Group #36352684 — subgroups

Subgroups — #36352684

Order 22144.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#26011863	C₂	1	1	yes	maximal, characteristic
2	#26011786	C₂	1	2	yes	maximal, complemented
3	#26011235	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#13063436	C₂	1	2	yes	maximal, complemented
5	#559×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583814	—	2	2	no	—
7	#19149679	C₂²	1	1	yes	characteristic
8	#19149612	—	2	2	no	—
9	C₄⋊#13063436	C₂²	1	2	yes	—
10	#19149603	—	2	2	no	—
11	C₄×#13063436	C₂²	1	1	yes	characteristic
12	#127×C₁₇₃	C₂²	1	1	yes	characteristic
13	#124×C₁₇₃	—	2	2	no	—
14	#123×C₁₇₃	—	2	2	no	—
15	C₄×Q₈×C₁₇₃	C₂²	1	2	yes	—
16	C₄×D₄×C₁₇₃	—	2	2	no	—
17	C₂×Dic₃₄₆	—	2	1	no	—
18	#14583814	—	4	4	no	—
19	C₂×#13063436	—	2	1	no	—
20	C₂×#13063436	—	2	2	no	—
21	C₄⋊Dic₁₇₃	—	2	2	no	—
22	C₂×Q₈×C₁₇₃	D₄	1	2	yes	—
23	C₂×D₄×C₁₇₃	—	2	2	no	—
24	C₂×D₄×C₁₇₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₇₃	—	2	2	no	—
26	C₄⋊C₄×C₁₇₃	—	2	2	no	—
27	C₄⋊C₄×C₁₇₃	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₇₃	—	2	4	no	—
29	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
30	C₄²×C₁₇₃	—	2	2	no	—
31	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
32	Dic₃₄₆	—	4	2	no	—
33	C₂×Dic₁₇₃	—	4	1	no	—
34	#13063436	—	2	2	no	—
35	#13063436	—	4	2	no	—
36	C₂³×C₁₇₃	—	2	2	no	—
37	Q₈×C₁₇₃	—	2	2	no	—
38	Q₈×C₁₇₃	—	2	2	no	—
39	D₄×C₁₇₃	—	4	2	no	—
40	D₄×C₁₇₃	—	4	2	no	—
41	D₄×C₁₇₃	—	4	2	no	—
42	C₂×C₄×C₁₇₃	—	2	2	no	—
43	C₂×C₄×C₁₇₃	—	2	2	no	—
44	C₂×C₄×C₁₇₃	—	8	2	no	—
45	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
47	C₂²×C₁₇₃	—	4	4	no	—
48	C₂²×C₁₇₃	—	4	2	no	—
49	C₂²×C₁₇₃	#120	1	1	yes	characteristic
50	C₄×C₁₇₃	—	2	1	no	—
51	C₄×C₁₇₃	—	4	2	no	—
52	C₄×C₁₇₃	—	4	2	no	—
53	C₄×C₁₇₃	#135	1	2	yes	—
54	C₄×C₁₇₃	#137	1	2	yes	—
55	Dic₁₇₃	—	8	1	no	—
56	C₂×C₁₇₃	—	8	2	no	—
57	C₂×C₁₇₃	#452	1	2	yes	—
58	C₂×C₁₇₃	#453	1	1	yes	characteristic
59	C₁₇₃	#1630	1	1	yes	characteristic, complemented, Sylow
60	#1630	—	173	1	no	maximal, Sylow
61	#559	D₁₇₃	1	1	yes	characteristic
62	#492	—	173	1	no	—
63	#477	—	173	2	no	—
64	#331	—	173	1	no	—
65	Q₈⋊C₈	—	173	2	no	—
66	C₂×SD₁₆	—	346	2	no	—
67	C₄⋊Q₈	—	173	1	no	—
68	#127	C₂×D₁₇₃	1	1	yes	characteristic
69	#124	—	2	2	no	—
70	#123	—	2	2	no	—
71	C₄×Q₈	C₂×D₁₇₃	1	2	yes	—
72	C₄×D₄	—	2	2	no	—
73	#107	—	346	2	no	—
74	C₄⋊C₈	—	173	2	no	—
75	#103	—	346	2	no	—
76	C₄×C₈	—	173	1	no	—
77	C₂×Q₈	—	346	1	no	—
78	C₂×Q₈	C₂²⋊D₁₇₃	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₁₇₃	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	SD₁₆	—	692	4	no	—
83	C₂×C₈	—	346	1	no	—
84	C₂×C₈	—	346	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	346	2	no	—
87	C₄⋊C₄	C₂²⋊D₁₇₃	1	2	yes	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₄²	—	2	2	no	—
91	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	692	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	692	1	no	—
103	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
105	C₈	—	346	2	no	—
106	C₈	—	692	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#19149648	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	1384	1	no	—
114	C₄	#19149699	1	2	yes	—
115	C₄	#19149709	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#26011653	1	2	yes	—
118	C₂	#26011659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26011863

