Group #36365367 — subgroups

Subgroups — #36365367

Order 22144.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#26012304	C₂	1	4	yes	maximal, complemented
2	#26011763	C₂	1	4	yes	maximal, complemented
3	#26011673	C₂	1	4	yes	maximal, complemented
4	#26011609	C₂	1	2	yes	maximal, complemented
5	#494×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19149727	C₂²	1	4	yes	—
7	#19149711	—	2	8	no	—
8	#19149710	C₂²	1	8	yes	—
9	#19149702	—	2	4	no	—
10	#19149701	C₂²	1	2	yes	—
11	Q₁₆×D₁₇₃	—	2	4	no	—
12	#19149685	C₂²	1	4	yes	—
13	#19149615	C₂²	1	2	yes	—
14	#19149607	C₂²	1	4	yes	—
15	#19149598	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₇₃	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
18	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
19	#14583827	—	2	8	no	—
20	#14583826	—	4	4	no	—
21	#14583826	C₂³	1	4	yes	—
22	Q₈×D₁₇₃	—	2	8	no	—
23	Dic₆₉₂	D₄	1	2	yes	—
24	#14583816	—	2	8	no	—
25	#14583815	D₄	1	4	yes	—
26	#14583813	—	2	8	no	—
27	D₁₃₈₄	D₄	1	2	yes	—
28	#14583810	C₂³	1	4	yes	—
29	#14583809	—	2	4	no	—
30	C₈×D₁₇₃	—	2	4	no	—
31	C₂×Q₈×C₁₇₃	C₂³	1	4	yes	—
32	Q₁₆×C₁₇₃	—	2	4	no	—
33	Q₁₆×C₁₇₃	—	2	4	no	—
34	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₇₃	—	4	2	no	—
36	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
37	Dic₃₄₆	—	4	4	no	—
38	Dic₃₄₆	C₂×D₄	1	4	yes	—
39	D₆₉₂	—	4	4	no	—
40	D₆₉₂	C₂×D₄	1	4	yes	—
41	C₂²⋊D₁₇₃	—	2	4	no	—
42	C₄×D₁₇₃	—	2	4	no	—
43	C₄×D₁₇₃	—	4	8	no	—
44	#13063436	—	2	4	no	—
45	Q₈×C₁₇₃	—	2	8	no	—
46	C₂×C₄×C₁₇₃	—	2	1	no	—
47	C₂×C₄×C₁₇₃	—	2	4	no	—
48	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₇₃	C₂×D₄	1	2	yes	—
50	C₈×C₁₇₃	C₂×D₄	1	2	yes	—
51	C₂²×C₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₇₃	—	2	4	no	—
53	C₄×C₁₇₃	—	2	2	no	—
54	C₄×C₁₇₃	—	4	4	no	—
55	C₄×C₁₇₃	C₂²×D₄	1	2	yes	—
56	Dic₁₇₃	—	2	4	no	—
57	C₂×C₁₇₃	—	2	1	no	—
58	C₂×C₁₇₃	D₄²	1	1	yes	characteristic
59	D₁₇₃	—	4	4	no	—
60	C₁₇₃	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	173	1	no	maximal, Sylow
62	#578	—	173	4	no	—
63	#494	D₁₇₃	1	1	yes	characteristic, complemented
64	#473	—	173	4	no	—
65	#456	—	173	4	no	—
66	#443	—	173	2	no	—
67	2₋¹⁺⁴	—	173	4	no	—
68	#137	—	346	8	no	—
69	#137	—	173	8	no	—
70	#135	—	346	4	no	—
71	#135	—	173	2	no	—
72	C₂×Q₁₆	—	346	4	no	—
73	C₂×Q₁₆	C₂×D₁₇₃	1	4	yes	—
74	#131	—	173	4	no	—
75	C₄⋊Q₈	C₂×D₁₇₃	1	2	yes	—
76	#108	—	173	2	no	—
77	#104	—	173	4	no	—
78	#101	—	173	4	no	—
79	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
80	#41	—	346	8	no	—
81	#41	—	173	4	no	—
82	#41	—	692	4	no	—
83	C₂×Q₈	—	346	8	no	—
84	C₂×Q₈	C₂²×D₁₇₃	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	346	8	no	—
88	Q₁₆	—	173	2	no	—
89	SD₁₆	—	346	8	no	—
90	SD₁₆	—	173	4	no	—
91	D₈	—	173	2	no	—
92	M₁₆	—	346	4	no	—
93	M₁₆	—	173	4	no	—
94	C₂×C₈	—	346	4	no	—
95	C₂×C₈	C₂²×D₁₇₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	173	4	no	—
100	Q₈	—	692	4	no	—
101	D₄	—	346	4	no	—
102	D₄	—	173	4	no	—
103	D₄	—	692	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	346	4	no	—
107	C₂×C₄	—	692	8	no	—
108	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
109	C₈	—	346	4	no	—
110	C₈	D₄×D₁₇₃	1	2	yes	—
111	C₈	D₄×D₁₇₃	1	2	yes	—
112	C₂²	—	346	4	no	—
113	C₂²	#19149724	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	346	4	no	—
117	C₄	C₂×D₄×D₁₇₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	692	4	no	—
120	C₂	#26012126	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26012304

