₄×#64436	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
19	#103×C₃₇	—	2	2	no	—
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×D₁₄₈	—	2	1	no	—
24	C₂×D₁₄₈	—	2	1	no	—
25	C₂×D₁₄₈	—	2	1	no	—
26	C₂×Dic₇₄	—	2	1	no	—
27	C₂×C₈×C₃₇	D₄	1	2	yes	—
28	C₄⋊C₄×C₃₇	—	2	1	no	—
29	C₄⋊C₄×C₃₇	—	2	1	no	—
30	C₄⋊C₄×C₃₇	—	2	1	no	—
31	C₄⋊C₄×C₃₇	—	2	1	no	—
32	C₄²×C₃₇	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₃₇	—	2	1	no	—
34	C₄⋊Dic₃₇	—	2	1	no	—
35	C₂×#64436	D₄	1	2	yes	—
36	C₂×#64436	D₄	1	2	yes	—
37	C₂²×D₃₇	—	4	1	no	—
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	D₁₄₈	—	4	2	no	—
47	D₁₄₈	—	4	2	no	—
48	D₁₄₈	—	4	2	no	—
49	Dic₇₄	—	4	2	no	—
50	C₈×C₃₇	—	2	2	no	—
51	#64436	—	2	2	no	—
52	#64436	—	2	2	no	—
53	C₂²×C₃₇	#127	1	1	yes	characteristic
54	C₂×D₃₇	—	8	1	no	—
55	C₂×D₃₇	—	8	1	no	—
56	C₂×D₃₇	—	8	1	no	—
57	C₄×C₃₇	—	8	1	no	—
58	C₄×C₃₇	—	8	1	no	—
59	C₄×C₃₇	#135	1	2	yes	—
60	C₄×C₃₇	#135	1	2	yes	—
61	C₄×C₃₇	#135	1	2	yes	—
62	Dic₃₇	—	8	1	no	—
63	#1711	—	37	1	no	maximal, Sylow
64	C₂×C₃₇	#495	1	1	yes	characteristic
65	C₂×C₃₇	#495	1	1	yes	characteristic
66	C₂×C₃₇	#495	1	1	yes	characteristic
67	D₃₇	—	16	1	no	—
68	#487	—	37	1	no	—
69	#487	—	37	1	no	—
70	#487	D₃₇	1	1	yes	characteristic, complemented
71	#486	—	37	1	no	—
72	#486	—	37	1	no	—
73	#486	—	37	1	no	—
74	C₈²	—	37	1	no	—
75	C₃₇	#1711	1	1	yes	characteristic, complemented, Sylow
76	C₄⋊Q₈	—	37	1	no	—
77	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
78	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
79	#127	—	37	1	no	—
80	#103	—	2	2	no	—
81	#103	—	74	2	no	—
82	#103	—	74	2	no	—
83	#102	—	74	2	no	—
84	#102	—	74	2	no	—
85	#102	—	74	2	no	—
86	C₄×C₈	—	37	1	no	—
87	C₄×C₈	—	37	1	no	—
88	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	74	1	no	—
92	C₂×D₄	—	74	1	no	—
93	C₂×D₄	—	74	1	no	—
94	C₂×D₄	—	74	1	no	—
95	C₂×C₈	—	37	2	no	—
96	C₂×C₈	—	37	2	no	—
97	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	74	1	no	—
103	C₄⋊C₄	—	74	1	no	—
104	C₄²	C₂²×D₃₇	1	1	yes	characteristic
105	C₂³	—	148	1	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	148	2	no	—
109	D₄	—	148	2	no	—
110	D₄	—	148	2	no	—
111	D₄	—	148	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	148	1	no	—
115	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
116	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	74	2	no	—
120	C₈	—	74	2	no	—
121	C₂²	—	296	1	no	—
122	C₂²	—	296	1	no	—
123	C₂²	—	296	1	no	—
124	C₂²	#11924532	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	296	1	no	—
128	C₄	#11924556	1	2	yes	—
129	C₄	#11924557	1	2	yes	—
130	C₄	#11924559	1	2	yes	—
131	C₂	—	592	1	no	—
132	C₂	#14557914	1	1	yes	characteristic
133	C₂	#14557915	1	1	yes	characteristic
134	C₂	#14557918	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557885

C₂

yes

maximal, characteristic, complemented

#14557883

C₂

yes

maximal, characteristic, complemented

#14557881

C₂

yes

maximal, characteristic

#14557881

C₂

yes

maximal, characteristic

#14557880

C₂

yes

maximal, characteristic

C₈×#64436

C₂

yes

maximal, characteristic, complemented

C₃₇×#487

C₂

yes

maximal, characteristic, complemented

#11924536

C₂²

yes

characteristic

#11924533

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#64436

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

#103×C₃₇

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×Dic₇₄

—

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

—

C₂×#64436

D₄

yes

—

C₂×#64436

D₄

yes

—

C₂²×D₃₇

—

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

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

#64436

—

C₂²×C₃₇

#127

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#135

yes

—

C₄×C₃₇

#135

yes

—

C₄×C₃₇

#135

yes

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

#487

—

#487

—

#487

D₃₇

yes

characteristic, complemented

#486

—

#486

—

#486

—

C₈²

—

C₃₇

#1711

yes

characteristic, complemented, Sylow

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×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂²×D₃₇

yes

characteristic

105

C₂³

—

148

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

148

—

109

D₄

—

148

—

110

D₄

—

148

—

111

D₄

—

148

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

148

—

115

C₂×C₄

D₄×D₃₇

yes

characteristic

116

C₂×C₄

D₄×D₃₇

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₂²

—

296

—

122

C₂²

—

296

—

123

C₂²

—

296

—

124

C₂²

#11924532

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

296

—

128

C₄

#11924556

yes

—

129

C₄

#11924557

yes

—

130

C₄

#11924559

yes

—

131

C₂

—

592

—

132

C₂

#14557914

yes

characteristic

133

C₂

#14557915

yes

characteristic

134

C₂

#14557918

yes

characteristic

	subgroup	G/H
center	C₂²	#11924532
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#487	C₂
socle	C₂²×C₃₇	#127
prime radical	#11924533	C₂²
hypercenter	#487	D₃₇
hypocenter	C₃₇	#1711
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#11924532

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#487

C₂

C₂²×C₃₇

C₂²

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
1. #14557885	C₂	semidirect
2. #14557883	C₂	semidirect
6. C₈×#64436	C₂	semidirect
7. C₃₇×#487	C₂	semidirect
70. #487	D₃₇	semidirect
75. C₃₇	#1711	semidirect

H ≅ G/N

type

1. #14557885

C₂

semidirect

2. #14557883

C₂

semidirect

6. C₈×#64436

C₂

semidirect

7. C₃₇×#487

C₂

semidirect

70. #487

D₃₇

semidirect

75. C₃₇

#1711

semidirect

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

subgroup	index
#14557885	2
#14557883	2
#14557881	2
#14557880	2
C₈×#64436	2
C₃₇×#487	2
#1711	37

subgroup

index

C₃₇×#487

Subgroups — #17994209

Conjugacy classes of subgroups

Complements

Maximal subgroups