Group #36364299 — subgroups

Subgroups — #36364299

Order 22144.

446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 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	#26012039	C₂	1	2	yes	maximal, complemented
2	#26011889	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011837	C₂	1	2	yes	maximal, complemented
4	#26011741	C₂	1	4	yes	maximal, complemented
5	#26011603	C₂	1	4	yes	maximal, complemented
6	C₄×#14583809	C₂	1	1	yes	maximal, characteristic, complemented
7	#493×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14583825	—	2	2	no	—
9	C₂×#14583809	C₂²	1	2	yes	—
10	#19149679	C₂²	1	2	yes	—
11	#19149666	—	2	2	no	—
12	#19149650	—	2	2	no	—
13	#19149650	C₂²	1	4	yes	—
14	D₄×Dic₁₇₃	—	2	2	no	—
15	D₄×Dic₁₇₃	C₂²	1	4	yes	—
16	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
17	#19149611	C₂²	1	4	yes	—
18	#19149600	C₂²	1	8	yes	—
19	#19149589	C₂²	1	2	yes	—
20	C₄×#13063436	C₂²	1	1	yes	characteristic
21	C₂×D₈×C₁₇₃	C₂²	1	4	yes	—
22	#127×C₁₇₃	C₂²	1	2	yes	—
23	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
24	#14583825	—	4	8	no	—
25	C₂×Dic₃₄₆	—	2	2	no	—
26	C₂×C₂²⋊D₁₇₃	—	2	4	no	—
27	C₂²×Dic₁₇₃	—	2	4	no	—
28	C₂×C₄×D₁₇₃	D₄	1	2	yes	—
29	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
30	#14583809	—	2	4	no	—
31	C₂×#13063436	C₂³	1	2	yes	—
32	C₄⋊Dic₁₇₃	—	2	2	no	—
33	C₄⋊Dic₁₇₃	C₂³	1	4	yes	—
34	C₂²⋊Dic₁₇₃	—	2	4	no	—
35	C₂²⋊Dic₁₇₃	—	2	4	no	—
36	C₄×Dic₁₇₃	D₄	1	2	yes	—
37	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₁₇₃	—	2	2	no	—
39	C₂×D₄×C₁₇₃	—	2	2	no	—
40	C₂×D₄×C₁₇₃	C₂³	1	4	yes	—
41	D₈×C₁₇₃	—	2	8	no	—
42	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
43	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
44	C₂²×D₁₇₃	C₂×D₄	1	1	yes	characteristic
45	Dic₃₄₆	—	4	4	no	—
46	C₂²⋊D₁₇₃	—	4	8	no	—
47	C₂×Dic₁₇₃	—	2	4	no	—
48	C₂×Dic₁₇₃	—	4	8	no	—
49	C₂×Dic₁₇₃	C₂×D₄	1	2	yes	—
50	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
51	C₄×D₁₇₃	—	2	4	no	—
52	C₄×D₁₇₃	—	2	2	no	—
53	#13063436	—	2	2	no	—
54	C₂³×C₁₇₃	—	2	4	no	—
55	D₄×C₁₇₃	—	2	8	no	—
56	D₄×C₁₇₃	—	4	4	no	—
57	D₄×C₁₇₃	—	4	4	no	—
58	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
60	C₈×C₁₇₃	#41	1	4	yes	complemented
61	C₂²×C₁₇₃	—	4	4	no	—
62	C₂²×C₁₇₃	—	4	4	no	—
63	C₂²×C₁₇₃	—	4	4	no	—
64	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₁₇₃	—	2	1	no	—
66	C₂×D₁₇₃	—	2	2	no	—
67	C₄×C₁₇₃	AGL(1,ℤ/8)	1	4	yes	—
68	C₄×C₁₇₃	C₂×#41	1	2	yes	—
69	Dic₁₇₃	—	2	1	no	—
70	Dic₁₇₃	—	2	2	no	—
71	Dic₁₇₃	—	4	4	no	—
72	C₂×C₁₇₃	—	8	4	no	—
73	C₂×C₁₇₃	#532	1	1	yes	characteristic
74	C₂×C₁₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
75	D₁₇₃	—	4	1	no	—
76	C₁₇₃	#3184	1	1	yes	characteristic, complemented, Sylow
77	#3184	—	173	1	no	maximal, Sylow
78	#532	—	173	2	no	—
79	#498	—	173	1	no	—
80	#493	D₁₇₃	1	1	yes	characteristic, complemented
81	#486	—	173	2	no	—
82	#469	—	173	4	no	—
83	#442	—	173	4	no	—
84	C₄×M₁₆	—	173	1	no	—
85	C₂×#41	—	346	2	no	—
86	C₂×D₈	C₂×D₁₇₃	1	4	yes	—
87	C₂×M₁₆	—	173	2	no	—
88	C₄⋊Q₈	—	173	2	no	—
89	#127	C₂×D₁₇₃	1	2	yes	—
90	#124	—	346	2	no	—
91	#121	—	346	2	no	—
92	#121	—	173	4	no	—
93	C₄×D₄	—	346	2	no	—
94	C₄×D₄	—	173	4	no	—
95	C₂×C₄²	—	173	1	no	—
96	#106	—	173	4	no	—
97	#102	—	173	8	no	—
98	#97	—	173	2	no	—
99	C₄×C₈	—	173	1	no	—
100	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
101	#41	—	692	8	no	—
102	C₂×Q₈	—	346	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	346	4	no	—
106	C₂×D₄	C₂²×D₁₇₃	1	4	yes	—
107	C₂²×C₄	—	346	4	no	—
108	C₂²×C₄	—	173	2	no	—
109	C₂²×C₄	—	173	1	no	—
110	D₈	—	2	8	no	—
111	M₁₆	—	346	4	no	—
112	C₂×C₈	—	173	2	no	—
113	C₂×C₈	C₂²×D₁₇₃	1	2	yes	—
114	C₄⋊C₄	—	346	2	no	—
115	C₄⋊C₄	—	173	4	no	—
116	C₂²⋊C₄	—	346	4	no	—
117	C₂²⋊C₄	—	346	4	no	—
118	C₄²	—	173	2	no	—
119	C₄²	—	173	1	no	—
120	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
121	C₂³	—	2	4	no	—
122	C₂³	—	173	1	no	—
123	Q₈	—	692	4	no	—
124	D₄	—	2	8	no	—
125	D₄	—	4	4	no	—
126	D₄	—	4	4	no	—
127	D₄	—	692	8	no	—
128	C₂×C₄	—	346	4	no	—
129	C₂×C₄	—	346	2	no	—
130	C₂×C₄	—	346	4	no	—
131	C₂×C₄	—	173	1	no	—
132	C₂×C₄	—	173	2	no	—
133	C₂×C₄	—	692	8	no	—
134	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
135	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
136	C₈	—	346	2	no	—
137	C₈	#14583825	1	4	yes	complemented
138	C₂²	—	4	4	no	—
139	C₂²	—	4	4	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	346	2	no	—
142	C₂²	—	346	1	no	—
143	C₂²	C₂×D₄×D₁₇₃	1	1	yes	characteristic
144	C₄	—	346	2	no	—
145	C₄	—	346	1	no	—
146	C₄	—	692	4	no	—
147	C₄	#19149703	1	4	yes	—
148	C₄	C₂×#14583825	1	2	yes	—
149	C₂	—	8	4	no	—
150	C₂	—	692	1	no	—
151	C₂	#26012039	1	1	yes	characteristic
152	C₂	C₂×#19149703	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26012039

