Group #36354395 — subgroups

Subgroups — #36354395

Order 22144.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#26012041	C₂	1	1	yes	maximal, characteristic
2	#26011515	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19149607	C₂	1	4	yes	maximal, complemented
4	#376×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14583826	C₂²	1	2	yes	—
6	C₂×#14583810	C₂²	1	2	yes	—
7	#19149679	C₄	1	1	yes	characteristic, complemented
8	#19149676	C₄	1	1	yes	characteristic, complemented
9	#19149667	C₄	1	2	yes	complemented
10	#19149653	—	2	2	no	—
11	C₄×D₆₉₂	—	2	2	no	—
12	C₄⋊#13063436	—	2	2	no	—
13	#19149607	—	2	8	no	—
14	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
16	#95×C₁₇₃	—	2	2	no	—
17	#14583826	—	2	4	no	—
18	#14583826	—	2	4	no	—
19	C₂×Dic₃₄₆	C₂×C₄	1	2	yes	—
20	C₂×D₆₉₂	—	2	2	no	—
21	C₂×D₆₉₂	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊D₁₇₃	—	2	2	no	—
23	C₂×C₄×D₁₇₃	—	2	2	no	—
24	#14583810	—	2	4	no	—
25	C₂×#13063436	—	2	2	no	—
26	C₄⋊Dic₁₇₃	—	2	2	no	—
27	#14583803	—	2	4	no	—
28	C₂²×C₄×C₁₇₃	—	2	2	no	—
29	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
30	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
31	C₄²×C₁₇₃	—	2	2	no	—
32	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₇₃	D₄	1	2	yes	—
35	C₄²×C₁₇₃	D₄	1	4	yes	complemented
36	C₂²×D₁₇₃	—	2	2	no	—
37	Dic₃₄₆	—	2	4	no	—
38	D₆₉₂	—	2	4	no	—
39	D₆₉₂	—	4	4	no	—
40	C₂²⋊D₁₇₃	—	4	4	no	—
41	C₂×Dic₁₇₃	—	2	2	no	—
42	C₄×D₁₇₃	—	4	2	no	—
43	C₄×D₁₇₃	—	4	2	no	—
44	#13063436	—	4	2	no	—
45	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇₃	—	2	4	no	—
47	C₂×C₄×C₁₇₃	—	4	2	no	—
48	C₂×C₄×C₁₇₃	—	4	4	no	—
49	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇₃	#41	1	2	yes	—
55	C₂²×C₁₇₃	—	2	2	no	—
56	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₇₃	#121	1	2	yes	—
58	C₂×D₁₇₃	—	4	2	no	—
59	C₂×D₁₇₃	—	4	4	no	—
60	C₄×C₁₇₃	—	2	2	no	—
61	C₄×C₁₇₃	—	2	2	no	—
62	C₄×C₁₇₃	—	4	4	no	—
63	C₄×C₁₇₃	C₄×D₄	1	2	yes	—
64	C₄×C₁₇₃	#124	1	1	yes	characteristic
65	C₄×C₁₇₃	#127	1	1	yes	characteristic
66	Dic₁₇₃	—	4	2	no	—
67	C₂×C₁₇₃	—	2	2	no	—
68	C₂×C₁₇₃	#390	1	1	yes	characteristic
