Group #28378586 — subgroups

Subgroups — #28378586

Order 13952.

184 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20296029	C₂	1	2	yes	maximal
2	#20295888	C₂	1	2	yes	maximal, complemented
3	#20295887	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182038	—	2	2	no	—
6	#16812766	—	2	2	no	—
7	C₄⋊#11724786	C₂²	1	4	yes	—
8	C₂²×#11724784	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
11	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	#13182038	—	4	4	no	—
16	C₂×#11724786	—	2	4	no	—
17	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
18	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
19	C₂×#11724784	—	2	2	no	—
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
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	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	—	2	2	no	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	—	2	2	no	—
36	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	SD₁₆	1	4	yes	—
39	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	#11724786	—	4	4	no	—
42	C₈×C₁₀₉	—	2	1	no	—
43	C₈×C₁₀₉	M₁₆	1	2	yes	complemented
44	#11724784	—	2	1	no	—
45	#11724784	M₁₆	1	2	yes	complemented
46	C₂²×C₁₀₉	—	2	1	no	—
47	C₂²×C₁₀₉	—	2	1	no	—
48	C₂²×C₁₀₉	#106	1	2	yes	—
49	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#106	1	2	yes	—
55	Dic₁₀₉	#108	1	2	yes	—
56	Dic₁₀₉	C₂×SD₁₆	1	2	yes	—
57	C₂×C₁₀₉	—	2	2	no	—
58	C₂×C₁₀₉	#423	1	1	yes	characteristic
59	C₂×C₁₀₉	C₂×#106	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
61	#1566	—	109	1	no	maximal, Sylow
62	C₁₀₉	#1566	1	1	yes	characteristic, complemented, Sylow
63	#423	—	109	2	no	—
64	C₂×C₄×C₈	—	109	1	no	—
65	#334	—	109	2	no	—
66	#334	—	109	2	no	—
67	C₂×M₁₆	—	218	2	no	—
68	C₂²×C₈	—	109	1	no	—
69	C₂²×C₈	#89137	1	1	yes	characteristic
70	C₂×C₄²	—	109	1	no	—
71	C₄⋊C₈	—	109	4	no	—
72	C₄⋊C₈	—	218	2	no	—
73	C₄×C₈	—	109	1	no	—
74	C₄×C₈	—	109	1	no	—
75	C₄×C₈	—	109	1	no	—
76	C₄×C₈	—	109	1	no	—
77	C₂²×C₄	—	109	1	no	—
78	C₂²×C₄	—	109	1	no	—
79	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
80	M₁₆	—	436	4	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	109	1	no	—
83	C₂×C₈	—	109	1	no	—
84	C₂×C₈	—	218	2	no	—
85	C₂×C₈	—	218	4	no	—
86	C₂×C₈	C₂×#89137	1	1	yes	characteristic
87	C₂×C₈	C₂×#89137	1	1	yes	characteristic
88	C₄²	—	109	1	no	—
89	C₄²	—	109	1	no	—
90	C₄²	—	109	1	no	—
91	C₄²	—	109	1	no	—
92	C₂³	C₄⋊#89137	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	109	4	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	218	2	no	—
101	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
102	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	109	2	no	—
105	C₈	—	436	4	no	—
106	C₈	—	218	1	no	—
107	C₈	#13182038	1	2	yes	complemented
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	#16812767	1	2	yes	—
111	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	109	2	no	—
114	C₄	—	109	2	no	—
115	C₄	—	109	2	no	—
116	C₄	—	218	1	no	—
117	C₄	C₂×#13182038	1	1	yes	characteristic
118	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
119	C₂	—	2	2	no	—
120	C₂	#20296029	1	1	yes	characteristic
121	C₂	C₂×#16812767	1	1	yes	characteristic
122	C₂	D₁₀₉⋊#108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296029

C₂

yes

maximal

#20295888

C₂

yes

maximal, complemented

#20295887

C₂

yes

maximal, complemented

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182038

—

#16812766

—

C₄⋊#11724786

C₂²

yes

—

C₂²×#11724784

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₄

yes

characteristic

#13182038

—

C₂×#11724786

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

Q₈

yes

characteristic

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

SD₁₆

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

characteristic

#11724786

—

C₈×C₁₀₉

—

C₈×C₁₀₉

M₁₆

yes

complemented

#11724784

—

#11724784

M₁₆

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#106

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

#106

yes

—

Dic₁₀₉

#108

yes

—

Dic₁₀₉

C₂×SD₁₆

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1566

yes

characteristic, complemented, Sylow

#423

—

109

—

C₂×C₄×C₈

—

109

—

#334

—

109

—

#334

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

109

—

C₂²×C₈

#89137

yes

characteristic

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

M₁₆

—

436

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₄⋊#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

218

—

101

C₂×C₄

C₄⋊#89137

yes

characteristic

102

C₂×C₄

C₂²×#89137

yes

characteristic

103

C₈

—

104

C₈

—

109

—

105

C₈

—

436

—

106

C₈

—

218

—

107

C₈

#13182038

yes

complemented

108

C₂²

—

109

C₂²

—

110

C₂²

#16812767

yes

—

111

C₂²

C₂×C₄⋊#89137

yes

characteristic

112

C₄

—

113

C₄

—

109

—

114

C₄

—

109

—

115

C₄

—

109

—

116

C₄

—

218

—

117

C₄

C₂×#13182038

yes

characteristic

118

C₄

D₁₀₉⋊M₁₆

yes

characteristic

119

C₂

—

120

C₂

#20296029

yes

characteristic

121

C₂

C₂×#16812767

yes

characteristic

122

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	C₂²×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	#1566
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂²×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₂²×C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
2. #20295888	C₂	semidirect
3. #20295887	C₂	semidirect
43. C₈×C₁₀₉	M₁₆	semidirect
45. #11724784	M₁₆	semidirect
62. C₁₀₉	#1566	semidirect
107. C₈	#13182038	semidirect

H ≅ G/N

type

2. #20295888

C₂

semidirect

3. #20295887

C₂

semidirect

43. C₈×C₁₀₉

M₁₆

semidirect

45. #11724784

M₁₆

semidirect

62. C₁₀₉

#1566

semidirect

107. C₈

#13182038

semidirect

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

subgroup	index
#20296029	2
#20295888	2
#20295887	2
C₂×C₈×Dic₁₀₉	2
#1566	109

subgroup

index

C₂×C₈×Dic₁₀₉

109

Subgroups — #28378586

Conjugacy classes of subgroups

Complements

Maximal subgroups