Group #36355117 — subgroups

Subgroups — #36355117

Order 22144.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26011540	C₂	1	1	yes	maximal, characteristic, complemented
2	#26011518	C₂	1	2	yes	maximal, complemented
3	#26011509	C₂	1	2	yes	maximal
4	#26011459	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14583826	C₂²	1	1	yes	characteristic
7	C₂×#14583810	C₂²	1	2	yes	—
8	C₂²×#13063436	C₂²	1	1	yes	characteristic
9	#19149615	—	2	2	no	—
10	C₄⋊#13063436	—	2	4	no	—
11	#19149607	—	2	4	no	—
12	#19149591	—	2	2	no	—
13	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
14	#117×C₁₇₃	C₂²	1	2	yes	—
15	#14583826	—	2	2	no	—
16	#14583826	—	2	2	no	—
17	C₂×Dic₃₄₆	Q₈	1	1	yes	characteristic
18	C₂×D₆₉₂	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₇₃	—	2	1	no	—
20	C₂×C₄×D₁₇₃	—	2	1	no	—
21	#14583810	—	2	2	no	—
22	#14583810	—	2	2	no	—
23	C₂×#13063436	—	2	1	no	—
24	C₂×#13063436	—	2	2	no	—
25	C₂×#13063436	—	2	2	no	—
26	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₇₃	—	2	2	no	—
28	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₇₃	—	2	2	no	—
30	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
31	C₄²×C₁₇₃	D₄	1	4	yes	—
32	C₂²×D₁₇₃	—	2	1	no	—
33	Dic₃₄₆	—	2	2	no	—
34	D₆₉₂	—	2	2	no	—
35	C₂²⋊D₁₇₃	—	4	2	no	—
36	C₂×Dic₁₇₃	—	2	1	no	—
37	C₄×D₁₇₃	—	4	1	no	—
38	C₄×D₁₇₃	—	4	1	no	—
39	#13063436	—	4	1	no	—
40	#13063436	—	4	2	no	—
41	C₂³×C₁₇₃	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₇₃	—	2	4	no	—
43	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇₃	#41	1	2	yes	—
47	C₈×C₁₇₃	—	2	2	no	—
48	C₂²×C₁₇₃	—	4	1	no	—
49	C₂²×C₁₇₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₇₃	#123	1	2	yes	—
51	C₂×D₁₇₃	—	4	2	no	—
52	C₂×D₁₇₃	—	4	1	no	—
53	C₄×C₁₇₃	—	4	4	no	—
54	C₄×C₁₇₃	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₇₃	#123	1	1	yes	characteristic
56	C₄×C₁₇₃	#127	1	1	yes	characteristic
57	Dic₁₇₃	—	4	1	no	—
58	C₂×C₁₇₃	—	2	2	no	—
59	C₂×C₁₇₃	—	2	1	no	—
60	C₂×C₁₇₃	#399	1	1	yes	characteristic
61	D₁₇₃	—	8	1	no	—
62	C₁₇₃	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	173	1	no	maximal, Sylow
64	#430	—	173	1	no	—
65	#424	—	173	2	no	—
66	#421	—	173	2	no	—
67	#408	—	173	1	no	—
68	H(3,ℤ/4)	D₁₇₃	1	1	yes	characteristic, complemented
69	C₂×#41	—	173	1	no	—
70	C₂×M₁₆	—	173	2	no	—
71	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
72	C₂²×C₈	—	173	1	no	—
73	#117	C₂×D₁₇₃	1	2	yes	—
74	#108	—	346	2	no	—
75	C₄⋊C₈	—	346	4	no	—
76	#104	—	346	4	no	—
77	C₂²⋊C₈	—	346	2	no	—
78	#41	—	346	2	no	—
79	#41	—	346	2	no	—
80	C₂×Q₈	—	173	1	no	—
81	C₂×D₄	—	346	1	no	—
82	C₂×D₄	—	173	1	no	—
83	C₂²×C₄	—	346	1	no	—
84	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	346	2	no	—
87	M₁₆	—	346	2	no	—
88	C₂×C₈	—	346	1	no	—
89	C₂×C₈	—	346	2	no	—
90	C₂×C₈	—	346	2	no	—
91	C₂×C₈	D₆₉₂	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₆₉₂	1	4	yes	—
95	C₂³	—	346	1	no	—
96	C₂³	#14583825	1	1	yes	characteristic
97	Q₈	—	346	2	no	—
98	D₄	—	346	2	no	—
99	D₄	—	692	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	346	1	no	—
102	C₂×C₄	—	692	1	no	—
103	C₂×C₄	—	692	1	no	—
104	C₂×C₄	C₂×D₆₉₂	1	2	yes	—
105	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₇₃	1	1	yes	characteristic
107	C₂×C₄	#14583825	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	692	1	no	—
110	C₈	—	692	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	692	2	no	—
113	C₂²	—	692	1	no	—
114	C₂²	#19149659	1	1	yes	characteristic
115	C₂²	#19149663	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	692	1	no	—
118	C₄	#19149659	1	2	yes	—
119	C₄	#19149663	1	1	yes	characteristic
120	C₄	#19149676	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1384	1	no	—
124	C₂	#26011428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26011540

