Group #36362493 — subgroups

Subgroups — #36362493

Order 22144.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#26012048	C₂	1	1	yes	maximal, characteristic
2	#26011618	C₂	1	4	yes	maximal, complemented
3	#26011615	C₂	1	4	yes	maximal, complemented
4	#26011519	C₂	1	2	yes	maximal
5	#26011449	C₂	1	2	yes	maximal
6	C₂×C₄×#13063436	C₂	1	1	yes	maximal, characteristic
7	#423×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13063436	C₂²	1	1	yes	characteristic
9	C₂²×#13063436	C₂²	1	1	yes	characteristic
10	#19149679	C₄	1	2	yes	—
11	#19149671	C₄	1	2	yes	—
12	#19149658	C₄	1	4	yes	—
13	C₄×Dic₃₄₆	C₂²	1	4	yes	—
14	#19149638	C₂²	1	2	yes	—
15	#19149608	C₂²	1	8	yes	—
16	#19149589	C₂²	1	4	yes	—
17	C₄×#13063436	C₂²	1	2	yes	—
18	C₄×#13063436	C₂²	1	2	yes	—
19	C₈×Dic₁₇₃	C₂²	1	4	yes	—
20	C₂×M₁₆×C₁₇₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₇₃	C₂²	1	4	yes	—
23	C₂×Dic₃₄₆	C₂×C₄	1	4	yes	—
24	C₂×#13063436	—	2	2	no	—
25	C₂×#13063436	Q₈	1	4	yes	complemented
26	C₂×#13063436	C₂³	1	2	yes	—
27	C₂×#13063436	C₂³	1	1	yes	characteristic
28	C₂×#13063436	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₇₃	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₁₇₃	C₂×C₄	1	4	yes	—
31	#14583805	C₂×C₄	1	8	yes	—
32	C₂²⋊Dic₁₇₃	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₇₃	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₇₃	—	2	4	no	—
37	C₂×C₈×C₁₇₃	C₂³	1	4	yes	—
38	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₇₃	C₂³	1	2	yes	—
41	Dic₃₄₆	—	2	8	no	—
42	C₂×Dic₁₇₃	C₂²×C₄	1	8	yes	—
43	#13063436	—	2	2	no	—
44	#13063436	C₂×Q₈	1	2	yes	—
45	#13063436	C₂×Q₈	1	2	yes	—
46	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇₃	—	2	4	no	—
48	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₁₇₃	#41	1	2	yes	—
53	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₇₃	—	2	4	no	—
55	C₂²×C₁₇₃	—	2	2	no	—
56	C₂²×C₁₇₃	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₇₃	—	2	2	no	—
59	C₄×C₁₇₃	C₄×Q₈	1	2	yes	—
60	C₄×C₁₇₃	#131	1	4	yes	—
61	C₄×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
63	Dic₁₇₃	—	2	8	no	—
64	C₂×C₁₇₃	—	2	2	no	—
65	C₂×C₁₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₇₃	C₂×#131	1	2	yes	—
67	C₁₇₃	#2960	1	1	yes	characteristic, complemented, Sylow
68	#2960	—	173	1	no	maximal, Sylow
69	#533	—	173	1	no	—
70	#446	—	173	4	no	—
71	C₈×Q₈	—	173	4	no	—
72	#424	—	173	2	no	—
73	#423	D₁₇₃	1	1	yes	characteristic
74	#405	—	173	2	no	—
75	C₂×C₄×C₈	—	173	1	no	—
76	C₂×M₁₆	C₂×D₁₇₃	1	2	yes	—
77	C₂²×C₈	—	173	1	no	—
78	C₂²×C₈	—	173	1	no	—
79	C₄⋊Q₈	—	173	2	no	—
80	#125	—	173	2	no	—
81	C₂²⋊Q₈	—	173	4	no	—
82	C₄×Q₈	—	173	4	no	—
83	#117	—	173	2	no	—
84	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
85	C₄⋊C₈	—	173	8	no	—
86	C₄⋊C₈	C₂×D₁₇₃	1	4	yes	—
87	#97	—	173	4	no	—
88	C₄×C₈	—	173	4	no	—
89	C₄×C₈	—	173	2	no	—
90	C₄×C₈	—	173	2	no	—
91	C₂×Q₈	—	173	4	no	—
92	C₂²×C₄	C₄×D₁₇₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
94	M₁₆	—	2	4	no	—
95	C₂×C₈	—	346	2	no	—
96	C₂×C₈	—	173	2	no	—
97	C₂×C₈	—	173	4	no	—
98	C₂×C₈	—	173	1	no	—
99	C₂×C₈	—	173	1	no	—
100	C₂×C₈	C₂²×D₁₇₃	1	4	yes	—
101	C₄⋊C₄	—	173	4	no	—
102	C₄⋊C₄	—	173	8	no	—
103	C₄⋊C₄	—	173	4	no	—
104	C₂²⋊C₄	—	173	4	no	—
105	C₄²	—	173	4	no	—
106	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
107	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
108	C₄²	C₂²×D₁₇₃	1	2	yes	—
109	C₂³	C₂×C₄×D₁₇₃	1	1	yes	characteristic
110	Q₈	—	346	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	173	8	no	—
113	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
116	C₂×C₄	Q₈×D₁₇₃	1	2	yes	—
117	C₂×C₄	#14583825	1	2	yes	—
118	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	346	2	no	—
121	C₈	—	173	2	no	—
122	C₈	—	173	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#19149644	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	346	8	no	—
128	C₄	#19149644	1	2	yes	—
129	C₄	#19149687	1	4	yes	—
130	C₄	C₂×Q₈×D₁₇₃	1	1	yes	characteristic
131	C₄	C₂×#14583825	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#19149644	1	1	yes	characteristic
134	C₂	C₂×#19149687	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26012048

