Group #37560156 — subgroups

Subgroups — #37560156

Order 23872.

236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 95 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	#26045788	C₂	1	1	yes	maximal, characteristic
2	#26045778	C₂	1	2	yes	maximal, complemented
3	#26045778	C₂	1	2	yes	maximal, complemented
4	#26045777	C₂	1	2	yes	maximal, complemented
5	#26045773	C₂	1	1	yes	maximal, characteristic, complemented
6	#26045757	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊D₃₇₃	C₂	1	2	yes	maximal, complemented
8	#26045752	C₂	1	2	yes	maximal, complemented
9	#123×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₇₄₆	C₂²	1	2	yes	—
11	C₂×D₁₄₉₂	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₃₇₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₃₇₃	C₂²	1	2	yes	—
14	C₂²×Dic₃₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₃₇₃	C₂²	1	2	yes	—
16	#20238589	C₂²	1	2	yes	—
17	#20238589	C₂²	1	2	yes	—
18	C₂²⋊Dic₃₇₃	C₂²	1	2	yes	—
19	C₂²⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
20	#20238587	C₂²	1	2	yes	—
21	#20238587	C₂²	1	2	yes	—
22	#20238587	C₂²	1	2	yes	—
23	C₄×Dic₃₇₃	C₂²	1	2	yes	—
24	C₄×Dic₃₇₃	C₂²	1	2	yes	—
25	C₄×Dic₃₇₃	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₃₇₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₃₇₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₃₇₃	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₃₇₃	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₃₇₃	C₂²	1	1	yes	characteristic
31	C₂²×D₃₇₃	C₂³	1	2	yes	—
32	Dic₇₄₆	—	2	4	no	—
33	D₁₄₉₂	—	4	1	no	—
34	C₂²⋊D₃₇₃	—	2	4	no	—
35	C₂²⋊D₃₇₃	—	2	4	no	—
36	C₂×Dic₃₇₃	—	4	1	no	—
37	C₂×Dic₃₇₃	C₂³	1	2	yes	—
38	C₂×Dic₃₇₃	C₂³	1	2	yes	—
39	C₂×Dic₃₇₃	C₂³	1	2	yes	—
40	C₄×D₃₇₃	—	4	2	no	—
41	C₂³×C₃₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₃₇₃	C₂³	1	1	yes	characteristic
43	D₄×C₃₇₃	—	4	1	no	—
44	C₂×C₄×C₃₇₃	—	4	1	no	—
45	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
46	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
48	C₂²×C₃₇₃	—	2	2	no	—
49	C₂²×C₃₇₃	—	2	2	no	—
50	C₂²×C₃₇₃	—	2	1	no	—
51	C₂²×C₃₇₃	—	2	1	no	—
52	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
53	C₂×D₃₇₃	—	2	2	no	—
54	C₂×D₃₇₃	—	2	2	no	—
55	C₂×D₃₇₃	—	2	2	no	—
56	C₄×C₃₇₃	—	2	2	no	—
57	C₄×C₃₇₃	—	2	2	no	—
58	C₄×C₃₇₃	—	2	1	no	—
59	Dic₃₇₃	—	2	2	no	—
60	Dic₃₇₃	—	2	2	no	—
61	Dic₃₇₃	#41	1	4	yes	—
62	C₂×C₃₇₃	—	4	1	no	—
63	C₂×C₃₇₃	—	4	1	no	—
64	C₂×C₃₇₃	C₂×#41	1	2	yes	—
65	C₂×C₃₇₃	H(5,𝔽₂)	1	1	yes	characteristic
66	D₃₇₃	—	4	2	no	—
67	C₃₇₃	#559	1	1	yes	characteristic, complemented, Sylow
68	#559	—	373	1	no	maximal, Sylow
69	#127	—	373	1	no	—
70	#124	—	373	2	no	—
71	#124	—	373	2	no	—
72	#123	—	373	1	no	—
73	#123	—	373	2	no	—
74	#123	D₃₇₃	1	1	yes	characteristic, complemented
75	C₄×Q₈	—	373	2	no	—
76	C₄×D₄	—	373	2	no	—
77	C₄×D₄	—	373	2	no	—
78	C₂×Q₈	—	373	2	no	—
79	C₂×D₄	—	373	2	no	—
80	C₂×D₄	—	373	2	no	—
81	C₂×D₄	—	373	1	no	—
82	C₂×D₄	C₂×D₃₇₃	1	1	yes	characteristic
83	C₂²×C₄	—	373	1	no	—
84	C₂²×C₄	—	373	2	no	—
85	C₂²×C₄	C₂×D₃₇₃	1	1	yes	characteristic
86	C₄⋊C₄	—	373	2	no	—
87	C₄⋊C₄	—	373	2	no	—
88	C₄⋊C₄	C₂×D₃₇₃	1	2	yes	—
89	C₂²⋊C₄	—	373	2	no	—
90	C₂²⋊C₄	—	373	1	no	—
91	C₂²⋊C₄	—	373	2	no	—
92	C₂²⋊C₄	—	373	2	no	—
93	C₂²⋊C₄	—	373	2	no	—
94	C₂²⋊C₄	C₂×D₃₇₃	1	2	yes	—
95	C₂²⋊C₄	C₂×D₃₇₃	1	1	yes	characteristic
96	C₄²	—	373	2	no	—
97	C₄²	—	373	2	no	—
98	C₄²	—	373	1	no	—
99	C₂³	—	373	2	no	—
100	C₂³	C₂²×D₃₇₃	1	1	yes	characteristic
101	C₂³	C₂²×D₃₇₃	1	1	yes	characteristic
102	Q₈	—	746	4	no	—
103	D₄	—	4	1	no	—
104	D₄	—	1492	1	no	—
105	D₄	—	746	4	no	—
106	D₄	—	746	4	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	373	2	no	—
109	C₂×C₄	—	373	2	no	—
110	C₂×C₄	—	373	2	no	—
111	C₂×C₄	—	1492	2	no	—
112	C₂×C₄	—	1492	1	no	—
113	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
114	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
115	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	746	2	no	—
121	C₂²	—	746	2	no	—
122	C₂²	—	746	2	no	—
123	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	373	4	no	—
128	C₄	—	746	2	no	—
129	C₄	—	746	2	no	—
130	C₂	—	4	1	no	—
131	C₂	—	4	1	no	—
132	C₂	—	1492	2	no	—
133	C₂	#41×D₃₇₃	1	2	yes	—
134	C₂	#26045837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26045788

