Group #36355122 — subgroups

Subgroups — #36355122

Order 22144.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26011389	C₂	1	1	yes	maximal, characteristic
2	#26011361	C₂	1	1	yes	maximal, characteristic
3	#26011254	C₂	1	2	yes	maximal
4	#26011254	C₂	1	2	yes	maximal
5	#336×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₇₃	—	2	1	no	—
7	C₂×C₄⋊Dic₁₇₃	—	2	1	no	—
8	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₇₃	C₂²	1	1	yes	characteristic
11	C₂×#14583805	—	2	1	no	—
12	#19149586	—	2	1	no	—
13	C₂²×C₈×C₁₇₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₇₃	—	2	1	no	—
16	C₂²×Dic₁₇₃	—	2	1	no	—
17	C₂²×Dic₁₇₃	—	2	2	no	—
18	C₄⋊Dic₁₇₃	—	2	1	no	—
19	C₄⋊Dic₁₇₃	—	2	2	no	—
20	C₄⋊Dic₁₇₃	—	2	4	no	—
21	C₄⋊Dic₁₇₃	—	4	2	no	—
22	C₄⋊Dic₁₇₃	—	4	2	no	—
23	C₄⋊Dic₁₇₃	Q₈	1	2	yes	—
24	#14583805	—	4	2	no	—
25	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₇₃	—	2	2	no	—
29	C₂×C₈×C₁₇₃	—	2	2	no	—
30	C₂×C₈×C₁₇₃	—	2	2	no	—
31	C₄²×C₁₇₃	—	2	2	no	—
32	C₂×Dic₁₇₃	—	2	2	no	—
33	C₂×Dic₁₇₃	—	4	1	no	—
34	C₂×Dic₁₇₃	—	4	1	no	—
35	C₂×Dic₁₇₃	—	4	1	no	—
36	C₂×Dic₁₇₃	—	4	2	no	—
37	C₂×Dic₁₇₃	—	4	1	no	—
38	C₂×Dic₁₇₃	—	4	2	no	—
39	C₂×Dic₁₇₃	—	4	2	no	—
40	C₂×Dic₁₇₃	—	4	1	no	—
41	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₃	—	2	1	no	—
43	C₂×C₄×C₁₇₃	—	2	1	no	—
44	C₂×C₄×C₁₇₃	—	2	1	no	—
45	C₂×C₄×C₁₇₃	—	2	1	no	—
46	C₂×C₄×C₁₇₃	D₈	1	2	yes	—
47	C₂×C₄×C₁₇₃	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇₃	#41	1	2	yes	—
50	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₁₇₃	#41	1	2	yes	—
52	C₈×C₁₇₃	—	4	2	no	—
53	C₂²×C₁₇₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₇₃	#123	1	1	yes	characteristic
55	C₂²×C₁₇₃	#124	1	1	yes	characteristic
56	C₂²×C₁₇₃	C₂×D₈	1	1	yes	characteristic
57	C₂²×C₁₇₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₁₇₃	#137	1	1	yes	characteristic
59	C₂²×C₁₇₃	#137	1	1	yes	characteristic
60	C₄×C₁₇₃	—	2	2	no	—
61	C₄×C₁₇₃	—	2	2	no	—
62	C₄×C₁₇₃	#125	1	2	yes	—
63	C₄×C₁₇₃	#126	1	2	yes	—
64	Dic₁₇₃	—	4	2	no	—
65	Dic₁₇₃	—	8	1	no	—
66	Dic₁₇₃	—	8	2	no	—
67	C₂×C₁₇₃	#400	1	1	yes	characteristic
68	C₂×C₁₇₃	#474	1	1	yes	characteristic
69	C₂×C₁₇₃	#475	1	1	yes	characteristic
70	C₂×C₁₇₃	#480	1	1	yes	characteristic
71	C₂×C₁₇₃	#483	1	1	yes	characteristic
