Group #36369521 — subgroups

Subgroups — #36369521

Order 22144.

252 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 66 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	#26012571	C₂	1	1	yes	maximal, characteristic
2	#26012506	C₂	1	2	yes	maximal, complemented
3	#26012466	C₂	1	1	yes	maximal, characteristic
4	#26012366	C₂	1	2	yes	maximal, complemented
5	#26011474	C₂	1	1	yes	maximal, characteristic
6	#19149767	—	2	2	no	—
7	#19149767	—	2	2	no	—
8	C₂×#14583836	C₂²	1	1	yes	characteristic
9	D₁₇₃⋊M₁₆	—	2	2	no	—
10	D₁₇₃⋊M₁₆	C₂²	1	2	yes	—
11	C₂×D₁₇₃⋊C₈	C₂²	1	1	yes	characteristic
12	#19149750	—	2	2	no	—
13	#19149749	—	2	2	no	—
14	#19149736	—	2	2	no	—
15	C₂×#14583827	C₂²	1	1	yes	characteristic
16	M₁₆×D₁₇₃	C₂²	1	2	yes	—
17	#19149598	C₄	1	1	yes	characteristic
18	#19149595	C₄	1	2	yes	—
19	#101×C₁₇₃	C₄	1	1	yes	characteristic
20	#14583836	—	2	2	no	—
21	#14583836	—	2	2	no	—
22	#14583836	—	2	2	no	—
23	#14583836	D₄	1	2	yes	—
24	#14583836	D₄	1	2	yes	—
25	#14583835	—	2	2	no	—
26	D₁₇₃⋊C₈	—	2	4	no	—
27	D₁₇₃⋊C₈	—	2	2	no	—
28	D₁₇₃⋊C₈	—	2	2	no	—
29	C₂×#13063443	—	2	1	no	—
30	C₂×#13063443	—	2	1	no	—
31	#14583827	—	2	2	no	—
32	#14583827	—	2	2	no	—
33	C₂×D₆₉₂	C₂×C₄	1	1	yes	characteristic
34	C₂×D₆₉₂	Q₈	1	2	yes	—
35	C₂×C₄×D₁₇₃	D₄	1	2	yes	—
36	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
37	#14583810	C₂×C₄	1	2	yes	—
38	#14583809	—	2	2	no	—
39	C₈×D₁₇₃	—	2	2	no	—
40	C₂×Q₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
41	M₁₆×C₁₇₃	C₂×C₄	1	2	yes	—
42	#13063443	—	2	2	no	—
43	#13063443	—	2	2	no	—
44	#13063443	—	4	1	no	—
45	#13063443	—	4	1	no	—
46	C₂²×D₁₇₃	C₄⋊C₄	1	2	yes	—
47	C₂²×D₁₇₃	C₂×Q₈	1	1	yes	characteristic
48	D₆₉₂	—	2	4	no	—
49	D₆₉₂	—	2	2	no	—
50	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
51	C₄×D₁₇₃	—	2	2	no	—
52	C₄×D₁₇₃	—	2	2	no	—
53	C₄×D₁₇₃	C₂×D₄	1	2	yes	—
54	C₄×D₁₇₃	#41	1	2	yes	—
55	#13063436	—	2	2	no	—
56	Q₈×C₁₇₃	—	2	2	no	—
57	C₂×C₄×C₁₇₃	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₈×C₁₇₃	—	2	2	no	—
60	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×D₁₇₃	—	2	2	no	—
62	C₂×D₁₇₃	—	4	1	no	—
63	C₂×D₁₇₃	—	4	2	no	—
64	C₂×D₁₇₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₇₃	—	2	2	no	—
66	C₄×C₁₇₃	C₄×D₄	1	2	yes	—
67	Dic₁₇₃	#120	1	1	yes	characteristic
68	Dic₁₇₃	#123	1	1	yes	characteristic
69	C₂×C₁₇₃	—	2	1	no	—
70	C₂×C₁₇₃	#385	1	1	yes	characteristic
71	D₁₇₃	—	2	2	no	—
72	D₁₇₃	—	4	2	no	—
73	C₁₇₃	#1858	1	1	yes	characteristic, complemented, Sylow
74	#1858	—	173	1	no	maximal, Sylow
75	#568	—	173	1	no	—
76	#430	—	173	2	no	—
77	#413	—	173	1	no	—
78	#408	—	173	1	no	—
79	#341	—	173	2	no	—
80	C₂×#41	—	173	1	no	—
81	#131	—	346	2	no	—
82	#131	—	346	2	no	—
83	C₂×M₁₆	—	346	2	no	—
84	C₂×M₁₆	—	173	1	no	—
85	C₂×M₁₆	—	173	2	no	—
86	C₂×M₁₆	—	173	2	no	—
87	C₂²×C₈	—	173	1	no	—
88	#108	—	346	2	no	—
89	#108	—	346	2	no	—
90	#101	—	173	1	no	—
91	#101	#10621924	1	1	yes	characteristic
92	#100	—	173	2	no	—
93	C₂²⋊C₈	—	346	2	no	—
94	#41	—	346	2	no	—
95	#41	—	346	2	no	—
96	C₂×Q₈	C₂×#10621924	1	1	yes	characteristic
97	C₂×D₄	—	173	2	no	—
98	C₂×D₄	—	173	1	no	—
99	C₂²×C₄	—	173	2	no	—
100	C₂²×C₄	—	173	1	no	—
101	M₁₆	—	346	2	no	—
102	M₁₆	—	346	2	no	—
103	M₁₆	—	346	2	no	—
104	M₁₆	—	346	2	no	—
105	M₁₆	—	346	2	no	—
106	M₁₆	—	173	2	no	—
107	M₁₆	—	173	2	no	—
108	M₁₆	—	173	2	no	—
109	M₁₆	C₂×#10621924	1	2	yes	—
110	C₂×C₈	—	346	2	no	—
111	C₂×C₈	—	346	4	no	—
112	C₂×C₈	—	346	2	no	—
113	C₂×C₈	—	346	1	no	—
114	C₂×C₈	—	346	1	no	—
115	C₂×C₈	—	346	2	no	—
116	C₂³	—	173	2	no	—
117	C₂³	—	173	1	no	—
118	Q₈	—	2	2	no	—
119	D₄	—	346	4	no	—
120	D₄	—	346	2	no	—
121	C₂×C₄	—	346	2	no	—
122	C₂×C₄	—	346	2	no	—
123	C₂×C₄	—	173	1	no	—
124	C₂×C₄	—	173	2	no	—
125	C₂×C₄	—	173	2	no	—
126	C₂×C₄	C₄⋊#10621924	1	2	yes	—
127	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	346	2	no	—
130	C₈	—	346	2	no	—
131	C₈	—	346	2	no	—
132	C₈	—	692	1	no	—
133	C₈	—	692	1	no	—
134	C₂²	—	346	2	no	—
135	C₂²	—	173	2	no	—
136	C₂²	—	692	1	no	—
137	C₂²	—	692	2	no	—
138	C₂²	C₂×C₄⋊#10621924	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	173	1	no	—
141	C₄	—	173	1	no	—
142	C₄	D₄×#10621924	1	2	yes	—
143	C₂	—	2	1	no	—
144	C₂	—	346	2	no	—
145	C₂	—	692	2	no	—
146	C₂	#26012438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26012571