C₂

yes

maximal, characteristic

#26011618

C₂

yes

maximal, complemented

#26011615

C₂

yes

maximal, complemented

#26011519

C₂

yes

maximal

#26011449

C₂

yes

maximal

C₂×C₄×#13063436

C₂

yes

maximal, characteristic

#423×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×#13063436

C₂²

yes

characteristic

C₂²×#13063436

C₂²

yes

characteristic

#19149679

C₄

yes

—

#19149671

C₄

yes

—

#19149658

C₄

yes

—

C₄×Dic₃₄₆

C₂²

yes

—

#19149638

C₂²

yes

—

#19149608

C₂²

yes

—

#19149589

C₂²

yes

—

C₄×#13063436

C₂²

yes

—

C₄×#13063436

C₂²

yes

—

C₈×Dic₁₇₃

C₂²

yes

—

C₂×M₁₆×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₃

C₂²

yes

—

C₂×Dic₃₄₆

C₂×C₄

yes

—

C₂×#13063436

—

C₂×#13063436

Q₈

yes

complemented

C₂×#13063436

C₂³

yes

—

C₂×#13063436

C₂³

yes

characteristic

C₂×#13063436

C₂³

yes

characteristic

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

#14583805

C₂×C₄

yes

—

C₂²⋊Dic₁₇₃

C₂×C₄

yes

—

C₄×Dic₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

M₁₆×C₁₇₃

—

C₂×C₈×C₁₇₃

C₂³

yes

—

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

C₄²×C₁₇₃

C₂³

yes

—

Dic₃₄₆

—

C₂×Dic₁₇₃

C₂²×C₄

yes

—

#13063436

—

#13063436

C₂×Q₈

yes

—

#13063436

C₂×Q₈

yes

—

C₂³×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂×C₄×C₁₇₃

C₂×Q₈

yes

—

C₂×C₄×C₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

C₂⁴

yes

characteristic

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₄×Q₈

yes

—

C₂²×C₁₇₃

C₂³×C₄

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

C₄×Q₈

yes

—

C₄×C₁₇₃

#131

yes

—

C₄×C₁₇₃

C₂²×Q₈

yes

characteristic

C₄×C₁₇₃

C₂×#41

yes

characteristic

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₇₃

C₂×#131

yes

—

C₁₇₃

#2960

yes

characteristic, complemented, Sylow

#2960

—

173

maximal, Sylow

#533

—

173

—

#446

—

173

—

C₈×Q₈

—

173

—

#424

—

173

—

#423

D₁₇₃

yes

characteristic

#405

—

173

—

C₂×C₄×C₈

—

173

—

C₂×M₁₆

C₂×D₁₇₃

yes

—

C₂²×C₈

—

173

—

C₂²×C₈

—

173

—

C₄⋊Q₈

—

173

—

#125

—

173

—

C₂²⋊Q₈

—

173

—

C₄×Q₈

—

173

—

#117

—

173

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

C₄⋊C₈

—

173

—

C₄⋊C₈

C₂×D₁₇₃

yes

—

#97

