Group #22601913 — subgroups

Subgroups — C₂×#17986728 #22601913

Order 9088.

1048 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 581 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	#17986728	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14556128	C₂	1	3	yes	maximal
3	C₂×#14556119	C₂	1	6	yes	maximal
4	C₂×Q₈×Dic₇₁	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₁₄₂	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₄₂	C₂²	1	3	yes	—
8	#14556128	C₂²	1	24	yes	—
9	#14556119	C₂²	1	48	yes	—
10	Q₈×Dic₇₁	C₂²	1	16	yes	—
11	C₄×Dic₁₄₂	C₂²	1	24	yes	—
12	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₇₁	C₂²	1	6	yes	—
14	C₂×C₄⋊Dic₇₁	C₂²	1	6	yes	—
15	C₂×Dic₇₁⋊C₄	C₂²	1	6	yes	—
16	C₂×C₄×Dic₇₁	C₂²	1	6	yes	—
17	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₇₁	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
20	C₂×C₄²×C₇₁	C₂²	1	3	yes	—
21	C₂×Dic₁₄₂	—	2	12	no	—
22	C₂×Dic₁₄₂	C₂³	1	12	yes	—
23	C₂²×Dic₇₁	C₂³	1	6	yes	—
24	C₂²×Dic₇₁	C₂³	1	2	yes	—
25	C₄⋊Dic₇₁	C₂³	1	4	yes	—
26	C₄⋊Dic₇₁	C₂³	1	24	yes	—
27	C₄⋊Dic₇₁	C₂³	1	24	yes	—
28	Dic₇₁⋊C₄	C₂³	1	24	yes	—
29	C₄×Dic₇₁	C₂³	1	24	yes	—
30	C₂×Q₈×C₇₁	Q₈	1	8	yes	complemented
31	C₂×Q₈×C₇₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₇₁	C₂³	1	3	yes	—
33	C₂²×C₄×C₇₁	C₂³	1	3	yes	—
34	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₇₁	C₂³	1	12	yes	—
36	C₄²×C₇₁	C₂³	1	12	yes	—
37	C₂³×C₇₁	C₂⁴	1	1	yes	characteristic
38	Q₈×C₇₁	C₂×Q₈	1	16	yes	complemented
39	C₂×C₄×C₇₁	—	2	6	no	—
40	C₂×C₄×C₇₁	C₂×Q₈	1	12	yes	—
41	C₂×C₄×C₇₁	#41	1	4	yes	—
42	C₂×C₄×C₇₁	C₂⁴	1	6	yes	—
43	C₂×C₄×C₇₁	C₂⁴	1	6	yes	—
44	C₂×C₄×C₇₁	C₂⁴	1	2	yes	—
45	C₂×Dic₇₁	—	2	12	no	—
46	C₂×Dic₇₁	—	2	4	no	—
47	C₂×Dic₇₁	C₂⁴	1	4	yes	—
48	C₂×Dic₇₁	C₂⁴	1	12	yes	—
49	Dic₁₄₂	—	2	24	no	—
50	C₂²×C₇₁	C₂²×Q₈	1	2	yes	—
51	C₂²×C₇₁	C₂×#41	1	2	yes	—
52	C₂²×C₇₁	2₋¹⁺⁴	1	2	yes	—
53	C₂²×C₇₁	C₂⁵	1	1	yes	characteristic
54	C₄×C₇₁	—	2	6	no	—
55	C₄×C₇₁	C₂²×Q₈	1	12	yes	—
56	C₄×C₇₁	C₂×#41	1	4	yes	—
57	Dic₇₁	—	2	12	no	—
58	Dic₇₁	—	2	4	no	—
59	C₂×C₇₁	#557	1	4	yes	complemented
