Group #28378972 — subgroups

Subgroups — #28378972

Order 13952.

220 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₂×#16812770	C₂	1	2	yes	maximal
2	C₂×#16812769	C₂	1	2	yes	maximal
3	#20296031	C₂	1	2	yes	maximal
4	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	C₂²	1	4	yes	—
6	#16812770	—	2	4	no	—
7	#16812769	—	2	4	no	—
8	#16812766	—	2	4	no	—
9	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×#11724784	C₄	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
15	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
16	#13182039	—	2	4	no	—
17	#13182039	—	2	4	no	—
18	C₂×#11724786	—	2	4	no	—
19	C₂×C₄×D₁₀₉	Q₈	1	2	yes	—
20	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
22	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
23	C₈×D₁₀₉	D₄	1	2	yes	—
24	C₈×D₁₀₉	D₄	1	2	yes	—
25	C₈×D₁₀₉	Q₈	1	2	yes	—
26	C₈×D₁₀₉	Q₈	1	2	yes	—
27	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₄×D₁₀₉	—	2	4	no	—
38	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
42	#11724786	—	4	4	no	—
43	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
44	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
45	#11724784	C₄⋊C₄	1	2	yes	—
46	#11724784	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₁₀₉	—	2	2	no	—
49	C₂×D₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×D₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₀₉	—	2	2	no	—
52	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	Dic₁₀₉	#108	1	4	yes	complemented
55	Dic₁₀₉	#127	1	1	yes	characteristic
56	Dic₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₁₀₉	#384	1	1	yes	characteristic
58	C₂×C₁₀₉	C₂×#108	1	2	yes	—
59	D₁₀₉	—	2	1	no	—
60	D₁₀₉	—	2	1	no	—
61	#1848	—	109	1	no	maximal, Sylow
62	C₁₀₉	#1848	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	—	109	2	no	—
64	C₂×#108	—	109	2	no	—
65	#423	—	109	2	no	—
66	C₂×C₄×C₈	—	109	1	no	—
67	C₂×M₁₆	—	109	4	no	—
68	C₂²×C₈	—	109	1	no	—
69	C₂²×C₈	—	109	1	no	—
70	C₂×C₄²	—	109	1	no	—
71	#108	—	218	4	no	—
72	#108	—	218	4	no	—
73	C₄⋊C₈	—	218	4	no	—
74	C₄×C₈	—	109	1	no	—
75	C₄×C₈	—	109	1	no	—
76	C₄×C₈	—	109	1	no	—
77	C₄×C₈	#89137	1	1	yes	characteristic
78	C₂²×C₄	—	109	1	no	—
79	C₂²×C₄	—	109	2	no	—
80	M₁₆	—	218	4	no	—
81	M₁₆	—	218	4	no	—
82	C₂×C₈	—	109	1	no	—
83	C₂×C₈	—	109	1	no	—
84	C₂×C₈	—	109	2	no	—
85	C₂×C₈	—	109	2	no	—
86	C₂×C₈	—	109	2	no	—
87	C₂×C₈	—	109	2	no	—
88	C₂×C₈	—	218	4	no	—
89	C₂×C₈	C₂×#89137	1	1	yes	characteristic
90	C₂×C₈	C₂×#89137	1	1	yes	characteristic
91	C₄²	—	109	2	no	—
92	C₄²	—	109	1	no	—
93	C₄²	C₂×#89137	1	1	yes	characteristic
94	C₂³	—	109	1	no	—
95	C₂×C₄	—	109	1	no	—
96	C₂×C₄	—	109	2	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	218	4	no	—
102	C₂×C₄	C₄⋊#89137	1	2	yes	—
103	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
104	C₈	—	109	2	no	—
105	C₈	—	109	2	no	—
106	C₈	—	436	4	no	—
107	C₈	C₄⋊#89137	1	2	yes	—
108	C₈	C₄⋊#89137	1	2	yes	—
109	C₂²	—	109	1	no	—
110	C₂²	—	109	1	no	—
111	C₂²	—	218	2	no	—
112	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	109	1	no	—
115	C₄	—	109	1	no	—
116	C₄	—	109	4	no	—
117	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
118	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
119	C₂	—	218	1	no	—
120	C₂	—	218	1	no	—
121	C₂	#20295972	1	1	yes	characteristic
122	C₂	D₁₀₉⋊#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812770

C₂

yes

maximal

C₂×#16812769

C₂

yes

maximal

#20296031

C₂

yes

maximal

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

#16812770

—

#16812769

—

#16812766

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

—

—

—

C₂×C₄×D₁₀₉

Q₈

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

—

C₈×D₁₀₉

D₄

yes

—

C₈×D₁₀₉

Q₈

yes

—

C₈×D₁₀₉

Q₈

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

C₂×Q₈

yes

characteristic

C₄×D₁₀₉

C₂×Q₈

yes

characteristic

#11724786

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

#11724784

C₄⋊C₄

yes

—

#11724784

C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊Q₈

yes

characteristic

C₂×D₁₀₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

Dic₁₀₉

yes

Dic₁₀₉

yes

Dic₁₀₉

C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

D₁₀₉

—

D₁₀₉

—

#1848

—

109

maximal, Sylow

C₁₀₉

#1848

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

C₂×#108

—

109

—

#423

—

109

—

C₂×C₄×C₈

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

#108

—

218

—

#108

—

218

—

C₄⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

218

—

102

C₂×C₄

C₄⋊#89137

yes

—

103

C₂×C₄

C₂²×#89137

yes

characteristic

104

C₈

—

109

—

105

C₈

—

109

—

106

C₈

—

436

—

107

C₈

C₄⋊#89137

yes

—

108

C₈

C₄⋊#89137

yes

—

109

C₂²

—

109

—

110

C₂²

—

109

—

111

C₂²

—

218

—

112

C₂²

C₂×C₄⋊#89137

yes

characteristic

113

C₄

—

114

C₄

—

109

—

115

C₄

—

109

—

116

C₄

—

109

—

117

C₄

C₂×C₄⋊#89137

yes

characteristic

118

C₄

C₂×C₄⋊#89137

yes

characteristic

119

C₂

—

218

—

120

C₂

—

218

—

121

C₂

#20295972

yes

characteristic

122

C₂

D₁₀₉⋊#108

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₂×Q₈
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1848
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₂×Q₈

cosocle

C₂×C₄×D₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
54. Dic₁₀₉	#108	semidirect
62. C₁₀₉	#1848	semidirect

H ≅ G/N

type

54. Dic₁₀₉

#108

semidirect

62. C₁₀₉

#1848

semidirect

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

subgroup	index
C₂×#16812770	2
C₂×#16812769	2
#20296031	2
C₄×C₈×D₁₀₉	2
#1848	109

subgroup

index

C₄×C₈×D₁₀₉

109

Subgroups — #28378972

Conjugacy classes of subgroups

Complements

Maximal subgroups