Group #22588495 — subgroups

Subgroups — #22588495

Order 9088.

280 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×#14556060	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556059	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985868	C₂	1	2	yes	maximal
5	C₂×#106×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	C₂×Dic₇₁⋊C₄	—	2	4	no	—
9	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
10	#14556060	—	2	2	no	—
11	#14556059	—	2	4	no	—
12	C₈×Dic₇₁	C₄	1	4	yes	complemented
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	#106×C₇₁	—	2	2	no	—
16	C₂²×Dic₇₁	—	2	2	no	—
17	C₂²×Dic₇₁	D₄	1	2	yes	—
18	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
20	C₂×C₇₁⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₇₁	—	2	2	no	—
22	C₄⋊Dic₇₁	—	2	2	no	—
23	Dic₇₁⋊C₄	—	4	8	no	—
24	C₄×Dic₇₁	C₂×C₄	1	2	yes	—
25	C₄×Dic₇₁	Q₈	1	2	yes	—
26	C₂²×C₄×C₇₁	—	2	2	no	—
27	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₁	D₄	1	2	yes	—
30	C₂×C₈×C₇₁	Q₈	1	2	yes	—
31	C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₄⋊C₄×C₇₁	—	2	2	no	—
33	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	—	4	2	no	—
35	C₂×C₄×C₇₁	—	4	4	no	—
36	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×Dic₇₁	—	4	2	no	—
43	C₂×Dic₇₁	—	4	4	no	—
44	C₂×Dic₇₁	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₇₁	SD₁₆	1	8	yes	—
46	C₈×C₇₁	C₄⋊C₄	1	4	yes	complemented
47	C₇₁⋊C₈	C₄⋊C₄	1	4	yes	complemented
48	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₇₁	#127	1	1	yes	characteristic
50	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₇₁	C₂×SD₁₆	1	4	yes	—
52	C₄×C₇₁	—	8	2	no	—
53	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₇₁	—	8	2	no	—
58	Dic₇₁	#106	1	8	yes	complemented
59	C₂×C₇₁	#384	1	1	yes	characteristic
60	C₂×C₇₁	C₂×#106	1	2	yes	—
61	C₂×C₇₁	#492	1	2	yes	—
62	C₂×C₇₁	#498	1	2	yes	—
63	#1844	—	71	1	no	maximal, Sylow
64	C₇₁	#1844	1	1	yes	characteristic, complemented, Sylow
65	C₂×#106	—	71	1	no	—
66	C₂×#106	—	71	2	no	—
67	C₂×#106	D₇₁	1	1	yes	characteristic
68	C₂×C₄×C₈	—	71	1	no	—
69	#384	—	71	2	no	—
70	C₂²×C₈	—	71	1	no	—
71	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
72	C₂×C₄⋊C₄	—	71	2	no	—
73	C₂×C₄⋊C₄	—	142	4	no	—
74	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
75	C₂×C₄²	—	71	1	no	—
76	#106	—	2	2	no	—
77	#106	—	142	4	no	—
78	#106	—	142	2	no	—
79	C₄×C₈	—	71	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	71	2	no	—
82	C₂²×C₄	—	142	2	no	—
83	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
84	C₂×C₈	—	71	2	no	—
85	C₂×C₈	—	71	2	no	—
86	C₂×C₈	—	71	2	no	—
87	C₂×C₈	Dic₁₄₂	1	2	yes	—
88	C₂×C₈	C₄×D₇₁	1	2	yes	—
89	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	284	8	no	—
93	C₄⋊C₄	—	142	2	no	—
94	C₄⋊C₄	—	142	2	no	—
95	C₄²	—	71	2	no	—
96	C₄²	—	71	2	no	—
97	C₂³	D₄×D₇₁	1	1	yes	characteristic
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	284	2	no	—
101	C₂×C₄	—	284	4	no	—
102	C₂×C₄	—	71	4	no	—
103	C₂×C₄	—	71	8	no	—
104	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
105	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
110	C₈	—	71	4	no	—
111	C₈	Dic₇₁⋊C₄	1	4	yes	complemented
112	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
113	C₂²	#14556124	1	1	yes	characteristic
114	C₂²	#14556127	1	1	yes	characteristic
115	C₂²	SD₁₆×D₇₁	1	4	yes	—
116	C₄	—	8	2	no	—
117	C₄	—	568	2	no	—
118	C₄	—	71	8	no	—
119	C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
120	C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
121	C₄	#14556126	1	1	yes	characteristic
122	C₄	#14556127	1	1	yes	characteristic
123	C₂	#17985868	1	1	yes	characteristic
124	C₂	#106×D₇₁	1	2	yes	—
125	C₂	#17986370	1	2	yes	—
126	C₂	#17986396	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556060

C₂

yes

maximal, characteristic

C₂×#14556059

C₂

yes

maximal

C₂×C₈×Dic₇₁

C₂

yes

maximal, characteristic

#17985868

C₂

yes

maximal

C₂×#106×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556060

—

#14556059

—

C₈×Dic₇₁

C₄

yes

complemented

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

#106×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

Q₈

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

SD₁₆

yes

—

C₈×C₇₁

C₄⋊C₄

yes

complemented

C₇₁⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#127

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#492

yes

—

C₂×C₇₁

#498

yes

—

#1844

—

maximal, Sylow

C₇₁

#1844

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×#106

—

C₂×#106

D₇₁

yes

characteristic

C₂×C₄×C₈

—

#384

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

#106

—

#106

—

142

—

#106

—

142

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₄₂

yes

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

—

C₄²

—

C₂³

D₄×D₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

284

—

101

C₂×C₄

—

284

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

105

C₂×C₄

D₄×D₇₁

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

107

C₂×C₄

Q₈×D₇₁

yes

characteristic

108

C₂×C₄

Q₈×D₇₁

yes

characteristic

109

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

110

C₈

—

111

C₈

Dic₇₁⋊C₄

yes

complemented

112

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

113

C₂²

#14556124

yes

characteristic

114

C₂²

#14556127

yes

characteristic

115

C₂²

SD₁₆×D₇₁

yes

—

116

C₄

—

117

C₄

—

568

—

118

C₄

—

119

C₄

C₄⋊C₄×D₇₁

yes

characteristic

120

C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

121

C₄

#14556126

yes

characteristic

122

C₄

#14556127

yes

characteristic

123

C₂

#17985868

yes

characteristic

124

C₂

#106×D₇₁

yes

—

125

C₂

#17986370

yes

—

126

C₂

#17986396

yes

—

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#106×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#106	D₇₁
hypocenter	C₇₁	#1844
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#106×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
12. C₈×Dic₇₁	C₄	semidirect
46. C₈×C₇₁	C₄⋊C₄	semidirect
47. C₇₁⋊C₈	C₄⋊C₄	semidirect
58. Dic₇₁	#106	semidirect
64. C₇₁	#1844	semidirect
111. C₈	Dic₇₁⋊C₄	semidirect

H ≅ G/N

type

12. C₈×Dic₇₁

C₄

semidirect

46. C₈×C₇₁

C₄⋊C₄

semidirect

47. C₇₁⋊C₈

C₄⋊C₄

semidirect

58. Dic₇₁

#106

semidirect

64. C₇₁

#1844

semidirect

111. C₈

Dic₇₁⋊C₄

semidirect

Subgroups — #22588495

Conjugacy classes of subgroups

Complements