Group #36361920 — subgroups

Subgroups — #36361920

Order 22144.

752 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 213 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	C₂×#19149679	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19149676	C₂	1	1	yes	maximal, characteristic
3	C₂×#19149667	C₂	1	4	yes	maximal, complemented
4	#26011393	C₂	1	8	yes	maximal, complemented
5	C₄³×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₃₄₆	C₂²	1	2	yes	—
7	C₂²×D₆₉₂	C₂²	1	4	yes	—
8	C₂²×D₆₉₂	C₂²	1	2	yes	—
9	#19149679	—	2	4	no	—
10	#19149676	—	2	4	no	—
11	#19149667	—	2	16	no	—
12	C₂×C₄⋊Dic₁₇₃	C₂²	1	4	yes	—
13	C₂×#14583803	C₂²	1	16	yes	—
14	C₂×C₄²×C₁₇₃	C₂²	1	4	yes	—
15	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
17	C₂³×D₁₇₃	C₂³	1	4	yes	—
18	C₂×Dic₃₄₆	—	2	8	no	—
19	C₂×Dic₃₄₆	—	2	4	no	—
20	C₂×D₆₉₂	—	2	8	no	—
21	C₂×D₆₉₂	—	2	8	no	—
22	C₂×D₆₉₂	—	2	4	no	—
23	C₂×D₆₉₂	—	2	16	no	—
24	C₂²×Dic₁₇₃	C₂³	1	4	yes	—
25	C₄⋊Dic₁₇₃	—	2	8	no	—
26	#14583803	—	2	32	no	—
27	C₂²×C₄×C₁₇₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
30	C₄²×C₁₇₃	—	2	8	no	—
31	C₄²×C₁₇₃	D₄	1	4	yes	—
32	C₄²×C₁₇₃	D₄	1	8	yes	complemented
33	C₂²×D₁₇₃	—	2	4	no	—
34	C₂²×D₁₇₃	—	2	8	no	—
35	C₂²×D₁₇₃	—	2	16	no	—
36	Dic₃₄₆	—	4	8	no	—
37	D₆₉₂	—	4	16	no	—
38	D₆₉₂	—	4	8	no	—
39	C₂×Dic₁₇₃	—	2	8	no	—
40	C₂×Dic₁₇₃	—	2	4	no	—
41	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₁₇₃	—	2	8	no	—
43	C₂×C₄×C₁₇₃	C₂×D₄	1	8	yes	—
44	C₂×C₄×C₁₇₃	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₇₃	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₁₇₃	#41	1	8	yes	—
48	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₁₇₃	C₂²×D₄	1	2	yes	—
50	C₂²×C₁₇₃	C₂×#41	1	4	yes	—
51	C₂×D₁₇₃	—	4	8	no	—
52	C₂×D₁₇₃	—	4	4	no	—
53	C₂×D₁₇₃	—	4	16	no	—
54	C₄×C₁₇₃	—	2	8	no	—
55	C₄×C₁₇₃	#124	1	8	yes	complemented
56	C₄×C₁₇₃	#127	1	4	yes	—
57	Dic₁₇₃	—	4	4	no	—
58	C₂×C₁₇₃	C₂×#124	1	2	yes	—
