Group #36353690 — subgroups

Subgroups — #36353690

Order 22144.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×C₄×#13063436	C₂	1	1	yes	maximal, characteristic
2	#26011361	C₂	1	1	yes	maximal, characteristic
3	#26011253	C₂	1	4	yes	maximal
4	#384×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13063436	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₇₃	—	2	1	no	—
7	C₂×C₄⋊Dic₁₇₃	—	2	1	no	—
8	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
9	C₄×#13063436	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₇₃	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₇₃	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₇₃	—	2	2	no	—
15	C₂×#13063436	—	2	2	no	—
16	C₂×#13063436	—	2	2	no	—
17	C₂×#13063436	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₇₃	—	2	4	no	—
19	C₄⋊Dic₁₇₃	—	4	2	no	—
20	C₄⋊Dic₁₇₃	—	4	2	no	—
21	C₂²×C₄×C₁₇₃	—	2	2	no	—
22	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₇₃	—	2	4	no	—
26	C₄⋊C₄×C₁₇₃	—	4	2	no	—
27	C₄⋊C₄×C₁₇₃	—	4	2	no	—
28	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
29	C₄²×C₁₇₃	Q₈	1	2	yes	—
30	C₂×Dic₁₇₃	—	4	2	no	—
31	C₂×Dic₁₇₃	—	4	2	no	—
32	C₂×Dic₁₇₃	—	4	2	no	—
33	#13063436	—	2	4	no	—
34	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇₃	—	4	2	no	—
36	C₂×C₄×C₁₇₃	—	4	2	no	—
37	C₂×C₄×C₁₇₃	—	4	2	no	—
38	C₂×C₄×C₁₇₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₇₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₇₃	D₈	1	2	yes	—
41	C₂×C₄×C₁₇₃	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₇₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₇₃	Q₁₆	1	2	yes	—
44	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇₃	#41	1	2	yes	—
47	C₂×C₄×C₁₇₃	#41	1	2	yes	—
48	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₇₃	#124	1	1	yes	characteristic
50	C₂²×C₁₇₃	#124	1	1	yes	characteristic
51	C₂²×C₁₇₃	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₇₃	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₇₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₁₇₃	—	8	2	no	—
56	C₄×C₁₇₃	#106	1	4	yes	—
57	C₄×C₁₇₃	#107	1	4	yes	—
58	C₄×C₁₇₃	#117	1	2	yes	—
59	C₄×C₁₇₃	#125	1	2	yes	—
60	Dic₁₇₃	—	8	2	no	—
61	C₂×C₁₇₃	#382	1	1	yes	characteristic
62	C₂×C₁₇₃	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₇₃	C₂×#107	1	1	yes	characteristic
64	C₂×C₁₇₃	#486	1	1	yes	characteristic
65	C₂×C₁₇₃	#486	1	1	yes	characteristic
66	C₂×C₁₇₃	#487	1	1	yes	characteristic
67	C₂×C₁₇₃	#487	1	1	yes	characteristic
68	C₁₇₃	#1834	1	1	yes	characteristic, complemented, Sylow
69	#1834	—	173	1	no	maximal, Sylow
70	C₂×C₄×C₈	—	173	1	no	—
71	#384	—	173	1	no	—
72	#384	D₁₇₃	1	1	yes	characteristic
73	#340	—	173	4	no	—
74	C₂²×C₈	—	173	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	346	1	no	—
78	C₂×C₄⋊C₄	—	346	1	no	—
79	C₂×C₄⋊C₄	—	173	2	no	—
80	C₂×C₄⋊C₄	C₂×D₁₇₃	1	2	yes	—
81	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
82	C₄×C₈	—	173	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	346	2	no	—
85	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
88	C₂×C₈	—	346	2	no	—
89	C₂×C₈	—	346	2	no	—
90	C₂×C₈	—	173	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	346	4	no	—
95	C₄⋊C₄	—	692	2	no	—
96	C₄⋊C₄	—	692	2	no	—
97	C₄²	C₄×D₁₇₃	1	2	yes	—
98	C₄²	Dic₃₄₆	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	692	2	no	—
104	C₂×C₄	—	692	2	no	—
105	C₂×C₄	—	692	2	no	—
106	C₂×C₄	#14583805	1	2	yes	—
107	C₂×C₄	#14583805	1	2	yes	—
108	C₂×C₄	#14583811	1	2	yes	—
109	C₂×C₄	#14583813	1	2	yes	—
110	C₂×C₄	#14583814	1	2	yes	—
111	C₂×C₄	#14583816	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₃₄₆	1	1	yes	characteristic
114	C₂×C₄	#14583825	1	2	yes	—
115	C₂×C₄	#14583827	1	2	yes	—
116	C₈	—	346	4	no	—
117	C₂²	C₂×#14583805	1	1	yes	characteristic
118	C₂²	#19149666	1	1	yes	characteristic
119	C₂²	#19149668	1	1	yes	characteristic
120	C₂²	C₂×#14583811	1	1	yes	characteristic
121	C₂²	C₂×#14583813	1	1	yes	characteristic
122	C₂²	C₂×#14583814	1	1	yes	characteristic
123	C₂²	C₂×#14583816	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	1384	2	no	—
126	C₄	#19149610	1	4	yes	—
127	C₄	#19149612	1	4	yes	—
128	C₄	#19149637	1	2	yes	—
129	C₄	#19149670	1	2	yes	—
130	C₂	#26011356	1	1	yes	characteristic
131	C₂	C₂×#19149610	1	1	yes	characteristic
132	C₂	C₂×#19149612	1	1	yes	characteristic
133	C₂	#26011837	1	1	yes	characteristic
134	C₂	#26011839	1	1	yes	characteristic
135	C₂	#26011841	1	1	yes	characteristic
136	C₂	#26011843	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#13063436