C₂

yes

maximal, complemented

#26011889

C₂

yes

maximal, characteristic, complemented

#26011837

C₂

yes

maximal, complemented

#26011741

C₂

yes

maximal, complemented

#26011603

C₂

yes

maximal, complemented

C₄×#14583809

C₂

yes

maximal, characteristic, complemented

#493×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14583825

—

C₂×#14583809

C₂²

yes

—

#19149679

C₂²

yes

—

#19149666

—

#19149650

—

#19149650

C₂²

yes

—

D₄×Dic₁₇₃

—

D₄×Dic₁₇₃

C₂²

yes

—

C₄²×D₁₇₃

C₂²

yes

characteristic

#19149611

C₂²

yes

—

#19149600

C₂²

yes

—

#19149589

C₂²

yes

—

C₄×#13063436

C₂²

yes

characteristic

C₂×D₈×C₁₇₃

C₂²

yes

—

#127×C₁₇₃

C₂²

yes

—

C₄×C₈×C₁₇₃

C₂²

yes

characteristic

#14583825

—

C₂×Dic₃₄₆

—

C₂×C₂²⋊D₁₇₃

—

C₂²×Dic₁₇₃

—

C₂×C₄×D₁₇₃

D₄

yes

—

C₂×C₄×D₁₇₃

C₂³

yes

characteristic

#14583809

—

C₂×#13063436

C₂³

yes

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

C₂³

yes

—

C₂²⋊Dic₁₇₃

—

C₂²⋊Dic₁₇₃

—

C₄×Dic₁₇₃

D₄

yes

—

C₄×Dic₁₇₃

C₂³

yes

characteristic

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

C₂³

yes

—

D₈×C₁₇₃

—

C₂×C₈×C₁₇₃

C₂³

yes

—

C₄²×C₁₇₃

C₂³

yes

characteristic

C₂²×D₁₇₃

C₂×D₄

yes

characteristic

Dic₃₄₆

—

C₂²⋊D₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

C₂×D₄

yes

—

C₂×Dic₁₇₃

C₂×D₄

yes

characteristic

C₄×D₁₇₃

—

C₄×D₁₇₃

—

#13063436

—

C₂³×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂⁴

yes

characteristic

C₈×C₁₇₃

#41

yes

complemented

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂²×D₄

yes

characteristic

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₇₃

C₂×#41

yes

—

Dic₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#532

yes

characteristic

C₂×C₁₇₃

C₂×AGL(1,ℤ/8)

