Group #22587700 — subgroups

Subgroups — #22587700

Order 9088.

188 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17986376	C₂	1	1	yes	maximal, characteristic
2	#17986199	C₂	1	2	yes	maximal, complemented
3	#17985752	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985742	C₂	1	2	yes	maximal, complemented
5	C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765852	—	2	2	no	—
7	C₂×D₅₆₈	—	2	1	no	—
8	C₂×D₅₆₈	—	2	1	no	—
9	#14556125	C₂²	1	2	yes	—
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	#14556048	—	2	2	no	—
12	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₇₁	—	2	1	no	—
14	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₇₁	—	2	1	no	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×D₂₈₄	—	2	2	no	—
18	C₂×D₂₈₄	—	2	2	no	—
19	C₂×D₂₈₄	—	2	2	no	—
20	#11765852	—	4	4	no	—
21	D₅₆₈	—	4	2	no	—
22	D₅₆₈	—	4	2	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	—	2	1	no	—
26	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₇₁	—	2	1	no	—
29	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
30	C₄²×C₇₁	—	2	1	no	—
31	C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₂²×D₇₁	—	4	2	no	—
33	Q₈×C₇₁	—	2	1	no	—
34	Q₈×C₇₁	D₈	1	2	yes	complemented
35	C₂×C₄×C₇₁	—	2	1	no	—
36	C₂×C₄×C₇₁	—	2	1	no	—
37	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
40	D₂₈₄	—	4	2	no	—
41	D₂₈₄	—	4	2	no	—
42	D₂₈₄	—	4	2	no	—
43	D₂₈₄	—	4	2	no	—
44	D₂₈₄	—	8	2	no	—
45	C₈×C₇₁	—	2	1	no	—
46	C₈×C₇₁	—	4	1	no	—
47	C₈×C₇₁	SD₁₆	1	2	yes	complemented
48	C₇₁⋊C₈	—	4	2	no	—
49	C₂²×C₇₁	#121	1	1	yes	characteristic
50	C₂×D₇₁	—	8	2	no	—
51	C₂×D₇₁	—	8	2	no	—
52	C₂×D₇₁	—	8	2	no	—
53	C₄×C₇₁	—	2	2	no	—
54	C₄×C₇₁	—	2	1	no	—
55	C₄×C₇₁	—	4	1	no	—
56	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
57	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
58	C₄×C₇₁	#135	1	1	yes	characteristic
59	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₇₁	#460	1	1	yes	characteristic
61	C₂×C₇₁	#466	1	1	yes	characteristic
62	C₂×C₇₁	#472	1	1	yes	characteristic
63	D₇₁	—	16	2	no	—
64	#1668	—	71	1	no	maximal, Sylow
65	C₇₁	#1668	1	1	yes	characteristic, complemented, Sylow
66	#493	—	71	1	no	—
67	#460	—	71	2	no	—
68	C₈×Q₈	D₇₁	1	1	yes	characteristic, complemented
69	#335	—	71	1	no	—
70	#331	—	71	2	no	—
71	C₂×SD₁₆	—	142	2	no	—
72	C₂×D₈	—	142	1	no	—
73	C₂×D₈	—	142	1	no	—
74	#127	—	71	2	no	—
75	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	—	71	2	no	—
78	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
79	#102	—	142	2	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
82	C₂×Q₈	D₂₈₄	1	1	yes	characteristic
83	C₂×D₄	—	142	2	no	—
84	C₂×D₄	—	142	2	no	—
85	C₂×D₄	—	142	2	no	—
86	SD₁₆	—	284	4	no	—
87	D₈	—	284	2	no	—
88	D₈	—	284	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	142	2	no	—
91	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
95	C₄²	—	2	1	no	—
96	C₄²	C₂²×D₇₁	1	1	yes	characteristic
97	C₂³	—	284	2	no	—
98	Q₈	—	2	1	no	—
99	Q₈	D₅₆₈	1	2	yes	complemented
100	D₄	—	284	2	no	—
101	D₄	—	284	2	no	—
102	D₄	—	284	2	no	—
103	D₄	—	284	2	no	—
104	D₄	—	568	2	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
108	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
109	C₂×C₄	#11765865	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	1	no	—
112	C₈	—	284	2	no	—
113	C₈	#11765852	1	2	yes	complemented
114	C₂²	—	568	2	no	—
115	C₂²	—	568	2	no	—
116	C₂²	—	568	2	no	—
117	C₂²	#14556102	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	1	no	—
121	C₄	C₂×D₅₆₈	1	1	yes	characteristic
122	C₄	C₂×#11765852	1	1	yes	characteristic
123	C₄	#14556150	1	1	yes	characteristic
124	C₄	#14556153	1	1	yes	characteristic
125	C₂	—	1136	2	no	—
126	C₂	#17986199	1	1	yes	characteristic
127	C₂	#17986235	1	1	yes	characteristic
128	C₂	#17986269	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986376

