Group #28379831 — subgroups

Subgroups — Q₈×#13182038 #28379831

Order 13952.

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	#20296077	C₂	1	6	yes	maximal, complemented
2	Q₈×#11724786	C₂	1	2	yes	maximal, complemented
3	#20296029	C₂	1	3	yes	maximal
4	C₄×#13182038	C₂	1	3	yes	maximal
5	C₂×Q₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182038	C₂²	1	3	yes	—
7	C₂×#13182038	C₂²	1	1	yes	characteristic
8	#16812766	C₂²	1	6	yes	—
9	C₄⋊#11724786	C₂²	1	6	yes	—
10	#16812754	C₂²	1	6	yes	—
11	C₄×#11724786	C₂²	1	6	yes	—
12	Q₈×Dic₁₀₉	C₄	1	3	yes	—
13	Q₈×Dic₁₀₉	C₄	1	1	yes	characteristic
14	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	Q₈×Dic₁₀₉	C₂²	1	3	yes	—
16	C₂×C₄⋊Dic₁₀₉	C₄	1	3	yes	—
17	C₂×C₄×Dic₁₀₉	C₂²	1	3	yes	—
18	C₂²×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
19	#13182038	—	2	6	no	—
20	#13182038	Q₈	1	4	yes	direct product factor
21	C₂×#11724786	C₂³	1	6	yes	—
22	C₂×#11724786	C₂³	1	2	yes	—
23	C₂²×Dic₁₀₉	C₂×C₄	1	3	yes	—
24	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₁₀₉	C₂×C₄	1	6	yes	—
26	C₄⋊Dic₁₀₉	C₂×C₄	1	3	yes	—
27	C₄⋊Dic₁₀₉	C₂×C₄	1	3	yes	—
28	C₄×Dic₁₀₉	C₂×C₄	1	3	yes	—
29	C₄×Dic₁₀₉	C₂×C₄	1	3	yes	—
30	C₄×Dic₁₀₉	C₂³	1	3	yes	—
31	C₄×Dic₁₀₉	C₂³	1	3	yes	—
32	C₂×Q₈×C₁₀₉	—	2	4	no	—
33	C₂×Q₈×C₁₀₉	C₂×C₄	1	3	yes	—
34	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₉	C₂×C₄	1	3	yes	—
36	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₀₉	—	2	6	no	—
38	Q₈×C₁₀₉	M₁₆	1	4	yes	complemented
39	C₂×C₄×C₁₀₉	—	2	6	no	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
41	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
42	C₂×Dic₁₀₉	—	4	3	no	—
43	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
44	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
45	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₁₀₉	C₂×Q₈	1	2	yes	—
47	C₂×Dic₁₀₉	#41	1	2	yes	—
48	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
49	#11724786	—	2	6	no	—
50	#11724786	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	—	2	1	no	—
53	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
54	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₁₀₉	—	2	3	no	—
56	C₄×C₁₀₉	C₂×M₁₆	1	6	yes	—
57	Dic₁₀₉	—	2	3	no	—
58	Dic₁₀₉	—	2	3	no	—
59	Dic₁₀₉	C₄×Q₈	1	2	yes	—
60	Dic₁₀₉	C₂²×Q₈	1	1	yes	characteristic
61	Dic₁₀₉	C₂×#41	1	1	yes	characteristic
62	C₂×C₁₀₉	—	2	2	no	—
63	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
64	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
65	C₂×C₁₀₉	#568	1	1	yes	characteristic
66	Q₈×M₁₆	—	109	1	no	maximal, Sylow
67	C₁₀₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×Q₈	—	109	1	no	—
69	#446	—	109	6	no	—
70	C₈×Q₈	—	109	2	no	—
71	#423	—	109	3	no	—
72	C₄×M₁₆	—	109	3	no	—
73	C₂²×Q₈	#89137	1	1	yes	characteristic
74	C₂×M₁₆	—	109	1	no	—
75	C₂×M₁₆	—	109	3	no	—
76	C₄×Q₈	—	109	3	no	—
77	C₄×Q₈	—	109	1	no	—
78	C₄×Q₈	—	109	1	no	—
79	C₄×Q₈	—	109	3	no	—
80	C₂×C₄⋊C₄	—	109	3	no	—
81	C₂×C₄²	—	109	3	no	—
82	C₄⋊C₈	—	109	6	no	—
83	C₄⋊C₈	—	109	6	no	—
84	#97	—	109	6	no	—
85	C₄×C₈	—	109	6	no	—
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	C₂×#89137	1	3	yes	—
88	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
89	C₂²×C₄	—	109	1	no	—
90	C₂²×C₄	—	109	3	no	—
91	C₂²×C₄	C₂×#89137	1	3	yes	—
92	M₁₆	—	109	4	no	—
93	M₁₆	—	218	6	no	—
94	C₂×C₈	—	109	2	no	—
95	C₂×C₈	—	109	6	no	—
96	C₄⋊C₄	—	109	6	no	—
97	C₄⋊C₄	—	109	3	no	—
98	C₄⋊C₄	—	109	3	no	—
99	C₄²	—	109	3	no	—
100	C₄²	—	109	3	no	—
101	C₄²	—	109	3	no	—
102	C₄²	—	109	3	no	—
103	C₂³	C₂²×#89137	1	1	yes	characteristic
104	Q₈	—	2	6	no	—
105	Q₈	#13182038	1	4	yes	direct product factor
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	109	2	no	—
108	C₂×C₄	—	109	2	no	—
109	C₂×C₄	—	109	1	no	—
110	C₂×C₄	—	109	1	no	—
111	C₂×C₄	—	109	3	no	—
112	C₂×C₄	—	109	3	no	—
113	C₂×C₄	—	436	3	no	—
114	C₂×C₄	C₂²×#89137	1	3	yes	—
115	C₂×C₄	C₂²×#89137	1	3	yes	—
116	C₈	—	109	4	no	—
117	C₈	—	218	6	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	Q₈×#89137	1	2	yes	—
121	C₂²	C₂³×#89137	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	109	1	no	—
124	C₄	—	109	1	no	—
125	C₄	—	109	2	no	—
126	C₄	—	218	3	no	—
127	C₄	—	218	3	no	—
128	C₄	C₂×#13182038	1	6	yes	—
129	C₂	—	2	2	no	—
130	C₂	C₂×Q₈×#89137	1	1	yes	characteristic
131	C₂	C₂²×#13182038	1	1	yes	characteristic
132	C₂	#20296106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296077

