Group #37560143 — subgroups

Subgroups — #37560143

Order 23872.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#26045791	C₂	1	1	yes	maximal, characteristic, complemented
2	#26045787	C₂	1	2	yes	maximal, complemented
3	#26045785	C₂	1	2	yes	maximal, complemented
4	#26045778	C₂	1	2	yes	maximal
5	#26045768	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#26045757	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊D₃₇₃	C₂	1	1	yes	maximal, characteristic
9	#26045751	C₂	1	1	yes	maximal, characteristic, complemented
10	#26045750	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₇₄₆	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₃₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₃₇₃	C₂²	1	2	yes	—
18	#20238589	C₂²	1	1	yes	characteristic
19	#20238589	C₂²	1	2	yes	—
20	#20238589	C₂²	1	2	yes	—
21	C₂²⋊Dic₃₇₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
23	#20238587	C₂²	1	2	yes	—
24	#20238587	C₂²	1	1	yes	characteristic
25	#20238587	C₂²	1	2	yes	—
26	C₄×Dic₃₇₃	C₂²	1	2	yes	—
27	C₄×Dic₃₇₃	C₂²	1	2	yes	—
28	C₄×Dic₃₇₃	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₇₃	C₂²	1	2	yes	—
30	C₂×Q₈×C₃₇₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₃₇₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₃₇₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₃₇₃	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₃₇₃	C₂²	1	2	yes	—
35	C₂²×D₃₇₃	C₂³	1	1	yes	characteristic
36	Dic₇₄₆	—	2	4	no	—
37	C₂²⋊D₃₇₃	—	2	2	no	—
38	C₂×Dic₃₇₃	C₂³	1	1	yes	characteristic
39	C₂×Dic₃₇₃	C₂³	1	2	yes	—
40	C₂×Dic₃₇₃	C₂³	1	2	yes	—
41	C₂×Dic₃₇₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₇₃	C₂³	1	1	yes	characteristic
43	C₄×D₃₇₃	—	2	2	no	—
44	C₂³×C₃₇₃	C₂³	1	1	yes	characteristic
45	Q₈×C₃₇₃	—	2	2	no	—
46	C₂×C₄×C₃₇₃	—	2	2	no	—
47	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
48	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
49	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
51	C₂²×C₃₇₃	—	2	1	no	—
52	C₂²×C₃₇₃	—	2	1	no	—
53	C₂²×C₃₇₃	—	2	1	no	—
54	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
55	C₂×D₃₇₃	—	2	1	no	—
56	C₂×D₃₇₃	—	2	1	no	—
57	C₂×D₃₇₃	—	2	1	no	—
58	C₄×C₃₇₃	—	2	1	no	—
59	C₄×C₃₇₃	—	2	2	no	—
60	C₄×C₃₇₃	—	2	2	no	—
61	C₄×C₃₇₃	#41	1	2	yes	—
62	Dic₃₇₃	—	2	1	no	—
63	Dic₃₇₃	—	2	2	no	—
64	Dic₃₇₃	—	2	2	no	—
65	Dic₃₇₃	—	2	1	no	—
66	Dic₃₇₃	#41	1	2	yes	—
67	C₂×C₃₇₃	—	4	1	no	—
68	C₂×C₃₇₃	C₂×#41	1	1	yes	characteristic
69	C₂×C₃₇₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₃₇₃	2₋¹⁺⁴	1	1	yes	characteristic
71	D₃₇₃	—	4	1	no	—
72	C₃₇₃	#556	1	1	yes	characteristic, complemented, Sylow
73	#556	—	373	1	no	maximal, Sylow
74	C₄⋊Q₈	—	373	1	no	—
75	#126	—	373	2	no	—
76	#126	—	373	2	no	—
77	#124	—	373	2	no	—
78	C₂²⋊Q₈	—	373	1	no	—
79	C₂²⋊Q₈	D₃₇₃	1	1	yes	characteristic, complemented
80	C₄×Q₈	—	373	1	no	—
81	C₄×Q₈	—	373	2	no	—
82	C₄×D₄	—	373	1	no	—
83	#117	—	373	1	no	—
84	#117	—	373	1	no	—
85	C₂×Q₈	—	373	2	no	—
86	C₂×Q₈	C₂×D₃₇₃	1	1	yes	characteristic
87	C₂×D₄	—	373	1	no	—
88	C₂²×C₄	—	373	1	no	—
89	C₂²×C₄	C₂×D₃₇₃	1	1	yes	characteristic
90	C₄⋊C₄	—	373	1	no	—
91	C₄⋊C₄	—	373	1	no	—
92	C₄⋊C₄	—	373	1	no	—
93	C₄⋊C₄	—	373	2	no	—
94	C₄⋊C₄	—	373	2	no	—
95	C₄⋊C₄	—	373	2	no	—
96	C₄⋊C₄	C₂×D₃₇₃	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₃₇₃	1	2	yes	—
98	C₂²⋊C₄	—	373	2	no	—
99	C₂²⋊C₄	—	373	1	no	—
100	C₂²⋊C₄	—	373	2	no	—
101	C₂²⋊C₄	—	373	1	no	—
102	C₂²⋊C₄	—	373	2	no	—
103	C₂²⋊C₄	C₂×D₃₇₃	1	2	yes	—
104	C₄²	—	373	2	no	—
105	C₄²	—	373	2	no	—
106	C₄²	—	373	1	no	—
107	C₄²	—	373	2	no	—
108	C₂³	—	373	1	no	—
109	C₂³	C₂²×D₃₇₃	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	—	746	4	no	—
112	D₄	—	746	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	373	1	no	—
115	C₂×C₄	—	373	2	no	—
116	C₂×C₄	—	373	1	no	—
117	C₂×C₄	—	373	1	no	—
118	C₂×C₄	—	373	2	no	—
119	C₂×C₄	—	746	2	no	—
120	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
122	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
123	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	746	1	no	—
128	C₂²	—	746	1	no	—
129	C₂²	—	746	1	no	—
130	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	373	2	no	—
135	C₄	—	746	2	no	—
136	C₄	—	746	1	no	—
137	C₄	—	746	2	no	—
138	C₄	—	746	1	no	—
139	C₄	#20238609	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	1492	1	no	—
142	C₂	#41×D₃₇₃	1	1	yes	characteristic
143	C₂	C₂×#20238609	1	1	yes	characteristic
144	C₂	#26045840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26045791

