Group #22588244 — subgroups

Subgroups — #108×Dic₇₁ #22588244

Order 9088.

228 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 121 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×#14556064	C₂	1	1	yes	maximal, characteristic
2	M₁₆×Dic₇₁	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985763	C₂	1	2	yes	maximal
5	C₂×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	2	yes	—
7	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
9	#14556064	C₄	1	4	yes	complemented
10	#14556038	—	2	2	no	—
11	C₈×Dic₇₁	—	2	2	no	—
12	C₈×Dic₇₁	C₄	1	4	yes	—
13	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#108×C₇₁	C₄	1	4	yes	complemented
16	C₂²×Dic₇₁	Q₈	1	2	yes	—
17	#11765849	—	2	2	no	—
18	#11765849	C₂×C₄	1	4	yes	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	—	2	1	no	—
21	C₂×C₇₁⋊C₈	—	2	1	no	—
22	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×Dic₇₁	C₂×C₄	1	2	yes	—
24	C₄×Dic₇₁	D₄	1	1	yes	characteristic
25	C₄×Dic₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₇₁	—	2	2	no	—
28	M₁₆×C₇₁	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₇₁	—	2	2	no	—
30	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
33	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
39	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
40	C₂×Dic₇₁	—	2	4	no	—
41	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
42	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
43	C₈×C₇₁	—	2	4	no	—
44	C₈×C₇₁	C₄²	1	4	yes	—
45	C₇₁⋊C₈	—	2	2	no	—
46	C₇₁⋊C₈	—	2	4	no	—
47	C₂²×C₇₁	—	2	2	no	—
48	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₇₁	C₄×Q₈	1	1	yes	characteristic
50	C₂²×C₇₁	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₇₁	#117	1	1	yes	characteristic
53	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
54	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
55	Dic₇₁	—	2	2	no	—
56	Dic₇₁	#108	1	4	yes	direct product factor
57	C₂×C₇₁	—	2	1	no	—
58	C₂×C₇₁	—	2	1	no	—
59	C₂×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₇₁	C₂×#108	1	2	yes	—
61	C₄×#108	—	71	1	no	maximal, Sylow
62	C₇₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	—	71	1	no	—
64	C₂×#108	D₇₁	1	1	yes	characteristic
65	C₄×M₁₆	—	71	2	no	—
66	C₂×C₄×C₈	—	71	1	no	—
67	#341	—	71	2	no	—
68	C₂×M₁₆	—	71	2	no	—
69	C₂×M₁₆	C₂×D₇₁	1	2	yes	—
70	C₂²×C₈	—	71	1	no	—
71	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
72	C₂×C₄²	—	71	1	no	—
73	#108	—	71	4	no	—
74	#108	Dic₇₁	1	4	yes	direct product factor
75	#97	—	142	2	no	—
76	C₄×C₈	—	71	4	no	—
77	C₄×C₈	—	142	2	no	—
78	C₂²×C₄	—	71	2	no	—
79	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	71	4	no	—
82	M₁₆	—	142	2	no	—
83	M₁₆	C₂×Dic₇₁	1	4	yes	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	71	2	no	—
86	C₂×C₈	—	142	2	no	—
87	C₂×C₈	—	142	1	no	—
88	C₂×C₈	—	142	1	no	—
89	C₂×C₈	C₄×D₇₁	1	2	yes	—
90	C₂×C₈	C₄×D₇₁	1	2	yes	—
91	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
92	C₄²	—	71	2	no	—
93	C₄²	—	71	1	no	—
94	C₄²	—	71	1	no	—
95	C₂³	Q₈×D₇₁	1	1	yes	characteristic
96	C₂×C₄	—	71	2	no	—
97	C₂×C₄	—	71	2	no	—
98	C₂×C₄	—	142	4	no	—
99	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
101	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
102	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
103	C₂×C₄	#11765864	1	1	yes	characteristic
104	C₂×C₄	#11765866	1	1	yes	characteristic
105	C₈	—	2	4	no	—
106	C₈	—	142	2	no	—
107	C₈	—	142	4	no	—
108	C₈	C₄×Dic₇₁	1	4	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
111	C₂²	#14556093	1	1	yes	characteristic
112	C₂²	Q₈×Dic₇₁	1	1	yes	characteristic
113	C₄	—	71	4	no	—
114	C₄	—	142	2	no	—
115	C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
116	C₄	#14556086	1	1	yes	characteristic
117	C₄	#14556090	1	1	yes	characteristic
118	C₄	D₄×Dic₇₁	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
122	C₂	#108×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556064

C₂

yes

maximal, characteristic

M₁₆×Dic₇₁

C₂

yes

maximal

C₂×C₈×Dic₇₁

C₂

yes

maximal, characteristic

#17985763

C₂

yes

maximal

C₂×#108×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556064

C₄

yes

complemented

#14556038

—

C₈×Dic₇₁

—

C₈×Dic₇₁

C₄

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

#108×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

Q₈

yes

—

#11765849

—

#11765849

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

D₄

yes

characteristic

C₄×Dic₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂³×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₄²

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₄×Q₈

yes

characteristic

C₂²×C₇₁

C₄×Q₈

yes

characteristic

C₄×C₇₁

C₂×C₄²

yes

characteristic

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

characteristic

Dic₇₁

—

Dic₇₁

#108

yes

direct product factor

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇₁

C₂×#108

yes

—

C₄×#108

—

maximal, Sylow

C₇₁

C₄×#108

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

D₇₁

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄²

—

#108

—

#108

Dic₇₁

yes

direct product factor

#97

—

142

—

C₄×C₈

—

C₄×C₈

—

142

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

142

—

M₁₆

C₂×Dic₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

100

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

101

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

102

C₂×C₄

D₄×D₇₁

yes

characteristic

103

C₂×C₄

#11765864

yes

characteristic

104

C₂×C₄

#11765866

yes

characteristic

105

C₈

—

106

C₈

—

142

—

107

C₈

—

142

—

108

C₈

C₄×Dic₇₁

yes

—

109

C₂²

—

110

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

111

C₂²

#14556093

yes

characteristic

112

C₂²

Q₈×Dic₇₁

yes

characteristic

113

C₄

—

114

C₄

—

142

—

115

C₄

C₂×C₄×Dic₇₁

yes

characteristic

116

C₄

#14556086

yes

characteristic

117

C₄

#14556090

yes

characteristic

118

C₄

D₄×Dic₇₁

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

C₄⋊C₄×Dic₇₁

yes

characteristic

122

C₂

#108×D₇₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#108×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	C₂×#108	D₇₁
hypocenter	C₇₁	C₄×#108
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₇₁

commutator

C₄×C₇₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#108×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #14556064	C₄	semidirect
15. #108×C₇₁	C₄	semidirect
56. Dic₇₁	#108	direct (N × H)
62. C₇₁	C₄×#108	semidirect
74. #108	Dic₇₁	direct (N × H)

H ≅ G/N

type

9. #14556064

C₄

semidirect

15. #108×C₇₁

C₄

semidirect

56. Dic₇₁

#108

direct (N × H)

62. C₇₁

C₄×#108

semidirect

74. #108

Dic₇₁

direct (N × H)

Subgroups — #108×Dic₇₁ #22588244

Conjugacy classes of subgroups

Complements

Subgroups — #108×Dic71 #22588244

Conjugacy classes of subgroups

Complements

Subgroups — #108×Dic₇₁ #22588244