Group #36353072 — subgroups

Subgroups — #36353072

Order 22144.

146 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 51 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	#26011887	C₂	1	1	yes	maximal, characteristic
2	#26011783	C₂	1	1	yes	maximal, characteristic
3	#26011702	C₂	1	1	yes	maximal, characteristic
4	#26011618	C₂	1	1	yes	maximal, characteristic
5	#26011243	C₂	1	1	yes	maximal, characteristic
6	#26011236	C₂	1	1	yes	maximal, characteristic
7	#333×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₉₂	—	2	1	no	—
9	#19149679	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₄₆	C₂²	1	1	yes	characteristic
11	#19149612	—	2	1	no	—
12	#19149610	—	2	1	no	—
13	#19149610	—	2	1	no	—
14	C₄⋊#13063436	C₂²	1	1	yes	characteristic
15	#19149608	—	2	1	no	—
16	#19149604	—	2	1	no	—
17	#19149602	—	2	1	no	—
18	#19149589	—	2	1	no	—
19	C₄×#13063436	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₇₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₄₆	—	2	1	no	—
24	C₂×Dic₃₄₆	D₄	1	1	yes	characteristic
25	Dic₆₉₂	—	4	2	no	—
26	C₂×#13063436	—	2	1	no	—
27	C₂×#13063436	D₄	1	1	yes	characteristic
28	C₂×#13063436	D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₇₃	—	2	1	no	—
30	C₄⋊Dic₁₇₃	—	2	1	no	—
31	C₄⋊Dic₁₇₃	D₄	1	1	yes	characteristic
32	#14583805	—	2	1	no	—
33	C₄×Dic₁₇₃	—	2	1	no	—
34	C₂×Q₈×C₁₇₃	—	2	1	no	—
35	C₂×C₈×C₁₇₃	—	2	1	no	—
36	C₂×C₈×C₁₇₃	—	2	1	no	—
37	C₄⋊C₄×C₁₇₃	—	2	1	no	—
38	C₄⋊C₄×C₁₇₃	—	2	1	no	—
39	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
40	Dic₃₄₆	—	2	1	no	—
41	Dic₃₄₆	—	2	1	no	—
42	Dic₃₄₆	—	4	1	no	—
43	Dic₃₄₆	—	4	1	no	—
44	C₂×Dic₁₇₃	—	2	1	no	—
45	C₂×Dic₁₇₃	—	2	1	no	—
46	C₂×Dic₁₇₃	—	4	1	no	—
47	#13063436	—	2	1	no	—
48	#13063436	—	4	1	no	—
49	#13063436	Q₁₆	1	2	yes	complemented
50	Q₈×C₁₇₃	—	8	1	no	—
51	C₂×C₄×C₁₇₃	—	4	1	no	—
52	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
55	C₈×C₁₇₃	—	4	1	no	—
56	C₈×C₁₇₃	—	4	1	no	—
57	C₂²×C₁₇₃	#121	1	1	yes	characteristic
58	C₄×C₁₇₃	—	2	1	no	—
59	C₄×C₁₇₃	—	8	1	no	—
60	C₄×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₁₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₄×C₁₇₃	#137	1	1	yes	characteristic
63	C₄×C₁₇₃	#137	1	1	yes	characteristic
64	Dic₁₇₃	—	4	1	no	—
65	Dic₁₇₃	—	4	1	no	—
66	Dic₁₇₃	—	8	1	no	—
67	C₂×C₁₇₃	#462	1	1	yes	characteristic
68	C₂×C₁₇₃	#468	1	1	yes	characteristic
69	C₂×C₁₇₃	#471	1	1	yes	characteristic
70	C₁₇₃	#1692	1	1	yes	characteristic, complemented, Sylow
71	#1692	—	173	1	no	maximal, Sylow
72	#498	—	173	1	no	—
73	#477	—	173	1	no	—
74	#462	—	173	1	no	—
75	#446	—	173	1	no	—
76	#335	—	173	1	no	—
77	#333	D₁₇₃	1	1	yes	characteristic
78	#332	—	173	1	no	—
79	C₂×Q₁₆	—	346	1	no	—
80	C₄⋊Q₈	—	173	1	no	—
81	C₄⋊Q₈	C₂×D₁₇₃	1	1	yes	characteristic
82	C₄×Q₈	—	173	1	no	—
83	#107	—	346	1	no	—
84	#106	—	346	1	no	—
85	#106	—	346	1	no	—
86	C₄⋊C₈	—	346	1	no	—
87	C₄⋊C₈	—	173	1	no	—
88	C₄⋊C₈	C₂×D₁₇₃	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₁₇₃	1	1	yes	characteristic
90	#103	—	346	1	no	—
91	#103	—	346	1	no	—
92	#97	—	346	1	no	—
93	C₄×C₈	—	173	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	346	1	no	—
96	C₂×Q₈	—	173	1	no	—
97	Q₁₆	—	692	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	346	1	no	—
101	C₂×C₈	—	173	1	no	—
102	C₂×C₈	—	173	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	346	1	no	—
106	C₄⋊C₄	—	346	1	no	—
107	C₄⋊C₄	—	346	1	no	—
108	C₄⋊C₄	—	173	1	no	—
109	C₄²	—	346	1	no	—
110	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
111	Q₈	—	8	1	no	—
112	Q₈	—	346	1	no	—
113	Q₈	—	346	1	no	—
114	Q₈	—	692	1	no	—
115	Q₈	—	692	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	346	1	no	—
118	C₂×C₄	—	346	1	no	—
119	C₂×C₄	—	692	1	no	—
120	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
121	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
122	C₂×C₄	#14583826	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	346	1	no	—
126	C₈	—	173	2	no	—
127	C₈	—	692	1	no	—
128	C₂²	#19149652	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	1384	1	no	—
132	C₄	—	692	1	no	—
133	C₄	—	692	1	no	—
134	C₄	Q₁₆×D₁₇₃	1	1	yes	characteristic
135	C₄	#19149706	1	1	yes	characteristic
136	C₄	#19149708	1	1	yes	characteristic
137	C₄	#19149711	1	1	yes	characteristic
138	C₂	#26011700	1	1	yes	characteristic
139	C₂	#26011740	1	1	yes	characteristic
140	C₂	#26011753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26011887

