Group #22588496 — subgroups

Subgroups — #22588496

Order 9088.

280 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556059	C₂	1	4	yes	maximal
2	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#17985869	C₂	1	1	yes	maximal, characteristic
4	#384×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₇₁	—	2	2	no	—
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	#14556059	—	2	8	no	—
9	C₄×C₇₁⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₇₁	—	2	2	no	—
14	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
15	C₂×C₇₁⋊C₈	D₄	1	4	yes	—
16	C₂×C₇₁⋊C₈	Q₈	1	4	yes	—
17	C₄⋊Dic₇₁	—	2	4	no	—
18	C₄⋊Dic₇₁	—	4	4	no	—
19	C₂²×C₄×C₇₁	—	2	2	no	—
20	C₂²×C₄×C₇₁	D₄	1	2	yes	—
21	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₇₁	—	2	4	no	—
23	C₄⋊C₄×C₇₁	—	4	4	no	—
24	C₄²×C₇₁	C₂×C₄	1	2	yes	—
25	C₄²×C₇₁	Q₈	1	2	yes	—
26	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₇₁	—	4	2	no	—
28	C₂×C₄×C₇₁	—	4	4	no	—
29	C₂×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
30	C₂×C₄×C₇₁	SD₁₆	1	4	yes	—
31	C₂×C₄×C₇₁	SD₁₆	1	4	yes	—
32	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×Dic₇₁	—	4	2	no	—
37	C₂×Dic₇₁	—	4	4	no	—
38	C₇₁⋊C₈	C₄⋊C₄	1	8	yes	complemented
39	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₇₁	#127	1	1	yes	characteristic
41	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₇₁	C₂×SD₁₆	1	2	yes	—
43	C₂²×C₇₁	C₂×SD₁₆	1	2	yes	—
44	C₄×C₇₁	—	8	2	no	—
45	C₄×C₇₁	#106	1	8	yes	complemented
46	C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
47	C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
48	Dic₇₁	—	8	2	no	—
49	C₂×C₇₁	#384	1	1	yes	characteristic
50	C₂×C₇₁	C₂×#106	1	2	yes	—
51	C₂×C₇₁	#492	1	1	yes	characteristic
52	C₂×C₇₁	#492	1	1	yes	characteristic
53	C₂×C₇₁	#498	1	2	yes	—
54	#1844	—	71	1	no	maximal, Sylow
55	C₇₁	#1844	1	1	yes	characteristic, complemented, Sylow
56	C₂×#106	—	71	4	no	—
57	C₂×C₄×C₈	—	71	1	no	—
58	#384	—	71	1	no	—
59	#384	D₇₁	1	1	yes	characteristic
60	C₂²×C₈	—	71	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	71	2	no	—
63	C₂×C₄⋊C₄	—	142	2	no	—
64	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
65	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
66	#106	—	142	8	no	—
67	C₄×C₈	—	71	4	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	142	2	no	—
70	C₂²×C₄	C₂²⋊D₇₁	1	2	yes	—
71	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
72	C₂×C₈	—	71	4	no	—
73	C₂×C₈	—	71	4	no	—
74	C₂×C₈	—	71	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	284	4	no	—
78	C₄⋊C₄	—	142	4	no	—
79	C₄²	Dic₁₄₂	1	2	yes	—
80	C₄²	C₄×D₇₁	1	2	yes	—
81	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	284	2	no	—
85	C₂×C₄	—	284	4	no	—
86	C₂×C₄	Dic₇₁⋊C₄	1	4	yes	—
87	C₂×C₄	#11765852	1	4	yes	—
88	C₂×C₄	#11765853	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
90	C₂×C₄	D₄×D₇₁	1	2	yes	—
91	C₂×C₄	Q₈×D₇₁	1	2	yes	—
92	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
93	C₈	—	71	8	no	—
94	C₂²	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
95	C₂²	#14556124	1	1	yes	characteristic
96	C₂²	#14556126	1	1	yes	characteristic
97	C₂²	C₂×#11765852	1	2	yes	—
98	C₂²	C₂×#11765853	1	2	yes	—
99	C₄	—	8	2	no	—
100	C₄	—	568	2	no	—
101	C₄	#14556059	1	8	yes	complemented
102	C₄	C₄⋊C₄×D₇₁	1	2	yes	—
103	C₄	#14556127	1	2	yes	—
104	C₂	#17985868	1	1	yes	characteristic
105	C₂	C₂×#14556059	1	2	yes	—
106	C₂	#17986371	1	1	yes	characteristic
107	C₂	#17986373	1	1	yes	characteristic
108	C₂	#17986395	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556059

C₂

yes

maximal

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985869

C₂

yes

maximal, characteristic

#384×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

#14556059

—

C₄×C₇₁⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

Q₈

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

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

—

C₂²×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

—

C₄×C₇₁

#106

yes

complemented

C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₄×C₇₁

C₄⋊Q₈

yes

—

Dic₇₁

—

C₂×C₇₁

#384

yes

characteristic

C₂×C₇₁

C₂×#106

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

#1844

—

maximal, Sylow

C₇₁

#1844

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×C₄×C₈

—

#384

—

#384

D₇₁

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#106

—

142

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄²

Dic₁₄₂

yes

—

C₄²

C₄×D₇₁

yes

—

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

Dic₇₁⋊C₄

yes

—

C₂×C₄

#11765852

yes

—

C₂×C₄

#11765853

yes

—

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

D₄×D₇₁

yes

—

C₂×C₄

Q₈×D₇₁

yes

—

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

C₈

—

C₂²

C₂×Dic₇₁⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×#11765853

yes

—

C₄

—

100

C₄

—

568

—

101

C₄

#14556059

yes

complemented

102

C₄

C₄⋊C₄×D₇₁

yes

—

103

C₄

#14556127

yes

—

104

C₂

#17985868

yes

characteristic

105

C₂

C₂×#14556059

yes

—

106

C₂

#17986371

yes

characteristic

107

C₂

#17986373

yes

characteristic

108

C₂

#17986395

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#384×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	#384	D₇₁
hypocenter	C₇₁	#1844
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#384×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×C₇₁⋊C₈	C₄	semidirect
38. C₇₁⋊C₈	C₄⋊C₄	semidirect
45. C₄×C₇₁	#106	semidirect
55. C₇₁	#1844	semidirect
101. C₄	#14556059	semidirect

H ≅ G/N

type

9. C₄×C₇₁⋊C₈

C₄

semidirect

38. C₇₁⋊C₈

C₄⋊C₄

semidirect

45. C₄×C₇₁

#106

semidirect

55. C₇₁

#1844

semidirect

101. C₄

#14556059

semidirect

Subgroups — #22588496

Conjugacy classes of subgroups

Complements