Group #22588243 — subgroups

Subgroups — C₄×#14556063 #22588243

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×#14556063	C₂	1	2	yes	maximal
2	C₄×#11765849	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#17985763	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	2	yes	—
7	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#14556063	C₄	1	8	yes	direct product factor
10	#14556039	—	2	1	no	—
11	C₄×C₇₁⋊C₈	—	2	1	no	—
12	C₄×C₇₁⋊C₈	C₄	1	4	yes	—
13	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	#97×C₇₁	—	2	1	no	—
16	C₄×C₈×C₇₁	—	2	1	no	—
17	#11765849	—	2	2	no	—
18	#11765849	C₂×C₄	1	4	yes	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₇₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₇₁	—	2	2	no	—
27	M₁₆×C₇₁	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₇₁	—	2	2	no	—
29	C₄²×C₇₁	C₂×C₄	1	2	yes	—
30	C₄²×C₇₁	D₄	1	1	yes	characteristic
31	C₄²×C₇₁	D₄	1	1	yes	characteristic
32	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₇₁	—	2	4	no	—
34	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	#41	1	2	yes	—
40	C₈×C₇₁	—	2	4	no	—
41	C₇₁⋊C₈	—	2	2	no	—
42	C₇₁⋊C₈	—	2	4	no	—
43	C₇₁⋊C₈	C₄²	1	4	yes	—
44	C₂²×C₇₁	—	2	2	no	—
45	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₇₁	C₄×Q₈	1	2	yes	—
47	C₄×C₇₁	—	2	2	no	—
48	C₄×C₇₁	#108	1	4	yes	complemented
49	C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₇₁	#117	1	1	yes	characteristic
51	C₄×C₇₁	C₄×D₄	1	2	yes	—
52	C₂×C₇₁	—	2	2	no	—
53	C₂×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₇₁	C₂×#108	1	2	yes	—
55	C₄×#108	—	71	1	no	maximal, Sylow
56	C₇₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
57	C₂×#108	—	71	2	no	—
58	C₄×M₁₆	—	71	1	no	—
59	C₄×M₁₆	D₇₁	1	1	yes	characteristic
60	C₂×C₄×C₈	—	71	1	no	—
61	#341	—	71	2	no	—
62	C₂×M₁₆	—	71	2	no	—
63	C₂×M₁₆	C₂×D₇₁	1	2	yes	—
64	C₂²×C₈	—	71	1	no	—
65	C₂²×C₈	—	71	1	no	—
66	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
67	#108	—	71	8	no	—
68	#97	—	2	1	no	—
69	#97	—	142	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	71	4	no	—
72	C₄×C₈	—	142	1	no	—
73	C₂²×C₄	Dic₁₄₂	1	2	yes	—
74	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	71	4	no	—
77	M₁₆	—	142	2	no	—
78	M₁₆	C₄×D₇₁	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	71	2	no	—
81	C₂×C₈	—	71	4	no	—
82	C₂×C₈	—	71	2	no	—
83	C₂×C₈	—	142	2	no	—
84	C₂×C₈	—	142	2	no	—
85	C₄²	C₄×D₇₁	1	2	yes	—
86	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
87	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
88	C₂³	C₂×Dic₁₄₂	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	Dic₇₁⋊C₄	1	2	yes	—
91	C₂×C₄	Dic₇₁⋊C₄	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₇₁	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
95	C₂×C₄	#11765865	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	71	4	no	—
98	C₈	—	142	2	no	—
99	C₈	—	142	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
102	C₂²	C₄×Dic₁₄₂	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	#14556063	1	4	yes	direct product factor
105	C₄	C₄²×D₇₁	1	1	yes	characteristic
106	C₄	#14556085	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
110	C₂	C₂×#14556063	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556063

C₂

yes

maximal

C₄×#11765849

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985763

C₂

yes

maximal

C₄×M₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

#14556063

C₄

yes

direct product factor

#14556039

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₄

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

C₄×C₈×C₇₁

—

#11765849

—

#11765849

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₂³×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄²

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

#108

yes

complemented

C₄×C₇₁

C₂×C₄²

yes

characteristic

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

—

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

—

C₄×M₁₆

—

C₄×M₁₆

D₇₁

yes

characteristic

C₂×C₄×C₈

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₁

yes

—

—

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

142

—

C₂²×C₄

Dic₁₄₂

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

142

—

M₁₆

C₄×D₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

C₄×D₇₁

yes

—

C₄²

C₂²⋊D₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

characteristic

C₂³

C₂×Dic₁₄₂

yes

characteristic

C₂×C₄

—

C₂×C₄

Dic₇₁⋊C₄

yes

—

C₂×C₄

Dic₇₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

C₂×C₄×D₇₁

yes

—

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

#11765865

yes

—

C₈

—

C₈

—

C₈

—

142

—

C₈

—

142

—

100

C₂²

—

101

C₂²

C₂×Dic₇₁⋊C₄

yes

characteristic

102

C₂²

C₄×Dic₁₄₂

yes

—

103

C₄

—

104

C₄

#14556063

yes

direct product factor

105

C₄

C₄²×D₇₁

yes

characteristic

106

C₄

#14556085

yes

characteristic

107

C₄

C₄×C₂²⋊D₇₁

yes

—

108

C₂

—

109

C₂

C₄×Dic₇₁⋊C₄

yes

characteristic

110

C₂

C₂×#14556063

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₄×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	C₄×#108
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²⋊D₇₁

commutator

C₄×C₇₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₄×M₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

hypercenter

C₄×M₁₆

D₇₁

hypocenter

C₇₁

C₄×#108

LCS

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #14556063	C₄	direct (N × H)
48. C₄×C₇₁	#108	semidirect
56. C₇₁	C₄×#108	semidirect
104. C₄	#14556063	direct (N × H)

H ≅ G/N

type

9. #14556063

C₄

direct (N × H)

48. C₄×C₇₁

#108

semidirect

56. C₇₁

C₄×#108

semidirect

104. C₄

#14556063

direct (N × H)

Subgroups — C₄×#14556063 #22588243

Conjugacy classes of subgroups

Complements

Subgroups — C4×#14556063 #22588243

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#14556063 #22588243