C₂

yes

maximal, characteristic

#26012506

C₂

yes

maximal, complemented

#26012466

C₂

yes

maximal, characteristic

#26012366

C₂

yes

maximal, complemented

#26011474

C₂

yes

maximal, characteristic

#19149767

—

#19149767

—

C₂×#14583836

C₂²

yes

characteristic

D₁₇₃⋊M₁₆

—

D₁₇₃⋊M₁₆

C₂²

yes

—

C₂×D₁₇₃⋊C₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

M₁₆×D₁₇₃

C₂²

yes

—

#19149598

C₄

yes

characteristic

#19149595

C₄

yes

—

#101×C₁₇₃

C₄

yes

—

—

—

D₄

yes

—

#14583836

D₄

yes

—

#14583835

—

D₁₇₃⋊C₈

—

D₁₇₃⋊C₈

—

D₁₇₃⋊C₈

—

—

—

—

—

C₂×D₆₉₂

C₂×C₄

yes

characteristic

C₂×D₆₉₂

Q₈

yes

—

C₂×C₄×D₁₇₃

D₄

yes

—

C₂×C₄×D₁₇₃

C₂³

yes

characteristic

#14583810

C₂×C₄

yes

—

#14583809

—

C₈×D₁₇₃

—

C₂×Q₈×C₁₇₃

C₂×C₄

yes

characteristic

M₁₆×C₁₇₃

C₂×C₄

yes

—

—

—

—

—

C₂²×D₁₇₃

C₄⋊C₄

yes

—

C₂²×D₁₇₃

C₂×Q₈

yes

characteristic

D₆₉₂

—

D₆₉₂

—

C₂×Dic₁₇₃

C₂×D₄

yes

characteristic

C₄×D₁₇₃

—

C₄×D₁₇₃

—

C₄×D₁₇₃

C₂×D₄

yes

—

C₄×D₁₇₃

#41

yes

—

#13063436

—

Q₈×C₁₇₃

—

C₂×C₄×C₁₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₈×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₂×D₁₇₃

