Group #22587416 — subgroups

Subgroups — #22587416

Order 9088.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17986729	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986196	C₂	1	2	yes	maximal
3	#17985748	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985731	C₂	1	2	yes	maximal, complemented
5	#492×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765853	—	2	2	no	—
7	#14556127	—	2	2	no	—
8	#14556126	—	2	2	no	—
9	C₄×Dic₁₄₂	C₂²	1	2	yes	—
10	#14556093	—	2	2	no	—
11	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
12	#14556049	—	2	2	no	—
13	C₂×SD₁₆×C₇₁	—	2	2	no	—
14	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	#127×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₂	—	2	2	no	—
18	C₂×Dic₁₄₂	D₄	1	2	yes	—
19	#11765853	—	4	4	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₇₁	D₄	1	2	yes	—
22	Dic₇₁⋊C₄	—	2	4	no	—
23	Dic₇₁⋊C₄	—	2	2	no	—
24	C₄×Dic₇₁	—	2	2	no	—
25	C₄×Dic₇₁	—	2	2	no	—
26	C₂×Q₈×C₇₁	D₄	1	2	yes	—
27	C₂×D₄×C₇₁	—	2	1	no	—
28	C₂×D₄×C₇₁	—	2	2	no	—
29	SD₁₆×C₇₁	—	4	4	no	—
30	C₂×C₈×C₇₁	—	2	1	no	—
31	C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₂³×C₇₁	—	4	1	no	—
34	Q₈×C₇₁	—	4	2	no	—
35	D₄×C₇₁	—	4	2	no	—
36	D₄×C₇₁	—	4	2	no	—
37	D₄×C₇₁	—	4	2	no	—
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₇₁	—	2	2	no	—
42	C₂×Dic₇₁	—	2	2	no	—
43	Dic₁₄₂	—	2	2	no	—
44	Dic₁₄₂	—	8	2	no	—
45	Dic₁₄₂	SD₁₆	1	4	yes	complemented
46	C₈×C₇₁	—	2	2	no	—
47	C₇₁⋊C₈	—	4	2	no	—
48	C₂²×C₇₁	—	8	2	no	—
49	C₂²×C₇₁	—	8	1	no	—
50	C₂²×C₇₁	#120	1	1	yes	characteristic
51	C₄×C₇₁	—	2	1	no	—
52	C₄×C₇₁	—	4	2	no	—
53	C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
54	C₄×C₇₁	#137	1	2	yes	—
55	Dic₇₁	—	2	4	no	—
56	Dic₇₁	—	4	2	no	—
57	C₂×C₇₁	—	16	1	no	—
58	C₂×C₇₁	#448	1	2	yes	—
59	C₂×C₇₁	#453	1	1	yes	characteristic
60	#1623	—	71	1	no	maximal, Sylow
61	C₇₁	#1623	1	1	yes	characteristic, complemented, Sylow
62	Q₈²	—	71	1	no	—
63	#492	D₇₁	1	1	yes	characteristic
64	#460	—	71	2	no	—
65	#333	—	71	1	no	—
66	Q₈⋊C₈	—	71	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	142	2	no	—
69	C₄⋊Q₈	—	142	2	no	—
70	C₄⋊Q₈	—	142	2	no	—
71	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
72	#127	C₂×D₇₁	1	1	yes	characteristic
73	C₄×Q₈	—	71	2	no	—
74	C₄×Q₈	—	142	2	no	—
75	C₄⋊C₈	—	71	2	no	—
76	#102	—	142	2	no	—
77	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
78	C₂×Q₈	—	71	2	no	—
79	C₂×Q₈	—	142	2	no	—
80	C₂×Q₈	C₂²⋊D₇₁	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	284	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	142	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	71	2	no	—
89	C₄⋊C₄	—	142	4	no	—
90	C₄⋊C₄	—	142	2	no	—
91	C₄²	—	142	2	no	—
92	C₄²	—	142	2	no	—
93	C₄²	C₂²×D₇₁	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	568	2	no	—
97	Q₈	—	71	4	no	—
98	Q₈	—	142	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	142	2	no	—
104	C₂×C₄	—	142	2	no	—
105	C₂×C₄	D₄×D₇₁	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	284	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#14556096	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	284	2	no	—
115	C₄	—	142	4	no	—
116	C₄	SD₁₆×D₇₁	1	2	yes	—
117	C₄	#14556157	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#17986136	1	2	yes	—
120	C₂	#17986166	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986729

C₂

yes

maximal, characteristic, complemented

#17986196

C₂

yes

maximal

#17985748

C₂

yes

maximal, characteristic, complemented

#17985731

C₂

yes

maximal, complemented

#492×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765853

—

#14556127

—

#14556126

—

C₄×Dic₁₄₂

C₂²

yes

—

#14556093

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556049

—

C₂×SD₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

#127×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

—

#11765853

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₄

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

D₄

yes

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

SD₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

SD₁₆

yes

complemented

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#120

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

#137

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#448

yes

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

Q₈²

—

#492

D₇₁

yes

characteristic

#460

—

#333

—

Q₈⋊C₈

—

C₂×SD₁₆

—

C₂×SD₁₆

—

142

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

#127

C₂×D₇₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

142

—

C₄⋊C₈

—

#102

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

C₂²⋊D₇₁

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

—

142

—

C₄²

—

142

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

568

—

Q₈

—

Q₈

—

142

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

142

—

104

C₂×C₄

—

142

—

105

C₂×C₄

D₄×D₇₁

yes

—

106

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

107

C₈

—

108

C₈

—

284

—

109

C₂²

—

110

C₂²

—

111

C₂²

#14556096

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

284

—

115

C₄

—

142

—

116

C₄

SD₁₆×D₇₁

yes

—

117

C₄

#14556157

yes

—

118

C₂

—

119

C₂

#17986136

yes

—

120

C₂

#17986166

yes

characteristic

	subgroup	G/H
center	C₂²	#14556096
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#492×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	#492	D₇₁
hypocenter	C₇₁	#1623
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14556096

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#492×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #17986729	C₂	semidirect
3. #17985748	C₂	semidirect
4. #17985731	C₂	semidirect
45. Dic₁₄₂	SD₁₆	semidirect
61. C₇₁	#1623	semidirect

H ≅ G/N

type

1. #17986729

C₂

semidirect

3. #17985748

C₂

semidirect

4. #17985731

C₂

semidirect

45. Dic₁₄₂

SD₁₆

semidirect

61. C₇₁

#1623

semidirect

Subgroups — #22587416

Conjugacy classes of subgroups

Complements