60	C₂×C₇₁	C₂³×Q₈	1	1	yes	characteristic
61	C₂×C₇₁	C₂²×#41	1	1	yes	characteristic
62	C₂×C₇₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
63	C₂×#557	—	71	1	no	maximal, Sylow
64	C₇₁	C₂×#557	1	1	yes	characteristic, complemented, Sylow
65	#557	—	71	16	no	—
66	C₂×C₄⋊Q₈	—	71	3	no	—
67	C₂×#125	—	71	6	no	—
68	C₂×C₄×Q₈	—	71	2	no	—
69	C₂×C₄×Q₈	—	71	3	no	—
70	C₂×C₄×Q₈	D₇₁	1	1	yes	characteristic
71	C₂²×Q₈	—	71	3	no	—
72	C₂²×Q₈	C₂×D₇₁	1	1	yes	characteristic
73	C₄⋊Q₈	—	71	24	no	—
74	#125	—	71	48	no	—
75	C₄×Q₈	—	71	16	no	—
76	C₄×Q₈	—	71	24	no	—
77	C₄×Q₈	C₂×D₇₁	1	8	yes	—
78	C₂×C₄⋊C₄	—	71	1	no	—
79	C₂×C₄⋊C₄	—	71	6	no	—
80	C₂×C₄⋊C₄	—	71	6	no	—
81	C₂×C₄⋊C₄	—	71	6	no	—
82	C₂×C₄⋊C₄	C₂×D₇₁	1	3	yes	—
83	C₂×C₄²	—	71	6	no	—
84	C₂×C₄²	C₂×D₇₁	1	3	yes	—
85	C₂×Q₈	—	71	12	no	—
86	C₂×Q₈	—	142	12	no	—
87	C₂×Q₈	Dic₁₄₂	1	8	yes	complemented
88	C₂×Q₈	C₂²×D₇₁	1	4	yes	—
89	C₂²×C₄	—	71	2	no	—
90	C₂²×C₄	—	71	6	no	—
91	C₂²×C₄	C₂²×D₇₁	1	3	yes	—
92	C₂²×C₄	C₂²×D₇₁	1	3	yes	—
93	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
94	C₄⋊C₄	—	71	24	no	—
95	C₄⋊C₄	—	71	4	no	—
96	C₄⋊C₄	—	71	24	no	—
97	C₄⋊C₄	—	71	24	no	—
98	C₄⋊C₄	C₂²×D₇₁	1	12	yes	—
99	C₄²	—	71	24	no	—
100	C₄²	C₂²×D₇₁	1	12	yes	—
101	C₂³	C₂³×D₇₁	1	1	yes	characteristic
102	Q₈	—	142	24	no	—
103	Q₈	C₂×Dic₁₄₂	1	16	yes	complemented
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	71	4	no	—
106	C₂×C₄	—	71	12	no	—
107	C₂×C₄	—	142	12	no	—
108	C₂×C₄	—	142	4	no	—
109	C₂×C₄	C₂×Dic₁₄₂	1	12	yes	—
110	C₂×C₄	#11765866	1	4	yes	—
111	C₂×C₄	C₂³×D₇₁	1	6	yes	—
112	C₂×C₄	C₂³×D₇₁	1	6	yes	—
113	C₂×C₄	C₂³×D₇₁	1	2	yes	—
114	C₂²	C₂²×Dic₁₄₂	1	2	yes	—
115	C₂²	C₂×#11765866	1	2	yes	—
116	C₂²	#14556175	1	2	yes	—
117	C₂²	C₂⁴×D₇₁	1	1	yes	characteristic
118	C₄	—	2	6	no	—
119	C₄	—	142	12	no	—
120	C₄	—	142	4	no	—
121	C₄	C₂²×Dic₁₄₂	1	12	yes	—
122	C₄	C₂×#11765866	1	4	yes	—
123	C₂	#17986728	1	4	yes	direct product factor
124	C₂	C₂³×Dic₁₄₂	1	1	yes	characteristic
125	C₂	C₂²×#11765866	1	1	yes	characteristic
126	C₂	C₂×#14556175	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986728

