Group #36352622 — subgroups

Subgroups — #36352622

Order 22144.

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

normal subgroups only

This group has 240 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	#26011867	C₂	1	1	yes	maximal, characteristic
2	#26011691	C₂	1	2	yes	maximal, complemented
3	#26011235	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#13063436	C₂	1	2	yes	maximal, complemented
5	#559×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14583813	—	2	2	no	—
7	C₂×#14583811	—	2	2	no	—
8	#19149676	C₂²	1	1	yes	characteristic
9	C₄⋊#13063436	C₂²	1	2	yes	—
10	#19149599	—	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×D₆₉₂	—	2	2	no	—
18	C₂×D₆₉₂	—	2	1	no	—
19	#14583813	—	4	4	no	—
20	#14583811	—	4	4	no	—
21	C₂×#13063436	—	2	1	no	—
22	C₂×#13063436	—	2	2	no	—
23	C₂×Q₈×C₁₇₃	D₄	1	2	yes	—
24	C₂×D₄×C₁₇₃	—	2	2	no	—
25	C₂×D₄×C₁₇₃	D₄	1	2	yes	—
26	C₂²×C₄×C₁₇₃	—	2	2	no	—
27	C₄⋊C₄×C₁₇₃	—	2	2	no	—
28	C₄⋊C₄×C₁₇₃	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₇₃	—	2	4	no	—
30	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
31	C₄²×C₁₇₃	—	2	2	no	—
32	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
33	C₂²×D₁₇₃	—	4	1	no	—
34	D₆₉₂	—	4	2	no	—
35	D₆₉₂	—	4	2	no	—
36	D₆₉₂	—	4	2	no	—
37	#13063436	—	2	2	no	—
38	#13063436	—	4	2	no	—
39	C₂³×C₁₇₃	—	2	2	no	—
40	Q₈×C₁₇₃	—	2	2	no	—
41	Q₈×C₁₇₃	—	2	2	no	—
42	D₄×C₁₇₃	—	4	2	no	—
43	D₄×C₁₇₃	—	4	2	no	—
44	D₄×C₁₇₃	—	4	2	no	—
45	C₂×C₄×C₁₇₃	—	2	2	no	—
46	C₂×C₄×C₁₇₃	—	2	2	no	—
47	C₂×C₄×C₁₇₃	—	8	2	no	—
48	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
50	C₂²×C₁₇₃	—	4	4	no	—
51	C₂²×C₁₇₃	—	4	2	no	—
52	C₂²×C₁₇₃	#120	1	1	yes	characteristic
53	C₂×D₁₇₃	—	8	2	no	—
54	C₂×D₁₇₃	—	8	1	no	—
55	C₄×C₁₇₃	—	2	1	no	—
56	C₄×C₁₇₃	—	4	2	no	—
57	C₄×C₁₇₃	—	4	2	no	—
58	C₄×C₁₇₃	#135	1	2	yes	—
59	C₄×C₁₇₃	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₁₇₃	—	8	2	no	—
61	C₂×C₁₇₃	#449	1	2	yes	—
62	C₂×C₁₇₃	#453	1	1	yes	characteristic
63	D₁₇₃	—	16	1	no	—
64	C₁₇₃	#1620	1	1	yes	characteristic, complemented, Sylow
65	#1620	—	173	1	no	maximal, Sylow
66	#559	D₁₇₃	1	1	yes	characteristic, complemented
67	#493	—	173	1	no	—
68	#460	—	173	2	no	—
69	#331	—	173	1	no	—
70	Q₈⋊C₈	—	173	2	no	—
71	C₂×SD₁₆	—	346	2	no	—
72	C₂×D₈	—	346	2	no	—
73	#127	—	173	1	no	—
74	#127	C₂×D₁₇₃	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₁₇₃	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	173	2	no	—
80	#102	—	346	2	no	—
81	C₄×C₈	—	173	1	no	—
82	C₂×Q₈	C₂²⋊D₁₇₃	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	346	2	no	—
85	C₂×D₄	—	346	1	no	—
86	C₂×D₄	C₂²⋊D₁₇₃	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	692	4	no	—
89	D₈	—	692	4	no	—
90	C₂×C₈	—	346	1	no	—
91	C₂×C₈	—	346	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₁₇₃	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	692	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	692	2	no	—
106	D₄	—	692	2	no	—
107	D₄	—	692	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
113	C₈	—	346	2	no	—
114	C₈	—	692	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	4	no	—
117	C₂²	—	1384	1	no	—
118	C₂²	—	1384	2	no	—
119	C₂²	#19149648	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#19149699	1	2	yes	—
124	C₄	#19149704	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	2768	1	no	—
127	C₂	#26011636	1	2	yes	—
128	C₂	#26011659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26011867

