Group #22626261 — subgroups

Subgroups — #22626261

Order 9344.

184 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 94 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×#14556664	C₂	1	2	yes	maximal
2	#17989508	C₂	1	2	yes	maximal, complemented
3	C₄⋊#11924031	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	#14556664	—	2	4	no	—
6	C₂×#11924031	C₂²	1	4	yes	—
7	C₂×C₈×D₇₃	C₄	1	1	yes	characteristic
8	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	C₄²×D₇₃	C₄	1	1	yes	characteristic
10	C₄×#10596831	C₂²	1	1	yes	characteristic
11	C₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
14	#11924031	—	2	4	no	—
15	C₂×C₄×D₇₃	C₈	1	1	yes	characteristic
16	C₂×C₄×D₇₃	C₈	1	1	yes	characteristic
17	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₂×#10596831	C₂³	1	1	yes	characteristic
20	C₈×D₇₃	—	2	2	no	—
21	C₈×D₇₃	D₄	1	2	yes	—
22	C₈×D₇₃	Q₈	1	2	yes	—
23	C₄×Dic₇₃	C₈	1	1	yes	characteristic
24	C₄×Dic₇₃	C₈	1	1	yes	characteristic
25	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
35	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
36	C₄×D₇₃	—	2	2	no	—
37	C₄×D₇₃	C₁₆	1	4	yes	complemented
38	C₄×D₇₃	C₄⋊C₄	1	2	yes	—
39	C₄×D₇₃	M₁₆	1	2	yes	—
40	C₈×C₇₃	—	2	1	no	—
41	C₈×C₇₃	C₄⋊C₄	1	2	yes	—
42	#10596831	—	2	1	no	—
43	#10596831	C₂×D₄	1	1	yes	characteristic
44	#10596831	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×D₇₃	C₄⋊C₈	1	2	yes	—
47	C₂×D₇₃	C₂×C₁₆	1	2	yes	—
48	C₂×D₇₃	M₃₂	1	2	yes	—
49	C₄×C₇₃	—	2	1	no	—
50	C₄×C₇₃	C₂×C₁₆	1	2	yes	complemented
51	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
53	Dic₇₃	—	2	1	no	—
54	Dic₇₃	C₄⋊C₈	1	2	yes	—
55	Dic₇₃	C₂×C₁₆	1	2	yes	complemented
56	C₂×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
58	C₂×C₇₃	C₂×M₃₂	1	1	yes	characteristic
59	D₇₃	C₄⋊C₁₆	1	4	yes	complemented
60	C₂×C₄⋊C₁₆	—	73	1	no	maximal, Sylow
61	C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₁₆	—	73	2	no	—
63	C₂×C₄×C₈	—	73	1	no	—
64	C₄⋊C₁₆	—	73	2	no	—
65	C₄⋊C₁₆	—	73	2	no	—
66	C₂²×C₈	—	73	1	no	—
67	C₂²×C₈	—	73	1	no	—
68	C₂×C₄²	—	73	1	no	—
69	C₂×C₁₆	—	73	4	no	—
70	C₂×C₁₆	—	146	4	no	—
71	C₄×C₈	—	73	1	no	—
72	C₄×C₈	—	73	1	no	—
73	C₄×C₈	—	73	1	no	—
74	C₄×C₈	#64408	1	1	yes	characteristic
75	C₂²×C₄	—	73	1	no	—
76	C₂²×C₄	—	73	1	no	—
77	C₂²×C₄	—	73	1	no	—
78	C₂×C₈	—	73	1	no	—
79	C₂×C₈	—	73	2	no	—
80	C₂×C₈	—	73	2	no	—
81	C₂×C₈	—	73	1	no	—
82	C₂×C₈	—	146	2	no	—
83	C₂×C₈	C₂×#64408	1	1	yes	characteristic
84	C₂×C₈	C₂×#64408	1	1	yes	characteristic
85	C₄²	—	73	1	no	—
86	C₄²	—	73	1	no	—
87	C₄²	—	73	1	no	—
88	C₄²	C₂×#64408	1	1	yes	characteristic
89	C₁₆	—	146	4	no	—
90	C₂³	—	73	1	no	—
91	C₂×C₄	—	73	1	no	—
92	C₂×C₄	—	73	1	no	—
93	C₂×C₄	—	73	2	no	—
94	C₂×C₄	—	73	4	no	—
95	C₂×C₄	—	73	2	no	—
96	C₂×C₄	—	73	1	no	—
97	C₂×C₄	—	146	2	no	—
98	C₂×C₄	#11924036	1	1	yes	characteristic
99	C₂×C₄	#11924036	1	1	yes	characteristic
100	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	73	1	no	—
103	C₈	—	73	1	no	—
104	C₈	—	146	1	no	—
105	C₈	C₄⋊#64408	1	2	yes	—
106	C₂²	—	73	2	no	—
107	C₂²	—	73	2	no	—
108	C₂²	—	73	2	no	—
109	C₂²	C₂×#11924036	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	73	2	no	—
112	C₄	—	73	2	no	—
113	C₄	—	146	1	no	—
114	C₄	#14556664	1	2	yes	complemented
115	C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
116	C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
117	C₂	—	73	4	no	—
118	C₂	#17989588	1	1	yes	characteristic
119	C₂	C₂×#14556664	1	1	yes	characteristic
120	C₂	#17989643	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556664

