Group #21456317 — subgroups

Subgroups — C₂×#16840159 #21456317

Order 7808.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14519776	C₂	1	2	yes	maximal
2	#16840159	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14519709	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519708	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₂₂	—	2	2	no	—
8	C₂²×#92418	C₂²	1	1	yes	characteristic
9	#14519776	—	2	4	no	—
10	#14519776	C₂²	1	8	yes	—
11	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	C₂×#11758196	—	2	2	no	—
13	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	#14519709	C₂²	1	4	yes	—
15	#14519708	C₂²	1	8	yes	—
16	C₈×Dic₆₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	#106×C₆₁	C₂²	1	4	yes	—
20	C₂²×C₄×C₆₁	—	2	2	no	—
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₆₁	—	2	2	no	—
23	C₂²×Dic₆₁	D₄	1	2	yes	—
24	C₂×Dic₁₂₂	—	2	8	no	—
25	C₂×Dic₁₂₂	—	4	8	no	—
26	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
27	C₂×C₈×C₆₁	C₂³	1	2	yes	—
28	C₄⋊C₄×C₆₁	—	2	2	no	—
29	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
30	C₄⋊Dic₆₁	—	2	2	no	—
31	C₄⋊Dic₆₁	C₂³	1	4	yes	—
32	#11758196	—	2	8	no	—
33	C₄×Dic₆₁	D₄	1	2	yes	—
34	C₄×Dic₆₁	C₂³	1	2	yes	—
35	C₂×#92418	Q₈	1	4	yes	complemented
36	C₂×#92418	C₂³	1	2	yes	—
37	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	—	2	4	no	—
39	C₂×C₄×C₆₁	—	4	4	no	—
40	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
44	C₂×Dic₆₁	—	2	4	no	—
45	C₂×Dic₆₁	—	4	4	no	—
46	C₂×Dic₆₁	SD₁₆	1	8	yes	—
47	C₂×Dic₆₁	C₂×D₄	1	4	yes	—
48	Dic₁₂₂	—	4	16	no	—
49	C₈×C₆₁	C₂×Q₈	1	4	yes	complemented
50	#92418	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
52	C₂²×C₆₁	C₂×SD₁₆	1	4	yes	—
53	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
54	C₄×C₆₁	—	4	4	no	—
55	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
56	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
57	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
58	Dic₆₁	—	4	4	no	—
59	Dic₆₁	C₂×SD₁₆	1	8	yes	—
60	C₂×#498	—	61	1	no	maximal, Sylow
61	C₂×C₆₁	#498	1	4	yes	complemented
62	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₆₁	C₂²×SD₁₆	1	2	yes	—
64	C₂×C₄⋊Q₈	—	61	2	no	—
65	#498	—	61	8	no	—
66	C₂×#106	—	61	1	no	—
67	C₂×#106	—	61	2	no	—
68	C₂×#106	D₆₁	1	1	yes	characteristic
69	C₂×C₄×C₈	—	61	1	no	—
70	C₆₁	C₂×#498	1	1	yes	characteristic, complemented, Sylow
71	C₂²×Q₈	—	122	2	no	—
72	C₂²×C₈	—	61	1	no	—
73	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
74	C₄⋊Q₈	—	61	8	no	—
75	C₄⋊Q₈	—	122	4	no	—
76	C₂×C₄⋊C₄	—	61	2	no	—
77	C₂×C₄⋊C₄	—	122	2	no	—
78	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
79	C₂×C₄²	—	61	1	no	—
80	#106	—	61	4	no	—
81	#106	—	61	8	no	—
82	#106	C₂×D₆₁	1	4	yes	—
83	C₄×C₈	—	61	4	no	—
84	C₂×Q₈	—	122	8	no	—
85	C₂×Q₈	—	244	8	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	61	2	no	—
88	C₂²×C₄	—	122	2	no	—
89	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
90	C₂×C₈	—	61	4	no	—
91	C₂×C₈	—	61	2	no	—
92	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
93	C₂×C₈	C₂²×D₆₁	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	61	4	no	—
96	C₄⋊C₄	—	122	8	no	—
97	C₄⋊C₄	—	122	2	no	—
98	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
99	C₄²	—	61	2	no	—
100	C₄²	—	61	2	no	—
101	C₂³	D₄×D₆₁	1	1	yes	characteristic
102	Q₈	—	244	16	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	—	61	4	no	—
106	C₂×C₄	—	61	8	no	—
107	C₂×C₄	—	122	4	no	—
108	C₂×C₄	—	244	4	no	—
109	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
110	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₆₁	1	2	yes	—
112	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
113	C₈	—	61	4	no	—
114	C₈	C₂×Dic₁₂₂	1	4	yes	complemented
115	C₂²	#14519776	1	2	yes	—
116	C₂²	SD₁₆×D₆₁	1	4	yes	—
117	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	61	8	no	—
120	C₄	—	244	4	no	—
121	C₄	#14519776	1	2	yes	—
122	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
123	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
124	C₂	#16840159	1	4	yes	direct product factor
125	C₂	C₂×#14519776	1	1	yes	characteristic
126	C₂	C₂×SD₁₆×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519776