C₂

yes

maximal, characteristic

#26011786

C₂

yes

maximal, complemented

#26011235

C₂

yes

maximal, characteristic

Q₈⋊#13063436

C₂

yes

maximal, complemented

#559×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583814

—

#19149679

C₂²

yes

characteristic

#19149612

—

C₄⋊#13063436

C₂²

yes

—

#19149603

—

C₄×#13063436

C₂²

yes

characteristic

#127×C₁₇₃

C₂²

yes

characteristic

#124×C₁₇₃

—

#123×C₁₇₃

—

C₄×Q₈×C₁₇₃

C₂²

yes

—

C₄×D₄×C₁₇₃

—

C₂×Dic₃₄₆

—

#14583814

—

C₂×#13063436

—

C₂×#13063436

—

C₄⋊Dic₁₇₃

—

C₂×Q₈×C₁₇₃

D₄

yes

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

D₄

yes

—

C₂²⋊C₄×C₁₇₃

—

C₂²⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

—

C₄²×C₁₇₃

C₂³

yes

characteristic

Dic₃₄₆

—

C₂×Dic₁₇₃

—

#13063436

—

#13063436

—

C₂³×C₁₇₃

—

Q₈×C₁₇₃

—

Q₈×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#120

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#135

yes

—

C₄×C₁₇₃

#137

yes

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#452

yes

—

C₂×C₁₇₃

#453

yes

characteristic

C₁₇₃

#1630

yes

characteristic, complemented, Sylow

#1630

—

173

maximal, Sylow

#559

D₁₇₃

yes

characteristic

#492

—

173

—

#477

—

173

—

#331

—

173

—

Q₈⋊C₈

—

173

—

C₂×SD₁₆

—

346

—

C₄⋊Q₈

—

173

—

#127

C₂×D₁₇₃

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₇₃

yes

—

C₄×D₄

—

#107

—

346

—

C₄⋊C₈

—

173

—

#103

—

346

—

C₄×C₈

—

173

—

C₂×Q₈

—

346

—

C₂×Q₈

C₂²⋊D₁₇₃

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₇₃

yes

—

C₂²×C₄

—

SD₁₆

—

692

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₄⋊C₄

—

C₄⋊C₄

—

346

—

C₄⋊C₄

C₂²⋊D₁₇₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₇₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

692

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

692

—

103

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

105

C₈

—

346

—

106

C₈

—

692

—

107

C₂²

—

108

C₂²

—

109

C₂²

#19149648

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

1384

—

114

C₄

#19149699

yes

—

115

C₄

#19149709

yes

—

116

C₂

—

117

C₂

#26011653

yes

—

118

C₂

#26011659

yes

characteristic

	subgroup	G/H
center	C₂²	#19149648
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#559×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#120
prime radical	#127×C₁₇₃	C₂²
hypercenter	#559	D₁₇₃
hypocenter	C₁₇₃	#1630
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#19149648

commutator, cosocle

C₄²×C₁₇₃

C₂³

Frattini

C₄²

C₂²×D₁₇₃

Fitting

#559×C₁₇₃

C₂

socle

C₂²×C₁₇₃

#120

prime radical

#127×C₁₇₃

C₂²

D₁₇₃

C₁₇₃

G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #26011786	C₂	semidirect
4. Q₈⋊#13063436	C₂	semidirect
59. C₁₇₃	#1630	semidirect

H ≅ G/N

type

2. #26011786

C₂

semidirect

4. Q₈⋊#13063436

C₂

semidirect

59. C₁₇₃

#1630

semidirect

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

subgroup	index
#26011863	2
#26011786	2
#26011235	2
Q₈⋊#13063436	2
#559×C₁₇₃	2
#1630	173

subgroup

index

#559×C₁₇₃

173

Subgroups — #36352684

Conjugacy classes of subgroups

Complements

Maximal subgroups