Group #22626834 — subgroups

Subgroups — #22626834

Order 9344.

364 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 168 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	#17989638	C₂	1	6	yes	maximal, complemented
2	Q₈×#10596833	C₂	1	2	yes	maximal, complemented
3	#17989608	C₂	1	3	yes	maximal
4	#17989604	C₂	1	3	yes	maximal
5	C₂×Q₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#14556658	C₂²	1	6	yes	—
7	C₄⋊#10596833	C₂²	1	6	yes	—
8	C₂²⋊#10596833	C₂²	1	1	yes	characteristic
9	C₂²⋊#10596833	C₂²	1	3	yes	—
10	#14556646	C₂²	1	6	yes	—
11	C₄×#10596833	C₂²	1	6	yes	—
12	Q₈×Dic₇₃	C₄	1	3	yes	—
13	Q₈×Dic₇₃	C₄	1	1	yes	characteristic
14	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	Q₈×Dic₇₃	C₂²	1	3	yes	—
16	C₂×C₄⋊Dic₇₃	C₄	1	3	yes	—
17	C₂×C₄×Dic₇₃	C₂²	1	3	yes	—
18	C₂²×Q₈×C₇₃	C₄	1	1	yes	characteristic
19	C₂×#10596833	C₂³	1	2	yes	—
20	C₂×#10596833	C₂³	1	6	yes	—
21	C₂²×Dic₇₃	C₂×C₄	1	3	yes	—
22	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₇₃	C₂×C₄	1	6	yes	—
24	C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
25	C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
26	C₄×Dic₇₃	C₂×C₄	1	3	yes	—
27	C₄×Dic₇₃	C₂×C₄	1	3	yes	—
28	C₄×Dic₇₃	C₂³	1	3	yes	—
29	C₄×Dic₇₃	C₂³	1	3	yes	—
30	C₂×Q₈×C₇₃	—	2	4	no	—
31	C₂×Q₈×C₇₃	C₂×C₄	1	3	yes	—
32	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	C₂×C₄	1	3	yes	—
34	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₇₃	—	2	6	no	—
36	Q₈×C₇₃	M₁₆	1	4	yes	—
37	C₂×C₄×C₇₃	—	2	6	no	—
38	C₂×C₄×C₇₃	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₇₃	C₂²×C₄	1	3	yes	—
40	C₂×Dic₇₃	—	2	6	no	—
41	C₂×Dic₇₃	—	2	2	no	—
42	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₇₃	C₂²×C₄	1	3	yes	—
44	C₂×Dic₇₃	C₂²×C₄	1	3	yes	—
45	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
46	#10596833	—	2	2	no	—
47	#10596833	—	2	6	no	—
48	C₂²×C₇₃	—	2	1	no	—
49	C₂²×C₇₃	—	2	1	no	—
50	C₂²×C₇₃	#131	1	2	yes	—
51	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₇₃	—	2	3	no	—
53	C₄×C₇₃	C₂×M₁₆	1	6	yes	—
54	Dic₇₃	—	2	1	no	—
55	Dic₇₃	—	2	3	no	—
56	Dic₇₃	—	2	3	no	—
57	Dic₇₃	2₋¹⁺⁴	1	2	yes	—
58	C₂×C₇₃	—	2	2	no	—
59	C₂×C₇₃	#519	1	1	yes	characteristic
60	C₂×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
61	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
62	#2984	—	73	1	no	maximal, Sylow
63	C₇₃	#2984	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄×Q₈	—	73	1	no	—
65	#446	—	73	6	no	—
66	C₈×Q₈	—	73	2	no	—
67	#432	—	73	3	no	—
68	#431	—	73	3	no	—
69	C₂²×Q₈	#64408	1	1	yes	characteristic
70	C₄×Q₈	—	73	3	no	—
71	C₄×Q₈	—	73	1	no	—
72	C₄×Q₈	—	73	1	no	—
73	C₄×Q₈	—	73	3	no	—
74	C₂×C₄⋊C₄	—	73	3	no	—
75	C₂×C₄²	—	73	3	no	—
76	C₄⋊C₈	—	73	6	no	—
77	C₄⋊C₈	—	73	6	no	—
78	C₂²⋊C₈	—	73	1	no	—
79	C₂²⋊C₈	—	73	3	no	—
80	#97	—	73	6	no	—
81	C₄×C₈	—	73	6	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	C₂×#64408	1	3	yes	—
84	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
85	C₂²×C₄	—	73	3	no	—
86	C₂²×C₄	—	73	1	no	—
87	C₂²×C₄	C₂×#64408	1	3	yes	—
88	C₂×C₈	—	73	2	no	—
89	C₂×C₈	—	73	6	no	—
90	C₄⋊C₄	—	73	6	no	—
91	C₄⋊C₄	—	73	3	no	—
92	C₄⋊C₄	—	73	3	no	—
93	C₄²	—	73	3	no	—
94	C₄²	—	73	3	no	—
95	C₄²	—	73	3	no	—
96	C₄²	—	73	3	no	—
97	C₂³	C₂²×#64408	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#11924037	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	73	1	no	—
102	C₂×C₄	—	73	1	no	—
103	C₂×C₄	—	73	3	no	—
104	C₂×C₄	—	73	3	no	—
105	C₂×C₄	—	146	2	no	—
106	C₂×C₄	—	146	6	no	—
107	C₂×C₄	C₂²×#64408	1	3	yes	—
108	C₂×C₄	C₂²×#64408	1	3	yes	—
109	C₈	—	146	2	no	—
110	C₈	—	146	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#14556679	1	2	yes	—
114	C₂²	C₂³×#64408	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	73	2	no	—
117	C₄	—	146	1	no	—
118	C₄	—	146	3	no	—
119	C₄	—	146	3	no	—
120	C₄	C₂×#11924037	1	6	yes	—
121	C₂	—	2	2	no	—
122	C₂	#17989655	1	1	yes	characteristic
123	C₂	C₂²×#11924037	1	1	yes	characteristic
124	C₂	C₂×#14556679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989638

