Group #23772354 — subgroups

Subgroups — Q₈×#11929608 #23772354

Order 10112.

388 conjugacy classes of subgroups; only the 241 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 241 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	#19098694	C₂	1	6	yes	maximal, complemented
2	Q₈×C₇₉⋊C₈	C₂	1	2	yes	maximal, complemented
3	#19098587	C₂	1	3	yes	maximal
4	C₄×#11929608	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11929608	C₂²	1	3	yes	—
7	C₂×#11929608	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₈)	C₂²	1	6	yes	—
9	C₄⋊(C₇₉⋊C₈)	C₂²	1	6	yes	—
10	#14579072	C₂²	1	6	yes	—
11	C₄×C₇₉⋊C₈	C₂²	1	6	yes	—
12	C₂²×Q₈×C₇₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₇₉	C₄	1	3	yes	—
14	C₄×Q₈×C₇₉	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₇₉	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₇₉	C₄	1	3	yes	—
18	C₂×C₄²×C₇₉	C₂²	1	3	yes	—
19	#11929608	—	2	6	no	—
20	#11929608	Q₈	1	4	yes	direct product factor
21	C₂×C₇₉⋊C₈	C₂³	1	6	yes	—
22	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
23	C₂×Q₈×C₇₉	—	2	4	no	—
24	C₂×Q₈×C₇₉	C₂×C₄	1	3	yes	—
25	C₂×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₉	C₂×C₄	1	3	yes	—
27	C₂²×C₄×C₇₉	C₂×C₄	1	3	yes	—
28	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₉	C₂×C₄	1	6	yes	—
30	C₄⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
31	C₄⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
32	C₄²×C₇₉	C₂×C₄	1	3	yes	—
33	C₄²×C₇₉	C₂×C₄	1	3	yes	—
34	C₄²×C₇₉	C₂³	1	3	yes	—
35	C₄²×C₇₉	C₂³	1	3	yes	—
36	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₇₉	—	2	6	no	—
38	Q₈×C₇₉	M₁₆	1	4	yes	complemented
39	C₂×C₄×C₇₉	—	2	6	no	—
40	C₂×C₄×C₇₉	—	4	3	no	—
41	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₇₉	#41	1	2	yes	—
48	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
49	C₇₉⋊C₈	—	2	6	no	—
50	C₇₉⋊C₈	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₇₉	—	2	1	no	—
52	C₂²×C₇₉	—	2	1	no	—
53	C₂²×C₇₉	C₄×Q₈	1	2	yes	—
54	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₇₉	—	2	3	no	—
56	C₄×C₇₉	—	2	3	no	—
57	C₄×C₇₉	—	2	3	no	—
58	C₄×C₇₉	C₄×Q₈	1	2	yes	—
59	C₄×C₇₉	C₂×M₁₆	1	6	yes	—
60	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
61	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
62	C₂×C₇₉	—	2	2	no	—
63	C₂×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
64	C₂×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
65	C₂×C₇₉	#568	1	1	yes	characteristic
66	Q₈×M₁₆	—	79	1	no	maximal, Sylow
67	C₇₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×Q₈	D₇₉	1	1	yes	characteristic
69	#446	—	79	6	no	—
70	C₈×Q₈	—	79	2	no	—
71	#423	—	79	3	no	—
72	C₄×M₁₆	—	79	3	no	—
73	C₂²×Q₈	Dic₇₉	1	1	yes	characteristic
74	C₂×M₁₆	—	79	1	no	—
75	C₂×M₁₆	—	79	3	no	—
76	C₄×Q₈	Dic₇₉	1	3	yes	—
77	C₄×Q₈	Dic₇₉	1	1	yes	characteristic
78	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
79	C₄×Q₈	C₂×D₇₉	1	3	yes	—
80	C₂×C₄⋊C₄	Dic₇₉	1	3	yes	—
81	C₂×C₄²	C₂×D₇₉	1	3	yes	—
82	C₄⋊C₈	—	79	6	no	—
83	C₄⋊C₈	—	79	6	no	—
84	#97	—	79	6	no	—
85	C₄×C₈	—	79	6	no	—
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	C₂×Dic₇₉	1	3	yes	—
88	C₂×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
89	C₂²×C₄	C₂×Dic₇₉	1	3	yes	—
90	C₂²×C₄	C₂×Dic₇₉	1	3	yes	—
91	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
92	M₁₆	—	79	4	no	—
93	M₁₆	—	158	6	no	—
94	C₂×C₈	—	79	2	no	—
95	C₂×C₈	—	79	6	no	—
96	C₄⋊C₄	C₂×Dic₇₉	1	6	yes	—
97	C₄⋊C₄	C₂×Dic₇₉	1	3	yes	—
98	C₄⋊C₄	C₂×Dic₇₉	1	3	yes	—
99	C₄²	C₂×Dic₇₉	1	3	yes	—
100	C₄²	C₂×Dic₇₉	1	3	yes	—
101	C₄²	C₂²×D₇₉	1	3	yes	—
102	C₄²	C₂²×D₇₉	1	3	yes	—
103	C₂³	C₂²×Dic₇₉	1	1	yes	characteristic
104	Q₈	—	2	6	no	—
105	Q₈	#11929608	1	4	yes	direct product factor
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	C₂²×Dic₇₉	1	3	yes	—
109	C₂×C₄	C₂²×Dic₇₉	1	3	yes	—
110	C₂×C₄	C₂²×Dic₇₉	1	3	yes	—
111	C₂×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₇₉	1	3	yes	—
113	C₂×C₄	Q₈×D₇₉	1	2	yes	—
114	C₂×C₄	#11929625	1	2	yes	—
115	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
116	C₈	—	79	4	no	—
117	C₈	—	158	6	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	Q₈×Dic₇₉	1	2	yes	—
121	C₂²	C₂³×Dic₇₉	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	2	3	no	—
124	C₄	—	2	3	no	—
125	C₄	Q₈×Dic₇₉	1	2	yes	—
126	C₄	C₂×#11929608	1	6	yes	—
127	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
128	C₄	C₂×#11929625	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
131	C₂	C₂²×#11929608	1	1	yes	characteristic
132	C₂	#19099330	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098694

