Group #36368759 — subgroups

Subgroups — #36368759

Order 22144.

248 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₄×D₁₇₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₇₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13063443⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₄×C₈×D₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₇₃⋊C₈	C₄	1	2	yes	—
6	C₂×D₁₇₃⋊C₈	C₄	1	2	yes	—
7	C₄×#13063443	C₄	1	2	yes	—
8	C₄×#13063443	C₄	1	2	yes	—
9	C₄×#13063443	C₂²	1	2	yes	—
10	C₄×#13063443	C₂²	1	2	yes	—
11	C₂×C₈×D₁₇₃	C₄	1	2	yes	—
12	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
13	C₄×#13063436	C₄	1	1	yes	characteristic
14	C₈×Dic₁₇₃	C₂²	1	2	yes	—
15	C₄×C₈×C₁₇₃	C₄	1	1	yes	characteristic
16	D₁₇₃⋊C₈	—	2	4	no	—
17	D₁₇₃⋊C₈	C₈	1	8	yes	complemented
18	C₂×#13063443	C₂×C₄	1	4	yes	—
19	C₂×#13063443	C₂×C₄	1	4	yes	—
20	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#13063436	C₂×C₄	1	2	yes	—
24	C₈×D₁₇₃	—	2	4	no	—
25	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
29	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
30	#13063443	—	2	4	no	—
31	#13063443	C₂×C₈	1	8	yes	complemented
32	C₂²×D₁₇₃	C₄²	1	1	yes	characteristic
33	C₂×Dic₁₇₃	C₄²	1	1	yes	characteristic
34	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₄×D₁₇₃	C₂×C₈	1	4	yes	—
37	C₄×D₁₇₃	M₁₆	1	4	yes	—
38	C₄×D₁₇₃	M₁₆	1	4	yes	—
39	#13063436	—	2	2	no	—
40	C₂×C₄×C₁₇₃	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₁₇₃	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₇₃	—	2	2	no	—
44	C₂²×C₁₇₃	C₂×C₄²	1	1	yes	characteristic
45	C₂×D₁₇₃	C₄×C₈	1	2	yes	—
46	C₂×D₁₇₃	#97	1	2	yes	—
47	C₂×D₁₇₃	#97	1	2	yes	—
48	C₄×C₁₇₃	C₄×C₈	1	2	yes	—
49	C₄×C₁₇₃	#97	1	2	yes	—
50	C₄×C₁₇₃	C₂×M₁₆	1	2	yes	—
51	Dic₁₇₃	#97	1	2	yes	—
52	Dic₁₇₃	C₂²×C₈	1	2	yes	—
53	Dic₁₇₃	C₂×M₁₆	1	2	yes	—
54	C₂×C₁₇₃	C₂×C₄×C₈	1	1	yes	characteristic
55	C₂×C₁₇₃	C₂×#97	1	1	yes	characteristic
56	C₂×C₁₇₃	C₂×#97	1	1	yes	characteristic
57	D₁₇₃	#322	1	4	yes	complemented
58	C₁₇₃	C₂×#322	1	1	yes	characteristic, complemented, Sylow
59	C₂×#322	—	173	1	no	maximal, Sylow
60	C₂×C₄×C₈	—	173	1	no	—
61	C₂×C₄×C₈	—	173	1	no	—
62	C₂×C₄×C₈	—	173	1	no	—
63	#322	—	173	4	no	—
64	C₂²×C₈	—	173	2	no	—
65	C₂²×C₈	—	173	2	no	—
66	C₂²×C₈	—	173	2	no	—
67	C₂×C₄²	—	173	1	no	—
68	C₄×C₈	—	173	2	no	—
69	C₄×C₈	—	173	2	no	—
70	C₄×C₈	—	173	2	no	—
71	C₄×C₈	—	173	2	no	—
72	C₄×C₈	—	173	2	no	—
73	C₄×C₈	—	173	1	no	—
74	C₄×C₈	#10621924	1	1	yes	characteristic
75	C₂²×C₄	—	173	1	no	—
76	C₂²×C₄	—	173	1	no	—
77	C₂²×C₄	—	173	1	no	—
78	C₂×C₈	—	346	4	no	—
79	C₂×C₈	—	346	4	no	—
80	C₂×C₈	—	173	4	no	—
81	C₂×C₈	—	173	2	no	—
82	C₂×C₈	—	173	4	no	—
83	C₂×C₈	—	173	8	no	—
84	C₂×C₈	C₂×#10621924	1	2	yes	—
85	C₄²	—	173	1	no	—
86	C₄²	—	173	1	no	—
87	C₄²	—	173	1	no	—
88	C₄²	C₂×#10621924	1	1	yes	characteristic
89	C₂³	—	173	1	no	—
90	C₂×C₄	—	173	1	no	—
91	C₂×C₄	—	173	1	no	—
92	C₂×C₄	—	173	1	no	—
93	C₂×C₄	—	173	4	no	—
94	C₂×C₄	—	173	4	no	—
95	C₂×C₄	—	173	4	no	—
96	C₂×C₄	C₄×#10621924	1	1	yes	characteristic
97	C₂×C₄	C₄×#10621924	1	1	yes	characteristic
98	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	346	4	no	—
101	C₈	—	346	2	no	—
102	C₈	—	173	8	no	—
103	C₂²	—	173	2	no	—
104	C₂²	—	173	2	no	—
105	C₂²	—	173	2	no	—
106	C₂²	C₂×C₄×#10621924	1	1	yes	characteristic
107	C₄	—	173	2	no	—
108	C₄	—	173	2	no	—
109	C₄	—	173	2	no	—
110	C₄	C₈×#10621924	1	2	yes	—
111	C₄	#19149733	1	2	yes	—
112	C₄	D₁₇₃⋊M₁₆	1	2	yes	—
113	C₂	—	173	4	no	—
114	C₂	C₂×C₈×#10621924	1	1	yes	characteristic
115	C₂	C₂×#19149733	1	1	yes	characteristic
116	C₂	#26012450	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₁₇₃⋊C₈