C₂

yes

maximal, complemented

Q₈×#10596833

C₂

yes

maximal, complemented

#17989608

C₂

yes

maximal

#17989604

C₂

yes

maximal

C₂×Q₈×Dic₇₃

C₂

yes

maximal, characteristic

#14556658

C₂²

yes

—

C₄⋊#10596833

C₂²

yes

—

C₂²⋊#10596833

C₂²

yes

characteristic

C₂²⋊#10596833

C₂²

yes

—

#14556646

C₂²

yes

—

C₄×#10596833

C₂²

yes

—

Q₈×Dic₇₃

C₄

yes

—

Q₈×Dic₇₃

C₄

yes

characteristic

Q₈×Dic₇₃

C₂²

yes

characteristic

Q₈×Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₄

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂²×Q₈×C₇₃

C₄

yes

characteristic

C₂×#10596833

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂³

yes

—

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂⁴

yes

characteristic

#10596833

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#131

yes

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

2₋¹⁺⁴

yes

—

C₂×C₇₃

—

C₂×C₇₃

#519

yes

characteristic

C₂×C₇₃

C₂²×M₁₆

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#446

—

C₈×Q₈

—

#432

—

#431

—

C₂²×Q₈

#64408

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#64408

yes

—

C₂×Q₈

C₂×#64408

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂²×#64408

yes

characteristic

Q₈

—

Q₈

#11924037

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

146

—

106

C₂×C₄

—

146

—

107

C₂×C₄

C₂²×#64408

yes

—

108

C₂×C₄

C₂²×#64408

yes

—

109

C₈

—

146

—

110

C₈

—

146

—

111

C₂²

—

112

C₂²

—

113

C₂²

#14556679

yes

—

114

C₂²

C₂³×#64408

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

146

—

118

C₄

—

146

—

119

C₄

—

146

—

120

C₄

C₂×#11924037

yes

—

121

C₂

—

122

C₂

#17989655

yes

characteristic

123

C₂

C₂²×#11924037

yes

characteristic

124

C₂

C₂×#14556679

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	C₂²×Q₈×C₇₃	C₄
hypercenter	C₂²×Q₈	#64408
hypocenter	C₇₃	#2984
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#64408

commutator, socle

C₂²×C₇₃

C₂³×C₄

Fitting

C₂²×Q₈×C₇₃

C₄

C₂²×Q₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. #17989638	C₂	semidirect
2. Q₈×#10596833	C₂	semidirect
63. C₇₃	#2984	semidirect

H ≅ G/N

type

1. #17989638

C₂

semidirect

2. Q₈×#10596833

C₂

semidirect

63. C₇₃

#2984

semidirect

Subgroups — #22626834

Conjugacy classes of subgroups

Complements