Group #22600144 — subgroups

Subgroups — #22600144

Order 9088.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#17986812	C₂	1	4	yes	maximal, complemented
2	#17986271	C₂	1	4	yes	maximal, complemented
3	#17986181	C₂	1	4	yes	maximal, complemented
4	#17986117	C₂	1	2	yes	maximal, complemented
5	#494×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#14556176	C₂²	1	4	yes	—
7	#14556160	—	2	8	no	—
8	#14556159	C₂²	1	8	yes	—
9	#14556151	—	2	4	no	—
10	#14556150	C₂²	1	2	yes	—
11	Q₁₆×D₇₁	—	2	4	no	—
12	#14556134	C₂²	1	4	yes	—
13	#14556064	C₂²	1	2	yes	—
14	#14556056	C₂²	1	4	yes	—
15	#14556047	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₇₁	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
18	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	#11765866	—	2	8	no	—
20	#11765865	—	4	4	no	—
21	#11765865	C₂³	1	4	yes	—
22	Q₈×D₇₁	—	2	8	no	—
23	Dic₂₈₄	D₄	1	2	yes	—
24	#11765855	—	2	8	no	—
25	#11765854	D₄	1	4	yes	—
26	#11765852	—	2	8	no	—
27	D₅₆₈	D₄	1	2	yes	—
28	#11765849	C₂³	1	4	yes	—
29	#11765848	—	2	4	no	—
30	C₈×D₇₁	—	2	4	no	—
31	C₂×Q₈×C₇₁	C₂³	1	4	yes	—
32	Q₁₆×C₇₁	—	2	4	no	—
33	Q₁₆×C₇₁	—	2	4	no	—
34	C₂×C₈×C₇₁	C₂³	1	2	yes	—
35	C₄⋊C₄×C₇₁	—	4	2	no	—
36	C₄²×C₇₁	C₂³	1	1	yes	characteristic
37	Q₈×C₇₁	—	2	8	no	—
38	C₂×C₄×C₇₁	—	2	1	no	—
39	C₂×C₄×C₇₁	—	2	4	no	—
40	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₇₁	—	2	4	no	—
42	D₂₈₄	—	4	4	no	—
43	D₂₈₄	C₂×D₄	1	4	yes	—
44	C₄×D₇₁	—	2	4	no	—
45	C₄×D₇₁	—	4	8	no	—
46	Dic₁₄₂	—	4	4	no	—
47	Dic₁₄₂	C₂×D₄	1	4	yes	—
48	C₈×C₇₁	C₂×D₄	1	2	yes	—
49	C₈×C₇₁	C₂×D₄	1	2	yes	—
50	C₇₁⋊C₈	—	2	4	no	—
51	C₂²×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₇₁	—	2	4	no	—
53	C₄×C₇₁	—	2	2	no	—
54	C₄×C₇₁	—	4	4	no	—
55	C₄×C₇₁	C₂²×D₄	1	2	yes	—
56	Dic₇₁	—	2	4	no	—
57	C₂×C₇₁	—	2	1	no	—
58	C₂×C₇₁	D₄²	1	1	yes	characteristic
59	D₇₁	—	4	4	no	—
60	#3293	—	71	1	no	maximal, Sylow
61	C₇₁	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	71	4	no	—
63	#494	D₇₁	1	1	yes	characteristic, complemented
64	#473	—	71	4	no	—
65	#456	—	71	4	no	—
66	#443	—	71	2	no	—
67	2₋¹⁺⁴	—	71	4	no	—
68	#137	—	71	8	no	—
69	#137	—	142	8	no	—
70	#135	—	71	2	no	—
71	#135	—	142	4	no	—
72	C₂×Q₁₆	—	142	4	no	—
73	C₂×Q₁₆	C₂×D₇₁	1	4	yes	—
74	#131	—	71	4	no	—
75	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
76	#108	—	71	2	no	—
77	#104	—	71	4	no	—
78	#101	—	71	4	no	—
79	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
80	#41	—	284	4	no	—
81	#41	—	71	4	no	—
82	#41	—	142	8	no	—
83	C₂×Q₈	—	142	8	no	—
84	C₂×Q₈	C₂²×D₇₁	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	71	2	no	—
88	Q₁₆	—	142	8	no	—
89	SD₁₆	—	71	4	no	—
90	SD₁₆	—	142	8	no	—
91	D₈	—	71	2	no	—
92	M₁₆	—	71	4	no	—
93	M₁₆	—	142	4	no	—
94	C₂×C₈	—	142	4	no	—
95	C₂×C₈	C₂²×D₇₁	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₇₁	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	284	4	no	—
100	Q₈	—	71	4	no	—
101	D₄	—	284	4	no	—
102	D₄	—	71	4	no	—
103	D₄	—	142	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	284	8	no	—
107	C₂×C₄	—	142	4	no	—
108	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
109	C₈	—	142	4	no	—
110	C₈	D₄×D₇₁	1	2	yes	—
111	C₈	D₄×D₇₁	1	2	yes	—
112	C₂²	—	142	4	no	—
113	C₂²	#14556173	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	142	4	no	—
117	C₄	C₂×D₄×D₇₁	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	284	4	no	—
120	C₂	#17986634	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986812