C₂

yes

maximal, complemented

Q₈×C₇₉⋊C₈

C₂

yes

maximal, complemented

#19098587

C₂

yes

maximal

C₄×#11929608

C₂

yes

maximal

C₂×C₄×Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₂²

yes

—

C₂×#11929608

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579072

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂²×Q₈×C₇₉

C₄

yes

characteristic

C₄×Q₈×C₇₉

C₄

yes

—

C₄×Q₈×C₇₉

C₄

yes

characteristic

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄×Q₈×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

C₄

yes

—

C₂×C₄²×C₇₉

C₂²

yes

—

#11929608

—

#11929608

Q₈

yes

direct product factor

C₂×C₇₉⋊C₈

C₂³

yes

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂×C₄

yes

—

C₂×Q₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂²×C₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

M₁₆

yes

complemented

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×Q₈

yes

—

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×Q₈

yes

—

C₄×C₇₉

C₂×M₁₆

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

C₂×#41

yes

characteristic

C₂×C₇₉

—

C₂×C₇₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₇₉

C₂²×M₁₆

yes

characteristic

C₂×C₇₉

#568

yes

characteristic

Q₈×M₁₆

—

maximal, Sylow

C₇₉

Q₈×M₁₆

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₇₉

yes

characteristic

#446

—

C₈×Q₈

—

#423

—

C₄×M₁₆

—

C₂²×Q₈

Dic₇₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₄×Q₈

Dic₇₉

yes

—

C₄×Q₈

Dic₇₉

yes

characteristic

C₄×Q₈

C₂×D₇₉

yes

characteristic

C₄×Q₈

C₂×D₇₉

yes

—

C₂×C₄⋊C₄

Dic₇₉

yes

—

C₂×C₄²

C₂×D₇₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₇₉

yes

—

C₂×Q₈

C₂×Dic₇₉

yes

characteristic

C₂²×C₄

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

M₁₆

—

M₁₆

—

158

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₇₉

yes

—

C₄⋊C₄

C₂×Dic₇₉

yes

—

C₄⋊C₄

C₂×Dic₇₉

yes

—

C₄²

C₂×Dic₇₉

yes

—

100

C₄²

C₂×Dic₇₉

yes

—

101

C₄²

C₂²×D₇₉

yes

—

102

C₄²

C₂²×D₇₉

yes

—

103

C₂³

C₂²×Dic₇₉

yes

characteristic

104

Q₈

—

105

Q₈

#11929608

yes

direct product factor

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×Dic₇₉

yes

—

109

C₂×C₄

C₂²×Dic₇₉

yes

—

110

C₂×C₄

C₂²×Dic₇₉

yes

—

111

C₂×C₄

C₂²×Dic₇₉

yes

characteristic

112

C₂×C₄

C₂²×Dic₇₉

yes

—

113

C₂×C₄

Q₈×D₇₉

yes

—

114

C₂×C₄

#11929625

yes

—

115

C₂×C₄

C₂³×D₇₉

yes

characteristic

116

C₈

—

117

C₈

—

158

—

118

C₂²

—

119

C₂²

—

120

C₂²

Q₈×Dic₇₉

yes

—

121

C₂²

C₂³×Dic₇₉

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

Q₈×Dic₇₉

yes

—

126

C₄

C₂×#11929608

yes

—

127

C₄

C₂×Q₈×D₇₉

yes

characteristic

128

C₄

C₂×#11929625

yes

characteristic

129

C₂

—

130

C₂

C₂×Q₈×Dic₇₉

yes

characteristic

131

C₂

C₂²×#11929608

yes

characteristic

132

C₂

#19099330

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₇₉	C₂
prime radical	C₂³×C₇₉	C₂²×C₄
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄×Q₈	D₇₉
hypocenter	C₇₉	Q₈×M₁₆
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₉

commutator, socle

C₂²×C₇₉

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₇₉

C₂

prime radical

C₂³×C₇₉

C₂²×C₄

cosocle

C₂×C₄×C₇₉

C₂⁴

hypercenter

C₂×C₄×Q₈

D₇₉

hypocenter

C₇₉

Q₈×M₁₆

LCS

G ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #19098694	C₂	semidirect
2. Q₈×C₇₉⋊C₈	C₂	semidirect
20. #11929608	Q₈	direct (N × H)
38. Q₈×C₇₉	M₁₆	semidirect
50. C₇₉⋊C₈	C₂×Q₈	semidirect
67. C₇₉	Q₈×M₁₆	semidirect
105. Q₈	#11929608	direct (N × H)

H ≅ G/N

type

1. #19098694

C₂

semidirect

2. Q₈×C₇₉⋊C₈

C₂

semidirect

20. #11929608

Q₈

direct (N × H)

38. Q₈×C₇₉

M₁₆

semidirect

50. C₇₉⋊C₈

C₂×Q₈

semidirect

67. C₇₉

Q₈×M₁₆

semidirect

105. Q₈

#11929608

direct (N × H)

Subgroups — Q₈×#11929608 #23772354

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11929608 #23772354

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11929608 #23772354