C₂

yes

maximal

#17989508

C₂

yes

maximal, complemented

C₄⋊#11924031

C₂

yes

maximal, complemented

C₄×C₈×D₇₃

C₂

yes

maximal, characteristic

#14556664

—

C₂×#11924031

C₂²

yes

—

C₂×C₈×D₇₃

C₄

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₄²×D₇₃

C₄

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₈×Dic₇₃

C₄

yes

characteristic

C₈×Dic₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₄

yes

characteristic

#11924031

—

C₂×C₄×D₇₃

C₈

yes

characteristic

C₂×C₄×D₇₃

C₈

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂³

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

D₄

yes

—

C₈×D₇₃

Q₈

yes

—

C₄×Dic₇₃

C₈

yes

characteristic

C₄×Dic₇₃

C₈

yes

characteristic

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₁₆

yes

complemented

C₄×D₇₃

C₄⋊C₄

yes

—

C₄×D₇₃

M₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₄⋊C₄

yes

—

#10596831

—

#10596831

C₂×D₄

yes

characteristic

#10596831

C₂×Q₈

yes

characteristic

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂×D₇₃

C₄⋊C₈

yes

—

C₂×D₇₃

C₂×C₁₆

yes

—

C₂×D₇₃

M₃₂

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₁₆

yes

complemented

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₂×M₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

C₄⋊C₈

yes

—

Dic₇₃

C₂×C₁₆

yes

complemented

C₂×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×C₇₃

C₂×M₃₂

yes

characteristic

D₇₃

C₄⋊C₁₆

yes

complemented

C₂×C₄⋊C₁₆

—

maximal, Sylow

C₇₃

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#64408

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#64408

yes

characteristic

C₁₆

—

146

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂²×#64408

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

146

—

105

C₈

C₄⋊#64408

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#11924036

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

146

—

114

C₄

#14556664

yes

complemented

115

C₄

C₂×C₄⋊#64408

yes

characteristic

116

C₄

D₇₃⋊M₁₆

yes

characteristic

117

C₂

—

118

C₂

#17989588

yes

characteristic

119

C₂

C₂×#14556664

yes

characteristic

120

C₂

#17989643

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#64408
commutator	C₂×C₇₃	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₈
hypercenter	C₄×C₈	#64408
hypocenter	C₇₃	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂×#64408

commutator

C₂×C₇₃

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄×C₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂²×C₈

hypercenter

C₄×C₈

#64408

hypocenter

C₇₃

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
2. #17989508	C₂	semidirect
3. C₄⋊#11924031	C₂	semidirect
37. C₄×D₇₃	C₁₆	semidirect
50. C₄×C₇₃	C₂×C₁₆	semidirect
55. Dic₇₃	C₂×C₁₆	semidirect
59. D₇₃	C₄⋊C₁₆	semidirect
61. C₇₃	C₂×C₄⋊C₁₆	semidirect
114. C₄	#14556664	semidirect

H ≅ G/N

type

2. #17989508

C₂

semidirect

3. C₄⋊#11924031

C₂

semidirect

37. C₄×D₇₃

C₁₆

semidirect

50. C₄×C₇₃

C₂×C₁₆

semidirect

55. Dic₇₃

C₂×C₁₆

semidirect

59. D₇₃

C₄⋊C₁₆

semidirect

61. C₇₃

C₂×C₄⋊C₁₆

semidirect

114. C₄

#14556664

semidirect

Subgroups — #22626261

Conjugacy classes of subgroups

Complements