Group #21461318 — subgroups

Subgroups — C₂×#16840622 #21461318

Order 7808.

256 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 90 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×#14519844	C₂	1	2	yes	maximal
2	#16840622	C₂	1	4	yes	maximal, direct product factor
3	C₂×#14519775	C₂	1	1	yes	maximal, characteristic
4	C₂²×#92420	—	2	2	no	—
5	#14519844	—	2	2	no	—
6	#14519844	C₂²	1	4	yes	—
7	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
8	#14519775	—	2	2	no	—
9	#14519775	C₄	1	2	yes	—
10	#14519775	C₂²	1	2	yes	—
11	C₂×#11758196	—	2	2	no	—
12	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₆₁	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₆₁	—	2	1	no	—
15	C₂×Q₈×C₆₁	—	4	2	no	—
16	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₆₁	—	2	1	no	—
18	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₆₁	—	2	1	no	—
20	C₂²×Dic₆₁	D₄	1	2	yes	—
21	C₂×Dic₁₂₂	—	2	1	no	—
22	C₂×Dic₁₂₂	—	4	2	no	—
23	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
24	C₂×#92420	—	2	4	no	—
25	C₂×#92420	—	4	4	no	—
26	#11758196	—	2	4	no	—
27	#11758196	—	2	4	no	—
28	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₆₁	D₄	1	2	yes	—
30	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
31	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
32	Q₈×C₆₁	—	4	2	no	—
33	Q₈×C₆₁	—	4	2	no	—
34	C₂×C₄×C₆₁	—	2	2	no	—
35	C₂×C₄×C₆₁	—	2	2	no	—
36	C₂×C₄×C₆₁	—	4	2	no	—
37	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₆₁	—	2	2	no	—
40	C₂×Dic₆₁	—	4	2	no	—
41	C₂×Dic₆₁	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₆₁	SD₁₆	1	4	yes	—
43	C₂×Dic₆₁	Q₁₆	1	4	yes	—
44	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
45	Dic₁₂₂	—	4	2	no	—
46	Dic₁₂₂	—	4	2	no	—
47	#92420	—	4	4	no	—
48	C₂²×C₆₁	#100	1	2	yes	—
49	C₂²×C₆₁	#103	1	4	yes	—
50	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₄×C₆₁	—	2	1	no	—
52	C₄×C₆₁	—	2	1	no	—
53	C₄×C₆₁	—	4	2	no	—
54	Dic₆₁	—	4	2	no	—
55	Dic₆₁	#103	1	4	yes	—
56	Dic₆₁	C₂×SD₁₆	1	2	yes	—
57	Dic₆₁	C₂×Q₁₆	1	2	yes	—
58	C₂×#333	—	61	1	no	maximal, Sylow
59	C₂×C₆₁	#333	1	4	yes	complemented
60	C₂×C₆₁	C₂×#100	1	1	yes	characteristic
61	C₂×C₆₁	C₂×#103	1	2	yes	—
62	C₂×C₄⋊Q₈	—	61	1	no	—
63	C₂×C₄⋊C₈	—	61	2	no	—
64	#333	—	61	4	no	—
65	C₆₁	C₂×#333	1	1	yes	characteristic, complemented, Sylow
66	C₂²×Q₈	—	61	1	no	—
67	C₂²×Q₈	#6923	1	1	yes	characteristic
68	C₂²×C₈	—	122	2	no	—
69	C₄⋊Q₈	—	61	2	no	—
70	C₄⋊Q₈	—	61	2	no	—
71	C₄⋊Q₈	—	122	2	no	—
72	C₂×C₄⋊C₄	—	122	2	no	—
73	C₂×C₄²	—	61	1	no	—
74	C₄⋊C₈	—	61	4	no	—
75	C₄⋊C₈	—	122	2	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	4	2	no	—
78	C₂×Q₈	—	61	2	no	—
79	C₂×Q₈	—	122	1	no	—
80	C₂×Q₈	—	244	2	no	—
81	C₂×Q₈	C₂×#6923	1	2	yes	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	61	2	no	—
84	C₂²×C₄	—	122	1	no	—
85	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
86	C₂×C₈	—	122	4	no	—
87	C₂×C₈	—	244	4	no	—
88	C₄⋊C₄	—	122	4	no	—
89	C₄⋊C₄	—	122	4	no	—
90	C₄²	—	61	2	no	—
91	C₄²	—	61	1	no	—
92	C₄²	—	61	1	no	—
93	C₂³	C₂²⋊#6923	1	1	yes	characteristic
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	244	2	no	—
97	Q₈	—	244	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	61	4	no	—
102	C₂×C₄	—	61	4	no	—
103	C₂×C₄	—	61	2	no	—
104	C₂×C₄	—	61	2	no	—
105	C₂×C₄	—	122	2	no	—
106	C₂×C₄	—	244	2	no	—
107	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
108	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
109	C₈	—	244	4	no	—
110	C₂²	#14519837	1	2	yes	—
111	C₂²	#14519840	1	4	yes	—
112	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	4	2	no	—
116	C₄	—	61	4	no	—
117	C₄	—	61	2	no	—
118	C₄	—	61	2	no	—
119	C₄	—	244	2	no	—
120	C₂	#16840622	1	4	yes	direct product factor
121	C₂	C₂×#14519837	1	1	yes	characteristic
122	C₂	C₂×#14519840	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519844