C₂

yes

maximal, characteristic

#26011361

C₂

yes

maximal, characteristic

#26011253

C₂

yes

maximal

#384×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×#13063436

C₂²

yes

—

C₂×C₄⋊Dic₁₇₃

—

C₂×C₄⋊Dic₁₇₃

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₄×#13063436

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₇₃

—

C₂×C₄⋊C₄×C₁₇₃

—

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₂²×Dic₁₇₃

—

C₂×#13063436

—

C₂×#13063436

—

C₂×#13063436

C₂×C₄

yes

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

D₄

yes

characteristic

C₂²×C₄×C₁₇₃

D₄

yes

characteristic

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

Q₈

yes

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₇₃

D₈

yes

—

C₂×C₄×C₁₇₃

SD₁₆

yes

—

C₂×C₄×C₁₇₃

SD₁₆

yes

—

C₂×C₄×C₁₇₃

Q₁₆

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

#41

yes

—

C₂²×C₁₇₃

C₂×C₄⋊C₄

yes

C₂²×C₁₇₃

yes

C₂²×C₁₇₃

yes

C₂²×C₁₇₃

C₂×D₈

yes

characteristic

C₂²×C₁₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₇₃

C₂×Q₁₆

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

#106

yes

—

C₄×C₁₇₃

#107

yes

—

C₄×C₁₇₃

#117

yes

—

C₄×C₁₇₃

#125

yes

—

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

#1834

—

173

maximal, Sylow

C₂×C₄×C₈

—

173

—

#384

—

173

—

#384

D₁₇₃

yes

characteristic

#340

—

173

—

C₂²×C₈

—

173

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

C₂×D₁₇₃

yes

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

C₄×C₈

—

173

—

C₂²×C₄

—

C₂²×C₄

—

346

—

C₂²×C₄

C₂²⋊D₁₇₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₇₃

yes

characteristic

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

173

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

692

—

C₄⋊C₄

—

692

—

C₄²

C₄×D₁₇₃

yes

—

C₄²

Dic₃₄₆

yes

—

C₂³

C₂×C₂²⋊D₁₇₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

692

—

104

C₂×C₄

—

692

—

105

C₂×C₄

—

692

—

106

C₂×C₄

#14583805

yes

—

107

C₂×C₄

#14583805

yes

—

108

C₂×C₄

#14583811

yes

—

109

C₂×C₄

#14583813

yes

—

110

C₂×C₄

#14583814

yes

—

111

C₂×C₄

#14583816

yes

—

112

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

113

C₂×C₄

C₂×Dic₃₄₆

yes

characteristic

114

C₂×C₄

#14583825

yes

—

115

C₂×C₄

#14583827

yes

—

116

C₈

—

346

—

117

C₂²

C₂×#14583805

yes

characteristic

118

C₂²

#19149666

yes

characteristic

119

C₂²

#19149668

yes

characteristic

120

C₂²

C₂×#14583811

yes

characteristic

121

C₂²

C₂×#14583813

yes

characteristic

122

C₂²

C₂×#14583814

yes

characteristic

123

C₂²

C₂×#14583816

yes

characteristic

124

C₄

—

125

C₄

—

1384

—

126

C₄

#19149610

yes

—

127

C₄

#19149612

yes

—

128

C₄

#19149637

yes

—

129

C₄

#19149670

yes

—

130

C₂

#26011356

yes

characteristic

131

C₂

C₂×#19149610

yes

characteristic

132

C₂

C₂×#19149612

yes

characteristic

133

C₂

#26011837

yes

characteristic

134

C₂

#26011839

yes

characteristic

135

C₂

#26011841

yes

characteristic

136

C₂

#26011843

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₃
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#384×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂×D₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#384	D₁₇₃
hypocenter	C₁₇₃	#1834
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₃

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

#384×C₁₇₃

C₂

socle, prime radical

C₂³×C₁₇₃

C₂×D₄

cosocle

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#13063436	C₄	semidirect
68. C₁₇₃	#1834	semidirect

H ≅ G/N

type

9. C₄×#13063436

C₄

semidirect

68. C₁₇₃

#1834

semidirect

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

subgroup	index
C₂×C₄×#13063436	2
#26011361	2
#26011253	2
#384×C₁₇₃	2
#1834	173

subgroup

index

#384×C₁₇₃

173

Subgroups — #36353690

Conjugacy classes of subgroups

Complements

Maximal subgroups