Group #28377896 — subgroups

Subgroups — #28377896

Order 13952.

136 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 44 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	#20296049	C₂	1	1	yes	maximal, characteristic
2	D₁₀₉⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20295081	C₂	1	1	yes	maximal, characteristic
4	C₂×D₁₀₉⋊C₈	C₄	1	2	yes	—
5	#16812766	—	2	1	no	—
6	#16812766	—	2	1	no	—
7	C₄⋊#11724786	—	2	1	no	—
8	#16812756	—	2	1	no	—
9	#16812754	C₄	1	2	yes	—
10	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
11	#16812628	—	2	1	no	—
12	#16812611	—	2	1	no	—
13	#16812610	C₄	1	1	yes	characteristic
14	#97×C₁₀₉	C₄	1	1	yes	characteristic
15	D₁₀₉⋊C₈	—	4	2	no	—
16	C₂×#11724786	—	2	2	no	—
17	C₂×#11724786	—	2	2	no	—
18	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
21	C₂×#11724784	—	2	1	no	—
22	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
24	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	—	2	1	no	—
26	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₄×D₁₀₉	—	2	2	no	—
35	C₄×D₁₀₉	—	2	1	no	—
36	C₄×D₁₀₉	—	2	1	no	—
37	C₄×D₁₀₉	—	4	1	no	—
38	#11724786	—	4	2	no	—
39	#11724786	—	4	2	no	—
40	C₈×C₁₀₉	—	4	1	no	—
41	#11724784	—	4	1	no	—
42	C₂²×C₁₀₉	#95	1	1	yes	characteristic
43	C₂×D₁₀₉	—	2	1	no	—
44	C₂×D₁₀₉	—	2	1	no	—
45	C₂×D₁₀₉	#108	1	2	yes	—
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	—	2	1	no	—
48	C₄×C₁₀₉	—	2	1	no	—
49	Dic₁₀₉	—	2	1	no	—
50	Dic₁₀₉	—	2	1	no	—
51	Dic₁₀₉	#100	1	1	yes	characteristic
52	Dic₁₀₉	#101	1	1	yes	characteristic
53	C₂×C₁₀₉	H(3,ℤ/4)	1	1	yes	characteristic
54	C₂×C₁₀₉	#341	1	1	yes	characteristic
55	C₂×C₁₀₉	#343	1	1	yes	characteristic
56	D₁₀₉	—	2	2	no	—
57	#1290	—	109	1	no	maximal, Sylow
58	C₁₀₉	#1290	1	1	yes	characteristic, complemented, Sylow
59	#432	—	109	1	no	—
60	#432	—	109	1	no	—
61	C₂×C₄⋊C₈	—	109	1	no	—
62	C₂²×C₈	—	109	2	no	—
63	C₂×C₄²	—	109	1	no	—
64	C₄⋊C₈	—	218	1	no	—
65	C₄⋊C₈	—	218	1	no	—
66	C₄⋊C₈	—	218	1	no	—
67	C₄⋊C₈	—	218	1	no	—
68	C₂²⋊C₈	—	218	1	no	—
69	C₂²⋊C₈	—	218	1	no	—
70	#97	—	109	2	no	—
71	#97	—	109	1	no	—
72	#97	#89137	1	1	yes	characteristic
73	C₂²×C₄	—	109	1	no	—
74	C₂²×C₄	—	109	1	no	—
75	C₂²×C₄	—	109	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	436	2	no	—
78	C₂×C₈	—	218	2	no	—
79	C₂×C₈	—	218	2	no	—
80	C₂×C₈	—	218	1	no	—
81	C₄²	—	109	1	no	—
82	C₄²	—	109	1	no	—
83	C₄²	—	109	1	no	—
84	C₄²	C₂×#89137	1	1	yes	characteristic
85	C₂³	—	109	1	no	—
86	C₂×C₄	—	109	1	no	—
87	C₂×C₄	—	109	1	no	—
88	C₂×C₄	—	109	1	no	—
89	C₂×C₄	—	436	1	no	—
90	C₂×C₄	—	218	1	no	—
91	C₂×C₄	—	218	1	no	—
92	C₂×C₄	—	218	2	no	—
93	C₂×C₄	C₄×#89137	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
95	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
96	C₈	—	4	1	no	—
97	C₈	—	436	1	no	—
98	C₈	—	436	2	no	—
99	C₈	—	436	2	no	—
100	C₂²	—	109	2	no	—
101	C₂²	—	218	1	no	—
102	C₂²	—	218	1	no	—
103	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	109	1	no	—
108	C₄	—	109	1	no	—
109	C₄	—	218	1	no	—
110	C₄	—	218	1	no	—
111	C₂	—	218	2	no	—
112	C₂	#20295893	1	1	yes	characteristic
113	C₂	#20295901	1	1	yes	characteristic
114	C₂	#20295905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296049

C₂

yes

maximal, characteristic

D₁₀₉⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

#20295081

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

C₄

yes

—

—

—

—

—

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

—

—

C₄

yes

characteristic

#97×C₁₀₉

C₄

yes

characteristic

D₁₀₉⋊C₈

—

C₂×#11724786

—

C₂×#11724786

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×#11724784

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

#11724786

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#108

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1290

—

109

maximal, Sylow

C₁₀₉

#1290

yes

characteristic, complemented, Sylow

#432

—

109

—

#432

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

#97

—

109

—

#97

—

109

—

#97

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

436

—

C₈

—

436

—

C₈

—

436

—

100

C₂²

—

109

—

101

C₂²

—

218

—

102

C₂²

—

218

—

103

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

109

—

108

C₄

—

109

—

109

C₄

—

218

—

110

C₄

—

218

—

111

C₂

—

218

—

112

C₂

#20295893

yes

characteristic

113

C₂

#20295901

yes

characteristic

114

C₂

#20295905

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#89137)⋊C₄
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₄²	C₂×#89137
Fitting	#97×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	#97	#89137
hypocenter	C₁₀₉	#1290
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

(C₂×#89137)⋊C₄

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₄²

C₂×#89137

Fitting

#97×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
58. C₁₀₉	#1290	semidirect

H ≅ G/N

type

58. C₁₀₉

#1290

semidirect

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

subgroup	index
#20296049	2
D₁₀₉⋊(C₄⋊C₈)	2
#20295081	2
#1290	109

subgroup

index

#20296049

D₁₀₉⋊(C₄⋊C₈)

#20295081

#1290

109

Subgroups — #28377896

Conjugacy classes of subgroups

Complements

Maximal subgroups