C₂

yes

maximal, characteristic

C₄×D₁₇₃⋊C₈

C₂

yes

maximal, characteristic

#13063443⋊C₈

C₂

yes

maximal, complemented

C₄×C₈×D₁₇₃

C₂

yes

maximal, characteristic

C₂×D₁₇₃⋊C₈

C₄

yes

—

C₂×D₁₇₃⋊C₈

C₄

yes

—

C₄×#13063443

C₄

yes

—

C₄×#13063443

C₄

yes

—

C₄×#13063443

C₂²

yes

—

C₄×#13063443

C₂²

yes

—

C₂×C₈×D₁₇₃

C₄

yes

—

C₄²×D₁₇₃

C₂²

yes

characteristic

C₄×#13063436

C₄

yes

characteristic

C₈×Dic₁₇₃

C₂²

yes

—

C₄×C₈×C₁₇₃

C₄

yes

characteristic

D₁₇₃⋊C₈

—

D₁₇₃⋊C₈

C₈

yes

complemented

C₂×#13063443

C₂×C₄

yes

—

C₂×#13063443

C₂×C₄

yes

—

C₂×C₄×D₁₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇₃

C₂×C₄

yes

characteristic

C₂×#13063436

C₂×C₄

yes

—

C₈×D₁₇₃

—

C₄×Dic₁₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₇₃

C₂³

yes

characteristic

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

C₂×C₄

yes

characteristic

#13063443

—

#13063443

C₂×C₈

yes

complemented

C₂²×D₁₇₃

C₄²

yes

characteristic

C₂×Dic₁₇₃

C₄²

yes

characteristic

C₂×Dic₁₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₇₃

C₂²×C₄

yes

characteristic

C₄×D₁₇₃

C₂×C₈

yes

—

C₄×D₁₇₃

M₁₆

yes

—

C₄×D₁₇₃

M₁₆

yes

—

#13063436

—

C₂×C₄×C₁₇₃

C₄²

yes

characteristic

C₂×C₄×C₁₇₃

C₄²

yes

characteristic

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₈×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₄²

