Group #36361830 — subgroups

Subgroups — #36361830

Order 22144.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#26011433	C₂	1	3	yes	maximal
2	#26011425	C₂	1	3	yes	maximal
3	#26011423	C₂	1	3	yes	maximal
4	#26011412	C₂	1	3	yes	maximal
5	#26011361	C₂	1	1	yes	maximal, characteristic
6	#26011359	C₂	1	1	yes	maximal, characteristic
7	#383×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₄₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇₃	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₁₇₃	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₁₇₃	C₂²	1	3	yes	—
12	C₂×#14583805	C₂²	1	3	yes	—
13	C₂×#14583805	C₂²	1	3	yes	—
14	#19149586	C₂²	1	3	yes	—
15	#19149586	C₂²	1	3	yes	—
16	#19149586	C₂²	1	3	yes	—
17	#19149586	C₂²	1	3	yes	—
18	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
19	#95×C₁₇₃	C₂²	1	3	yes	—
20	#95×C₁₇₃	C₂²	1	3	yes	—
21	C₂×Dic₃₄₆	—	4	3	no	—
22	C₂²×Dic₁₇₃	C₂³	1	3	yes	—
23	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₇₃	C₂³	1	3	yes	—
26	C₄⋊Dic₁₇₃	—	4	3	no	—
27	C₄⋊Dic₁₇₃	—	4	3	no	—
28	C₄⋊Dic₁₇₃	—	4	3	no	—
29	#14583805	—	4	3	no	—
30	#14583805	—	4	3	no	—
31	C₂²×C₄×C₁₇₃	C₂³	1	3	yes	—
32	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₇₃	C₂³	1	3	yes	—
34	C₄²×C₁₇₃	Q₈	1	4	yes	complemented
35	Dic₃₄₆	—	4	4	no	—
36	C₂×Dic₁₇₃	—	2	3	no	—
37	C₂×Dic₁₇₃	—	2	3	no	—
38	C₂×Dic₁₇₃	—	2	3	no	—
39	C₂×Dic₁₇₃	—	2	3	no	—
40	C₂×Dic₁₇₃	—	2	3	no	—
41	C₂×Dic₁₇₃	—	2	3	no	—
42	C₂×Dic₁₇₃	—	2	6	no	—
43	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₁₇₃	—	2	3	no	—
45	C₂×C₄×C₁₇₃	—	2	3	no	—
46	C₂×C₄×C₁₇₃	—	2	3	no	—
47	C₂×C₄×C₁₇₃	—	2	3	no	—
48	C₂×C₄×C₁₇₃	—	2	6	no	—
49	C₂×C₄×C₁₇₃	C₂×Q₈	1	6	yes	—
50	C₂²×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₇₃	H(5,𝔽₂)	1	3	yes	—
52	C₂²×C₁₇₃	2₋¹⁺⁴	1	3	yes	—
53	C₄×C₁₇₃	—	2	6	no	—
54	C₄×C₁₇₃	—	4	1	no	—
55	C₄×C₁₇₃	—	4	3	no	—
56	Dic₁₇₃	—	4	3	no	—
57	Dic₁₇₃	—	4	1	no	—
58	Dic₁₇₃	—	4	1	no	—
59	Dic₁₇₃	—	4	3	no	—
60	C₂×C₁₇₃	#544	1	3	yes	—
61	C₂×C₁₇₃	#560	1	1	yes	characteristic
62	C₂×C₁₇₃	#563	1	3	yes	—
63	C₁₇₃	#2843	1	1	yes	characteristic, complemented, Sylow
64	#2843	—	173	1	no	maximal, Sylow
65	#400	—	173	3	no	—
66	#398	—	173	3	no	—
67	#398	—	173	3	no	—
68	#395	—	173	3	no	—
69	#384	—	173	1	no	—
70	#383	—	173	1	no	—
71	#383	D₁₇₃	1	1	yes	characteristic
72	C₂²×Q₈	—	173	1	no	—
73	C₂×C₄⋊C₄	—	173	3	no	—
74	C₂×C₄⋊C₄	—	173	3	no	—
75	C₂×C₄⋊C₄	—	173	3	no	—
76	C₂×C₄⋊C₄	—	173	3	no	—
77	C₂×C₄⋊C₄	—	173	3	no	—
78	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
79	#95	—	173	3	no	—
80	#95	—	173	3	no	—
81	#95	—	173	3	no	—
82	#95	—	173	3	no	—
83	#95	C₂×D₁₇₃	1	3	yes	—
84	#95	C₂×D₁₇₃	1	3	yes	—
85	C₂×Q₈	—	692	3	no	—
86	C₂²×C₄	—	173	3	no	—
87	C₂²×C₄	—	173	1	no	—
88	C₂²×C₄	—	173	1	no	—
89	C₂²×C₄	—	173	3	no	—
90	C₂²×C₄	C₂²×D₁₇₃	1	3	yes	—
91	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₁₇₃	1	3	yes	—
93	C₄⋊C₄	—	692	3	no	—
94	C₄⋊C₄	—	692	3	no	—
95	C₄⋊C₄	—	692	3	no	—
96	C₄⋊C₄	—	692	3	no	—
97	C₄⋊C₄	—	692	3	no	—
98	C₄²	Dic₃₄₆	1	4	yes	complemented
99	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
100	Q₈	—	692	4	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	346	3	no	—
107	C₂×C₄	—	346	3	no	—
108	C₂×C₄	—	346	3	no	—
109	C₂×C₄	—	346	3	no	—
110	C₂×C₄	—	346	3	no	—
111	C₂×C₄	—	346	3	no	—
112	C₂×C₄	—	346	6	no	—
113	C₂×C₄	C₂×Dic₃₄₆	1	6	yes	—
114	C₂²	C₂²×Dic₃₄₆	1	1	yes	characteristic
115	C₂²	#19149725	1	3	yes	—
116	C₂²	#19149726	1	3	yes	—
117	C₄	—	2	6	no	—
118	C₄	—	4	1	no	—
119	C₄	—	4	3	no	—
120	C₄	—	692	1	no	—
121	C₄	—	692	1	no	—
122	C₄	—	692	3	no	—
123	C₄	—	692	3	no	—
124	C₂	#26012119	1	3	yes	—
125	C₂	#26012232	1	1	yes	characteristic
126	C₂	#26012243	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26011433