C₂

yes

maximal, direct product factor

C₂×#14556128

C₂

yes

maximal

C₂×#14556119

C₂

yes

maximal

C₂×Q₈×Dic₇₁

C₂

yes

maximal

C₂×C₄×Dic₁₄₂

C₂

yes

maximal

C₂×C₄×Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

—

#14556128

C₂²

yes

—

#14556119

C₂²

yes

—

Q₈×Dic₇₁

C₂²

yes

—

C₄×Dic₁₄₂

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×Dic₇₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇₁

C₂²

yes

—

C₂²×Q₈×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂³

yes

—

C₂²×Dic₇₁

C₂³

yes

—

C₂²×Dic₇₁

C₂³

yes

—

C₄⋊Dic₇₁

C₂³

yes

—

C₄⋊Dic₇₁

C₂³

yes

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

C₂³

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

Q₈

yes

complemented

C₂×Q₈×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂⁴

yes

characteristic

Q₈×C₇₁

C₂×Q₈

yes

complemented

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂⁴

yes

—

C₂×Dic₇₁

C₂⁴

yes

—

Dic₁₄₂

—

C₂²×C₇₁

C₂²×Q₈

yes

—

C₂²×C₇₁

C₂×#41

yes

—

C₂²×C₇₁

2₋¹⁺⁴

yes

—

C₂²×C₇₁

C₂⁵

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₂²×Q₈

yes

—

C₄×C₇₁

C₂×#41

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#557

yes

complemented

C₂×C₇₁

C₂³×Q₈

yes

characteristic

C₂×C₇₁

C₂²×#41

yes

characteristic

C₂×C₇₁

C₂×2₋¹⁺⁴

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#557

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₇₁

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

Dic₁₄₂

yes

complemented

C₂×Q₈

C₂²×D₇₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₇₁

yes

—

C₄²

—

100

C₄²

C₂²×D₇₁

yes

—

101

C₂³

C₂³×D₇₁

yes

characteristic

102

Q₈

—

142

—

103

Q₈

C₂×Dic₁₄₂

yes

complemented

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

142

—

108

C₂×C₄

—

142

—

109

C₂×C₄

C₂×Dic₁₄₂

yes

—

110

C₂×C₄

#11765866

yes

—

111

C₂×C₄

C₂³×D₇₁

yes

—

112

C₂×C₄

C₂³×D₇₁

yes

—

113

C₂×C₄

C₂³×D₇₁

yes

—

114

C₂²

C₂²×Dic₁₄₂

yes

—

115

C₂²

C₂×#11765866

yes

—

116

C₂²

#14556175

yes

—

117

C₂²

C₂⁴×D₇₁

yes

characteristic

118

C₄

—

119

C₄

—

142

—

120

C₄

—

142

—

121

C₄

C₂²×Dic₁₄₂

yes

—

122

C₄

C₂×#11765866

yes

—

123

C₂

#17986728

yes

direct product factor

124

C₂

C₂³×Dic₁₄₂

yes

characteristic

125

C₂

C₂²×#11765866

yes

characteristic

126

C₂

C₂×#14556175

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₇₁
commutator	C₂²×C₇₁	C₂⁵
Frattini	C₂²	C₂⁴×D₇₁
Fitting	C₂×C₄×Q₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂⁴
hypercenter	C₂×C₄×Q₈	D₇₁
hypocenter	C₇₁	C₂×#557
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂³×D₇₁

commutator

C₂²×C₇₁

C₂⁵

Frattini

C₂²

C₂⁴×D₇₁

Fitting

C₂×C₄×Q₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂⁴

hypercenter

C₂×C₄×Q₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #17986728	C₂	direct (N × H)
30. C₂×Q₈×C₇₁	Q₈	semidirect
38. Q₈×C₇₁	C₂×Q₈	semidirect
59. C₂×C₇₁	#557	semidirect
64. C₇₁	C₂×#557	semidirect
87. C₂×Q₈	Dic₁₄₂	semidirect
103. Q₈	C₂×Dic₁₄₂	semidirect
123. C₂	#17986728	direct (N × H)

H ≅ G/N

type

1. #17986728

C₂

direct (N × H)

30. C₂×Q₈×C₇₁

Q₈

semidirect

38. Q₈×C₇₁

C₂×Q₈

semidirect

59. C₂×C₇₁

#557

semidirect

64. C₇₁

C₂×#557

semidirect

87. C₂×Q₈

Dic₁₄₂

semidirect

103. Q₈

C₂×Dic₁₄₂

semidirect

123. C₂

#17986728

direct (N × H)

Subgroups — C₂×#17986728 #22601913

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17986728 #22601913

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17986728 #22601913