C₂

yes

maximal, characteristic, complemented

#26011518

C₂

yes

maximal, complemented

#26011509

C₂

yes

maximal

#26011459

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14583826

C₂²

yes

characteristic

C₂×#14583810

C₂²

yes

—

C₂²×#13063436

C₂²

yes

—

—

—

—

C₂×M₁₆×C₁₇₃

C₂²

yes

characteristic

#117×C₁₇₃

C₂²

yes

—

#14583826

—

#14583826

—

C₂×Dic₃₄₆

Q₈

yes

characteristic

C₂×D₆₉₂

Q₈

yes

characteristic

C₂×C₂²⋊D₁₇₃

—

C₂×C₄×D₁₇₃

—

—

—

—

—

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

M₁₆×C₁₇₃

—

C₂×C₈×C₁₇₃

D₄

yes

—

C₄⋊C₄×C₁₇₃

—

C₂²⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

D₄

yes

—

C₂²×D₁₇₃

—

Dic₃₄₆

—

D₆₉₂

—

C₂²⋊D₁₇₃

—

C₂×Dic₁₇₃

—

C₄×D₁₇₃

—

C₄×D₁₇₃

—

#13063436

—

#13063436

—

C₂³×C₁₇₃

#41

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇₃

#123

yes

—

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂²⋊Q₈

yes

—

C₄×C₁₇₃

yes

C₄×C₁₇₃

yes

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#399

yes

characteristic

D₁₇₃

—

C₁₇₃

#2098

yes

characteristic, complemented, Sylow

#2098

—

173

maximal, Sylow

#430

—

173

—

#424

—

173

—

#421

—

173

—

#408

—

173

—

H(3,ℤ/4)

D₁₇₃

yes

characteristic, complemented

C₂×#41

—

173

—

C₂×M₁₆

—

173

—

C₂×M₁₆

C₂×D₁₇₃

yes

characteristic

C₂²×C₈

—

173

—

#117

C₂×D₁₇₃

yes

—

#108

—

346

—

C₄⋊C₈

—

346

—

#104

—

346

—

C₂²⋊C₈

—

346

—

#41

—

346

—

#41

—

346

—

C₂×Q₈

—

173

—

C₂×D₄

—

346

—

C₂×D₄

—

173

—

C₂²×C₄

—

346

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

M₁₆

—

M₁₆

—

346

—

M₁₆

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

D₆₉₂

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₆₉₂

yes

—

C₂³

—

346

—

C₂³

#14583825

yes

characteristic

Q₈

—

346

—

D₄

—

346

—

D₄

—

692

—

100

C₂×C₄

—

101

C₂×C₄

—

346

—

102

C₂×C₄

—

692

—

103

C₂×C₄

—

692

—

104

C₂×C₄

C₂×D₆₉₂

yes

—

105

C₂×C₄

C₂×D₆₉₂

yes

characteristic

106

C₂×C₄

Q₈×D₁₇₃

yes

characteristic

107

C₂×C₄

#14583825

yes

—

108

C₈

—

109

C₈

—

692

—

110

C₈

—

692

—

111

C₂²

—

112

C₂²

—

692

—

113

C₂²

—

692

—

114

C₂²

#19149659

yes

characteristic

115

C₂²

#19149663

yes

—

116

C₄

—

117

C₄

—

692

—

118

C₄

#19149659

yes

—

119

C₄

#19149663

yes

characteristic

120

C₄

#19149676

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1384

—

124

C₂

#26011428

yes

characteristic

	subgroup	G/H
center	C₄	#19149676
commutator, cosocle	C₂²×C₄×C₁₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	H(3,ℤ/4)×C₁₇₃	C₂
socle	C₂×C₁₇₃	#399
prime radical	C₂×#14583826	C₂²
hypercenter	H(3,ℤ/4)	D₁₇₃
hypocenter	C₁₇₃	#2098
LCS	G ▷ C₂²×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#19149676

commutator, cosocle

C₂²×C₄×C₁₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

H(3,ℤ/4)×C₁₇₃

C₂

C₂×C₁₇₃

C₂²

D₁₇₃

C₁₇₃

G ▷ C₂²×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #26011540	C₂	semidirect
2. #26011518	C₂	semidirect
5. H(3,ℤ/4)×C₁₇₃	C₂	semidirect
62. C₁₇₃	#2098	semidirect
68. H(3,ℤ/4)	D₁₇₃	semidirect

H ≅ G/N

type

1. #26011540

C₂

semidirect

2. #26011518

C₂

semidirect

5. H(3,ℤ/4)×C₁₇₃

C₂

semidirect

62. C₁₇₃

#2098

semidirect

68. H(3,ℤ/4)

D₁₇₃

semidirect

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

subgroup	index
#26011540	2
#26011518	2
#26011509	2
#26011459	2
H(3,ℤ/4)×C₁₇₃	2
#2098	173

subgroup

index

H(3,ℤ/4)×C₁₇₃

173

Subgroups — #36355117

Conjugacy classes of subgroups

Complements

Maximal subgroups