C₂

yes

maximal, characteristic

#26011783

C₂

yes

maximal, characteristic

#26011702

C₂

yes

maximal, characteristic

#26011618

C₂

yes

maximal, characteristic

#26011243

C₂

yes

maximal, characteristic

#26011236

C₂

yes

maximal, characteristic

#333×C₁₇₃

C₂

yes

maximal, characteristic

C₂×Dic₆₉₂

—

#19149679

C₂²

yes

characteristic

C₄×Dic₃₄₆

C₂²

yes

—

—

—

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₁₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₃

C₂²

yes

characteristic

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

D₄

yes

characteristic

Dic₆₉₂

—

C₂×#13063436

—

C₂×#13063436

D₄

yes

characteristic

C₂×#13063436

D₄

yes

characteristic

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

D₄

yes

characteristic

#14583805

—

C₄×Dic₁₇₃

—

C₂×Q₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

C₂³

yes

characteristic

Dic₃₄₆

—

Dic₃₄₆

—

Dic₃₄₆

—

Dic₃₄₆

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

#13063436

—

#13063436

Q₁₆

yes

complemented

Q₈×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

characteristic

C₈×C₁₇₃

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

#121

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂×Q₁₆

yes

C₄×C₁₇₃

yes

C₄×C₁₇₃

yes

C₄×C₁₇₃

yes

Dic₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₁₇₃

yes

characteristic, complemented, Sylow

#1692

—

173

maximal, Sylow

#498

—

173

—

#477

—

173

—

#462

—

173

—

#446

—

173

—

#335

—

173

—

#333

D₁₇₃

yes

characteristic

#332

—

173

—

C₂×Q₁₆

—

346

—

C₄⋊Q₈

—

173

—

C₄⋊Q₈

C₂×D₁₇₃

yes

characteristic

C₄×Q₈

—

173

—

#107

—

346

—

#106

—

346

—

#106

—

346

—

C₄⋊C₈

—

346

—

C₄⋊C₈

—

173

—

C₄⋊C₈

C₂×D₁₇₃

yes

characteristic

C₄⋊C₈

C₂×D₁₇₃

yes

characteristic

#103

—

346

—

#103

—

346

—

#97

—

346

—

C₄×C₈

—

173

—

C₂×Q₈

—

C₂×Q₈

—

346

—

C₂×Q₈

—

173

—

Q₁₆

—

692

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

346

—

101

C₂×C₈

—

173

—

102

C₂×C₈

—

173

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

346

—

106

C₄⋊C₄

—

346

—

107

C₄⋊C₄

—

346

—

108

C₄⋊C₄

—

173

—

109

C₄²

—

346

—

110

C₄²

C₂²×D₁₇₃

yes

characteristic

111

Q₈

—

112

Q₈

—

346

—

113

Q₈

—

346

—

114

Q₈

—

692

—

115

Q₈

—

692

—

116

C₂×C₄

—

117

C₂×C₄

—

346

—

118

C₂×C₄

—

346

—

119

C₂×C₄

—

692

—

120

C₂×C₄

D₄×D₁₇₃

yes

characteristic

121

C₂×C₄

D₄×D₁₇₃

yes

characteristic

122

C₂×C₄

#14583826

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

346

—

126

C₈

—

173

—

127

C₈

—

692

—

128

C₂²

#19149652

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

1384

—

132

C₄

—

692

—

133

C₄

—

692

—

134

C₄

Q₁₆×D₁₇₃

yes

characteristic

135

C₄

#19149706

yes

characteristic

136

C₄

#19149708

yes

characteristic

137

C₄

#19149711

yes

characteristic

138

C₂

#26011700

yes

characteristic

139

C₂

#26011740

yes

characteristic

140

C₂

#26011753

yes

characteristic

	subgroup	G/H
center	C₂²	#19149652
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#333×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#121
hypercenter	#333	D₁₇₃
hypocenter	C₁₇₃	#1692
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#19149652

commutator, cosocle

C₄²×C₁₇₃

C₂³

Frattini

C₄²

C₂²×D₁₇₃

Fitting

#333×C₁₇₃

C₂

C₂²×C₁₇₃

D₁₇₃

C₁₇₃

G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
49. #13063436	Q₁₆	semidirect
70. C₁₇₃	#1692	semidirect

H ≅ G/N

type

49. #13063436

Q₁₆

semidirect

70. C₁₇₃

#1692

semidirect

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

subgroup	index
#26011887	2
#26011783	2
#26011702	2
#26011618	2
#26011243	2
#26011236	2
#333×C₁₇₃	2
#1692	173

subgroup

index

#333×C₁₇₃

173

Subgroups — #36353072

Conjugacy classes of subgroups

Complements

Maximal subgroups