69	C₂×C₁₇₃	#421	1	2	yes	—
70	D₁₇₃	—	8	2	no	—
71	C₁₇₃	#1963	1	1	yes	characteristic, complemented, Sylow
72	#1963	—	173	1	no	maximal, Sylow
73	#532	—	173	1	no	—
74	#423	—	173	1	no	—
75	C₂×#104	—	173	4	no	—
76	#376	D₁₇₃	1	1	yes	characteristic, complemented
77	C₂×#41	—	173	2	no	—
78	C₂×M₁₆	—	173	2	no	—
79	C₄⋊Q₈	—	173	1	no	—
80	#127	—	173	1	no	—
81	#124	—	173	2	no	—
82	#121	—	346	2	no	—
83	C₄×D₄	—	346	2	no	—
84	C₂×C₄²	C₂×D₁₇₃	1	2	yes	—
85	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
86	C₄⋊C₈	—	346	2	no	—
87	#104	—	346	8	no	—
88	#95	—	2	2	no	—
89	#41	—	346	4	no	—
90	#41	—	346	4	no	—
91	C₂×Q₈	—	173	2	no	—
92	C₂×D₄	—	346	2	no	—
93	C₂×D₄	—	346	2	no	—
94	C₂×D₄	—	173	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	346	2	no	—
97	C₂²×C₄	C₂²⋊D₁₇₃	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
99	M₁₆	—	346	4	no	—
100	C₂×C₈	—	346	2	no	—
101	C₄⋊C₄	—	346	2	no	—
102	C₂²⋊C₄	—	346	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
105	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
106	C₄²	D₆₉₂	1	2	yes	—
107	C₄²	D₆₉₂	1	4	yes	complemented
108	C₂³	—	346	2	no	—
109	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
110	Q₈	—	346	4	no	—
111	D₄	—	346	4	no	—
112	D₄	—	692	4	no	—
113	D₄	—	692	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	346	2	no	—
118	C₂×C₄	—	692	2	no	—
119	C₂×C₄	—	692	2	no	—
120	C₂×C₄	#14583803	1	2	yes	—
121	C₂×C₄	#14583803	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
123	C₂×C₄	C₂×D₆₉₂	1	2	yes	—
124	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
125	C₂×C₄	#14583826	1	2	yes	—
126	C₈	—	692	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	692	2	no	—
129	C₂²	—	692	4	no	—
130	C₂²	C₂×#14583803	1	1	yes	characteristic
131	C₂²	#19149653	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	692	2	no	—
136	C₄	C₄×D₆₉₂	1	2	yes	—
137	C₄	#19149667	1	1	yes	characteristic
138	C₄	#19149676	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	1384	2	no	—
141	C₂	#26011393	1	1	yes	characteristic
142	C₂	#26011509	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26012041

