Group #36369750 — subgroups

Subgroups — #36369750

Order 22144.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#26012495	C₂	1	1	yes	maximal, characteristic
2	#26012427	C₂	1	1	yes	maximal, characteristic
3	#26012361	C₂	1	4	yes	maximal
4	#26012047	C₂	1	1	yes	maximal, characteristic
5	C₂×#14583836	C₂²	1	2	yes	—
6	C₂×C₄×#10621924	C₂²	1	2	yes	—
7	C₄⋊#13063443	—	2	2	no	—
8	(C₂×#10621924)⋊C₄	—	2	2	no	—
9	#19149678	C₄	1	2	yes	complemented
10	#19149671	C₄	1	1	yes	characteristic, complemented
11	#19149657	—	2	2	no	—
12	#19149644	—	2	2	no	—
13	#19149637	C₂²	1	2	yes	—
14	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
15	#125×C₁₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#10621924	—	2	2	no	—
17	#14583836	—	2	4	no	—
18	C₂×#13063443	—	2	2	no	—
19	C₄×#10621924	—	2	2	no	—
20	C₄×#10621924	Q₈	1	4	yes	complemented
21	C₂×Dic₃₄₆	—	2	2	no	—
22	C₂×C₄×D₁₇₃	D₄	1	2	yes	—
23	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
25	#14583805	—	2	2	no	—
26	#14583805	—	2	2	no	—
27	#14583803	—	2	2	no	—
28	C₄×Dic₁₇₃	—	2	2	no	—
29	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₇₃	D₄	1	2	yes	—
31	C₄⋊C₄×C₁₇₃	—	2	2	no	—
32	C₄⋊C₄×C₁₇₃	C₂×C₄	1	2	yes	—
33	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#10621924	—	2	4	no	—
35	C₂×#10621924	—	4	4	no	—
36	#13063443	—	4	2	no	—
37	C₂²×D₁₇₃	C₂×D₄	1	1	yes	characteristic
38	Dic₃₄₆	—	4	4	no	—
39	C₂×Dic₁₇₃	—	2	2	no	—
40	C₂×Dic₁₇₃	C₂²⋊C₄	1	2	yes	—
41	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
42	C₄×D₁₇₃	—	4	2	no	—
43	C₄×D₁₇₃	C₂×Q₈	1	2	yes	—
44	C₄×D₁₇₃	#41	1	2	yes	—
45	C₂×C₄×C₁₇₃	—	2	2	no	—
46	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₁₇₃	—	2	2	no	—
50	C₂×D₁₇₃	C₂²⋊Q₈	1	2	yes	—
51	#10621924	—	4	4	no	—
52	C₄×C₁₇₃	—	2	2	no	—
53	C₄×C₁₇₃	—	4	2	no	—
54	C₄×C₁₇₃	C₄×Q₈	1	2	yes	—
55	Dic₁₇₃	—	2	2	no	—
56	Dic₁₇₃	—	4	2	no	—
57	Dic₁₇₃	#124	1	1	yes	characteristic
58	Dic₁₇₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₁₇₃	#391	1	1	yes	characteristic
60	C₂×C₁₇₃	#421	1	2	yes	—
61	D₁₇₃	—	2	2	no	—
62	C₁₇₃	#1995	1	1	yes	characteristic, complemented, Sylow
63	#1995	—	173	1	no	maximal, Sylow
64	#533	—	173	1	no	—
65	#423	—	173	1	no	—
66	#376	—	173	1	no	—
67	#339	—	173	4	no	—
68	C₂×M₁₆	—	173	2	no	—
69	C₄⋊Q₈	—	173	2	no	—
70	#125	—	173	1	no	—
71	#125	#10621924	1	1	yes	characteristic, complemented
72	C₂²⋊Q₈	—	346	2	no	—
73	C₄×Q₈	—	346	2	no	—
74	#117	—	173	2	no	—
75	C₂×C₄²	—	173	2	no	—
76	C₂×C₄²	—	173	1	no	—
77	C₄⋊C₈	—	346	2	no	—
78	#95	—	346	2	no	—
79	C₂×Q₈	—	346	2	no	—
80	C₂²×C₄	—	346	2	no	—
81	C₂²×C₄	—	173	1	no	—
82	C₂²×C₄	—	173	2	no	—
83	M₁₆	—	346	4	no	—
84	C₂×C₈	—	346	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	346	2	no	—
87	C₄⋊C₄	—	346	2	no	—
88	C₄⋊C₄	—	173	2	no	—
89	C₄⋊C₄	C₂×#10621924	1	2	yes	—
90	C₂²⋊C₄	—	346	2	no	—
91	C₄²	—	346	2	no	—
92	C₄²	—	346	2	no	—
93	C₄²	—	173	4	no	—
94	C₄²	—	173	2	no	—
95	C₄²	—	173	1	no	—
96	C₄²	C₂×#10621924	1	1	yes	characteristic
97	C₂³	—	173	1	no	—
98	Q₈	—	692	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	346	4	no	—
101	C₂×C₄	—	346	2	no	—
102	C₂×C₄	—	173	2	no	—
103	C₂×C₄	—	173	1	no	—
104	C₂×C₄	—	173	2	no	—
105	C₂×C₄	—	173	2	no	—
106	C₂×C₄	—	692	2	no	—
107	C₂×C₄	—	692	4	no	—
108	C₂×C₄	C₂²⋊#10621924	1	2	yes	—
109	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
110	C₈	—	692	2	no	—
111	C₂²	—	346	2	no	—
112	C₂²	—	173	2	no	—
113	C₂²	C₂×C₂²⋊#10621924	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	346	2	no	—
117	C₄	—	173	1	no	—
118	C₄	—	173	1	no	—
119	C₄	—	692	4	no	—
120	C₄	—	692	2	no	—
121	C₄	Q₈×#10621924	1	2	yes	—
122	C₂	—	346	2	no	—
123	C₂	(C₂×#10621924)⋊Q₈	1	1	yes	characteristic
124	C₂	#26012488	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26012495