C₂

yes

maximal, characteristic

#26045778

C₂

yes

maximal, complemented

#26045778

C₂

yes

maximal, complemented

#26045777

C₂

yes

maximal, complemented

#26045773

C₂

yes

maximal, characteristic, complemented

#26045757

C₂

yes

maximal, complemented

C₄×C₂²⋊D₃₇₃

C₂

yes

maximal, complemented

#26045752

C₂

yes

maximal, complemented

#123×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₄₆

C₂²

yes

—

C₂×D₁₄₉₂

C₂²

yes

characteristic

C₂×C₂²⋊D₃₇₃

C₂²

yes

—

C₂×C₂²⋊D₃₇₃

C₂²

yes

—

C₂²×Dic₃₇₃

C₂²

yes

characteristic

C₂×C₄×D₃₇₃

C₂²

yes

—

#20238589

C₂²

yes

—

#20238589

C₂²

yes

—

C₂²⋊Dic₃₇₃

C₂²

yes

—

C₂²⋊Dic₃₇₃

C₂²

yes

characteristic

#20238587

C₂²

yes

—

#20238587

C₂²

yes

—

#20238587

C₂²

yes

—

C₄×Dic₃₇₃

C₂²

yes

—

C₄×Dic₃₇₃

C₂²

yes

—

C₄×Dic₃₇₃

C₂²

yes

characteristic

C₂×D₄×C₃₇₃

C₂²

yes

characteristic

C₂²×C₄×C₃₇₃

C₂²

yes

characteristic

C₄⋊C₄×C₃₇₃

C₂²

yes

—

C₂²⋊C₄×C₃₇₃

C₂²

yes

—

C₂²⋊C₄×C₃₇₃

C₂²

yes

characteristic

C₂²×D₃₇₃

C₂³

yes

—

Dic₇₄₆

—

D₁₄₉₂

—

C₂²⋊D₃₇₃

—

C₂²⋊D₃₇₃

—

C₂×Dic₃₇₃

—

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

—

C₄×D₃₇₃

—

C₂³×C₃₇₃

C₂³

yes

characteristic

C₂³×C₃₇₃

C₂³

yes

characteristic

D₄×C₃₇₃

—

C₂×C₄×C₃₇₃

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

C₂⁴

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

#41

yes

—

C₂×C₃₇₃

—

C₂×C₃₇₃

—

C₂×C₃₇₃

C₂×#41

yes

—

C₂×C₃₇₃

H(5,𝔽₂)