C₂

yes

maximal, characteristic

#26011691

C₂

yes

maximal, complemented

#26011235

C₂

yes

maximal, characteristic, complemented

Q₈⋊#13063436

C₂

yes

maximal, complemented

#559×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14583813

—

C₂×#14583811

—

#19149676

C₂²

yes

characteristic

C₄⋊#13063436

C₂²

yes

—

#19149599

—

C₄×#13063436

C₂²

yes

characteristic

#127×C₁₇₃

C₂²

yes

characteristic

#124×C₁₇₃

—

#123×C₁₇₃

—

C₄×Q₈×C₁₇₃

C₂²

yes

—

C₄×D₄×C₁₇₃

—

C₂×D₆₉₂

—

C₂×D₆₉₂

—

—

—

—

—

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

C₂²×D₁₇₃

—

D₆₉₂

—

D₆₉₂

—

D₆₉₂

—

#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

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#135

yes

—

C₄×C₁₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#449

yes

—

C₂×C₁₇₃

#453

yes

characteristic

D₁₇₃

—

C₁₇₃

#1620

yes

characteristic, complemented, Sylow

#1620

—

173

maximal, Sylow

#559

D₁₇₃

yes

characteristic, complemented

#493

—

173

—

#460

—

173

—

#331

—

173

—

Q₈⋊C₈

—

173

—

C₂×SD₁₆

—

346

—

C₂×D₈

—

346

—

#127

—

173

—

#127

C₂×D₁₇₃

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₇₃

yes

—

C₄×D₄

—

C₄⋊C₈

—

173

—

#102

—

346

—

C₄×C₈

—

173

—

C₂×Q₈

C₂²⋊D₁₇₃

yes

—

C₂×D₄

—

C₂×D₄

—

346

—

C₂×D₄

—

346

—

C₂×D₄

C₂²⋊D₁₇₃

yes

—

C₂²×C₄

—

SD₁₆

—

692

—

D₈

—

692

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₇₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₇₃

yes

characteristic

C₂³

—

C₂³

—

692

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

692

—

106

D₄

—

692

—

107

D₄

—

692

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

113

C₈

—

346

—

114

C₈

—

692

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

1384

—

118

C₂²

—

1384

—

119

C₂²

#19149648

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#19149699

yes

—

124

C₄

#19149704

yes

—

125

C₂

—

126

C₂

—

2768

—

127

C₂

#26011636

yes

—

128

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	#26011867	C₂
hypercenter	#559	D₁₇₃
hypocenter	C₁₇₃	#1620
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

C₂²×C₁₇₃

C₂

D₁₇₃

C₁₇₃

G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #26011691	C₂	semidirect
3. #26011235	C₂	semidirect
4. Q₈⋊#13063436	C₂	semidirect
5. #559×C₁₇₃	C₂	semidirect
64. C₁₇₃	#1620	semidirect
66. #559	D₁₇₃	semidirect

H ≅ G/N

type

2. #26011691

C₂

semidirect

3. #26011235

C₂

semidirect

4. Q₈⋊#13063436

C₂

semidirect

5. #559×C₁₇₃

C₂

semidirect

64. C₁₇₃

#1620

semidirect

66. #559

D₁₇₃

semidirect

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

subgroup	index
#26011867	2
#26011691	2
#26011235	2
Q₈⋊#13063436	2
#559×C₁₇₃	2
#1620	173

subgroup

index

#559×C₁₇₃

173

Subgroups — #36352622

Conjugacy classes of subgroups

Complements

Maximal subgroups