C₂²⋊Q₈

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₄×D₄

yes

—

Dic₁₇₃

yes

Dic₁₇₃

yes

C₂×C₁₇₃

—

C₂×C₁₇₃

#385

yes

characteristic

D₁₇₃

—

D₁₇₃

—

C₁₇₃

#1858

yes

characteristic, complemented, Sylow

#1858

—

173

maximal, Sylow

#568

—

173

—

#430

—

173

—

#413

—

173

—

#408

—

173

—

#341

—

173

—

C₂×#41

—

173

—

#131

—

346

—

#131

—

346

—

C₂×M₁₆

—

346

—

C₂×M₁₆

—

173

—

C₂×M₁₆

—

173

—

C₂×M₁₆

—

173

—

C₂²×C₈

—

173

—

#108

—

346

—

#108

—

346

—

#101

—

173

—

yes

—

173

—

C₂²⋊C₈

—

346

—

#41

—

346

—

#41

—

346

—

C₂×Q₈

C₂×#10621924

yes

characteristic

C₂×D₄

—

173

—

C₂×D₄

—

173

—

C₂²×C₄

—

173

—

100

C₂²×C₄

—

173

—

101

M₁₆

—

346

—

102

M₁₆

—

346

—

103

M₁₆

—

346

—

104

M₁₆

—

346

—

105

M₁₆

—

346

—

106

M₁₆

—

173

—

107

M₁₆

—

173

—

108

M₁₆

—

173

—

109

M₁₆

C₂×#10621924

yes

—

110

C₂×C₈

—

346

—

111

C₂×C₈

—

346

—

112

C₂×C₈

—

346

—

113

C₂×C₈

—

346

—

114

C₂×C₈

—

346

—

115

C₂×C₈

—

346

—

116

C₂³

—

173

—

117

C₂³

—

173

—

118

Q₈

—

119

D₄

—

346

—

120

D₄

—

346

—

121

C₂×C₄

—

346

—

122

C₂×C₄

—

346

—

123

C₂×C₄

—

173

—

124

C₂×C₄

—

173

—

125

C₂×C₄

—

173

—

126

C₂×C₄

C₄⋊#10621924

yes

—

127

C₂×C₄

C₂²×#10621924

yes

characteristic

128

C₈

—

129

C₈

—

346

—

130

C₈

—

346

—

131

C₈

—

346

—

132

C₈

—

692

—

133

C₈

—

692

—

134

C₂²

—

346

—

135

C₂²

—

173

—

136

C₂²

—

692

—

137

C₂²

—

692

—

138

C₂²

C₂×C₄⋊#10621924

yes

characteristic

139

C₄

—

140

C₄

—

173

—

141

C₄

—

173

—

142

C₄

D₄×#10621924

yes

—

143

C₂

—

144

C₂

—

346

—

145

C₂

—

692

—

146

C₂

#26012438

yes

characteristic

	subgroup	G/H
center	C₂	#26012438
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10621924
Fitting	#101×C₁₇₃	C₄
socle	C₂×C₁₇₃	#385
prime radical	C₂×#14583827	C₂²
cosocle	C₂×C₄×D₁₇₃	C₂³
hypercenter	#101	#10621924
hypocenter	C₁₇₃	#1858
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

#26012438

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10621924

Fitting

#101×C₁₇₃

C₄

C₂×C₁₇₃

C₂²

C₂×C₄×D₁₇₃

C₂³

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
2. #26012506	C₂	semidirect
4. #26012366	C₂	semidirect
73. C₁₇₃	#1858	semidirect

H ≅ G/N

type

2. #26012506

C₂

semidirect

4. #26012366

C₂

semidirect

73. C₁₇₃

#1858

semidirect

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

subgroup	index
#26012571	2
#26012506	2
#26012466	2
#26012366	2
#26011474	2
#1858	173

subgroup

index

173

Subgroups — #36369521

Conjugacy classes of subgroups

Complements

Maximal subgroups