C₂

yes

maximal, characteristic, complemented

#26045787

C₂

yes

maximal, complemented

#26045785

C₂

yes

maximal, complemented

#26045778

C₂

yes

maximal

#26045768

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₃₇₃

C₂

yes

maximal, characteristic, complemented

#26045757

C₂

yes

maximal, complemented

C₄×C₂²⋊D₃₇₃

C₂

yes

maximal, characteristic

#26045751

C₂

yes

maximal, characteristic, complemented

#26045750

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₄₆

C₂²

yes

—

C₂×C₂²⋊D₃₇₃

C₂²

yes

characteristic

C₂×C₄×D₃₇₃

C₂²

yes

characteristic

C₄⋊Dic₃₇₃

C₂²

yes

characteristic

C₄⋊Dic₃₇₃

C₂²

yes

characteristic

C₄⋊Dic₃₇₃

C₂²

yes

—

#20238589

C₂²

yes

characteristic

#20238589

C₂²

yes

—

#20238589

C₂²

yes

—

C₂²⋊Dic₃₇₃

C₂²

yes

—

C₂²⋊Dic₃₇₃

C₂²

yes

characteristic

#20238587

C₂²

yes

—

#20238587

C₂²

yes

characteristic

#20238587

C₂²

yes

—

C₄×Dic₃₇₃

C₂²

yes

—

C₄×Dic₃₇₃

C₂²

yes

—

C₄×Dic₃₇₃

C₂²

yes

characteristic

C₄×Dic₃₇₃

C₂²

yes

—

C₂×Q₈×C₃₇₃

C₂²

yes

characteristic

C₂²×C₄×C₃₇₃

C₂²

yes

characteristic

C₄⋊C₄×C₃₇₃

C₂²

yes

characteristic

C₄⋊C₄×C₃₇₃

C₂²

yes

—

C₂²⋊C₄×C₃₇₃

C₂²

yes

—

C₂²×D₃₇₃

C₂³

yes

characteristic

Dic₇₄₆

—

C₂²⋊D₃₇₃

—

C₂×Dic₃₇₃

C₂³

yes

characteristic

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

characteristic

C₂×Dic₃₇₃

C₂³

yes

characteristic

C₄×D₃₇₃

—

C₂³×C₃₇₃

C₂³

yes

characteristic

Q₈×C₃₇₃

—

C₂×C₄×C₃₇₃

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂²×C₃₇₃

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

C₂⁴

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

#41

yes

—

Dic₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

#41

yes

—

C₂×C₃₇₃

—

C₂×C₃₇₃

C₂×#41

yes

characteristic

C₂×C₃₇₃

C₂×#41

yes

characteristic

C₂×C₃₇₃

2₋¹⁺⁴

yes

characteristic

D₃₇₃

—

C₃₇₃

#556

yes

characteristic, complemented, Sylow

#556

—

373

maximal, Sylow

C₄⋊Q₈

—

373

—

#126

—

373

—

#126

—

373

—

#124

—

373

—

C₂²⋊Q₈

—

373

—

C₂²⋊Q₈

D₃₇₃

yes

characteristic, complemented

C₄×Q₈

—

373

—

C₄×Q₈

—

373

—

C₄×D₄

—

373

—

#117

—

373

—

#117

—

373

—

C₂×Q₈

—

373

—

C₂×Q₈

C₂×D₃₇₃

yes

characteristic

C₂×D₄

—

373

—

C₂²×C₄

—

373

—

C₂²×C₄

C₂×D₃₇₃

yes

characteristic

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

C₂×D₃₇₃

yes

characteristic

C₄⋊C₄

C₂×D₃₇₃

yes

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