59	C₂×C₁₇₃	C₂×#127	1	1	yes	characteristic
60	C₂×C₁₇₃	#532	1	4	yes	—
61	D₁₇₃	—	8	4	no	—
62	C₁₇₃	#2858	1	1	yes	characteristic, complemented, Sylow
63	#2858	—	173	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	173	1	no	—
65	C₂×#127	—	173	1	no	—
66	C₂×#124	—	173	4	no	—
67	#390	—	173	8	no	—
68	C₄³	D₁₇₃	1	1	yes	characteristic, complemented
69	C₂²×Q₈	—	173	2	no	—
70	C₂²×D₄	—	173	2	no	—
71	C₂²×D₄	—	173	4	no	—
72	C₄⋊Q₈	—	346	4	no	—
73	#127	—	346	4	no	—
74	#124	—	346	16	no	—
75	C₂×C₄⋊C₄	—	173	4	no	—
76	C₂×C₂²⋊C₄	—	173	16	no	—
77	C₂×C₄²	C₂×D₁₇₃	1	4	yes	—
78	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₁₇₃	1	2	yes	—
80	C₂⁴	—	173	4	no	—
81	C₂×Q₈	—	346	8	no	—
82	C₂×Q₈	—	346	4	no	—
83	C₂×D₄	—	346	8	no	—
84	C₂×D₄	—	346	8	no	—
85	C₂×D₄	—	346	4	no	—
86	C₂×D₄	—	346	16	no	—
87	C₂²×C₄	—	173	4	no	—
88	C₂²×C₄	C₂²×D₁₇₃	1	4	yes	—
89	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
91	C₄⋊C₄	—	346	8	no	—
92	C₂²⋊C₄	—	346	32	no	—
93	C₄²	—	2	8	no	—
94	C₄²	D₆₉₂	1	4	yes	—
95	C₄²	D₆₉₂	1	8	yes	complemented
96	C₂³	—	346	8	no	—
97	C₂³	—	346	4	no	—
98	C₂³	—	346	16	no	—
99	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
100	Q₈	—	692	8	no	—
101	D₄	—	692	16	no	—
102	D₄	—	692	8	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	346	8	no	—
105	C₂×C₄	—	346	4	no	—
106	C₂×C₄	C₂×D₆₉₂	1	8	yes	—
107	C₂×C₄	C₂×D₆₉₂	1	4	yes	—
108	C₂×C₄	C₂×D₆₉₂	1	2	yes	—
109	C₂×C₄	C₂×D₆₉₂	1	4	yes	—
110	C₂×C₄	#14583826	1	8	yes	—
111	C₂²	—	692	8	no	—
112	C₂²	—	692	4	no	—
113	C₂²	—	692	16	no	—
114	C₂²	C₂²×D₆₉₂	1	1	yes	characteristic
115	C₂²	C₂²×D₆₉₂	1	2	yes	—
116	C₂²	C₂×#14583826	1	4	yes	—
117	C₄	—	2	8	no	—
118	C₄	—	692	4	no	—
119	C₄	#19149667	1	8	yes	complemented
120	C₄	#19149676	1	4	yes	—
121	C₂	—	1384	4	no	—
122	C₂	C₂×#19149667	1	2	yes	—
123	C₂	C₂×#19149676	1	1	yes	characteristic
124	C₂	#26012041	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149679