72	C₂×C₁₇₃	#486	1	1	yes	characteristic
73	C₂×C₁₇₃	#491	1	1	yes	characteristic
74	C₁₇₃	#2099	1	1	yes	characteristic, complemented, Sylow
75	#2099	—	173	1	no	maximal, Sylow
76	#389	—	173	1	no	—
77	#384	—	173	1	no	—
78	#340	—	173	2	no	—
79	#340	—	173	2	no	—
80	#336	D₁₇₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₁₇₃	1	2	yes	—
82	C₂×C₄⋊C₄	—	346	1	no	—
83	C₂×C₄⋊C₄	—	346	1	no	—
84	C₂×C₄⋊C₄	—	346	1	no	—
85	C₂×C₄⋊C₄	—	173	2	no	—
86	C₂×C₄⋊C₄	—	173	1	no	—
87	C₂×C₄⋊C₄	—	173	1	no	—
88	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
89	#95	—	346	1	no	—
90	C₂²×C₄	—	346	1	no	—
91	C₂²×C₄	—	346	1	no	—
92	C₂²×C₄	—	346	2	no	—
93	C₂²×C₄	D₆₉₂	1	1	yes	characteristic
94	C₂²×C₄	D₆₉₂	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	346	1	no	—
100	C₄⋊C₄	—	346	2	no	—
101	C₄⋊C₄	—	346	4	no	—
102	C₄⋊C₄	—	173	2	no	—
103	C₄⋊C₄	—	692	2	no	—
104	C₄⋊C₄	—	692	2	no	—
105	C₄⋊C₄	—	692	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₆₉₂	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	346	2	no	—
113	C₂×C₄	—	692	1	no	—
114	C₂×C₄	—	692	2	no	—
115	C₂×C₄	—	692	1	no	—
116	C₂×C₄	—	692	2	no	—
117	C₂×C₄	—	692	2	no	—
118	C₂×C₄	—	692	1	no	—
119	C₂×C₄	—	692	1	no	—
120	C₂×C₄	—	692	1	no	—
121	C₂×C₄	D₁₃₈₄	1	2	yes	—
122	C₂×C₄	#14583815	1	2	yes	—
123	C₂×C₄	Q₈×D₁₇₃	1	1	yes	characteristic
124	C₂×C₄	#14583825	1	2	yes	—
125	C₂×C₄	#14583825	1	1	yes	characteristic
126	C₂×C₄	#14583826	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₂²	#19149659	1	1	yes	characteristic
129	C₂²	#19149663	1	1	yes	characteristic
130	C₂²	#19149667	1	1	yes	characteristic
131	C₂²	C₂×D₁₃₈₄	1	1	yes	characteristic
132	C₂²	C₂×#14583815	1	1	yes	characteristic
133	C₂²	#19149712	1	1	yes	characteristic
134	C₂²	#19149712	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	1384	1	no	—
138	C₄	—	1384	2	no	—
139	C₄	—	692	2	no	—
140	C₄	#19149669	1	2	yes	—
141	C₄	#19149672	1	2	yes	—
142	C₂	#26011431	1	1	yes	characteristic
143	C₂	#26011772	1	1	yes	characteristic
144	C₂	#26011777	1	1	yes	characteristic
145	C₂	#26011805	1	1	yes	characteristic
146	C₂	#26011822	1	1	yes	characteristic
147	C₂	#26011838	1	1	yes	characteristic
148	C₂	#26011860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26011389