C₂

yes

maximal, characteristic

#17986199

C₂

yes

maximal, complemented

#17985752

C₂

yes

maximal, characteristic, complemented

#17985742

C₂

yes

maximal, complemented

C₈×Q₈×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765852

—

C₂×D₅₆₈

—

C₂×D₅₆₈

—

#14556125

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556048

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

#11765852

—

D₅₆₈

—

D₅₆₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

D₈

yes

complemented

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₁₆

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1668

—

maximal, Sylow

C₇₁

#1668

yes

characteristic, complemented, Sylow

#493

—

#460

—

C₈×Q₈

D₇₁

yes

characteristic, complemented

#335

—

#331

—

C₂×SD₁₆

—

142

—

C₂×D₈

—

142

—

C₂×D₈

—

142

—

#127

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#102

—

142

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

D₂₈₄

yes

characteristic

C₂×D₄

—

142

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

SD₁₆

—

284

—

D₈

—

284

—

D₈

—

284

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₂₈₄

yes

characteristic

C₄²

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

284

—

Q₈

—

Q₈

D₅₆₈

yes

complemented

100

D₄

—

284

—

101

D₄

—

284

—

102

D₄

—

284

—

103

D₄

—

284

—

104

D₄

—

568

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

108

C₂×C₄

C₂×D₂₈₄

yes

characteristic

109

C₂×C₄

#11765865

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

284

—

113

C₈

#11765852

yes

complemented

114

C₂²

—

568

—

115

C₂²

—

568

—

116

C₂²

—

568

—

117

C₂²

#14556102

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×D₅₆₈

yes

characteristic

122

C₄

C₂×#11765852

yes

characteristic

123

C₄

#14556150

yes

characteristic

124

C₄

#14556153

yes

characteristic

125

C₂

—

1136

—

126

C₂

#17986199

yes

characteristic

127

C₂

#17986235

yes

characteristic

128

C₂

#17986269

yes

characteristic

	subgroup	G/H
center	C₂²	#14556102
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	C₈×Q₈×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	C₈×Q₈	D₇₁
hypocenter	C₇₁	#1668
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14556102

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

C₈×Q₈×C₇₁

C₂

socle

C₂²×C₇₁

#121

hypercenter

C₈×Q₈

D₇₁

hypocenter

C₇₁

#1668

LCS

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
2. #17986199	C₂	semidirect
3. #17985752	C₂	semidirect
4. #17985742	C₂	semidirect
5. C₈×Q₈×C₇₁	C₂	semidirect
34. Q₈×C₇₁	D₈	semidirect
47. C₈×C₇₁	SD₁₆	semidirect
65. C₇₁	#1668	semidirect
68. C₈×Q₈	D₇₁	semidirect
99. Q₈	D₅₆₈	semidirect
113. C₈	#11765852	semidirect

H ≅ G/N

type

2. #17986199

C₂

semidirect

3. #17985752

C₂

semidirect

4. #17985742

C₂

semidirect

5. C₈×Q₈×C₇₁

C₂

semidirect

34. Q₈×C₇₁

D₈

semidirect

47. C₈×C₇₁

SD₁₆

semidirect

65. C₇₁

#1668

semidirect

68. C₈×Q₈

D₇₁

semidirect

99. Q₈

D₅₆₈

semidirect

113. C₈

#11765852

semidirect

Subgroups — #22587700

Conjugacy classes of subgroups

Complements