Group #21462619 — subgroups

Subgroups — Q₈×#11758213 #21462619

Order 7808.

388 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 182 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	#16840812	C₂	1	6	yes	maximal, complemented
2	Q₈×#92420	C₂	1	2	yes	maximal, complemented
3	#16840766	C₂	1	3	yes	maximal
4	C₄×#11758213	C₂	1	3	yes	maximal
5	C₄×Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758213	C₂²	1	3	yes	—
7	C₂×#11758213	C₂²	1	1	yes	characteristic
8	C₄⋊#92420	C₂²	1	6	yes	—
9	C₄⋊#92420	C₂²	1	6	yes	—
10	#14519832	C₂²	1	6	yes	—
11	C₄×#92420	C₂²	1	6	yes	—
12	C₂×Q₈×D₆₁	C₄	1	1	yes	characteristic
13	Q₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₂₂	C₄	1	3	yes	—
15	#14519742	C₂²	1	3	yes	—
16	C₄⋊C₄×D₆₁	C₄	1	3	yes	—
17	C₄²×D₆₁	C₂²	1	3	yes	—
18	C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
19	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	Q₈×D₆₁	—	2	4	no	—
21	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
22	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₂₂	C₂×C₄	1	3	yes	—
25	C₂×#92420	C₂³	1	6	yes	—
26	C₂×#92420	C₂³	1	2	yes	—
27	#11758213	—	2	6	no	—
28	#11758213	Q₈	1	4	yes	direct product factor
29	C₄⋊C₄×C₆₁	C₂×C₄	1	3	yes	—
30	C₄²×C₆₁	C₂×C₄	1	3	yes	—
31	C₄⋊Dic₆₁	C₂×C₄	1	3	yes	—
32	#11758196	C₂×C₄	1	6	yes	—
33	C₄×Dic₆₁	C₂×C₄	1	3	yes	—
34	C₄×Dic₆₁	C₂³	1	3	yes	—
35	C₄×Dic₆₁	C₂³	1	3	yes	—
36	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₆₁	M₁₆	1	4	yes	complemented
38	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
41	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
42	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
43	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
44	C₄×D₆₁	—	2	6	no	—
45	C₄×D₆₁	—	4	3	no	—
46	C₄×D₆₁	C₂×Q₈	1	2	yes	—
47	C₄×D₆₁	#41	1	2	yes	—
48	Dic₁₂₂	—	2	6	no	—
49	#92420	—	2	6	no	—
50	#92420	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
52	C₂×D₆₁	—	2	1	no	—
53	C₂×D₆₁	—	2	1	no	—
54	C₂×D₆₁	C₄×Q₈	1	2	yes	—
55	C₄×C₆₁	—	2	3	no	—
56	C₄×C₆₁	C₄×Q₈	1	2	yes	—
57	C₄×C₆₁	C₂×M₁₆	1	6	yes	—
58	Dic₆₁	—	2	3	no	—
59	Dic₆₁	—	2	3	no	—
60	Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
61	Dic₆₁	C₂×#41	1	1	yes	characteristic
62	Q₈×M₁₆	—	61	1	no	maximal, Sylow
63	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
64	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
65	C₂×C₆₁	#568	1	1	yes	characteristic
66	D₆₁	—	2	2	no	—
67	C₂×C₄×Q₈	—	61	1	no	—
68	#446	—	61	6	no	—
69	C₈×Q₈	—	61	2	no	—
70	#423	—	61	3	no	—
71	C₄×M₁₆	—	61	3	no	—
72	C₆₁	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	61	1	no	—
74	C₂×M₁₆	—	61	1	no	—
75	C₂×M₁₆	—	61	3	no	—
76	C₄×Q₈	—	61	1	no	—
77	C₄×Q₈	—	61	3	no	—
78	C₄×Q₈	—	61	3	no	—
79	C₄×Q₈	#6923	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	61	3	no	—
81	C₂×C₄²	—	61	3	no	—
82	C₄⋊C₈	—	61	6	no	—
83	C₄⋊C₈	—	61	6	no	—
84	#97	—	61	6	no	—
85	C₄×C₈	—	61	6	no	—
86	C₂×Q₈	—	61	3	no	—
87	C₂×Q₈	—	122	4	no	—
88	C₂×Q₈	C₂×#6923	1	1	yes	characteristic
89	C₂²×C₄	—	61	1	no	—
90	C₂²×C₄	—	61	3	no	—
91	C₂²×C₄	—	61	3	no	—
92	M₁₆	—	61	4	no	—
93	M₁₆	—	122	6	no	—
94	C₂×C₈	—	61	2	no	—
95	C₂×C₈	—	61	6	no	—
96	C₄⋊C₄	—	61	6	no	—
97	C₄⋊C₄	—	61	3	no	—
98	C₄⋊C₄	C₂×#6923	1	3	yes	—
99	C₄²	—	61	3	no	—
100	C₄²	—	61	3	no	—
101	C₄²	—	61	3	no	—
102	C₄²	C₂×#6923	1	3	yes	—
103	C₂³	—	61	1	no	—
104	Q₈	—	122	6	no	—
105	Q₈	#11758213	1	4	yes	direct product factor
106	C₂×C₄	—	61	1	no	—
107	C₂×C₄	—	61	2	no	—
108	C₂×C₄	—	61	2	no	—
109	C₂×C₄	—	61	3	no	—
110	C₂×C₄	—	61	3	no	—
111	C₂×C₄	—	122	6	no	—
112	C₂×C₄	—	244	3	no	—
113	C₂×C₄	C₂²×#6923	1	3	yes	—
114	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
115	C₂×C₄	C₂²×#6923	1	3	yes	—
116	C₈	—	61	4	no	—
117	C₈	—	122	6	no	—
118	C₂²	—	61	2	no	—
119	C₂²	—	122	1	no	—
120	C₂²	—	122	1	no	—
121	C₂²	C₂³×#6923	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	61	1	no	—
124	C₄	—	61	1	no	—
125	C₄	—	122	3	no	—
126	C₄	—	122	3	no	—
127	C₄	Q₈×#6923	1	2	yes	—
128	C₄	C₂×#11758213	1	6	yes	—
129	C₂	—	122	2	no	—
130	C₂	C₂×Q₈×#6923	1	1	yes	characteristic
131	C₂	C₂²×#11758213	1	1	yes	characteristic
132	C₂	#16840841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840812

