Group #22587418 — subgroups

Subgroups — #22587418

Order 9088.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17986373	C₂	1	1	yes	maximal, characteristic
2	#17986199	C₂	1	2	yes	maximal
3	#17985748	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal, complemented
5	Q₈²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765852	—	2	2	no	—
7	C₂×#11765852	—	2	2	no	—
8	#14556125	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	#14556048	—	2	2	no	—
11	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₇₁	—	2	2	no	—
13	C₄⋊Q₈×C₇₁	—	2	2	no	—
14	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₇₁	—	2	2	no	—
16	C₄×Q₈×C₇₁	C₂²	1	2	yes	—
17	C₂×D₂₈₄	—	2	2	no	—
18	C₂×D₂₈₄	—	2	1	no	—
19	#11765852	—	4	4	no	—
20	#11765852	—	4	4	no	—
21	C₂×C₇₁⋊C₈	—	2	1	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₂×Q₈×C₇₁	—	2	2	no	—
24	C₂×Q₈×C₇₁	D₄	1	2	yes	—
25	C₂×Q₈×C₇₁	D₄	1	2	yes	—
26	C₄⋊C₄×C₇₁	—	2	4	no	—
27	C₄⋊C₄×C₇₁	—	2	2	no	—
28	C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₄⋊C₄×C₇₁	D₄	1	2	yes	—
30	C₄²×C₇₁	—	2	2	no	—
31	C₄²×C₇₁	—	2	2	no	—
32	C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₂²×D₇₁	—	4	1	no	—
34	Q₈×C₇₁	—	2	2	no	—
35	Q₈×C₇₁	—	4	2	no	—
36	Q₈×C₇₁	—	8	2	no	—
37	Q₈×C₇₁	SD₁₆	1	4	yes	complemented
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	—	2	2	no	—
40	C₂×C₄×C₇₁	—	2	2	no	—
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
43	D₂₈₄	—	4	2	no	—
44	D₂₈₄	—	4	2	no	—
45	D₂₈₄	—	4	2	no	—
46	C₇₁⋊C₈	—	2	2	no	—
47	C₇₁⋊C₈	—	4	2	no	—
48	C₂²×C₇₁	#120	1	1	yes	characteristic
49	C₂×D₇₁	—	8	2	no	—
50	C₂×D₇₁	—	8	1	no	—
51	C₄×C₇₁	—	2	4	no	—
52	C₄×C₇₁	—	2	1	no	—
53	C₄×C₇₁	—	4	2	no	—
54	C₄×C₇₁	—	4	2	no	—
55	C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
56	C₄×C₇₁	#137	1	2	yes	—
57	C₂×C₇₁	#448	1	2	yes	—
58	C₂×C₇₁	#453	1	1	yes	characteristic
59	D₇₁	—	16	1	no	—
60	#1623	—	71	1	no	maximal, Sylow
61	C₇₁	#1623	1	1	yes	characteristic, complemented, Sylow
62	Q₈²	D₇₁	1	1	yes	characteristic, complemented
63	#492	—	71	1	no	—
64	#460	—	71	2	no	—
65	#333	—	71	1	no	—
66	Q₈⋊C₈	—	71	2	no	—
67	C₂×SD₁₆	—	142	2	no	—
68	C₂×SD₁₆	—	142	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
72	#127	—	71	1	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	C₂×D₇₁	1	2	yes	—
75	C₄⋊C₈	—	71	2	no	—
76	#102	—	142	2	no	—
77	C₄×C₈	—	71	1	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	C₂²⋊D₇₁	1	2	yes	—
80	C₂×Q₈	C₂²⋊D₇₁	1	2	yes	—
81	C₂×D₄	—	142	2	no	—
82	C₂×D₄	—	142	1	no	—
83	SD₁₆	—	284	4	no	—
84	SD₁₆	—	284	4	no	—
85	C₂×C₈	—	142	2	no	—
86	C₂×C₈	—	142	1	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	C₂²⋊D₇₁	1	2	yes	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	C₂²×D₇₁	1	1	yes	characteristic
94	C₂³	—	284	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	4	2	no	—
97	Q₈	—	8	2	no	—
98	Q₈	#11765852	1	4	yes	complemented
99	D₄	—	284	2	no	—
100	D₄	—	284	2	no	—
101	D₄	—	284	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
107	C₈	—	284	2	no	—
108	C₈	—	142	2	no	—
109	C₂²	—	568	2	no	—
110	C₂²	—	568	1	no	—
111	C₂²	#14556097	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂×#11765852	1	2	yes	—
117	C₄	#14556159	1	2	yes	—
118	C₂	—	1136	1	no	—
119	C₂	#17986138	1	2	yes	—
120	C₂	#17986167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986373

C₂

yes

maximal, characteristic

#17986199

C₂

yes

maximal

#17985748

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₇₁⋊C₈)

C₂

yes

maximal, complemented

Q₈²×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765852

—

C₂×#11765852

—

#14556125

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556048

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

C₂²

yes

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

#11765852

—

#11765852

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

—

C₂×Q₈×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

SD₁₆

yes

complemented

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#120

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

#137

yes

—

C₂×C₇₁

#448

yes

—

C₂×C₇₁

#453

yes

characteristic

D₇₁

—

#1623

—

maximal, Sylow

C₇₁

#1623

yes

characteristic, complemented, Sylow

Q₈²

D₇₁

yes

characteristic, complemented

#492

—

#460

—

#333

—

Q₈⋊C₈

—

C₂×SD₁₆

—

142

—

C₂×SD₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇₁

yes

—

C₄⋊C₈

—

#102

—

142

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₇₁

yes

—

C₂×Q₈

C₂²⋊D₇₁

yes

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

SD₁₆

—

284

—

SD₁₆

—

284

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₇₁

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

284

—

Q₈

—

Q₈

—

Q₈

—

Q₈

#11765852

yes

complemented

D₄

—

284

—

100

D₄

—

284

—

101

D₄

—

284

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

107

C₈

—

284

—

108

C₈

—

142

—

109

C₂²

—

568

—

110

C₂²

—

568

—

111

C₂²

#14556097

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂×#11765852

yes

—

117

C₄

#14556159

yes

—

118

C₂

—

1136

—

119

C₂

#17986138

yes

—

120

C₂

#17986167

yes

characteristic

	subgroup	G/H
center	C₂²	#14556097
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	Q₈²×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	Q₈²	D₇₁
hypocenter	C₇₁	#1623
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#14556097

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

Q₈²×C₇₁

C₂

socle

C₂²×C₇₁

#120

hypercenter

Q₈²

D₇₁

hypocenter

C₇₁

#1623

LCS

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
3. #17985748	C₂	semidirect
4. Q₈⋊(C₇₁⋊C₈)	C₂	semidirect
5. Q₈²×C₇₁	C₂	semidirect
37. Q₈×C₇₁	SD₁₆	semidirect
61. C₇₁	#1623	semidirect
62. Q₈²	D₇₁	semidirect
98. Q₈	#11765852	semidirect

H ≅ G/N

type

3. #17985748

C₂

semidirect

4. Q₈⋊(C₇₁⋊C₈)

C₂

semidirect

5. Q₈²×C₇₁

C₂

semidirect

37. Q₈×C₇₁

SD₁₆

semidirect

61. C₇₁

#1623

semidirect

62. Q₈²

D₇₁

semidirect

98. Q₈

#11765852

semidirect

Subgroups — #22587418

Conjugacy classes of subgroups

Complements