C₂

yes

maximal, characteristic

#26012427

C₂

yes

maximal, characteristic

#26012361

C₂

yes

maximal

#26012047

C₂

yes

maximal, characteristic

C₂×#14583836

C₂²

yes

—

C₂×C₄×#10621924

C₂²

yes

—

C₄⋊#13063443

—

(C₂×#10621924)⋊C₄

—

#19149678

C₄

yes

complemented

#19149671

C₄

yes

characteristic, complemented

#19149657

—

#19149644

—

#19149637

C₂²

yes

—

C₄²×D₁₇₃

C₂²

yes

characteristic

#125×C₁₇₃

C₄

yes

characteristic, complemented

—

—

—

—

Q₈

yes

complemented

C₂×Dic₃₄₆

—

C₂×C₄×D₁₇₃

D₄

yes

—

C₂×C₄×D₁₇₃

C₂³

yes

characteristic

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

#14583805

—

#14583805

—

#14583803

—

C₄×Dic₁₇₃

—

C₄×Dic₁₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₇₃

D₄

yes

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

C₂×C₄

yes

—

—

—

C₂²×D₁₇₃

C₂×D₄

yes

characteristic

Dic₃₄₆

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

C₂²⋊C₄

yes

—

C₂×Dic₁₇₃

C₂×D₄

yes

characteristic

C₄×D₁₇₃

—

C₄×D₁₇₃

C₂×Q₈

yes

—

C₄×D₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇₃

—

C₂×D₁₇₃

C₂²⋊Q₈

yes

—

#10621924

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₄×Q₈

yes

—

Dic₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

#124

yes

characteristic

Dic₁₇₃

C₄⋊Q₈

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

—

D₁₇₃

—

C₁₇₃

#1995

yes

characteristic, complemented, Sylow

#1995

—

173

maximal, Sylow

#533

—

173

—

#423

—

173

—

#376

—

173

—

#339

—

173

—

C₂×M₁₆

—

173

—

C₄⋊Q₈

—

173

—

#125

—

173

—

#125

#10621924

yes

characteristic, complemented

C₂²⋊Q₈

—

346

—

C₄×Q₈

—

346

—

#117

—

173

—

C₂×C₄²

—

173

—

C₂×C₄²

—

173

—

C₄⋊C₈

—

346

—

#95

—

346

—

C₂×Q₈

—

346

—

C₂²×C₄

—

346

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

M₁₆

—

346

—

C₂×C₈

—

346

—

C₄⋊C₄

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

173

—

C₄⋊C₄

C₂×#10621924

yes

—

C₂²⋊C₄

—

346

—

C₄²

—

346

—

C₄²

—

346

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

C₂×#10621924

yes

characteristic

C₂³

—

173

—

Q₈

—

692

—

C₂×C₄

—

100

C₂×C₄

—

346

—

101

C₂×C₄

—

346

—

102

C₂×C₄

—

173

—

103

C₂×C₄

—

173

—

104

C₂×C₄

—

173

—

105

C₂×C₄

—

173

—

106

C₂×C₄

—

692

—

107

C₂×C₄

—

692

—

108

C₂×C₄

C₂²⋊#10621924

yes

—

109

C₂×C₄

C₂²×#10621924

yes

characteristic

110

C₈

—

692

—

111

C₂²

—

346

—

112

C₂²

—

173

—

113

C₂²

C₂×C₂²⋊#10621924

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

346

—

117

C₄

—

173

—

118

C₄

—

173

—

119

C₄

—

692

—

120

C₄

—

692

—

121

C₄

Q₈×#10621924

yes

—

122

C₂

—

346

—

123

C₂

(C₂×#10621924)⋊Q₈

yes

characteristic

124

C₂

#26012488

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#10621924
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10621924
Fitting	#125×C₁₇₃	C₄
socle	C₂²×C₁₇₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₇₃	C₂×D₄
cosocle	C₂×C₄×D₁₇₃	C₂³
hypercenter	#125	#10621924
hypocenter	C₁₇₃	#1995
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#10621924

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10621924

Fitting

#125×C₁₇₃

C₄

socle

C₂²×C₁₇₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₇₃

C₂×D₄

cosocle

C₂×C₄×D₁₇₃

C₂³

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #19149678	C₄	semidirect
10. #19149671	C₄	semidirect
15. #125×C₁₇₃	C₄	semidirect
20. C₄×#10621924	Q₈	semidirect
62. C₁₇₃	#1995	semidirect
71. #125	#10621924	semidirect

H ≅ G/N

type

9. #19149678

C₄

semidirect

10. #19149671

C₄

semidirect

15. #125×C₁₇₃

C₄

semidirect

20. C₄×#10621924

Q₈

semidirect

62. C₁₇₃

#1995

semidirect

71. #125

#10621924

semidirect

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

subgroup	index
#26012495	2
#26012427	2
#26012361	2
#26012047	2
#1995	173

subgroup

index

173

Subgroups — #36369750

Conjugacy classes of subgroups

Complements

Maximal subgroups