Group #22590509 — subgroups

Subgroups — #22590509

Order 9088.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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	#17986396	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986116	C₂	1	1	yes	maximal, characteristic
3	#17985822	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985821	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985819	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985809	C₂	1	1	yes	maximal, characteristic
7	#360×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556135	—	2	1	no	—
9	#14556127	—	2	1	no	—
10	#14556068	C₂²	1	1	yes	characteristic
11	#14556064	C₂²	1	1	yes	characteristic
12	#14556060	C₂²	1	1	yes	characteristic
13	#14556059	—	2	1	no	—
14	#14556055	—	2	1	no	—
15	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
16	#135×C₇₁	C₂²	1	1	yes	characteristic
17	M₃₂×C₇₁	C₂²	1	1	yes	characteristic
18	#106×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄₂	—	4	1	no	—
20	#11765849	—	2	1	no	—
21	#11765849	—	4	1	no	—
22	C₂×C₇₁⋊C₈	—	4	1	no	—
23	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₇₁	—	2	1	no	—
25	Dic₇₁⋊C₄	—	4	1	no	—
26	C₄×Dic₇₁	D₄	1	1	yes	characteristic
27	C₇₁⋊C₁₆	—	2	1	no	—
28	#41×C₇₁	—	2	1	no	—
29	Q₁₆×C₇₁	Q₈	1	1	yes	characteristic, complemented
30	SD₁₆×C₇₁	—	2	1	no	—
31	D₈×C₇₁	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₇₁	—	2	1	no	—
34	C₁₆×C₇₁	—	2	1	no	—
35	Q₈×C₇₁	—	2	1	no	—
36	D₄×C₇₁	—	2	1	no	—
37	D₄×C₇₁	—	4	1	no	—
38	C₂×C₄×C₇₁	—	4	1	no	—
39	C₂×C₄×C₇₁	—	4	1	no	—
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₇₁	—	2	1	no	—
42	C₂×Dic₇₁	—	4	1	no	—
43	Dic₁₄₂	—	8	2	no	—
44	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₇₁	#41	1	1	yes	characteristic
46	C₇₁⋊C₈	—	4	1	no	—
47	C₇₁⋊C₈	SD₁₆	1	1	yes	characteristic
48	C₇₁⋊C₈	SD₁₆	1	1	yes	characteristic
49	C₂²×C₇₁	—	4	1	no	—
50	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₇₁	—	4	1	no	—
52	C₄×C₇₁	—	8	1	no	—
53	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₇₁	—	2	2	no	—
56	Dic₇₁	—	8	1	no	—
57	C₂×C₇₁	—	2	1	no	—
58	C₂×C₇₁	—	8	1	no	—
59	C₂×C₇₁	#476	1	1	yes	characteristic
60	#2230	—	71	1	no	maximal, Sylow
61	C₇₁	#2230	1	1	yes	characteristic, complemented, Sylow
62	#498	—	71	1	no	—
63	#443	—	71	1	no	—
64	#365	—	71	1	no	—
65	#365	—	71	1	no	—
66	#364	—	71	1	no	—
67	#360	—	71	1	no	—
68	#360	D₇₁	1	1	yes	characteristic
69	#135	C₂×D₇₁	1	1	yes	characteristic
70	#131	—	142	1	no	—
71	C₄⋊Q₈	—	142	1	no	—
72	M₃₂	—	71	1	no	—
73	M₃₂	C₂×D₇₁	1	1	yes	characteristic
74	#108	—	71	1	no	—
75	#106	—	71	1	no	—
76	#106	—	142	1	no	—
77	#106	C₂×D₇₁	1	1	yes	characteristic
78	#104	—	142	1	no	—
79	C₄×C₈	—	71	1	no	—
80	#41	—	2	1	no	—
81	C₂×Q₈	—	284	1	no	—
82	Q₁₆	Dic₁₄₂	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	Dic₁₄₂	1	1	yes	characteristic, complemented
85	M₁₆	—	284	1	no	—
86	M₁₆	—	142	1	no	—
87	C₂×C₈	—	284	1	no	—
88	C₂×C₈	—	71	1	no	—
89	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	284	1	no	—
92	C₄⋊C₄	—	142	1	no	—
93	C₄²	—	71	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	142	1	no	—
96	Q₈	—	2	1	no	—
97	Q₈	—	568	2	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	284	1	no	—
103	C₂×C₄	—	142	1	no	—
104	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
105	C₈	—	284	1	no	—
106	C₈	—	71	1	no	—
107	C₈	—	71	1	no	—
108	C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
109	C₈	#11765864	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#14556152	1	1	yes	characteristic
112	C₄	—	4	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	568	1	no	—
115	C₄	—	142	2	no	—
116	C₄	#14556105	1	1	yes	characteristic
117	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#17986287	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986396