C₂

yes

maximal, complemented

#26011763

C₂

yes

maximal, complemented

#26011673

C₂

yes

maximal, complemented

#26011609

C₂

yes

maximal, complemented

#494×C₁₇₃

C₂

yes

maximal, characteristic, complemented

#19149727

C₂²

yes

—

#19149711

—

#19149710

C₂²

yes

—

#19149702

—

#19149701

C₂²

yes

—

Q₁₆×D₁₇₃

—

#19149685

C₂²

yes

—

#19149615

C₂²

yes

—

#19149607

C₂²

yes

—

#19149598

C₂²

yes

—

C₂×Q₁₆×C₁₇₃

C₂²

yes

—

C₄⋊Q₈×C₁₇₃

C₂²

yes

—

C₄×C₈×C₁₇₃

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₇₃

—

Dic₆₉₂

D₄

yes

—

#14583816

—

#14583815

D₄

yes

—

#14583813

—

D₁₃₈₄

D₄

yes

—

#14583810

C₂³

yes

—

#14583809

—

C₈×D₁₇₃

—

C₂×Q₈×C₁₇₃

C₂³

yes

—

Q₁₆×C₁₇₃

—

Q₁₆×C₁₇₃

—

C₂×C₈×C₁₇₃

C₂³

yes

—

C₄⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

C₂³

yes

characteristic

Dic₃₄₆

—

Dic₃₄₆

C₂×D₄

yes

—

D₆₉₂

—

D₆₉₂

C₂×D₄

yes

—

C₂²⋊D₁₇₃

—

C₄×D₁₇₃

—

C₄×D₁₇₃

—

#13063436

—

Q₈×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂⁴

yes

characteristic

C₈×C₁₇₃

C₂×D₄

yes

—

C₈×C₁₇₃

C₂×D₄

yes

—

C₂²×C₁₇₃

H(5,𝔽₂)

yes

characteristic

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂²×D₄

yes

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

D₄²

yes

characteristic

D₁₇₃

—

C₁₇₃

#3293

yes

characteristic, complemented, Sylow

#3293

—

173

maximal, Sylow

#578

—

173

—

#494

D₁₇₃

yes

characteristic, complemented

#473

—

173

—

#456

—

173

—

#443

—

173

—

2₋¹⁺⁴

—

173

—

#137

—

346

—

#137

—

173

—

#135

—

346

—

#135

—

173

—

C₂×Q₁₆

—

346

—

C₂×Q₁₆

C₂×D₁₇₃

yes

—

#131

—

173

—

C₄⋊Q₈

C₂×D₁₇₃

yes

—

#108

—

173

—

#104

—

173

—

#101

—

173

—

C₄×C₈

C₂×D₁₇₃

yes

characteristic

#41

—

346

—

#41

—

173

—

#41

—

692

—

C₂×Q₈

—

346

—

C₂×Q₈

C₂²×D₁₇₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

346

—

Q₁₆

—

173

—

SD₁₆

—

346

—

SD₁₆

—

173

—

D₈

—

173

—

M₁₆

—

346

—

M₁₆

—

173

—

C₂×C₈

—

346

—

C₂×C₈

C₂²×D₁₇₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₇₃

yes

characteristic

Q₈

—

Q₈

—

173

—

100

Q₈

—

692

—

101

D₄

—

346

—

102

D₄

—

173

—

103

D₄

—

692

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

346

—

107

C₂×C₄

—

692

—

108

C₂×C₄

C₂³×D₁₇₃

yes

characteristic

109

C₈

—

346

—

110

C₈

D₄×D₁₇₃

yes

—

111

C₈

D₄×D₁₇₃

yes

—

112

C₂²

—

346

—

113

C₂²

#19149724

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

346

—

117

C₄

C₂×D₄×D₁₇₃

yes

—

118

C₂

—

119

C₂

—

692

—

120

C₂

#26012126

yes

characteristic

	subgroup	G/H
center	C₂	#26012126
commutator, cosocle	C₂×C₄×C₁₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	#494×C₁₇₃	C₂
socle	C₂×C₁₇₃	D₄²
hypercenter	#494	D₁₇₃
hypocenter	C₁₇₃	#3293
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#26012126

commutator, cosocle

C₂×C₄×C₁₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇₃

Fitting

#494×C₁₇₃

C₂

socle

C₂×C₁₇₃

D₄²

D₁₇₃

C₁₇₃

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #26012304	C₂	semidirect
2. #26011763	C₂	semidirect
3. #26011673	C₂	semidirect
4. #26011609	C₂	semidirect
5. #494×C₁₇₃	C₂	semidirect
60. C₁₇₃	#3293	semidirect
63. #494	D₁₇₃	semidirect

H ≅ G/N

type

1. #26012304

C₂

semidirect

2. #26011763

C₂

semidirect

3. #26011673

C₂

semidirect

4. #26011609

C₂

semidirect

5. #494×C₁₇₃

C₂

semidirect

60. C₁₇₃

#3293

semidirect

63. #494

D₁₇₃

semidirect

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

subgroup	index
#26012304	2
#26011763	2
#26011673	2
#26011609	2
#494×C₁₇₃	2
#3293	173

subgroup

index

#494×C₁₇₃

173

Subgroups — #36365367

Conjugacy classes of subgroups

Complements

Maximal subgroups