Group #21456234 — subgroups

Subgroups — C₂×#16840141 #21456234

Order 7808.

552 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂²×#11758202	C₂	1	4	yes	maximal
2	C₂×#14519777	C₂	1	1	yes	maximal, characteristic
3	#16840141	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×#92418	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	2	yes	—
7	C₂×#11758202	—	2	16	no	—
8	C₂×#11758202	C₂²	1	16	yes	—
9	C₂²×#92418	C₂²	1	2	yes	—
10	#14519777	—	2	2	no	—
11	#14519777	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₆₁	—	2	2	no	—
13	C₄×#92418	C₂²	1	4	yes	—
14	C₂²×Q₈×C₆₁	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₆₁	—	2	2	no	—
16	C₄⋊Q₈×C₆₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
18	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₆₁	—	2	2	no	—
20	C₂×Q₈×C₆₁	—	2	8	no	—
21	C₂×Q₈×C₆₁	C₂³	1	4	yes	—
22	C₂²×C₄×C₆₁	—	2	2	no	—
23	C₂²×C₄×C₆₁	D₄	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₂²×Dic₆₁	—	2	2	no	—
26	C₂×Dic₁₂₂	—	2	8	no	—
27	C₂×Dic₁₂₂	—	2	2	no	—
28	C₂×Dic₁₂₂	C₂³	1	4	yes	—
29	C₄⋊C₄×C₆₁	—	2	8	no	—
30	C₄²×C₆₁	D₄	1	2	yes	—
31	C₄²×C₆₁	C₂³	1	2	yes	—
32	#11758202	—	2	32	no	—
33	C₄⋊Dic₆₁	—	2	8	no	—
34	C₂×#92418	D₄	1	8	yes	—
35	C₂×#92418	C₂³	1	4	yes	—
36	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₆₁	—	2	8	no	—
38	Q₈×C₆₁	—	4	4	no	—
39	C₂×C₄×C₆₁	—	2	4	no	—
40	C₂×C₄×C₆₁	—	4	4	no	—
41	C₂×C₄×C₆₁	Q₁₆	1	8	yes	complemented
42	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
44	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₁	—	2	4	no	—
47	C₂×Dic₆₁	—	4	4	no	—
48	Dic₁₂₂	—	2	8	no	—
49	Dic₁₂₂	—	4	4	no	—
50	#92418	C₂×D₄	1	8	yes	—
51	C₂²×C₆₁	#127	1	2	yes	—
52	C₂²×C₆₁	C₂×Q₁₆	1	4	yes	—
53	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
54	C₄×C₆₁	—	4	4	no	—
55	C₄×C₆₁	#127	1	2	yes	—
56	C₄×C₆₁	C₂×Q₁₆	1	8	yes	complemented
57	C₄×C₆₁	C₂²×D₄	1	2	yes	—
58	Dic₆₁	—	4	4	no	—
59	C₂×#494	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	#494	1	4	yes	complemented
61	C₂×C₆₁	C₂×#127	1	1	yes	characteristic
62	C₂×C₆₁	C₂²×Q₁₆	1	2	yes	—
63	C₂²×Q₁₆	—	61	4	no	—
64	C₂×C₄⋊Q₈	—	61	1	no	—
65	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
66	#494	—	61	8	no	—
67	C₂×C₄×C₈	—	61	1	no	—
68	C₆₁	C₂×#494	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	—	61	2	no	—
70	C₂²×Q₈	C₂×D₆₁	1	2	yes	—
71	C₂×Q₁₆	—	61	16	no	—
72	C₂×Q₁₆	—	122	16	no	—
73	C₂²×C₈	—	61	2	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	61	4	no	—
76	C₄⋊Q₈	—	122	2	no	—
77	C₄⋊Q₈	C₂×D₆₁	1	4	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄⋊C₄	—	122	2	no	—
80	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
81	C₄×C₈	—	61	4	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	8	no	—
84	C₂×Q₈	—	61	4	no	—
85	C₂×Q₈	—	122	2	no	—
86	C₂×Q₈	—	122	8	no	—
87	C₂×Q₈	C₂²×D₆₁	1	4	yes	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	122	2	no	—
90	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
92	Q₁₆	—	122	32	no	—
93	C₂×C₈	—	61	8	no	—
94	C₂×C₈	—	61	4	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	122	8	no	—
97	C₄²	C₂²⋊D₆₁	1	2	yes	—
98	C₄²	C₂²×D₆₁	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
100	Q₈	—	2	8	no	—
101	Q₈	—	4	4	no	—
102	Q₈	—	122	8	no	—
103	Q₈	—	244	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	122	4	no	—
107	C₂×C₄	—	244	4	no	—
108	C₂×C₄	#11758202	1	8	yes	complemented
109	C₂×C₄	D₄×D₆₁	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₆₁	1	4	yes	—
112	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
113	C₈	—	61	8	no	—
114	C₂²	#14519773	1	2	yes	—
115	C₂²	C₂×#11758202	1	4	yes	—
116	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
117	C₄	—	4	4	no	—
118	C₄	—	244	4	no	—
119	C₄	#14519773	1	2	yes	—
120	C₄	C₂×#11758202	1	8	yes	complemented
121	C₄	C₂×D₄×D₆₁	1	2	yes	—
122	C₂	#16840141	1	4	yes	direct product factor
123	C₂	C₂×#14519773	1	1	yes	characteristic
124	C₂	C₂²×#11758202	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11758202

