Group #36353612 — subgroups

Subgroups — #36353612

Order 22144.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#19149687	C₂	1	1	yes	maximal, characteristic
2	C₂×#19149614	C₂	1	2	yes	maximal, complemented
3	#26011473	C₂	1	2	yes	maximal
4	#26011249	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14583826	C₂²	1	1	yes	characteristic
7	#19149687	—	2	4	no	—
8	C₂×#14583810	C₂²	1	2	yes	—
9	C₂×#14583809	C₄	1	2	yes	—
10	C₂²×#13063436	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₇₃	C₄	1	2	yes	—
12	#19149614	—	2	4	no	—
13	#19149597	—	2	2	no	—
14	#19149595	—	2	2	no	—
15	C₂×M₁₆×C₁₇₃	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
17	#14583826	—	2	2	no	—
18	#14583826	—	2	2	no	—
19	C₂×Dic₃₄₆	D₄	1	1	yes	characteristic
20	C₂×D₆₉₂	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₇₃	Q₈	1	2	yes	—
22	C₂×C₄×D₁₇₃	C₂×C₄	1	2	yes	—
23	#14583810	—	2	2	no	—
24	#14583810	—	2	2	no	—
25	#14583809	—	2	4	no	—
26	C₂×#13063436	—	2	2	no	—
27	C₂×#13063436	C₂×C₄	1	1	yes	characteristic
28	C₂×#13063436	C₂×C₄	1	1	yes	characteristic
29	C₂×#13063436	D₄	1	4	yes	—
30	C₈×D₁₇₃	—	2	4	no	—
31	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₇₃	—	2	2	no	—
33	M₁₆×C₁₇₃	—	2	2	no	—
34	M₁₆×C₁₇₃	—	2	2	no	—
35	C₂×C₈×C₁₇₃	—	2	2	no	—
36	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
37	C₂²×D₁₇₃	C₄⋊C₄	1	2	yes	—
38	Dic₃₄₆	—	2	2	no	—
39	D₆₉₂	—	2	2	no	—
40	C₂²⋊D₁₇₃	—	2	4	no	—
41	C₂×Dic₁₇₃	C₄⋊C₄	1	2	yes	—
42	C₄×D₁₇₃	—	2	2	no	—
43	C₄×D₁₇₃	—	2	2	no	—
44	#13063436	—	4	2	no	—
45	#13063436	C₂²⋊C₄	1	2	yes	—
46	#13063436	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇₃	#41	1	2	yes	—
52	C₈×C₁₇₃	—	2	2	no	—
53	C₈×C₁₇₃	—	4	2	no	—
54	C₂²×C₁₇₃	—	4	1	no	—
55	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₇₃	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₁₇₃	—	2	2	no	—
58	C₂×D₁₇₃	—	4	2	no	—
59	C₄×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₇₃	#117	1	1	yes	characteristic
61	C₄×C₁₇₃	#121	1	2	yes	—
62	Dic₁₇₃	—	2	2	no	—
63	C₂×C₁₇₃	—	2	1	no	—
64	C₂×C₁₇₃	—	2	2	no	—
65	C₂×C₁₇₃	#380	1	1	yes	characteristic
66	D₁₇₃	—	4	2	no	—
67	C₁₇₃	#1813	1	1	yes	characteristic, complemented, Sylow
68	#1813	—	173	1	no	maximal, Sylow
69	C₂×#131	—	173	1	no	—
70	C₂×#108	—	173	2	no	—
71	#413	—	173	2	no	—
72	#338	—	173	1	no	—
73	#338	D₁₇₃	1	1	yes	characteristic, complemented
74	C₂×#41	—	173	1	no	—
75	#131	—	346	4	no	—
76	C₂×M₁₆	—	173	2	no	—
77	C₂×M₁₆	—	173	2	no	—
78	C₂×M₁₆	C₂×D₁₇₃	1	2	yes	—
79	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
80	C₂²×C₈	—	173	2	no	—
81	C₂²×C₈	—	173	1	no	—
82	#108	—	346	4	no	—
83	#101	—	346	2	no	—
84	#100	—	346	2	no	—
85	#41	—	346	2	no	—
86	#41	—	346	2	no	—
87	C₂×Q₈	—	173	1	no	—
88	C₂×D₄	—	173	2	no	—
89	C₂×D₄	—	173	1	no	—
90	C₂²×C₄	—	173	2	no	—
91	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	346	4	no	—
96	M₁₆	—	346	2	no	—
97	M₁₆	—	346	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	346	4	no	—
100	C₂×C₈	—	346	2	no	—
101	C₂×C₈	—	173	1	no	—
102	C₂×C₈	—	173	1	no	—
103	C₂×C₈	—	173	4	no	—
104	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
105	C₂³	—	173	2	no	—
106	C₂³	Q₈×D₁₇₃	1	1	yes	characteristic
107	Q₈	—	346	2	no	—
108	D₄	—	346	4	no	—
109	D₄	—	346	2	no	—
110	C₂×C₄	—	346	2	no	—
111	C₂×C₄	—	346	2	no	—
112	C₂×C₄	—	173	2	no	—
113	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
115	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
116	C₂×C₄	#14583826	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	173	2	no	—
120	C₈	—	173	2	no	—
121	C₈	—	692	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	346	2	no	—
124	C₂²	—	692	2	no	—
125	C₂²	C₄⋊C₄×D₁₇₃	1	1	yes	characteristic
126	C₂²	#19149657	1	2	yes	—
127	C₄	—	346	2	no	—
128	C₄	C₂²⋊C₄×D₁₇₃	1	1	yes	characteristic
129	C₄	#19149636	1	1	yes	characteristic
130	C₄	#19149652	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	692	2	no	—
134	C₂	#26011350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149687

