Group #28379934 — subgroups

Subgroups — #28379934

Order 13952.

364 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 166 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	#20296076	C₂	1	6	yes	maximal, complemented
2	Q₈×#11724786	C₂	1	2	yes	maximal, complemented
3	#20296048	C₂	1	3	yes	maximal
4	#20296045	C₂	1	3	yes	maximal
5	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11724786	C₂²	1	6	yes	—
7	C₄⋊#11724786	C₂²	1	6	yes	—
8	#16812756	C₂²	1	1	yes	characteristic
9	#16812756	C₂²	1	3	yes	—
10	#16812754	C₂²	1	6	yes	—
11	C₄×#11724786	C₂²	1	6	yes	—
12	C₂×Q₈×D₁₀₉	C₄	1	1	yes	characteristic
13	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×Dic₂₁₈	C₄	1	3	yes	—
15	#16812664	C₂²	1	3	yes	—
16	C₄⋊C₄×D₁₀₉	C₄	1	3	yes	—
17	C₄²×D₁₀₉	C₂²	1	3	yes	—
18	C₄×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
19	C₂×#11724786	C₂³	1	2	yes	—
20	C₂×#11724786	C₂³	1	6	yes	—
21	C₂×Dic₂₁₈	C₂×C₄	1	3	yes	—
22	Q₈×D₁₀₉	—	2	4	no	—
23	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₀₉	C₂×C₄	1	3	yes	—
27	#13182008	C₂×C₄	1	6	yes	—
28	C₄×Dic₁₀₉	C₂×C₄	1	3	yes	—
29	C₄×Dic₁₀₉	C₂³	1	3	yes	—
30	C₄×Dic₁₀₉	C₂³	1	3	yes	—
31	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₉	C₂×C₄	1	3	yes	—
33	C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
34	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₀₉	M₁₆	1	4	yes	—
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
39	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
40	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
41	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
42	C₄×D₁₀₉	—	2	6	no	—
43	C₄×D₁₀₉	—	2	6	no	—
44	C₄×D₁₀₉	—	2	2	no	—
45	Dic₂₁₈	—	2	6	no	—
46	#11724786	—	2	2	no	—
47	#11724786	—	2	6	no	—
48	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
49	C₂×D₁₀₉	—	2	1	no	—
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	#131	1	2	yes	—
52	C₄×C₁₀₉	—	2	1	no	—
53	C₄×C₁₀₉	—	2	3	no	—
54	C₄×C₁₀₉	C₂×M₁₆	1	6	yes	—
55	Dic₁₀₉	—	2	3	no	—
56	Dic₁₀₉	—	2	3	no	—
57	Dic₁₀₉	2₋¹⁺⁴	1	2	yes	—
58	C₂×C₁₀₉	#519	1	1	yes	characteristic
59	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
61	D₁₀₉	—	2	2	no	—
62	#2984	—	109	1	no	maximal, Sylow
63	C₁₀₉	#2984	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄×Q₈	—	109	1	no	—
65	#446	—	109	6	no	—
66	C₈×Q₈	—	109	2	no	—
67	#432	—	109	3	no	—
68	#431	—	109	3	no	—
69	C₂²×Q₈	—	109	1	no	—
70	C₄×Q₈	—	109	1	no	—
71	C₄×Q₈	—	109	3	no	—
72	C₄×Q₈	—	109	3	no	—
73	C₄×Q₈	#89137	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	109	3	no	—
75	C₂×C₄²	—	109	3	no	—
76	C₄⋊C₈	—	109	6	no	—
77	C₄⋊C₈	—	109	6	no	—
78	C₂²⋊C₈	—	109	1	no	—
79	C₂²⋊C₈	—	109	3	no	—
80	#97	—	109	6	no	—
81	C₄×C₈	—	109	6	no	—
82	C₂×Q₈	—	109	3	no	—
83	C₂×Q₈	—	218	4	no	—
84	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
85	C₂²×C₄	—	109	3	no	—
86	C₂²×C₄	—	109	3	no	—
87	C₂²×C₄	—	109	1	no	—
88	C₂×C₈	—	109	2	no	—
89	C₂×C₈	—	109	6	no	—
90	C₄⋊C₄	—	109	6	no	—
91	C₄⋊C₄	—	109	3	no	—
92	C₄⋊C₄	C₂×#89137	1	3	yes	—
93	C₄²	—	109	3	no	—
94	C₄²	—	109	3	no	—
95	C₄²	—	109	3	no	—
96	C₄²	C₂×#89137	1	3	yes	—
97	C₂³	—	109	1	no	—
98	Q₈	—	218	6	no	—
99	Q₈	#13182039	1	4	yes	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	109	3	no	—
102	C₂×C₄	—	109	3	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	—	218	6	no	—
105	C₂×C₄	—	218	6	no	—
106	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
107	C₂×C₄	C₂²×#89137	1	3	yes	—
108	C₂×C₄	C₂²×#89137	1	3	yes	—
109	C₈	—	218	2	no	—
110	C₈	—	218	6	no	—
111	C₂²	—	109	2	no	—
112	C₂²	—	218	1	no	—
113	C₂²	—	218	1	no	—
114	C₂²	C₂³×#89137	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	109	2	no	—
118	C₄	—	218	3	no	—
119	C₄	—	218	3	no	—
120	C₄	C₂×#13182039	1	6	yes	—
121	C₂	—	218	2	no	—
122	C₂	#20296094	1	1	yes	characteristic
123	C₂	C₂²×#13182039	1	1	yes	characteristic
124	C₂	D₁₀₉⋊#131	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296076