100

C₂²⋊C₄

—

373

—

101

C₂²⋊C₄

—

373

—

102

C₂²⋊C₄

—

373

—

103

C₂²⋊C₄

C₂×D₃₇₃

yes

—

104

C₄²

—

373

—

105

C₄²

—

373

—

106

C₄²

—

373

—

107

C₄²

—

373

—

108

C₂³

—

373

—

109

C₂³

C₂²×D₃₇₃

yes

characteristic

110

Q₈

—

111

Q₈

—

746

—

112

D₄

—

746

—

113

C₂×C₄

—

114

C₂×C₄

—

373

—

115

C₂×C₄

—

373

—

116

C₂×C₄

—

373

—

117

C₂×C₄

—

373

—

118

C₂×C₄

—

373

—

119

C₂×C₄

—

746

—

120

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

121

C₂×C₄

C₂²×D₃₇₃

yes

—

122

C₂×C₄

C₂²×D₃₇₃

yes

—

123

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

746

—

128

C₂²

—

746

—

129

C₂²

—

746

—

130

C₂²

C₂³×D₃₇₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

373

—

135

C₄

—

746

—

136

C₄

—

746

—

137

C₄

—

746

—

138

C₄

—

746

—

139

C₄

#20238609

yes

—

140

C₂

—

141

C₂

—

1492

—

142

C₂

#41×D₃₇₃

yes

characteristic

143

C₂

C₂×#20238609

yes

characteristic

144

C₂

#26045840

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	C₂²⋊Q₈×C₃₇₃	C₂
prime radical	C₂×C₂²⋊D₃₇₃	C₂²
hypercenter	C₂²⋊Q₈	D₃₇₃
hypocenter	C₃₇₃	#556
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇₃

commutator, socle, cosocle

C₂²×C₃₇₃

C₂⁴

Fitting

C₂²⋊Q₈×C₃₇₃

C₂

prime radical

C₂×C₂²⋊D₃₇₃

C₂²

hypercenter

C₂²⋊Q₈

D₃₇₃

C₃₇₃

G ▷ C₂²×C₃₇₃ ▷ C₃₇₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #26045791	C₂	semidirect
2. #26045787	C₂	semidirect
3. #26045785	C₂	semidirect
5. #26045768	C₂	semidirect
6. Q₈×Dic₃₇₃	C₂	semidirect
7. #26045757	C₂	semidirect
9. #26045751	C₂	semidirect
10. #26045750	C₂	semidirect
11. C₂²⋊Q₈×C₃₇₃	C₂	semidirect
72. C₃₇₃	#556	semidirect
79. C₂²⋊Q₈	D₃₇₃	semidirect

H ≅ G/N

type

1. #26045791

C₂

semidirect

2. #26045787

C₂

semidirect

3. #26045785

C₂

semidirect

5. #26045768

C₂

semidirect

6. Q₈×Dic₃₇₃

C₂

semidirect

7. #26045757

C₂

semidirect

9. #26045751

C₂

semidirect

10. #26045750

C₂

semidirect

11. C₂²⋊Q₈×C₃₇₃

C₂

semidirect

72. C₃₇₃

#556

semidirect

79. C₂²⋊Q₈

D₃₇₃

semidirect

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

subgroup	index
#26045791	2
#26045787	2
#26045785	2
#26045778	2
#26045768	2
Q₈×Dic₃₇₃	2
#26045757	2
C₄×C₂²⋊D₃₇₃	2
#26045751	2
#26045750	2
C₂²⋊Q₈×C₃₇₃	2
#556	373

subgroup

index

Q₈×Dic₃₇₃

C₄×C₂²⋊D₃₇₃

C₂²⋊Q₈×C₃₇₃

373

Subgroups — #37560143

Conjugacy classes of subgroups

Complements

Maximal subgroups