Group #36352224 — subgroups

Subgroups — #36352224

Order 22144.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26012042	C₂	1	1	yes	maximal, characteristic
2	#26011543	C₂	1	2	yes	maximal, complemented
3	#26011240	C₂	1	1	yes	maximal, characteristic, complemented
4	#26011234	C₂	1	2	yes	maximal, complemented
5	#333×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14583827	C₄	1	2	yes	—
7	#19149676	C₂²	1	1	yes	characteristic
8	#19149668	C₄	1	2	yes	—
9	#19149654	—	2	2	no	—
10	#19149641	—	2	2	no	—
11	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊#13063436	C₂²	1	2	yes	—
13	#19149591	—	2	2	no	—
14	C₈×Dic₁₇₃	—	2	2	no	—
15	C₄⋊Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₇₃	C₂²	1	2	yes	—
17	#14583827	—	4	4	no	—
18	C₂×D₆₉₂	—	2	2	no	—
19	C₂×D₆₉₂	—	2	2	no	—
20	C₂×D₆₉₂	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₁₇₃	—	2	2	no	—
22	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₇₃	D₄	1	2	yes	—
24	C₂×#13063436	—	2	2	no	—
25	#14583803	—	2	4	no	—
26	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₇₃	D₄	1	2	yes	—
28	C₂×Q₈×C₁₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₇₃	—	2	2	no	—
30	C₄⋊C₄×C₁₇₃	—	2	2	no	—
31	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
32	C₂²×D₁₇₃	—	2	2	no	—
33	C₂²×D₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
34	D₆₉₂	—	4	2	no	—
35	D₆₉₂	—	4	4	no	—
36	D₆₉₂	—	4	2	no	—
37	D₆₉₂	—	4	2	no	—
38	C₂×Dic₁₇₃	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×D₁₇₃	—	2	2	no	—
41	C₄×D₁₇₃	—	2	2	no	—
42	C₄×D₁₇₃	—	2	2	no	—
43	C₄×D₁₇₃	—	4	4	no	—
44	#13063436	—	4	2	no	—
45	Q₈×C₁₇₃	—	4	2	no	—
46	C₂×C₄×C₁₇₃	—	2	2	no	—
47	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
49	C₈×C₁₇₃	—	4	2	no	—
50	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×D₁₇₃	—	2	1	no	—
52	C₂×D₁₇₃	—	2	2	no	—
53	C₂×D₁₇₃	—	4	2	no	—
54	C₂×D₁₇₃	—	4	4	no	—
55	C₄×C₁₇₃	—	2	1	no	—
56	C₄×C₁₇₃	—	4	2	no	—
57	C₄×C₁₇₃	#135	1	2	yes	—
58	C₄×C₁₇₃	#135	1	2	yes	—
59	Dic₁₇₃	—	2	2	no	—
60	Dic₁₇₃	#100	1	2	yes	—
61	C₂×C₁₇₃	C₂×#100	1	1	yes	characteristic
62	C₂×C₁₇₃	#416	1	2	yes	—
63	D₁₇₃	—	4	1	no	—
64	D₁₇₃	—	8	2	no	—
65	C₁₇₃	#1543	1	1	yes	characteristic, complemented, Sylow
66	#1543	—	173	1	no	maximal, Sylow
67	#532	—	173	1	no	—
68	#431	—	173	2	no	—
69	#333	—	173	1	no	—
70	#333	D₁₇₃	1	1	yes	characteristic, complemented
71	#331	—	173	2	no	—
72	C₂×#41	—	173	2	no	—
73	C₄⋊Q₈	C₂×D₁₇₃	1	1	yes	characteristic
74	#127	—	173	1	no	—
75	#124	—	173	2	no	—
76	#121	—	346	2	no	—
77	C₄×D₄	—	346	2	no	—
78	C₂×C₄²	—	173	1	no	—
79	C₄⋊C₈	—	173	2	no	—
80	C₄⋊C₈	C₂×D₁₇₃	1	2	yes	—
81	C₂²⋊C₈	—	346	2	no	—
82	C₄×C₈	—	346	2	no	—
83	#41	—	692	4	no	—
84	C₂×Q₈	C₄×D₁₇₃	1	2	yes	—
85	C₂×D₄	—	346	2	no	—
86	C₂×D₄	—	346	2	no	—
87	C₂×D₄	—	173	2	no	—
88	C₂²×C₄	—	346	2	no	—
89	C₂²×C₄	—	173	1	no	—
90	C₂²×C₄	—	173	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	346	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₂²⋊C₄	—	346	4	no	—
95	C₄²	—	173	1	no	—
96	C₄²	—	173	2	no	—
97	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
98	C₂³	—	346	2	no	—
99	C₂³	—	173	1	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	692	2	no	—
102	D₄	—	692	4	no	—
103	D₄	—	692	2	no	—
104	D₄	—	692	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	346	2	no	—
107	C₂×C₄	—	346	2	no	—
108	C₂×C₄	—	346	2	no	—
109	C₂×C₄	—	173	2	no	—
110	C₂×C₄	—	173	1	no	—
111	C₂×C₄	—	692	4	no	—
112	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
113	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	692	2	no	—
116	C₂²	—	346	1	no	—
117	C₂²	—	346	2	no	—
118	C₂²	—	692	2	no	—
119	C₂²	—	692	4	no	—
120	C₂²	C₂²⋊C₄×D₁₇₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	4	2	no	—
123	C₄	—	346	2	no	—
124	C₄	—	173	2	no	—
125	C₄	#19149700	1	2	yes	—
126	C₄	#19149702	1	2	yes	—
127	C₂	—	1384	2	no	—
128	C₂	—	692	1	no	—
129	C₂	#100×D₁₇₃	1	1	yes	characteristic
130	C₂	#26011486	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26012042

