Group #22586616 — subgroups

Subgroups — C₂×#17985721 #22586616

Order 9088.

248 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 173 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×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#17985721	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₈	C₄	1	2	yes	—
6	C₂²×C₇₁⋊C₈	C₄	1	2	yes	—
7	C₄×C₇₁⋊C₈	C₄	1	2	yes	—
8	C₄×C₇₁⋊C₈	C₄	1	2	yes	—
9	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
10	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₈×C₇₁	C₄	1	2	yes	—
12	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₇₁	C₄	1	2	yes	—
14	C₄×C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×C₇₁⋊C₈	—	2	4	no	—
16	C₂×C₇₁⋊C₈	C₈	1	8	yes	complemented
17	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₇₁	—	2	4	no	—
23	C₂×C₈×C₇₁	C₂×C₄	1	4	yes	—
24	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₇₁	C₂³	1	1	yes	characteristic
28	C₂³×C₇₁	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂×C₈	1	4	yes	—
33	C₂×C₄×C₇₁	M₁₆	1	4	yes	—
34	C₂×C₄×C₇₁	M₁₆	1	4	yes	—
35	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₇₁	—	2	4	no	—
39	C₇₁⋊C₈	—	2	4	no	—
40	C₇₁⋊C₈	C₂×C₈	1	8	yes	complemented
41	C₂²×C₇₁	C₄×C₈	1	2	yes	—
42	C₂²×C₇₁	#97	1	2	yes	—
43	C₂²×C₇₁	#97	1	2	yes	—
44	C₂²×C₇₁	C₂×C₄²	1	1	yes	characteristic
45	C₄×C₇₁	C₄×C₈	1	2	yes	—
46	C₄×C₇₁	#97	1	2	yes	—
47	C₄×C₇₁	#97	1	2	yes	—
48	C₄×C₇₁	C₂²×C₈	1	2	yes	—
49	C₄×C₇₁	C₂×M₁₆	1	2	yes	—
50	C₄×C₇₁	C₂×M₁₆	1	2	yes	—
51	C₂×C₇₁	#322	1	4	yes	complemented
52	C₂×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
53	C₂×C₇₁	C₂×#97	1	1	yes	characteristic
54	C₂×C₇₁	C₂×#97	1	1	yes	characteristic
55	C₂×#322	—	71	1	no	maximal, Sylow
56	C₇₁	C₂×#322	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄×C₈	—	71	1	no	—
58	C₂×C₄×C₈	—	71	1	no	—
59	C₂×C₄×C₈	D₇₁	1	1	yes	characteristic
60	#322	—	71	4	no	—
61	C₂²×C₈	—	71	2	no	—
62	C₂²×C₈	—	71	2	no	—
63	C₂²×C₈	Dic₇₁	1	2	yes	—
64	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
65	C₄×C₈	—	71	2	no	—
66	C₄×C₈	—	71	2	no	—
67	C₄×C₈	—	71	2	no	—
68	C₄×C₈	—	71	2	no	—
69	C₄×C₈	Dic₇₁	1	2	yes	—
70	C₄×C₈	C₂×D₇₁	1	2	yes	—
71	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
73	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	71	8	no	—
76	C₂×C₈	—	71	4	no	—
77	C₂×C₈	—	71	4	no	—
78	C₂×C₈	—	142	4	no	—
79	C₂×C₈	C₂×Dic₇₁	1	4	yes	—
80	C₄²	C₄×D₇₁	1	1	yes	characteristic
81	C₄²	C₄×D₇₁	1	1	yes	characteristic
82	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
83	C₄²	C₂²×D₇₁	1	1	yes	characteristic
84	C₂³	C₄×Dic₇₁	1	1	yes	characteristic
85	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
86	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
87	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
88	C₂×C₄	C₈×D₇₁	1	4	yes	—
89	C₂×C₄	#11765848	1	4	yes	—
90	C₂×C₄	#11765849	1	4	yes	—
91	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
92	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
93	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
94	C₈	—	2	4	no	—
95	C₈	—	71	8	no	—
96	C₈	—	142	4	no	—
97	C₂²	C₈×Dic₇₁	1	2	yes	—
98	C₂²	#14556038	1	2	yes	—
99	C₂²	#14556039	1	2	yes	—
100	C₂²	C₂×C₄×Dic₇₁	1	1	yes	characteristic
101	C₄	C₈×Dic₇₁	1	2	yes	—
102	C₄	#14556038	1	2	yes	—
103	C₄	#14556039	1	2	yes	—
104	C₄	C₂×C₈×D₇₁	1	2	yes	—
105	C₄	C₂×#11765848	1	2	yes	—
106	C₄	C₂×#11765849	1	2	yes	—
107	C₂	#17985721	1	4	yes	direct product factor
108	C₂	C₂×C₈×Dic₇₁	1	1	yes	characteristic
109	C₂	C₂×#14556038	1	1	yes	characteristic
110	C₂	C₂×#14556039	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985721