yes

characteristic

D₃₇₃

—

C₃₇₃

#559

yes

characteristic, complemented, Sylow

#559

—

373

maximal, Sylow

#127

—

373

—

#124

—

373

—

#124

—

373

—

#123

—

373

—

#123

—

373

—

#123

D₃₇₃

yes

characteristic, complemented

C₄×Q₈

—

373

—

C₄×D₄

—

373

—

C₄×D₄

—

373

—

C₂×Q₈

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

C₂×D₃₇₃

yes

characteristic

C₂²×C₄

—

373

—

C₂²×C₄

—

373

—

C₂²×C₄

C₂×D₃₇₃

yes

characteristic

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

C₂×D₃₇₃

yes

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

C₂×D₃₇₃

yes

—

C₂²⋊C₄

C₂×D₃₇₃

yes

characteristic

C₄²

—

373

—

C₄²

—

373

—

C₄²

—

373

—

C₂³

—

373

—

100

C₂³

C₂²×D₃₇₃

yes

characteristic

101

C₂³

C₂²×D₃₇₃

yes

characteristic

102

Q₈

—

746

—

103

D₄

—

104

D₄

—

1492

—

105

D₄

—

746

—

106

D₄

—

746

—

107

C₂×C₄

—

108

C₂×C₄

—

373

—

109

C₂×C₄

—

373

—

110

C₂×C₄

—

373

—

111

C₂×C₄

—

1492

—

112

C₂×C₄

—

1492

—

113

C₂×C₄

C₂²×D₃₇₃

yes

—

114

C₂×C₄

C₂²×D₃₇₃

yes

—

115

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

746

—

121

C₂²

—

746

—

122

C₂²

—

746

—

123

C₂²

C₂³×D₃₇₃

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

373

—

128

C₄

—

746

—

129

C₄

—

746

—

130

C₂

—

131

C₂

—

132

C₂

—

1492

—

133

C₂

#41×D₃₇₃

yes

—

134

C₂

#26045837

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	#123×C₃₇₃	C₂
prime radical	#26045788	C₂
hypercenter	#123	D₃₇₃
hypocenter	C₃₇₃	#559
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇₃

commutator, socle, cosocle

C₂²×C₃₇₃

C₂⁴

Fitting

#123×C₃₇₃

C₂

C₂

D₃₇₃

C₃₇₃

G ▷ C₂²×C₃₇₃ ▷ C₃₇₃

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
2. #26045778	C₂	semidirect
3. #26045778	C₂	semidirect
4. #26045777	C₂	semidirect
5. #26045773	C₂	semidirect
6. #26045757	C₂	semidirect
7. C₄×C₂²⋊D₃₇₃	C₂	semidirect
8. #26045752	C₂	semidirect
9. #123×C₃₇₃	C₂	semidirect
67. C₃₇₃	#559	semidirect
74. #123	D₃₇₃	semidirect

H ≅ G/N

type

2. #26045778

C₂

semidirect

3. #26045778

C₂

semidirect

4. #26045777

C₂

semidirect

5. #26045773

C₂

semidirect

6. #26045757

C₂

semidirect

7. C₄×C₂²⋊D₃₇₃

C₂

semidirect

8. #26045752

C₂

semidirect

9. #123×C₃₇₃

C₂

semidirect

67. C₃₇₃

#559

semidirect

74. #123

D₃₇₃

semidirect

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

subgroup	index
#26045788	2
#26045778	2
#26045777	2
#26045773	2
#26045757	2
C₄×C₂²⋊D₃₇₃	2
#26045752	2
#123×C₃₇₃	2
#559	373

subgroup

index

C₄×C₂²⋊D₃₇₃

#123×C₃₇₃

373

Subgroups — #37560156

Conjugacy classes of subgroups

Complements

Maximal subgroups