C₂

yes

maximal, characteristic, complemented

C₂×#19149676

C₂

yes

maximal, characteristic

C₂×#19149667

C₂

yes

maximal, complemented

#26011393

C₂

yes

maximal, complemented

C₄³×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₃₄₆

C₂²

yes

—

C₂²×D₆₉₂

C₂²

yes

—

C₂²×D₆₉₂

C₂²

yes

—

#19149679

—

#19149676

—

#19149667

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×#14583803

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₂×C₄²×C₁₇₃

C₂²

yes

—

C₂³×D₁₇₃

C₂³

yes

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

C₂²×Dic₁₇₃

C₂³

yes

—

C₄⋊Dic₁₇₃

—

#14583803

—

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₄²×C₁₇₃

—

C₄²×C₁₇₃

D₄

yes

—

C₄²×C₁₇₃

D₄

yes

complemented

C₂²×D₁₇₃

—

C₂²×D₁₇₃

—

C₂²×D₁₇₃

—

Dic₃₄₆

—

D₆₉₂

—

D₆₉₂

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂³×C₁₇₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

#41

yes

—

C₂²×C₁₇₃

C₂²×D₄

yes

characteristic

C₂²×C₁₇₃

C₂²×D₄

yes

—

C₂²×C₁₇₃

C₂×#41

yes

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#124

yes

complemented

C₄×C₁₇₃

#127

yes

—

Dic₁₇₃

—

C₂×C₁₇₃

C₂×#124

yes

—

C₂×C₁₇₃

C₂×#127

yes

characteristic

C₂×C₁₇₃

#532

yes

—

D₁₇₃

—

C₁₇₃

#2858

yes

characteristic, complemented, Sylow

#2858

—

173

maximal, Sylow

C₂×C₄⋊Q₈

—

173

—

C₂×#127

—

173

—

C₂×#124

—

173

—

#390

—

173

—

C₄³

D₁₇₃

yes

characteristic, complemented

C₂²×Q₈

—

173

—

C₂²×D₄

—

173

—

C₂²×D₄

—

173

—

C₄⋊Q₈

—

346

—

#127

—

346

—

#124

—

346

—

C₂×C₄⋊C₄

—

173

—

C₂×C₂²⋊C₄

—

173

—

C₂×C₄²

C₂×D₁₇₃

yes

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

C₂×C₄²

C₂×D₁₇₃

yes

—

C₂⁴

—

173

—

C₂×Q₈

—

346

—

C₂×Q₈

—

346

—

C₂×D₄

—

346

—

C₂×D₄

—

346

—

C₂×D₄

—

346

—

C₂×D₄

—

346

—

C₂²×C₄

—

173

—

C₂²×C₄

C₂²×D₁₇₃

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂²×C₄

C₂²×D₁₇₃

yes

—

C₄⋊C₄

—

346

—

C₂²⋊C₄

—

346

—

C₄²

—

C₄²

D₆₉₂

yes

—

C₄²

D₆₉₂

yes

complemented

C₂³

—

346

—

C₂³

—

346

—

C₂³

—

346

—

C₂³

C₂³×D₁₇₃

yes

characteristic

100

Q₈

—

692

—

101

D₄

—

692

—

102

D₄

—

692

—

103

C₂×C₄

—

104

C₂×C₄

—

346

—

105

C₂×C₄

—

346

—

106

C₂×C₄

C₂×D₆₉₂

yes

—

107

C₂×C₄

C₂×D₆₉₂

yes

—

108

C₂×C₄

C₂×D₆₉₂

yes

—

109

C₂×C₄

C₂×D₆₉₂

yes

—

110

C₂×C₄

#14583826

yes

—

111

C₂²

—

692

—

112

C₂²

—

692

—

113

C₂²

—

692

—

114

C₂²

C₂²×D₆₉₂

yes

characteristic

115

C₂²

C₂²×D₆₉₂

yes

—

116

C₂²

C₂×#14583826

yes

—

117

C₄

—

118

C₄

—

692

—

119

C₄

#19149667

yes

complemented

120

C₄

#19149676

yes

—

121

C₂

—

1384

—

122

C₂

C₂×#19149667

yes

—

123

C₂

C₂×#19149676

yes

characteristic

124

C₂

#26012041

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇₃
commutator, socle, cosocle	C₂³×C₁₇₃	C₂⁴
Fitting	C₄³×C₁₇₃	C₂
prime radical	C₂×#19149676	C₂
hypercenter	C₄³	D₁₇₃
hypocenter	C₁₇₃	#2858
LCS	G ▷ C₂³×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇₃

commutator, socle, cosocle

C₂³×C₁₇₃

C₂⁴

Fitting

C₄³×C₁₇₃

C₂

prime radical

C₂×#19149676

C₂

hypercenter

C₄³

D₁₇₃

C₁₇₃

G ▷ C₂³×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
1. C₂×#19149679	C₂	semidirect
3. C₂×#19149667	C₂	semidirect
4. #26011393	C₂	semidirect
5. C₄³×C₁₇₃	C₂	semidirect
32. C₄²×C₁₇₃	D₄	semidirect
55. C₄×C₁₇₃	#124	semidirect
62. C₁₇₃	#2858	semidirect
68. C₄³	D₁₇₃	semidirect
95. C₄²	D₆₉₂	semidirect
119. C₄	#19149667	semidirect

H ≅ G/N

type

1. C₂×#19149679

C₂

semidirect

3. C₂×#19149667

C₂

semidirect

4. #26011393

C₂

semidirect

5. C₄³×C₁₇₃

C₂

semidirect

32. C₄²×C₁₇₃

D₄

semidirect

55. C₄×C₁₇₃

#124

semidirect

62. C₁₇₃

#2858

semidirect

68. C₄³

D₁₇₃

semidirect

95. C₄²

D₆₉₂

semidirect

119. C₄

#19149667

semidirect

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

subgroup	index
C₂×#19149679	2
C₂×#19149676	2
C₂×#19149667	2
#26011393	2
C₄³×C₁₇₃	2
#2858	173

subgroup

index

C₄³×C₁₇₃

173

Subgroups — #36361920

Conjugacy classes of subgroups

Complements

Maximal subgroups