C₂

yes

maximal

#16840159

C₂

yes

maximal, direct product factor

C₂×#14519709

C₂

yes

maximal, characteristic

C₂×#14519708

C₂

yes

maximal

C₂×C₈×Dic₆₁

C₂

yes

maximal, characteristic

C₂×#106×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

—

C₂²×#92418

C₂²

yes

characteristic

#14519776

—

#14519776

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519709

C₂²

yes

—

#14519708

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

Q₈

yes

complemented

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

SD₁₆

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

complemented

#92418

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

C₂×SD₁₆

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

C₂×SD₁₆

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂²×SD₁₆

yes

—

C₂×C₄⋊Q₈

—

—

—

—

D₆₁

yes

characteristic

C₂×C₄×C₈

—

C₆₁

C₂×#498

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

#106

—

#106

—

#106

C₂×D₆₁

yes

—

C₄×C₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₂₂

yes

complemented

C₂×C₈

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

C₂²×D₆₁

yes

—

C₄²

—

100

C₄²

—

101

C₂³

D₄×D₆₁

yes

characteristic

102

Q₈

—

244

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

122

—

108

C₂×C₄

—

244

—

109

C₂×C₄

C₂×Dic₁₂₂

yes

—

110

C₂×C₄

D₄×D₆₁

yes

characteristic

111

C₂×C₄

Q₈×D₆₁

yes

—

112

C₂×C₄

C₂³×D₆₁

yes

characteristic

113

C₈

—

114

C₈

C₂×Dic₁₂₂

yes

complemented

115

C₂²

#14519776

yes

—

116

C₂²

SD₁₆×D₆₁

yes

—

117

C₂²

C₂×D₄×D₆₁

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

244

—

121

C₄

#14519776

yes

—

122

C₄

C₂×Q₈×D₆₁

yes

characteristic

123

C₄

C₂²×Dic₁₂₂

yes

characteristic

124

C₂

#16840159

yes

direct product factor

125

C₂

C₂×#14519776

yes

characteristic

126

C₂

C₂×SD₁₆×D₆₁

yes

—

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#106×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#106	D₆₁
hypocenter	C₆₁	C₂×#498
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×#106×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #16840159	C₂	direct (N × H)
26. C₂×C₈×C₆₁	Q₈	semidirect
35. C₂×#92418	Q₈	semidirect
49. C₈×C₆₁	C₂×Q₈	semidirect
50. #92418	C₂×Q₈	semidirect
61. C₂×C₆₁	#498	semidirect
70. C₆₁	C₂×#498	semidirect
92. C₂×C₈	Dic₁₂₂	semidirect
114. C₈	C₂×Dic₁₂₂	semidirect
124. C₂	#16840159	direct (N × H)

H ≅ G/N

type

2. #16840159

C₂

direct (N × H)

26. C₂×C₈×C₆₁

Q₈

semidirect

35. C₂×#92418

Q₈

semidirect

49. C₈×C₆₁

C₂×Q₈

semidirect

50. #92418

C₂×Q₈

semidirect

61. C₂×C₆₁

#498

semidirect

70. C₆₁

C₂×#498

semidirect

92. C₂×C₈

Dic₁₂₂

semidirect

114. C₈

C₂×Dic₁₂₂

semidirect

124. C₂

#16840159

direct (N × H)

Subgroups — C₂×#16840159 #21456317

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840159 #21456317

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840159 #21456317