Group #28377921 — subgroups

Subgroups — #28377921

Order 13952.

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

normal subgroups only

This group has 172 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	#20296048	C₂	1	1	yes	maximal, characteristic
2	#20295962	C₂	1	1	yes	maximal, characteristic
3	#20294978	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#89137	—	2	1	no	—
5	C₄⋊#11724786	C₄	1	2	yes	complemented
6	#16812756	—	2	1	no	—
7	#16812754	—	2	1	no	—
8	(C₂×#89137)⋊C₄	—	2	1	no	—
9	(C₂×#89137)⋊C₄	C₄	1	2	yes	—
10	C₂×#13182012	—	2	1	no	—
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	#16812610	C₄	1	1	yes	characteristic, complemented
13	C₈×Dic₁₀₉	—	2	1	no	—
14	#97×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×#89137	—	2	2	no	—
16	C₂×#11724786	—	2	2	no	—
17	C₄×#89137	—	2	4	no	—
18	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
21	#13182012	—	2	4	no	—
22	C₂×#11724784	—	2	1	no	—
23	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
25	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₁₀₉	—	2	1	no	—
27	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×#89137	—	2	4	no	—
30	C₂×#89137	—	4	2	no	—
31	C₂×#89137	—	4	2	no	—
32	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₄×D₁₀₉	—	2	2	no	—
39	C₄×D₁₀₉	—	2	2	no	—
40	C₄×D₁₀₉	—	4	1	no	—
41	#11724786	—	4	2	no	—
42	C₈×C₁₀₉	—	2	2	no	—
43	#11724784	—	2	2	no	—
44	C₂²×C₁₀₉	#95	1	1	yes	characteristic
45	C₂×D₁₀₉	—	2	1	no	—
46	C₂×D₁₀₉	—	2	1	no	—
47	C₂×D₁₀₉	#104	1	2	yes	—
48	#89137	—	4	4	no	—
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	—	2	1	no	—
51	C₄×C₁₀₉	—	2	1	no	—
52	Dic₁₀₉	—	2	1	no	—
53	Dic₁₀₉	—	2	1	no	—
54	Dic₁₀₉	#104	1	2	yes	—
55	C₂×C₁₀₉	#339	1	1	yes	characteristic
56	C₂×C₁₀₉	#339	1	1	yes	characteristic
57	C₂×C₁₀₉	#344	1	1	yes	characteristic
58	D₁₀₉	—	2	2	no	—
59	#1300	—	109	1	no	maximal, Sylow
60	C₁₀₉	#1300	1	1	yes	characteristic, complemented, Sylow
61	#432	—	109	1	no	—
62	C₄×M₁₆	—	109	1	no	—
63	#376	—	109	1	no	—
64	C₂×M₁₆	—	218	1	no	—
65	C₂×C₄²	—	109	1	no	—
66	C₂×C₄²	—	218	1	no	—
67	C₄⋊C₈	—	109	2	no	—
68	C₂²⋊C₈	—	218	1	no	—
69	#97	—	109	1	no	—
70	#97	—	218	1	no	—
71	#97	#89137	1	1	yes	characteristic, complemented
72	C₄×C₈	—	218	1	no	—
73	#95	—	109	2	no	—
74	#95	—	218	1	no	—
75	C₂²×C₄	—	109	1	no	—
76	C₂²×C₄	—	109	1	no	—
77	C₂²×C₄	—	109	1	no	—
78	C₂²×C₄	—	218	2	no	—
79	M₁₆	—	218	4	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	218	2	no	—
82	C₂×C₈	—	218	1	no	—
83	C₄²	—	109	1	no	—
84	C₄²	—	109	1	no	—
85	C₄²	—	109	1	no	—
86	C₄²	—	218	4	no	—
87	C₄²	C₂×#89137	1	1	yes	characteristic
88	C₂³	—	109	1	no	—
89	C₂×C₄	—	109	1	no	—
90	C₂×C₄	—	109	1	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	436	1	no	—
93	C₂×C₄	—	436	2	no	—
94	C₂×C₄	—	436	2	no	—
95	C₂×C₄	—	218	2	no	—
96	C₂×C₄	—	218	2	no	—
97	C₂×C₄	—	218	4	no	—
98	C₂×C₄	C₄×#89137	1	1	yes	characteristic
99	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
100	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	436	2	no	—
103	C₈	—	218	2	no	—
104	C₂²	—	109	2	no	—
105	C₂²	—	218	1	no	—
106	C₂²	—	218	1	no	—
107	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	109	2	no	—
112	C₄	—	436	4	no	—
113	C₄	—	218	1	no	—
114	C₄	—	218	1	no	—
115	C₂	—	218	2	no	—
116	C₂	#20295897	1	1	yes	characteristic
117	C₂	#20295898	1	1	yes	characteristic
118	C₂	#20295909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296048

C₂

yes

maximal, characteristic

#20295962

C₂

yes

maximal, characteristic

#20294978

C₂

yes

maximal, characteristic

C₂×C₄×#89137

—

C₄⋊#11724786

C₄

yes

—

—

—

C₄

yes

—

C₂×#13182012

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812610

C₄

yes

characteristic, complemented

C₈×Dic₁₀₉

—

#97×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

C₂×#11724786

—

C₄×#89137

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

#13182012

—

C₂×#11724784

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

—

—

—

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

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#104

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#104

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1300

—

109

maximal, Sylow

C₁₀₉

#1300

yes

characteristic, complemented, Sylow

#432

—

109

—

C₄×M₁₆

—

109

—

#376

—

109

—

C₂×M₁₆

—

218

—

C₂×C₄²

—

109

—

C₂×C₄²

—

218

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

218

—

#97

—

109

—

#97

—

218

—

#97

#89137

yes

characteristic, complemented

C₄×C₈

—

218

—

#95

—

109

—

#95

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₄⋊#89137

yes

characteristic

101

C₈

—

102

C₈

—

436

—

103

C₈

—

218

—

104

C₂²

—

109

—

105

C₂²

—

218

—

106

C₂²

—

218

—

107

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

109

—

112

C₄

—

436

—

113

C₄

—

218

—

114

C₄

—

218

—

115

C₂

—

218

—

116

C₂

#20295897

yes

characteristic

117

C₂

#20295898

yes

characteristic

118

C₂

#20295909

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	C₂²⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	#97	#89137
hypocenter	C₁₀₉	#1300
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×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

C₂²⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
5. C₄⋊#11724786	C₄	semidirect
12. #16812610	C₄	semidirect
14. #97×C₁₀₉	C₄	semidirect
60. C₁₀₉	#1300	semidirect
71. #97	#89137	semidirect

H ≅ G/N

type

5. C₄⋊#11724786

C₄

semidirect

12. #16812610

C₄

semidirect

14. #97×C₁₀₉

C₄

semidirect

60. C₁₀₉

#1300

semidirect

71. #97

#89137

semidirect

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

subgroup	index
#20296048	2
#20295962	2
#20294978	2
#1300	109

subgroup

index

109

Subgroups — #28377921

Conjugacy classes of subgroups

Complements

Maximal subgroups