C₂

yes

maximal, complemented

Q₈×#11724786

C₂

yes

maximal, complemented

#20296029

C₂

yes

maximal

C₄×#13182038

C₂

yes

maximal

C₂×Q₈×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182038

C₂²

yes

—

C₂×#13182038

C₂²

yes

characteristic

#16812766

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

—

#16812754

C₂²

yes

—

C₄×#11724786

C₂²

yes

—

Q₈×Dic₁₀₉

C₄

yes

—

Q₈×Dic₁₀₉

C₄

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₄

yes

characteristic

#13182038

—

#13182038

Q₈

yes

direct product factor

C₂×#11724786

C₂³

yes

—

C₂×#11724786

C₂³

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

M₁₆

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

—

C₂×Dic₁₀₉

#41

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

characteristic

#11724786

—

#11724786

C₂×Q₈

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₄×Q₈

yes

—

Dic₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₀₉

C₂²×M₁₆

yes

characteristic

C₂×C₁₀₉

#568

yes

characteristic

Q₈×M₁₆

—

109

maximal, Sylow

C₁₀₉

Q₈×M₁₆

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

109

—

#446

—

109

—

C₈×Q₈

—

109

—

#423

—

109

—

C₄×M₁₆

—

109

—

C₂²×Q₈

#89137

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

C₂×Q₈

C₂×#89137

yes

—

C₂×Q₈

C₂×#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄²

—

109

—

100

C₄²

—

109

—

101

C₄²

—

109

—

102

C₄²

—

109

—

103

C₂³

C₂²×#89137

yes

characteristic

104

Q₈

—

105

Q₈

#13182038

yes

direct product factor

106

C₂×C₄

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

109

—

111

C₂×C₄

—

109

—

112

C₂×C₄

—

109

—

113

C₂×C₄

—

436

—

114

C₂×C₄

C₂²×#89137

yes

—

115

C₂×C₄

C₂²×#89137

yes

—

116

C₈

—

109

—

117

C₈

—

218

—

118

C₂²

—

119

C₂²

—

120

C₂²

Q₈×#89137

yes

—

121

C₂²

C₂³×#89137

yes

characteristic

122

C₄

—

123

C₄

—

109

—

124

C₄

—

109

—

125

C₄

—

109

—

126

C₄

—

218

—

127

C₄

—

218

—

128

C₄

C₂×#13182038

yes

—

129

C₂

—

130

C₂

C₂×Q₈×#89137

yes

characteristic

131

C₂

C₂²×#13182038

yes

characteristic

132

C₂

#20296106

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₂²×Q₈×C₁₀₉	C₄
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	C₂²×Q₈	#89137
hypocenter	C₁₀₉	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#89137

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

C₂²×Q₈×C₁₀₉

C₄

prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂×Dic₁₀₉

C₂⁴

hypercenter

C₂²×Q₈

#89137

hypocenter

C₁₀₉

Q₈×M₁₆

LCS

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. #20296077	C₂	semidirect
2. Q₈×#11724786	C₂	semidirect
20. #13182038	Q₈	direct (N × H)
38. Q₈×C₁₀₉	M₁₆	semidirect
50. #11724786	C₂×Q₈	semidirect
67. C₁₀₉	Q₈×M₁₆	semidirect
105. Q₈	#13182038	direct (N × H)

H ≅ G/N

type

1. #20296077

C₂

semidirect

2. Q₈×#11724786

C₂

semidirect

20. #13182038

Q₈

direct (N × H)

38. Q₈×C₁₀₉

M₁₆

semidirect

50. #11724786

C₂×Q₈

semidirect

67. C₁₀₉

Q₈×M₁₆

semidirect

105. Q₈

#13182038

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20296077	2
Q₈×#11724786	2
#20296029	2
C₄×#13182038	2
C₂×Q₈×Dic₁₀₉	2
Q₈×M₁₆	109

subgroup

index

C₂×Q₈×Dic₁₀₉

Q₈×M₁₆

109

Subgroups — Q₈×#13182038 #28379831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×#13182038 #28379831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13182038 #28379831