yes

—

D₁₇₃

—

C₁₇₃

#3184

yes

characteristic, complemented, Sylow

#3184

—

173

maximal, Sylow

#532

—

173

—

#498

—

173

—

#493

D₁₇₃

yes

characteristic, complemented

#486

—

173

—

#469

—

173

—

#442

—

173

—

C₄×M₁₆

—

173

—

C₂×#41

—

346

—

C₂×D₈

C₂×D₁₇₃

yes

—

C₂×M₁₆

—

173

—

C₄⋊Q₈

—

173

—

#127

C₂×D₁₇₃

yes

—

#124

—

346

—

#121

—

346

—

#121

—

173

—

C₄×D₄

—

346

—

C₄×D₄

—

173

—

C₂×C₄²

—

173

—

#106

—

173

—

#102

—

173

—

#97

—

173

—

C₄×C₈

—

173

—

100

C₄×C₈

C₂×D₁₇₃

yes

characteristic

101

#41

—

692

—

102

C₂×Q₈

—

346

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

346

—

106

C₂×D₄

C₂²×D₁₇₃

yes

—

107

C₂²×C₄

—

346

—

108

C₂²×C₄

—

173

—

109

C₂²×C₄

—

173

—

110

D₈

—

111

M₁₆

—

346

—

112

C₂×C₈

—

173

—

113

C₂×C₈

C₂²×D₁₇₃

yes

—

114

C₄⋊C₄

—

346

—

115

C₄⋊C₄

—

173

—

116

C₂²⋊C₄

—

346

—

117

C₂²⋊C₄

—

346

—

118

C₄²

—

173

—

119

C₄²

—

173

—

120

C₄²

C₂²×D₁₇₃

yes

characteristic

121

C₂³

—

122

C₂³

—

173

—

123

Q₈

—

692

—

124

D₄

—

125

D₄

—

126

D₄

—

127

D₄

—

692

—

128

C₂×C₄

—

346

—

129

C₂×C₄

—

346

—

130

C₂×C₄

—

346

—

131

C₂×C₄

—

173

—

132

C₂×C₄

—

173

—

133

C₂×C₄

—

692

—

134

C₂×C₄

D₄×D₁₇₃

yes

—

135

C₂×C₄

C₂³×D₁₇₃

yes

characteristic

136

C₈

—

346

—

137

C₈

#14583825

yes

complemented

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

346

—

142

C₂²

—

346

—

143

C₂²

C₂×D₄×D₁₇₃

yes

characteristic

144

C₄

—

346

—

145

C₄

—

346

—

146

C₄

—

692

—

147

C₄

#19149703

yes

—

148

C₄

C₂×#14583825

yes

—

149

C₂

—

150

C₂

—

692

—

151

C₂

#26012039

yes

characteristic

152

C₂

C₂×#19149703

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇₃
commutator, cosocle	C₂×C₄×C₁₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	#493×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂²×D₄
hypercenter	#493	D₁₇₃
hypocenter	C₁₇₃	#3184
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₁₇₃

commutator, cosocle

C₂×C₄×C₁₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇₃

Fitting

#493×C₁₇₃

C₂

socle

C₂²×C₁₇₃

C₂²×D₄

D₁₇₃

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #26012039	C₂	semidirect
2. #26011889	C₂	semidirect
3. #26011837	C₂	semidirect
4. #26011741	C₂	semidirect
5. #26011603	C₂	semidirect
6. C₄×#14583809	C₂	semidirect
7. #493×C₁₇₃	C₂	semidirect
60. C₈×C₁₇₃	#41	semidirect
76. C₁₇₃	#3184	semidirect
80. #493	D₁₇₃	semidirect
137. C₈	#14583825	semidirect

H ≅ G/N

type

1. #26012039

C₂

semidirect

2. #26011889

C₂

semidirect

3. #26011837

C₂

semidirect

4. #26011741

C₂

semidirect

5. #26011603

C₂

semidirect

6. C₄×#14583809

C₂

semidirect

7. #493×C₁₇₃

C₂

semidirect

60. C₈×C₁₇₃

#41

semidirect

76. C₁₇₃

#3184

semidirect

80. #493

D₁₇₃

semidirect

137. C₈

#14583825

semidirect

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

subgroup	index
#26012039	2
#26011889	2
#26011837	2
#26011741	2
#26011603	2
C₄×#14583809	2
#493×C₁₇₃	2
#3184	173

subgroup

index

#493×C₁₇₃

173

Subgroups — #36364299

Conjugacy classes of subgroups

Complements

Maximal subgroups