C₂

yes

maximal

#16840622

C₂

yes

maximal, direct product factor

C₂×#14519775

C₂

yes

maximal, characteristic

C₂²×#92420

—

#14519844

—

#14519844

C₂²

yes

—

C₂²×Dic₁₂₂

C₄

yes

characteristic

#14519775

—

#14519775

C₄

yes

—

#14519775

C₂²

yes

—

C₂×#11758196

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂²×Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

—

—

—

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

—

C₂×Dic₆₁

SD₁₆

yes

—

C₂×Dic₆₁

Q₁₆

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

#100

yes

—

C₂²×C₆₁

#103

yes

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

#103

yes

—

Dic₆₁

C₂×SD₁₆

yes

—

Dic₆₁

C₂×Q₁₆

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

#333

—

C₆₁

C₂×#333

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

#6923

yes

characteristic

C₂²×C₈

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂×Q₈

C₂×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂×#6923

yes

characteristic

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂²⋊#6923

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

244

—

Q₈

—

244

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

122

—

106

C₂×C₄

—

244

—

107

C₂×C₄

C₂²⋊#6923

yes

characteristic

108

C₂×C₄

C₂²×#6923

yes

characteristic

109

C₈

—

244

—

110

C₂²

#14519837

yes

—

111

C₂²

#14519840

yes

—

112

C₂²

C₂×C₂²⋊#6923

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

244

—

120

C₂

#16840622

yes

direct product factor

121

C₂

C₂×#14519837

yes

characteristic

122

C₂

C₂×#14519840

yes

—

	subgroup	G/H
center	C₂³	C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₂²×Q₈×C₆₁	C₄
socle	C₂³×C₆₁	C₂²⋊C₄
hypercenter	C₂²×Q₈	#6923
hypocenter	C₆₁	C₂×#333
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂³

C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₂²×Q₈×C₆₁

C₄

socle

C₂³×C₆₁

C₂²⋊C₄

C₂²×Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #16840622	C₂	direct (N × H)
59. C₂×C₆₁	#333	semidirect
65. C₆₁	C₂×#333	semidirect
120. C₂	#16840622	direct (N × H)

H ≅ G/N

type

2. #16840622

C₂

direct (N × H)

59. C₂×C₆₁

#333

semidirect

65. C₆₁

C₂×#333

semidirect

120. C₂

#16840622

direct (N × H)

Subgroups — C₂×#16840622 #21461318

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840622 #21461318

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840622 #21461318