Group #28378593 — subgroups

Subgroups — #28378593

Order 13952.

168 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 70 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	#20296048	C₂	1	2	yes	maximal
2	#20295889	C₂	1	2	yes	maximal, complemented
3	#20295887	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11724786	C₂²	1	4	yes	—
6	#16812756	—	2	2	no	—
7	#16812754	—	2	2	no	—
8	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
10	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₄×#11724784	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	C₂×#11724786	—	2	4	no	—
16	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
18	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
19	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₈×D₁₀₉	—	2	2	no	—
22	C₈×D₁₀₉	—	2	2	no	—
23	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
24	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
25	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
36	C₄×D₁₀₉	—	2	2	no	—
37	C₄×D₁₀₉	—	2	1	no	—
38	C₄×D₁₀₉	—	2	1	no	—
39	C₄×D₁₀₉	—	2	2	no	—
40	#11724786	—	4	4	no	—
41	C₈×C₁₀₉	M₁₆	1	2	yes	—
42	C₈×C₁₀₉	M₁₆	1	2	yes	—
43	#11724784	—	2	1	no	—
44	#11724784	—	2	1	no	—
45	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×D₁₀₉	—	2	1	no	—
47	C₂×D₁₀₉	—	2	1	no	—
48	C₂×D₁₀₉	#108	1	2	yes	—
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	—	2	1	no	—
51	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
52	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
53	Dic₁₀₉	—	2	1	no	—
54	Dic₁₀₉	—	2	1	no	—
55	Dic₁₀₉	#135	1	2	yes	—
56	C₂×C₁₀₉	#423	1	1	yes	characteristic
57	C₂×C₁₀₉	#427	1	1	yes	characteristic
58	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
59	D₁₀₉	—	2	2	no	—
60	#1568	—	109	1	no	maximal, Sylow
61	C₁₀₉	#1568	1	1	yes	characteristic, complemented, Sylow
62	#432	—	109	2	no	—
63	C₂×C₄×C₈	—	109	1	no	—
64	#335	—	109	2	no	—
65	#334	—	109	2	no	—
66	C₂²×C₈	—	109	1	no	—
67	C₂²×C₈	—	109	1	no	—
68	C₂×C₄²	—	109	1	no	—
69	C₄⋊C₈	—	109	4	no	—
70	C₂²⋊C₈	—	218	2	no	—
71	#97	—	218	2	no	—
72	C₄×C₈	—	109	1	no	—
73	C₄×C₈	—	109	1	no	—
74	C₄×C₈	—	109	1	no	—
75	C₄×C₈	#89137	1	1	yes	characteristic
76	C₂²×C₄	—	109	1	no	—
77	C₂²×C₄	—	109	1	no	—
78	C₂²×C₄	—	109	1	no	—
79	C₂×C₈	—	109	1	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	218	4	no	—
82	C₂×C₈	—	218	2	no	—
83	C₂×C₈	—	218	2	no	—
84	C₂×C₈	C₂×#89137	1	1	yes	characteristic
85	C₂×C₈	C₂×#89137	1	1	yes	characteristic
86	C₄²	—	109	1	no	—
87	C₄²	—	109	1	no	—
88	C₄²	—	109	1	no	—
89	C₄²	C₂×#89137	1	1	yes	characteristic
90	C₂³	—	109	1	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	109	1	no	—
93	C₂×C₄	—	109	1	no	—
94	C₂×C₄	—	218	2	no	—
95	C₂×C₄	—	218	2	no	—
96	C₂×C₄	—	218	1	no	—
97	C₂×C₄	—	218	1	no	—
98	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
99	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
100	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
101	C₈	—	436	4	no	—
102	C₈	—	218	1	no	—
103	C₈	—	218	1	no	—
104	C₈	#13182039	1	2	yes	—
105	C₈	#13182039	1	2	yes	—
106	C₂²	—	109	2	no	—
107	C₂²	—	218	1	no	—
108	C₂²	—	218	1	no	—
109	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	109	2	no	—
113	C₄	—	218	1	no	—
114	C₄	—	218	1	no	—
115	C₄	C₂×#13182039	1	1	yes	characteristic
116	C₄	C₂×#13182039	1	1	yes	characteristic
117	C₂	—	218	2	no	—
118	C₂	#20296030	1	1	yes	characteristic
119	C₂	#20296036	1	1	yes	characteristic
120	C₂	D₁₀₉⋊#108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296048

C₂

yes

maximal

#20295889

C₂

yes

maximal, complemented

#20295887

C₂

yes

maximal, complemented

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₄⋊#11724786

C₂²

yes

—

#16812756

—

#16812754

—

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

characteristic

C₂×#11724786

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

#11724786

—

C₈×C₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

M₁₆

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#108

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#135

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1568

—

109

maximal, Sylow

C₁₀₉

#1568

yes

characteristic, complemented, Sylow

#432

—

109

—

C₂×C₄×C₈

—

109

—

#335

—

109

—

#334

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

218

—

#97

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂²×#89137

yes

characteristic

101

C₈

—

436

—

102

C₈

—

218

—

103

C₈

—

218

—

104

C₈

#13182039

yes

—

105

C₈

#13182039

yes

—

106

C₂²

—

109

—

107

C₂²

—

218

—

108

C₂²

—

218

—

109

C₂²

C₂×C₄⋊#89137

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

109

—

113

C₄

—

218

—

114

C₄

—

218

—

115

C₄

C₂×#13182039

yes

characteristic

116

C₄

C₂×#13182039

yes

characteristic

117

C₂

—

218

—

118

C₂

#20296030

yes

characteristic

119

C₂

#20296036

yes

characteristic

120

C₂

D₁₀₉⋊#108

yes

characteristic

	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	C₄⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1568
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

C₄⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. #20295889	C₂	semidirect
3. #20295887	C₂	semidirect
61. C₁₀₉	#1568	semidirect

H ≅ G/N

type

2. #20295889

C₂

semidirect

3. #20295887

C₂

semidirect

61. C₁₀₉

#1568

semidirect

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

subgroup	index
#20296048	2
#20295889	2
#20295887	2
C₄×C₈×D₁₀₉	2
#1568	109

subgroup

index

C₄×C₈×D₁₀₉

109

Subgroups — #28378593

Conjugacy classes of subgroups

Complements

Maximal subgroups