Group #22626827 — subgroups

Subgroups — #22626827

Order 9344.

356 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 164 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	8	yes	maximal, complemented
2	#17989607	C₂	1	4	yes	maximal
3	#17989603	C₂	1	2	yes	maximal
4	C₂×Q₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	#14556658	C₂²	1	8	yes	—
6	C₄⋊#10596833	C₂²	1	4	yes	—
7	C₂²⋊#10596833	C₂²	1	4	yes	—
8	#14556646	C₂²	1	8	yes	—
9	C₄×#10596833	C₂²	1	4	yes	—
10	Q₈×Dic₇₃	C₄	1	4	yes	—
11	Q₈×Dic₇₃	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₇₃	C₄	1	2	yes	—
13	C₂×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
16	C₂²×Q₈×C₇₃	C₄	1	1	yes	characteristic
17	C₂×#10596833	C₂³	1	8	yes	—
18	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
20	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₇₃	C₂×C₄	1	8	yes	—
22	C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
23	C₄×Dic₇₃	C₂×C₄	1	4	yes	—
24	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
25	C₄×Dic₇₃	C₂³	1	2	yes	—
26	C₄×Dic₇₃	C₂³	1	4	yes	—
27	C₂×Q₈×C₇₃	—	2	4	no	—
28	C₂×Q₈×C₇₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
32	Q₈×C₇₃	—	2	8	no	—
33	C₂×C₄×C₇₃	—	2	4	no	—
34	C₂×C₄×C₇₃	M₁₆	1	4	yes	—
35	C₂×C₄×C₇₃	C₂²×C₄	1	4	yes	—
36	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
37	C₂×Dic₇₃	—	2	2	no	—
38	C₂×Dic₇₃	—	4	1	no	—
39	C₂×Dic₇₃	—	4	2	no	—
40	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
42	C₂×Dic₇₃	C₂²×C₄	1	4	yes	—
43	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
44	#10596833	—	2	8	no	—
45	C₂²×C₇₃	—	2	1	no	—
46	C₂²×C₇₃	—	2	1	no	—
47	C₂²×C₇₃	C₂×M₁₆	1	2	yes	—
48	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
49	C₄×C₇₃	—	2	4	no	—
50	C₄×C₇₃	C₂×M₁₆	1	4	yes	—
51	Dic₇₃	—	2	1	no	—
52	Dic₇₃	—	2	2	no	—
53	Dic₇₃	—	2	4	no	—
54	Dic₇₃	2₋¹⁺⁴	1	2	yes	—
55	C₂×C₇₃	—	2	2	no	—
56	C₂×C₇₃	#519	1	1	yes	characteristic
57	C₂×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
58	C₂×C₇₃	#568	1	1	yes	characteristic
59	#2983	—	73	1	no	maximal, Sylow
60	C₇₃	#2983	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₄×Q₈	—	73	1	no	—
62	#446	—	73	8	no	—
63	#432	—	73	4	no	—
64	#431	—	73	2	no	—
65	C₂²×Q₈	#64408	1	1	yes	characteristic
66	C₄×Q₈	—	73	4	no	—
67	C₄×Q₈	—	73	4	no	—
68	C₂×C₄⋊C₄	—	73	2	no	—
69	C₂×C₄⋊C₄	—	73	1	no	—
70	C₂×C₄²	—	73	1	no	—
71	C₂×C₄²	—	73	2	no	—
72	C₄⋊C₈	—	73	4	no	—
73	C₄⋊C₈	—	73	8	no	—
74	C₂²⋊C₈	—	73	4	no	—
75	#97	—	73	8	no	—
76	C₄×C₈	—	73	4	no	—
77	C₂×Q₈	—	2	4	no	—
78	C₂×Q₈	C₂×#64408	1	4	yes	—
79	C₂²×C₄	—	73	1	no	—
80	C₂²×C₄	—	73	2	no	—
81	C₂²×C₄	—	73	1	no	—
82	C₂²×C₄	C₂×#64408	1	2	yes	—
83	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
84	C₂×C₈	—	73	8	no	—
85	C₄⋊C₄	—	73	8	no	—
86	C₄⋊C₄	—	73	4	no	—
87	C₄²	—	73	4	no	—
88	C₄²	—	73	2	no	—
89	C₄²	—	73	2	no	—
90	C₄²	—	73	4	no	—
91	C₂³	C₂²×#64408	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	292	1	no	—
95	C₂×C₄	—	292	2	no	—
96	C₂×C₄	—	73	1	no	—
97	C₂×C₄	—	73	1	no	—
98	C₂×C₄	—	73	2	no	—
99	C₂×C₄	—	73	4	no	—
100	C₂×C₄	—	146	2	no	—
101	C₂×C₄	#11924037	1	4	yes	—
102	C₂×C₄	C₂²×#64408	1	4	yes	—
103	C₂×C₄	C₂²×#64408	1	2	yes	—
104	C₈	—	146	8	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×#11924037	1	2	yes	—
108	C₂²	C₂³×#64408	1	1	yes	characteristic
109	C₄	—	2	4	no	—
110	C₄	—	73	2	no	—
111	C₄	—	146	1	no	—
112	C₄	—	146	2	no	—
113	C₄	—	146	4	no	—
114	C₄	C₂×#11924037	1	4	yes	—
115	C₂	—	2	2	no	—
116	C₂	#17989655	1	1	yes	characteristic
117	C₂	C₂²×#11924037	1	1	yes	characteristic
118	C₂	#17989667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989638

C₂

yes

maximal, complemented

#17989607

C₂

yes

maximal

#17989603

C₂

yes

maximal

C₂×Q₈×Dic₇₃

C₂

yes

maximal, characteristic

#14556658

C₂²

yes

—

C₄⋊#10596833

C₂²

yes

—

C₂²⋊#10596833

C₂²

yes

—

#14556646

C₂²

yes

—

C₄×#10596833

C₂²

yes

—

Q₈×Dic₇₃

C₄

yes

—

Q₈×Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₄

yes

—

C₂×C₄⋊Dic₇₃

C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂²×Q₈×C₇₃

C₄

yes

characteristic

C₂×#10596833

C₂³

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

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

C₂³

yes

—

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

C₂²×C₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

—

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

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×M₁₆

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

—

#432

—

#431

—

C₂²×Q₈

#64408

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#64408

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

—

C₂²×C₄

C₂×#64408

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂²×#64408

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

146

—

101

C₂×C₄

#11924037

yes

—

102

C₂×C₄

C₂²×#64408

yes

—

103

C₂×C₄

C₂²×#64408

yes

—

104

C₈

—

146

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×#11924037

yes

—

108

C₂²

C₂³×#64408

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

146

—

112

C₄

—

146

—

113

C₄

—

146

—

114

C₄

C₂×#11924037

yes

—

115

C₂

—

116

C₂

#17989655

yes

characteristic

117

C₂

C₂²×#11924037

yes

characteristic

118

C₂

#17989667

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	#2983
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
60. C₇₃	#2983	semidirect

H ≅ G/N

type

1. #17989638

C₂

semidirect

60. C₇₃

#2983

semidirect

Subgroups — #22626827

Conjugacy classes of subgroups

Complements