C₂

yes

maximal, complemented

#17986271

C₂

yes

maximal, complemented

#17986181

C₂

yes

maximal, complemented

#17986117

C₂

yes

maximal, complemented

#494×C₇₁

C₂

yes

maximal, characteristic, complemented

#14556176

C₂²

yes

—

#14556160

—

#14556159

C₂²

yes

—

#14556151

—

#14556150

C₂²

yes

—

Q₁₆×D₇₁

—

#14556134

C₂²

yes

—

#14556064

C₂²

yes

—

#14556056

C₂²

yes

—

#14556047

C₂²

yes

—

C₂×Q₁₆×C₇₁

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₇₁

—

Dic₂₈₄

D₄

yes

—

#11765855

—

#11765854

D₄

yes

—

#11765852

—

D₅₆₈

D₄

yes

—

#11765849

C₂³

yes

—

#11765848

—

C₈×D₇₁

—

C₂×Q₈×C₇₁

C₂³

yes

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂²⋊D₇₁

—

D₂₈₄

—

D₂₈₄

C₂×D₄

yes

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

C₂×D₄

yes

—

C₈×C₇₁

C₂×D₄

yes

—

C₈×C₇₁

C₂×D₄

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

H(5,𝔽₂)

yes

characteristic

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²×D₄

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

D₄²

yes

characteristic

D₇₁

—

#3293

—

maximal, Sylow

C₇₁

#3293

yes

characteristic, complemented, Sylow

#578

—

#494

D₇₁

yes

characteristic, complemented

#473

—

#456

—

#443

—

2₋¹⁺⁴

—

#137

—

#137

—

142

—

#135

—

#135

—

142

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

C₂×D₇₁

yes

—

#131

—

C₄⋊Q₈

C₂×D₇₁

yes

—

#108

—

#104

—

#101

—

C₄×C₈

C₂×D₇₁

yes

characteristic

#41

—

284

—

#41

—

#41

—

142

—

C₂×Q₈

—

142

—

C₂×Q₈

C₂²×D₇₁

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

142

—

SD₁₆

—

SD₁₆

—

142

—

D₈

—

M₁₆

—

M₁₆

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₂²×D₇₁

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₇₁

yes

characteristic

Q₈

—

Q₈

—

284

—

100

Q₈

—

101

D₄

—

284

—

102

D₄

—

103

D₄

—

142

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

284

—

107

C₂×C₄

—

142

—

108

C₂×C₄

C₂³×D₇₁

yes

characteristic

109

C₈

—

142

—

110

C₈

D₄×D₇₁

yes

—

111

C₈

D₄×D₇₁

yes

—

112

C₂²

—

142

—

113

C₂²

#14556173

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

142

—

117

C₄

C₂×D₄×D₇₁

yes

—

118

C₂

—

119

C₂

—

284

—

120

C₂

#17986634

yes

characteristic

	subgroup	G/H
center	C₂	#17986634
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#494×C₇₁	C₂
socle	C₂×C₇₁	D₄²
hypercenter	#494	D₇₁
hypocenter	C₇₁	#3293
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#17986634

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

#494×C₇₁

C₂

socle

C₂×C₇₁

D₄²

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #17986812	C₂	semidirect
2. #17986271	C₂	semidirect
3. #17986181	C₂	semidirect
4. #17986117	C₂	semidirect
5. #494×C₇₁	C₂	semidirect
61. C₇₁	#3293	semidirect
63. #494	D₇₁	semidirect

H ≅ G/N

type

1. #17986812

C₂

semidirect

2. #17986271

C₂

semidirect

3. #17986181

C₂

semidirect

4. #17986117

C₂

semidirect

5. #494×C₇₁

C₂

semidirect

61. C₇₁

#3293

semidirect

63. #494

D₇₁

semidirect

Subgroups — #22600144

Conjugacy classes of subgroups

Complements