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₄

yes

—

C₂²×C₇₁⋊C₈

C₄

yes

—

C₄×C₇₁⋊C₈

C₄

yes

—

C₄×C₇₁⋊C₈

C₄

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₂²×C₈×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₄

yes

—

C₄×C₈×C₇₁

C₂²

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₈

yes

complemented

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂×C₈

yes

complemented

C₂²×C₇₁

C₄×C₈

yes

—

C₂²×C₇₁

#97

yes

—

C₂²×C₇₁

#97

yes

—

C₂²×C₇₁

C₂×C₄²

yes

characteristic

C₄×C₇₁

C₄×C₈

yes

—

C₄×C₇₁

#97

yes

—

C₄×C₇₁

#97

yes

—

C₄×C₇₁

C₂²×C₈

yes

—

C₄×C₇₁

C₂×M₁₆

yes

—

C₄×C₇₁

C₂×M₁₆

yes

—

C₂×C₇₁

#322

yes

complemented

C₂×C₇₁

C₂×C₄×C₈

yes

characteristic

C₂×C₇₁

C₂×#97

yes

characteristic

C₂×C₇₁

C₂×#97

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₇₁

yes

characteristic

#322

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₇₁

yes

—

C₄×C₈

C₂×D₇₁

yes

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₄²

C₄×D₇₁

yes

characteristic

C₄²

C₄×D₇₁

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

C₄×Dic₇₁

yes

characteristic

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

C₈×D₇₁

yes

—

C₂×C₄

#11765848

yes

—

C₂×C₄

#11765849

yes

—

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

C₈

—

C₈

—

C₈

—

142

—

C₂²

C₈×Dic₇₁

yes

—

C₂²

#14556038

yes

—

C₂²

#14556039

yes

—

100

C₂²

C₂×C₄×Dic₇₁

yes

characteristic

101

C₄

C₈×Dic₇₁

yes

—

102

C₄

#14556038

yes

—

103

C₄

#14556039

yes

—

104

C₄

C₂×C₈×D₇₁

yes

—

105

C₄

C₂×#11765848

yes

—

106

C₄

C₂×#11765849

yes

—

107

C₂

#17985721

yes

direct product factor

108

C₂

C₂×C₈×Dic₇₁

yes

characteristic

109

C₂

C₂×#14556038

yes

characteristic

110

C₂

C₂×#14556039

yes

characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₇₁
commutator	C₂×C₇₁	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₇₁
Fitting	C₂×C₄×C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₄²
hypercenter	C₂×C₄×C₈	D₇₁
hypocenter	C₇₁	C₂×#322
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄²

C₂×D₇₁

commutator

C₂×C₇₁

C₂×C₄×C₈

Frattini

C₄²

C₂²×D₇₁

Fitting

C₂×C₄×C₈×C₇₁

C₂

socle

C₂³×C₇₁

C₄²

hypercenter

C₂×C₄×C₈

D₇₁

C₇₁

G ▷ C₂×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #17985721	C₂	direct (N × H)
16. C₂×C₇₁⋊C₈	C₈	semidirect
40. C₇₁⋊C₈	C₂×C₈	semidirect
51. C₂×C₇₁	#322	semidirect
56. C₇₁	C₂×#322	semidirect
107. C₂	#17985721	direct (N × H)

H ≅ G/N

type

3. #17985721

C₂

direct (N × H)

16. C₂×C₇₁⋊C₈

C₈

semidirect

40. C₇₁⋊C₈

C₂×C₈

semidirect

51. C₂×C₇₁

#322

semidirect

56. C₇₁

C₂×#322

semidirect

107. C₂

#17985721

direct (N × H)

Subgroups — C₂×#17985721 #22586616

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17985721 #22586616

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17985721 #22586616