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₂×Q₈

—

173

—

C₂²×C₄

C₄×D₁₇₃

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

M₁₆

—

C₂×C₈

—

346

—

C₂×C₈

—

173

—

C₂×C₈

—

173

—

C₂×C₈

—

173

—

C₂×C₈

—

173

—

100

C₂×C₈

C₂²×D₁₇₃

yes

—

101

C₄⋊C₄

—

173

—

102

C₄⋊C₄

—

173

—

103

C₄⋊C₄

—

173

—

104

C₂²⋊C₄

—

173

—

105

C₄²

—

173

—

106

C₄²

C₄×D₁₇₃

yes

characteristic

107

C₄²

C₄×D₁₇₃

yes

characteristic

108

C₄²

C₂²×D₁₇₃

yes

—

109

C₂³

C₂×C₄×D₁₇₃

yes

characteristic

110

Q₈

—

346

—

111

C₂×C₄

—

112

C₂×C₄

—

173

—

113

C₂×C₄

C₂×C₄×D₁₇₃

yes

—

114

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

115

C₂×C₄

C₂×C₄×D₁₇₃

yes

—

116

C₂×C₄

Q₈×D₁₇₃

yes

—

117

C₂×C₄

#14583825

yes

—

118

C₂×C₄

C₂³×D₁₇₃

yes

characteristic

119

C₈

—

120

C₈

—

346

—

121

C₈

—

173

—

122

C₈

—

173

—

123

C₂²

—

124

C₂²

#19149644

yes

—

125

C₂²

C₂²×C₄×D₁₇₃

yes

characteristic

126

C₄

—

127

C₄

—

346

—

128

C₄

#19149644

yes

—

129

C₄

#19149687

yes

—

130

C₄

C₂×Q₈×D₁₇₃

yes

characteristic

131

C₄

C₂×#14583825

yes

characteristic

132

C₂

—

133

C₂

C₂×#19149644

yes

characteristic

134

C₂

C₂×#19149687

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇₃
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting	#423×C₁₇₃	C₂
prime radical	C₂³×C₁₇₃	C₂²×C₄
cosocle	C₂×C₄×C₁₇₃	C₂⁴
hypercenter	#423	D₁₇₃
hypocenter	C₁₇₃	#2960
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₇₃

commutator, socle

C₂²×C₁₇₃

C₂³×C₄

Fitting

#423×C₁₇₃

C₂

prime radical

C₂³×C₁₇₃

C₂²×C₄

cosocle

C₂×C₄×C₁₇₃

C₂⁴

D₁₇₃

C₁₇₃

G ▷ C₂²×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #26011618	C₂	semidirect
3. #26011615	C₂	semidirect
25. C₂×#13063436	Q₈	semidirect
67. C₁₇₃	#2960	semidirect

H ≅ G/N

type

2. #26011618

C₂

semidirect

3. #26011615

C₂

semidirect

25. C₂×#13063436

Q₈

semidirect

67. C₁₇₃

#2960

semidirect

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

subgroup	index
#26012048	2
#26011618	2
#26011615	2
#26011519	2
#26011449	2
C₂×C₄×#13063436	2
#423×C₁₇₃	2
#2960	173

subgroup

index

#423×C₁₇₃

173

Subgroups — #36362493

Conjugacy classes of subgroups

Complements

Maximal subgroups