C₂

yes

maximal, complemented

Q₈×#92420

C₂

yes

maximal, complemented

#16840766

C₂

yes

maximal

C₄×#11758213

C₂

yes

maximal

C₄×Q₈×D₆₁

C₂

yes

maximal, characteristic

C₂×#11758213

C₂²

yes

—

C₂×#11758213

C₂²

yes

characteristic

C₄⋊#92420

C₂²

yes

—

C₄⋊#92420

C₂²

yes

—

#14519832

C₂²

yes

—

C₄×#92420

C₂²

yes

—

C₂×Q₈×D₆₁

C₄

yes

characteristic

Q₈×Dic₆₁

C₂²

yes

characteristic

C₄×Dic₁₂₂

C₄

yes

—

#14519742

C₂²

yes

—

C₄⋊C₄×D₆₁

C₄

yes

—

C₄²×D₆₁

C₂²

yes

—

C₄×Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

Q₈×D₆₁

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×#92420

C₂³

yes

—

C₂×#92420

C₂³

yes

—

#11758213

—

#11758213

Q₈

yes

direct product factor

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂²×D₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

M₁₆

yes

complemented

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂⁴

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×Q₈

yes

—

C₄×D₆₁

#41

yes

—

Dic₁₂₂

—

#92420

—

#92420

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₄×Q₈

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

C₂×M₁₆

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

C₂×#41

yes

characteristic

Q₈×M₁₆

—

maximal, Sylow

C₂×C₆₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₁

C₂²×M₁₆

yes

characteristic

C₂×C₆₁

#568

yes

characteristic

D₆₁

—

C₂×C₄×Q₈

—

#446

—

C₈×Q₈

—

#423

—

C₄×M₁₆

—

C₆₁

Q₈×M₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

#6923

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

C₂×#6923

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#6923

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂×#6923

yes

—

103

C₂³

—

104

Q₈

—

122

—

105

Q₈

#11758213

yes

direct product factor

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

122

—

112

C₂×C₄

—

244

—

113

C₂×C₄

C₂²×#6923

yes

—

114

C₂×C₄

C₂²×#6923

yes

characteristic

115

C₂×C₄

C₂²×#6923

yes

—

116

C₈

—

117

C₈

—

122

—

118

C₂²

—

119

C₂²

—

122

—

120

C₂²

—

122

—

121

C₂²

C₂³×#6923

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

122

—

126

C₄

—

122

—

127

C₄

Q₈×#6923

yes

—

128

C₄

C₂×#11758213

yes

—

129

C₂

—

122

—

130

C₂

C₂×Q₈×#6923

yes

characteristic

131

C₂

C₂²×#11758213

yes

characteristic

132

C₂

#16840841

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₄×Q₈×C₆₁	C₄
hypercenter	C₄×Q₈	#6923
hypocenter	C₆₁	Q₈×M₁₆
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#6923

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₄×Q₈×C₆₁

C₄

hypercenter

C₄×Q₈

#6923

hypocenter

C₆₁

Q₈×M₁₆

LCS

G ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #16840812	C₂	semidirect
2. Q₈×#92420	C₂	semidirect
28. #11758213	Q₈	direct (N × H)
37. Q₈×C₆₁	M₁₆	semidirect
50. #92420	C₂×Q₈	semidirect
72. C₆₁	Q₈×M₁₆	semidirect
105. Q₈	#11758213	direct (N × H)

H ≅ G/N

type

1. #16840812

C₂

semidirect

2. Q₈×#92420

C₂

semidirect

28. #11758213

Q₈

direct (N × H)

37. Q₈×C₆₁

M₁₆

semidirect

50. #92420

C₂×Q₈

semidirect

72. C₆₁

Q₈×M₁₆

semidirect

105. Q₈

#11758213

direct (N × H)

Subgroups — Q₈×#11758213 #21462619

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11758213 #21462619

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11758213 #21462619