yes

characteristic

C₂×D₁₇₃

C₄×C₈

yes

—

C₂×D₁₇₃

#97

yes

—

C₂×D₁₇₃

#97

yes

—

C₄×C₁₇₃

C₄×C₈

yes

—

C₄×C₁₇₃

#97

yes

—

C₄×C₁₇₃

C₂×M₁₆

yes

—

Dic₁₇₃

#97

yes

—

Dic₁₇₃

C₂²×C₈

yes

—

Dic₁₇₃

C₂×M₁₆

yes

—

C₂×C₁₇₃

C₂×C₄×C₈

yes

characteristic

C₂×C₁₇₃

C₂×#97

yes

characteristic

C₂×C₁₇₃

C₂×#97

yes

D₁₇₃

yes

C₁₇₃

yes

characteristic, complemented, Sylow

C₂×#322

—

173

maximal, Sylow

C₂×C₄×C₈

—

173

—

C₂×C₄×C₈

—

173

—

C₂×C₄×C₈

—

173

—

#322

—

173

—

C₂²×C₈

—

173

—

C₂²×C₈

—

173

—

C₂²×C₈

—

173

—

C₂×C₄²

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₄×C₈

#10621924

yes

characteristic

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

173

—

C₂×C₈

—

173

—

C₂×C₈

—

173

—

C₂×C₈

—

173

—

C₂×C₈

C₂×#10621924

yes

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

C₂×#10621924

yes

characteristic

C₂³

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

100

C₈

—

346

—

101

C₈

—

346

—

102

C₈

—

173

—

103

C₂²

—

173

—

104

C₂²

—

173

—

105

C₂²

—

173

—

106

C₂²

C₂×C₄×#10621924

yes

characteristic

107

C₄

—

173

—

108

C₄

—

173

—

109

C₄

—

173

—

110

C₄

C₈×#10621924

yes

—

111

C₄

#19149733

yes

—

112

C₄

D₁₇₃⋊M₁₆

yes

—

113

C₂

—

173

—

114

C₂

C₂×C₈×#10621924

yes

characteristic

115

C₂

C₂×#19149733

yes

characteristic

116

C₂

#26012450

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×#10621924
commutator	C₂×C₁₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10621924
Fitting	C₄×C₈×C₁₇₃	C₄
socle	C₂²×C₁₇₃	C₂×C₄²
prime radical	C₂²×D₁₇₃	C₄²
cosocle	C₄×Dic₁₇₃	C₂³
hypercenter	C₄×C₈	#10621924
hypocenter	C₁₇₃	C₂×#322
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₄²

C₂×#10621924

commutator

C₂×C₁₇₃

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#10621924

Fitting

C₄×C₈×C₁₇₃

C₄

socle

C₂²×C₁₇₃

C₂×C₄²

prime radical

C₂²×D₁₇₃

C₄²

cosocle

C₄×Dic₁₇₃

C₂³

C₄×C₈

C₁₇₃

G ▷ C₂×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
3. #13063443⋊C₈	C₂	semidirect
17. D₁₇₃⋊C₈	C₈	semidirect
31. #13063443	C₂×C₈	semidirect
57. D₁₇₃	#322	semidirect
58. C₁₇₃	C₂×#322	semidirect

H ≅ G/N

type

3. #13063443⋊C₈

C₂

semidirect

17. D₁₇₃⋊C₈

C₈

semidirect

31. #13063443

C₂×C₈

semidirect

57. D₁₇₃

#322

semidirect

58. C₁₇₃

C₂×#322

semidirect

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

subgroup	index
C₄×D₁₇₃⋊C₈	2
#13063443⋊C₈	2
C₄×C₈×D₁₇₃	2
C₂×#322	173

subgroup

index

C₄×D₁₇₃⋊C₈

#13063443⋊C₈

C₄×C₈×D₁₇₃

C₂×#322

173

Subgroups — #36368759

Conjugacy classes of subgroups

Complements

Maximal subgroups