C₂

yes

maximal, characteristic

#26011515

C₂

yes

maximal, characteristic, complemented

C₂×#19149607

C₂

yes

maximal, complemented

#376×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14583826

C₂²

yes

—

C₂×#14583810

C₂²

yes

—

#19149679

C₄

yes

characteristic, complemented

#19149676

C₄

yes

characteristic, complemented

#19149667

C₄

yes

complemented

#19149653

—

C₄×D₆₉₂

—

C₄⋊#13063436

—

#19149607

—

C₂×C₄²×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

#95×C₁₇₃

—

#14583826

—

#14583826

—

C₂×Dic₃₄₆

C₂×C₄

yes

—

C₂×D₆₉₂

—

C₂×D₆₉₂

C₂×C₄

yes

—

C₂×C₂²⋊D₁₇₃

—

C₂×C₄×D₁₇₃

—

#14583810

—

C₂×#13063436

—

C₄⋊Dic₁₇₃

—

#14583803

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₄²×C₁₇₃

—

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

C₄²×C₁₇₃

D₄

yes

—

C₄²×C₁₇₃

D₄

yes

complemented

C₂²×D₁₇₃

—

Dic₃₄₆

—

D₆₉₂

—

D₆₉₂

—

C₂²⋊D₁₇₃

—

C₂×Dic₁₇₃

—

C₄×D₁₇₃

—

C₄×D₁₇₃

—

#13063436

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₃

#121

yes

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₄×D₄

yes

—

C₄×C₁₇₃

yes

C₄×C₁₇₃

yes

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#390

yes

characteristic

C₂×C₁₇₃

#421

yes

—

D₁₇₃

—

C₁₇₃

#1963

yes

characteristic, complemented, Sylow

#1963

—

173

maximal, Sylow

#532

—

173

—

#423

—

173

—

C₂×#104

—

173

—

#376

D₁₇₃

yes

characteristic, complemented

C₂×#41

—

173

—

C₂×M₁₆

—

173

—

C₄⋊Q₈

—

173

—

#127

—

173

—

#124

—

173

—

#121

—

346

—

C₄×D₄

—

346

—

C₂×C₄²

C₂×D₁₇₃

yes

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

C₄⋊C₈

—

346

—

#104

—

346

—

#95

—

#41

—

346

—

#41

—

346

—

C₂×Q₈

—

173

—

C₂×D₄

—

346

—

C₂×D₄

—

346

—

C₂×D₄

—

173

—

C₂²×C₄

—

C₂²×C₄

—

346

—

C₂²×C₄

C₂²⋊D₁₇₃

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

M₁₆

—

346

—

100

C₂×C₈

—

346

—

101

C₄⋊C₄

—

346

—

102

C₂²⋊C₄

—

346

—

103

C₄²

—

104

C₄²

C₄×D₁₇₃

yes

characteristic

105

C₄²

C₄×D₁₇₃

yes

characteristic

106

C₄²

D₆₉₂

yes

—

107

C₄²

D₆₉₂

yes

complemented

108

C₂³

—

346

—

109

C₂³

C₂×C₂²⋊D₁₇₃

yes

characteristic

110

Q₈

—

346

—

111

D₄

—

346

—

112

D₄

—

692

—

113

D₄

—

692

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

346

—

118

C₂×C₄

—

692

—

119

C₂×C₄

—

692

—

120

C₂×C₄

#14583803

yes

—

121

C₂×C₄

#14583803

yes

—

122

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

123

C₂×C₄

C₂×D₆₉₂

yes

—

124

C₂×C₄

C₂×D₆₉₂

yes

characteristic

125

C₂×C₄

#14583826

yes

—

126

C₈

—

692

—

127

C₂²

—

128

C₂²

—

692

—

129

C₂²

—

692

—

130

C₂²

C₂×#14583803

yes

characteristic

131

C₂²

#19149653

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

692

—

136

C₄

C₄×D₆₉₂

yes

—

137

C₄

#19149667

yes

characteristic

138

C₄

#19149676

yes

characteristic

139

C₂

—

140

C₂

—

1384

—

141

C₂

#26011393

yes

characteristic

142

C₂

#26011509

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₉₂
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#376×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂×C₂²⋊C₄
prime radical	#26012041	C₂
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#376	D₁₇₃
hypocenter	C₁₇₃	#1963
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₆₉₂

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

#376×C₁₇₃

C₂

socle

C₂²×C₁₇₃

C₂×C₂²⋊C₄

prime radical

#26012041

C₂

cosocle

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #26011515	C₂	semidirect
3. C₂×#19149607	C₂	semidirect
4. #376×C₁₇₃	C₂	semidirect
7. #19149679	C₄	semidirect
8. #19149676	C₄	semidirect
9. #19149667	C₄	semidirect
35. C₄²×C₁₇₃	D₄	semidirect
71. C₁₇₃	#1963	semidirect
76. #376	D₁₇₃	semidirect
107. C₄²	D₆₉₂	semidirect

H ≅ G/N

type

2. #26011515

C₂

semidirect

3. C₂×#19149607

C₂

semidirect

4. #376×C₁₇₃

C₂

semidirect

7. #19149679

C₄

semidirect

8. #19149676

C₄

semidirect

9. #19149667

C₄

semidirect

35. C₄²×C₁₇₃

D₄

semidirect

71. C₁₇₃

#1963

semidirect

76. #376

D₁₇₃

semidirect

107. C₄²

D₆₉₂

semidirect

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

subgroup	index
#26012041	2
#26011515	2
C₂×#19149607	2
#376×C₁₇₃	2
#1963	173

subgroup

index

#376×C₁₇₃

173

Subgroups — #36354395

Conjugacy classes of subgroups

Complements

Maximal subgroups