C₂

yes

maximal, characteristic

C₂×#19149614

C₂

yes

maximal, complemented

#26011473

C₂

yes

maximal

#26011249

C₂

yes

maximal, characteristic, complemented

#338×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14583826

C₂²

yes

characteristic

#19149687

—

C₂×#14583810

C₂²

yes

—

C₂×#14583809

C₄

yes

—

C₂²×#13063436

C₂²

yes

characteristic

C₂×C₈×D₁₇₃

C₄

yes

—

#19149614

—

#19149597

—

#19149595

—

C₂×M₁₆×C₁₇₃

C₂²

yes

—

C₂×M₁₆×C₁₇₃

C₂²

yes

characteristic

#14583826

—

#14583826

—

C₂×Dic₃₄₆

D₄

yes

characteristic

C₂×D₆₉₂

D₄

yes

characteristic

C₂×C₂²⋊D₁₇₃

Q₈

yes

—

C₂×C₄×D₁₇₃

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#13063436

C₂×C₄

yes

characteristic

C₂×#13063436

D₄

yes

—

C₈×D₁₇₃

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

M₁₆×C₁₇₃

—

M₁₆×C₁₇₃

—

M₁₆×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂²×D₁₇₃

C₄⋊C₄

yes

—

Dic₃₄₆

—

D₆₉₂

—

C₂²⋊D₁₇₃

—

C₂×Dic₁₇₃

C₄⋊C₄

yes

—

C₄×D₁₇₃

—

C₄×D₁₇₃

—

#13063436

—

#13063436

C₂²⋊C₄

yes

—

#13063436

C₂²⋊C₄

yes

—

C₂³×C₁₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₈×C₁₇₃

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇₃

C₂²⋊Q₈

yes

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

C₂×C₂²⋊C₄

yes

C₄×C₁₇₃

yes

C₄×C₁₇₃

yes

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#380

yes

characteristic

D₁₇₃

—

C₁₇₃

#1813

yes

characteristic, complemented, Sylow

#1813

—

173

maximal, Sylow

C₂×#131

—

173

—

C₂×#108

—

173

—

#413

—

173

—

#338

—

173

—

#338

D₁₇₃

yes

characteristic, complemented

C₂×#41

—

173

—

#131

—

346

—

C₂×M₁₆

—

173

—

C₂×M₁₆

—

173

—

C₂×M₁₆

C₂×D₁₇₃

yes

—

C₂×M₁₆

C₂×D₁₇₃

yes

characteristic

C₂²×C₈

—

173

—

C₂²×C₈

—

173

—

#108

—

346

—

#101

—

346

—

#100

—

346

—

#41

—

346

—

#41

—

346

—

C₂×Q₈

—

173

—

C₂×D₄

—

173

—

C₂×D₄

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

346

—

M₁₆

—

346

—

M₁₆

—

346

—

C₂×C₈

—

C₂×C₈

—

346

—

100

C₂×C₈

—

346

—

101

C₂×C₈

—

173

—

102

C₂×C₈

—

173

—

103

C₂×C₈

—

173

—

104

C₂×C₈

C₄×D₁₇₃

yes

—

105

C₂³

—

173

—

106

C₂³

Q₈×D₁₇₃

yes

characteristic

107

Q₈

—

346

—

108

D₄

—

346

—

109

D₄

—

346

—

110

C₂×C₄

—

346

—

111

C₂×C₄

—

346

—

112

C₂×C₄

—

173

—

113

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

114

C₂×C₄

D₄×D₁₇₃

yes

—

115

C₂×C₄

D₄×D₁₇₃

yes

characteristic

116

C₂×C₄

#14583826

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

173

—

120

C₈

—

173

—

121

C₈

—

692

—

122

C₂²

—

123

C₂²

—

346

—

124

C₂²

—

692

—

125

C₂²

C₄⋊C₄×D₁₇₃

yes

characteristic

126

C₂²

#19149657

yes

—

127

C₄

—

346

—

128

C₄

C₂²⋊C₄×D₁₇₃

yes

characteristic

129

C₄

#19149636

yes

characteristic

130

C₄

#19149652

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

692

—

134

C₂

#26011350

yes

characteristic

	subgroup	G/H
center	C₄	#19149636
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#338×C₁₇₃	C₂
socle	C₂×C₁₇₃	#380
prime radical	C₂×#14583826	C₂²
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#338	D₁₇₃
hypocenter	C₁₇₃	#1813
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#19149636

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

#338×C₁₇₃

C₂

C₂×C₁₇₃

C₂²

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#19149614	C₂	semidirect
4. #26011249	C₂	semidirect
5. #338×C₁₇₃	C₂	semidirect
67. C₁₇₃	#1813	semidirect
73. #338	D₁₇₃	semidirect

H ≅ G/N

type

2. C₂×#19149614

C₂

semidirect

4. #26011249

C₂

semidirect

5. #338×C₁₇₃

C₂

semidirect

67. C₁₇₃

#1813

semidirect

73. #338

D₁₇₃

semidirect

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

subgroup	index
C₂×#19149687	2
C₂×#19149614	2
#26011473	2
#26011249	2
#338×C₁₇₃	2
#1813	173

subgroup

index

#338×C₁₇₃

173

Subgroups — #36353612

Conjugacy classes of subgroups

Complements

Maximal subgroups