C₂

yes

maximal, characteristic

#26011543

C₂

yes

maximal, complemented

#26011240

C₂

yes

maximal, characteristic, complemented

#26011234

C₂

yes

maximal, complemented

#333×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14583827

C₄

yes

—

#19149676

C₂²

yes

characteristic

#19149668

C₄

yes

—

#19149654

—

#19149641

—

C₄²×D₁₇₃

C₂²

yes

characteristic

C₄⋊#13063436

C₂²

yes

—

#19149591

—

C₈×Dic₁₇₃

—

C₄⋊Q₈×C₁₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₃

C₂²

yes

—

#14583827

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

C₂×D₆₉₂

C₂×C₄

yes

—

C₂×C₄×D₁₇₃

—

C₂×C₄×D₁₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇₃

D₄

yes

—

C₂×#13063436

—

#14583803

—

C₄×Dic₁₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₇₃

D₄

yes

—

C₂×Q₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

C₂³

yes

characteristic

C₂²×D₁₇₃

—

C₂²×D₁₇₃

C₂²⋊C₄

yes

characteristic

D₆₉₂

—

D₆₉₂

—

D₆₉₂

—

D₆₉₂

—

C₂×Dic₁₇₃

C₂²⋊C₄

yes

—

C₂×Dic₁₇₃

C₂²⋊C₄

yes

characteristic

C₄×D₁₇₃

—

C₄×D₁₇₃

—

C₄×D₁₇₃

—

C₄×D₁₇₃

—

#13063436

—

Q₈×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#135

yes

—

C₄×C₁₇₃

#135

yes

—

Dic₁₇₃

—

Dic₁₇₃

#100

yes

—

C₂×C₁₇₃

C₂×#100

yes

characteristic

C₂×C₁₇₃

#416

yes

—

D₁₇₃

—

D₁₇₃

—

C₁₇₃

#1543

yes

characteristic, complemented, Sylow

#1543

—

173

maximal, Sylow

#532

—

173

—

#431

—

173

—

#333

—

173

—

#333

D₁₇₃

yes

characteristic, complemented

#331

—

173

—

C₂×#41

—

173

—

C₄⋊Q₈

C₂×D₁₇₃

yes

characteristic

#127

—

173

—

#124

—

173

—

#121

—

346

—

C₄×D₄

—

346

—

C₂×C₄²

—

173

—

C₄⋊C₈

—

173

—

C₄⋊C₈

C₂×D₁₇₃

yes

—

C₂²⋊C₈

—

346

—

C₄×C₈

—

346

—

#41

—

692

—

C₂×Q₈

C₄×D₁₇₃

yes

—

C₂×D₄

—

346

—

C₂×D₄

—

346

—

C₂×D₄

—

173

—

C₂²×C₄

—

346

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂×C₈

—

C₂×C₈

—

346

—

C₄⋊C₄

—

C₂²⋊C₄

—

346

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

C₂²×D₁₇₃

yes

characteristic

C₂³

—

346

—

C₂³

—

173

—

100

Q₈

—

101

D₄

—

692

—

102

D₄

—

692

—

103

D₄

—

692

—

104

D₄

—

692

—

105

C₂×C₄

—

106

C₂×C₄

—

346

—

107

C₂×C₄

—

346

—

108

C₂×C₄

—

346

—

109

C₂×C₄

—

173

—

110

C₂×C₄

—

173

—

111

C₂×C₄

—

692

—

112

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

113

C₂×C₄

D₄×D₁₇₃

yes

—

114

C₈

—

115

C₈

—

692

—

116

C₂²

—

346

—

117

C₂²

—

346

—

118

C₂²

—

692

—

119

C₂²

—

692

—

120

C₂²

C₂²⋊C₄×D₁₇₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

346

—

124

C₄

—

173

—

125

C₄

#19149700

yes

—

126

C₄

#19149702

yes

—

127

C₂

—

1384

—

128

C₂

—

692

—

129

C₂

#100×D₁₇₃

yes

characteristic

130

C₂

#26011486

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₇₃
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#333×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂×C₂²⋊C₄
prime radical	#26012042	C₂
cosocle	C₄²×C₁₇₃	C₂³
hypercenter	#333	D₁₇₃
hypocenter	C₁₇₃	#1543
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₇₃

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₁₇₃

Fitting

#333×C₁₇₃

C₂

socle

C₂²×C₁₇₃

C₂×C₂²⋊C₄

prime radical

#26012042

C₂

cosocle

C₄²×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #26011543	C₂	semidirect
3. #26011240	C₂	semidirect
4. #26011234	C₂	semidirect
5. #333×C₁₇₃	C₂	semidirect
65. C₁₇₃	#1543	semidirect
70. #333	D₁₇₃	semidirect

H ≅ G/N

type

2. #26011543

C₂

semidirect

3. #26011240

C₂

semidirect

4. #26011234

C₂

semidirect

5. #333×C₁₇₃

C₂

semidirect

65. C₁₇₃

#1543

semidirect

70. #333

D₁₇₃

semidirect

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

subgroup	index
#26012042	2
#26011543	2
#26011240	2
#26011234	2
#333×C₁₇₃	2
#1543	173

subgroup

index

#333×C₁₇₃

173

Subgroups — #36352224

Conjugacy classes of subgroups

Complements

Maximal subgroups