C₂

yes

maximal

#26011425

C₂

yes

maximal

#26011423

C₂

yes

maximal

#26011412

C₂

yes

maximal

#26011361

C₂

yes

maximal, characteristic

#26011359

C₂

yes

maximal, characteristic

#383×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₄₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×#14583805

C₂²

yes

—

C₂×#14583805

C₂²

yes

—

#19149586

C₂²

yes

—

#19149586

C₂²

yes

—

#19149586

C₂²

yes

—

#19149586

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

#95×C₁₇₃

C₂²

yes

—

#95×C₁₇₃

C₂²

yes

—

C₂×Dic₃₄₆

—

C₂²×Dic₁₇₃

C₂³

yes

—

C₂²×Dic₁₇₃

C₂³

yes

characteristic

C₂²×Dic₁₇₃

C₂³

yes

characteristic

C₂²×Dic₁₇₃

C₂³

yes

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

#14583805

—

#14583805

—

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₄²×C₁₇₃

Q₈

yes

complemented

Dic₃₄₆

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂³×C₁₇₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×Q₈

yes

—

C₂²×C₁₇₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₇₃

H(5,𝔽₂)

yes

—

C₂²×C₁₇₃

2₋¹⁺⁴

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

#544

yes

—

C₂×C₁₇₃

#560

yes

characteristic

C₂×C₁₇₃

#563

yes

—

C₁₇₃

#2843

yes

characteristic, complemented, Sylow

#2843

—

173

maximal, Sylow

#400

—

173

—

#398

—

173

—

#398

—

173

—

#395

—

173

—

#384

—

173

—

#383

—

173

—

#383

D₁₇₃

yes

characteristic

C₂²×Q₈

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

#95

—

173

—

#95

—

173

—

#95

—

173

—

#95

—

173

—

#95

C₂×D₁₇₃

yes

—

#95

C₂×D₁₇₃

yes

—

C₂×Q₈

—

692

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

C₂²×D₁₇₃

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂²×C₄

C₂²×D₁₇₃

yes

—

C₄⋊C₄

—

692

—

C₄⋊C₄

—

692

—

C₄⋊C₄

—

692

—

C₄⋊C₄

—

692

—

C₄⋊C₄

—

692

—

C₄²

Dic₃₄₆

yes

complemented

C₂³

C₂³×D₁₇₃

yes

characteristic

100

Q₈

—

692

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

346

—

107

C₂×C₄

—

346

—

108

C₂×C₄

—

346

—

109

C₂×C₄

—

346

—

110

C₂×C₄

—

346

—

111

C₂×C₄

—

346

—

112

C₂×C₄

—

346

—

113

C₂×C₄

C₂×Dic₃₄₆

yes

—

114

C₂²

C₂²×Dic₃₄₆

yes

characteristic

115

C₂²

#19149725

yes

—

116

C₂²

#19149726

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

692

—

121

C₄

—

692

—

122

C₄

—

692

—

123

C₄

—

692

—

124

C₂

#26012119

yes

—

125

C₂

#26012232

yes

characteristic

126

C₂

#26012243

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇₃
commutator, socle, prime radical, cosocle	C₂³×C₁₇₃	C₂⁴
Fitting	#383×C₁₇₃	C₂
hypercenter	#383	D₁₇₃
hypocenter	C₁₇₃	#2843
LCS	G ▷ C₂³×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇₃

commutator, socle, prime radical, cosocle

C₂³×C₁₇₃

C₂⁴

Fitting

#383×C₁₇₃

C₂

D₁₇₃

C₁₇₃

G ▷ C₂³×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
34. C₄²×C₁₇₃	Q₈	semidirect
63. C₁₇₃	#2843	semidirect
98. C₄²	Dic₃₄₆	semidirect

H ≅ G/N

type

34. C₄²×C₁₇₃

Q₈

semidirect

63. C₁₇₃

#2843

semidirect

98. C₄²

Dic₃₄₆

semidirect

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

subgroup	index
#26011433	2
#26011425	2
#26011423	2
#26011412	2
#26011361	2
#26011359	2
#383×C₁₇₃	2
#2843	173

subgroup

index

#383×C₁₇₃

173

Subgroups — #36361830

Conjugacy classes of subgroups

Complements

Maximal subgroups