C₂

yes

maximal, characteristic

#26011361

C₂

yes

maximal, characteristic

#26011254

C₂

yes

maximal

#26011254

C₂

yes

maximal

#336×C₁₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₇₃

—

C₂×C₄⋊Dic₁₇₃

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇₃

C₂²

yes

characteristic

C₂×#14583805

—

#19149586

—

C₂²×C₈×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

Q₈

yes

—

#14583805

—

C₂²×C₄×C₁₇₃

D₄

yes

characteristic

C₂²×C₄×C₁₇₃

D₄

yes

characteristic

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₄²×C₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

D₈

yes

—

C₂×C₄×C₁₇₃

SD₁₆

yes

—

C₂×C₄×C₁₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

#41

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

C₂²⋊Q₈

yes

C₂²×C₁₇₃

yes

C₂²×C₁₇₃

yes

C₂²×C₁₇₃

C₂×D₈

yes

characteristic

C₂²×C₁₇₃

C₂×SD₁₆

yes

C₂²×C₁₇₃

yes

C₂²×C₁₇₃

yes

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#125

yes

—

C₄×C₁₇₃

#126

yes

—

Dic₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₁₇₃

yes

characteristic, complemented, Sylow

#2099

—

173

maximal, Sylow

#389

—

173

—

#384

—

173

—

#340

—

173

—

#340

—

173

—

#336

D₁₇₃

yes

characteristic

C₂²×C₈

C₂×D₁₇₃

yes

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

#95

—

346

—

C₂²×C₄

—

346

—

C₂²×C₄

—

346

—

C₂²×C₄

—

346

—

C₂²×C₄

D₆₉₂

yes

characteristic

C₂²×C₄

D₆₉₂

yes

characteristic

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

346

—

100

C₄⋊C₄

—

346

—

101

C₄⋊C₄

—

346

—

102

C₄⋊C₄

—

173

—

103

C₄⋊C₄

—

692

—

104

C₄⋊C₄

—

692

—

105

C₄⋊C₄

—

692

—

106

C₄²

—

107

C₂³

C₂×D₆₉₂

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

346

—

113

C₂×C₄

—

692

—

114

C₂×C₄

—

692

—

115

C₂×C₄

—

692

—

116

C₂×C₄

—

692

—

117

C₂×C₄

—

692

—

118

C₂×C₄

—

692

—

119

C₂×C₄

—

692

—

120

C₂×C₄

—

692

—

121

C₂×C₄

D₁₃₈₄

yes

—

122

C₂×C₄

#14583815

yes

—

123

C₂×C₄

Q₈×D₁₇₃

yes

characteristic

124

C₂×C₄

#14583825

yes

—

125

C₂×C₄

#14583825

yes

characteristic

126

C₂×C₄

#14583826

yes

—

127

C₈

—

128

C₂²

#19149659

yes

characteristic

129

C₂²

#19149663

yes

characteristic

130

C₂²

#19149667

yes

characteristic

131

C₂²

C₂×D₁₃₈₄

yes

characteristic

132

C₂²

C₂×#14583815

yes

characteristic

133

C₂²

#19149712

yes

characteristic

134

C₂²

#19149712

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

1384

—

138

C₄

—

1384

—

139

C₄

—

692

—

140

C₄

#19149669

yes

—

141

C₄

#19149672

yes

—

142

C₂

#26011431

yes

characteristic

143

C₂

#26011772

yes

characteristic

144

C₂

#26011777

yes

characteristic

145

C₂

#26011805

yes

characteristic

146

C₂

#26011822

yes

characteristic

147

C₂

#26011838

yes

characteristic

148

C₂

#26011860

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₉₂
commutator, cosocle	C₂²×C₄×C₁₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#336×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂×D₄
hypercenter	#336	D₁₇₃
hypocenter	C₁₇₃	#2099
LCS	G ▷ C₂²×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₆₉₂

commutator, cosocle

C₂²×C₄×C₁₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

#336×C₁₇₃

C₂

socle, prime radical

C₂³×C₁₇₃

C₂×D₄

D₁₇₃

C₁₇₃

G ▷ C₂²×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
74. C₁₇₃	#2099	semidirect

H ≅ G/N

type

74. C₁₇₃

#2099

semidirect

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

subgroup	index
#26011389	2
#26011361	2
#26011254	2
#336×C₁₇₃	2
#2099	173

subgroup

index

#336×C₁₇₃

173

Subgroups — #36355122

Conjugacy classes of subgroups

Complements

Maximal subgroups