C₂

yes

maximal, characteristic, complemented

#17986116

C₂

yes

maximal, characteristic

#17985822

C₂

yes

maximal, characteristic, complemented

#17985821

C₂

yes

maximal, characteristic, complemented

#17985819

C₂

yes

maximal, characteristic, complemented

#17985809

C₂

yes

maximal, characteristic

#360×C₇₁

C₂

yes

maximal, characteristic

#14556135

—

#14556127

—

#14556068

C₂²

yes

characteristic

#14556064

C₂²

yes

characteristic

#14556060

C₂²

yes

characteristic

#14556059

—

#14556055

—

C₈×Dic₇₁

C₂²

yes

characteristic

#135×C₇₁

C₂²

yes

characteristic

M₃₂×C₇₁

C₂²

yes

characteristic

#106×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

#41×C₇₁

—

Q₁₆×C₇₁

Q₈

yes

characteristic, complemented

SD₁₆×C₇₁

—

D₈×C₇₁

Q₈

yes

characteristic, complemented

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

SD₁₆

yes

characteristic

C₇₁⋊C₈

SD₁₆

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic

#135

C₂×D₇₁

yes

characteristic

#131

—

142

—

C₄⋊Q₈

—

142

—

M₃₂

—

M₃₂

C₂×D₇₁

yes

—

—

—

142

—

#106

C₂×D₇₁

yes

characteristic

#104

—

142

—

C₄×C₈

—

#41

—

C₂×Q₈

—

284

—

Q₁₆

Dic₁₄₂

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₁₄₂

yes

characteristic, complemented

M₁₆

—

284

—

M₁₆

—

142

—

C₂×C₈

—

284

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄²

—

C₁₆

—

C₁₆

—

142

—

Q₈

—

Q₈

—

568

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

284

—

103

C₂×C₄

—

142

—

104

C₂×C₄

D₄×D₇₁

yes

characteristic

105

C₈

—

284

—

106

C₈

—

107

C₈

—

108

C₈

C₂×Dic₁₄₂

yes

characteristic

109

C₈

#11765864

yes

characteristic

110

C₂²

—

111

C₂²

#14556152

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

568

—

115

C₄

—

142

—

116

C₄

#14556105

yes

characteristic

117

C₄

SD₁₆×D₇₁

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#17986287

yes

characteristic

	subgroup	G/H
center	C₂	#17986287
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#360×C₇₁	C₂
socle	C₂×C₇₁	#476
hypercenter	#360	D₇₁
hypocenter	C₇₁	#2230
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#17986287

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#360×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
1. #17986396	C₂	semidirect
3. #17985822	C₂	semidirect
4. #17985821	C₂	semidirect
5. #17985819	C₂	semidirect
29. Q₁₆×C₇₁	Q₈	semidirect
31. D₈×C₇₁	Q₈	semidirect
61. C₇₁	#2230	semidirect
82. Q₁₆	Dic₁₄₂	semidirect
84. D₈	Dic₁₄₂	semidirect

H ≅ G/N

type

1. #17986396

C₂

semidirect

3. #17985822

C₂

semidirect

4. #17985821

C₂

semidirect

5. #17985819

C₂

semidirect

29. Q₁₆×C₇₁

Q₈

semidirect

31. D₈×C₇₁

Q₈

semidirect

61. C₇₁

#2230

semidirect

82. Q₁₆

Dic₁₄₂

semidirect

84. D₈

Dic₁₄₂

semidirect

Subgroups — #22590509

Conjugacy classes of subgroups

Complements