C₂

yes

maximal, complemented

Q₈×#11724786

C₂

yes

maximal, complemented

#20296048

C₂

yes

maximal

#20296045

C₂

yes

maximal

C₄×Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₄⋊#11724786

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

—

#16812756

C₂²

yes

characteristic

#16812756

C₂²

yes

—

#16812754

C₂²

yes

—

C₄×#11724786

C₂²

yes

—

C₂×Q₈×D₁₀₉

C₄

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×Dic₂₁₈

C₄

yes

—

#16812664

C₂²

yes

—

C₄⋊C₄×D₁₀₉

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

C₄

yes

characteristic

C₂×#11724786

C₂³

yes

—

C₂×#11724786

C₂³

yes

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

C₂²×C₄

yes

characteristic

Q₈×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#131

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

2₋¹⁺⁴

yes

—

C₂×C₁₀₉

#519

yes

characteristic

C₂×C₁₀₉

C₂²×M₁₆

yes

characteristic

C₂×C₁₀₉

C₂×#131

yes

characteristic

D₁₀₉

—

#2984

—

109

maximal, Sylow

C₁₀₉

#2984

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

109

—

#446

—

109

—

C₈×Q₈

—

109

—

#432

—

109

—

#431

—

109

—

C₂²×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

#89137

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

—

C₂³

—

109

—

Q₈

—

218

—

Q₈

#13182039

yes

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

218

—

104

C₂×C₄

—

218

—

105

C₂×C₄

—

218

—

106

C₂×C₄

C₂²×#89137

yes

characteristic

107

C₂×C₄

C₂²×#89137

yes

—

108

C₂×C₄

C₂²×#89137

yes

—

109

C₈

—

218

—

110

C₈

—

218

—

111

C₂²

—

109

—

112

C₂²

—

218

—

113

C₂²

—

218

—

114

C₂²

C₂³×#89137

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

109

—

118

C₄

—

218

—

119

C₄

—

218

—

120

C₄

C₂×#13182039

yes

—

121

C₂

—

218

—

122

C₂

#20296094

yes

characteristic

123

C₂

C₂²×#13182039

yes

characteristic

124

C₂

D₁₀₉⋊#131

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₄×Q₈×C₁₀₉	C₄
prime radical	C₂²×D₁₀₉	C₂²×C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	C₄×Q₈	#89137
hypocenter	C₁₀₉	#2984
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#89137

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

C₄×Q₈×C₁₀₉

C₄

prime radical

C₂²×D₁₀₉

C₂²×C₄

cosocle

C₂×Dic₁₀₉

C₂⁴

C₄×Q₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #20296076	C₂	semidirect
2. Q₈×#11724786	C₂	semidirect
63. C₁₀₉	#2984	semidirect

H ≅ G/N

type

1. #20296076

C₂

semidirect

2. Q₈×#11724786

C₂

semidirect

63. C₁₀₉

#2984

semidirect

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

subgroup	index
#20296076	2
Q₈×#11724786	2
#20296048	2
#20296045	2
C₄×Q₈×D₁₀₉	2
#2984	109

subgroup

index

C₄×Q₈×D₁₀₉

109

Subgroups — #28379934

Conjugacy classes of subgroups

Complements

Maximal subgroups