C₂

yes

maximal

C₂×#14519777

C₂

yes

maximal, characteristic

#16840141

C₂

yes

maximal, direct product factor

C₂×C₄×#92418

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

C₂×#11758202

—

C₂×#11758202

C₂²

yes

—

C₂²×#92418

C₂²

yes

—

#14519777

—

#14519777

C₂²

yes

—

C₂×C₄⋊Dic₆₁

—

C₄×#92418

C₂²

yes

—

C₂²×Q₈×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758202

—

C₄⋊Dic₆₁

—

C₂×#92418

D₄

yes

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

Q₁₆

yes

complemented

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

#92418

C₂×D₄

yes

—

C₂²×C₆₁

#127

yes

—

C₂²×C₆₁

C₂×Q₁₆

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#127

yes

—

C₄×C₆₁

C₂×Q₁₆

yes

complemented

C₄×C₆₁

C₂²×D₄

yes

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

C₂²×Q₁₆

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₆₁

yes

characteristic

#494

—

C₂×C₄×C₈

—

C₆₁

C₂×#494

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₁

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

C₂²×D₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

Q₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄²

C₂²⋊D₆₁

yes

—

C₄²

C₂²×D₆₁

yes

—

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

122

—

103

Q₈

—

244

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

122

—

107

C₂×C₄

—

244

—

108

C₂×C₄

#11758202

yes

complemented

109

C₂×C₄

D₄×D₆₁

yes

—

110

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

112

C₂×C₄

C₂³×D₆₁

yes

characteristic

113

C₈

—

114

C₂²

#14519773

yes

—

115

C₂²

C₂×#11758202

yes

—

116

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

117

C₄

—

118

C₄

—

244

—

119

C₄

#14519773

yes

—

120

C₄

C₂×#11758202

yes

complemented

121

C₄

C₂×D₄×D₆₁

yes

—

122

C₂

#16840141

yes

direct product factor

123

C₂

C₂×#14519773

yes

characteristic

124

C₂

C₂²×#11758202

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	C₂×#494
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₁

hypocenter

C₆₁

C₂×#494

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #16840141	C₂	direct (N × H)
41. C₂×C₄×C₆₁	Q₁₆	semidirect
56. C₄×C₆₁	C₂×Q₁₆	semidirect
60. C₂×C₆₁	#494	semidirect
68. C₆₁	C₂×#494	semidirect
108. C₂×C₄	#11758202	semidirect
120. C₄	C₂×#11758202	semidirect
122. C₂	#16840141	direct (N × H)

H ≅ G/N

type

3. #16840141

C₂

direct (N × H)

41. C₂×C₄×C₆₁

Q₁₆

semidirect

56. C₄×C₆₁

C₂×Q₁₆

semidirect

60. C₂×C₆₁

#494

semidirect

68. C₆₁

C₂×#494

semidirect

108. C₂×C₄

#11758202

semidirect

120. C₄

C₂×#11758202

semidirect

122. C₂

#16840141

direct (N × H)

Subgroups — C₂×#16840141 #21456234

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840141 #21456234

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840141 #21456234