Group #28377910 — subgroups

Subgroups — #28377910

Order 13952.

188 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 76 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×#16812766	C₂	1	2	yes	maximal
2	#20294895	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11724786	—	2	2	no	—
4	#16812766	—	2	2	no	—
5	#16812766	C₄	1	4	yes	complemented
6	C₂×C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#13182008	—	2	2	no	—
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
10	C₂×#11724786	—	2	4	no	—
11	C₂×#11724786	—	4	2	no	—
12	C₂×#11724786	—	4	2	no	—
13	C₂²×Dic₁₀₉	—	2	1	no	—
14	C₂²×Dic₁₀₉	D₄	1	2	yes	—
15	C₄⋊Dic₁₀₉	—	4	1	no	—
16	#13182008	—	4	2	no	—
17	#13182008	—	4	2	no	—
18	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
19	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
20	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	—	2	1	no	—
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₁₀₉	—	4	1	no	—
24	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	—	4	1	no	—
28	C₂×C₄×C₁₀₉	—	4	2	no	—
29	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	—	4	1	no	—
32	C₂×Dic₁₀₉	—	4	2	no	—
33	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
34	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₁₀₉	D₈	1	2	yes	—
36	C₂×Dic₁₀₉	SD₁₆	1	2	yes	—
37	C₂×Dic₁₀₉	SD₁₆	1	2	yes	—
38	C₂×Dic₁₀₉	Q₁₆	1	2	yes	—
39	#11724786	—	4	4	no	—
40	C₂²×C₁₀₉	#95	1	1	yes	characteristic
41	C₂²×C₁₀₉	#100	1	1	yes	characteristic
42	C₂²×C₁₀₉	#101	1	1	yes	characteristic
43	C₂²×C₁₀₉	#102	1	2	yes	—
44	C₂²×C₁₀₉	#103	1	2	yes	—
45	C₄×C₁₀₉	—	2	1	no	—
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	—	8	1	no	—
48	Dic₁₀₉	—	8	1	no	—
49	Dic₁₀₉	#102	1	2	yes	—
50	Dic₁₀₉	#103	1	2	yes	—
51	Dic₁₀₉	#106	1	2	yes	—
52	Dic₁₀₉	#107	1	2	yes	—
53	C₂×C₁₀₉	#331	1	1	yes	characteristic
54	C₂×C₁₀₉	#332	1	2	yes	—
55	C₂×C₁₀₉	#333	1	1	yes	characteristic
56	C₂×C₁₀₉	#340	1	2	yes	—
57	C₂×C₁₀₉	#343	1	1	yes	characteristic
58	#1296	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1296	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊C₈	—	109	2	no	—
61	#384	—	109	1	no	—
62	C₂²×C₈	—	218	2	no	—
63	C₂×C₄⋊C₄	—	109	1	no	—
64	C₂×C₄⋊C₄	—	218	2	no	—
65	C₂×C₄⋊C₄	#89137	1	1	yes	characteristic
66	C₂×C₄²	—	109	1	no	—
67	C₄⋊C₈	—	109	4	no	—
68	C₄⋊C₈	—	218	2	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	109	2	no	—
71	C₂²×C₄	—	218	1	no	—
72	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
73	C₂×C₈	—	436	2	no	—
74	C₂×C₈	—	436	2	no	—
75	C₂×C₈	—	218	4	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	436	1	no	—
78	C₄⋊C₄	—	436	2	no	—
79	C₄⋊C₄	—	436	2	no	—
80	C₄²	—	109	2	no	—
81	C₄²	—	109	1	no	—
82	C₄²	—	109	1	no	—
83	C₂³	C₂²⋊#89137	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	109	2	no	—
89	C₂×C₄	—	109	2	no	—
90	C₂×C₄	—	109	2	no	—
91	C₂×C₄	—	109	2	no	—
92	C₂×C₄	—	109	2	no	—
93	C₂×C₄	—	109	2	no	—
94	C₂×C₄	—	436	1	no	—
95	C₂×C₄	—	436	2	no	—
96	C₂×C₄	C₄×#89137	1	1	yes	characteristic
97	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
98	C₈	—	436	4	no	—
99	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
100	C₂²	#16812759	1	1	yes	characteristic
101	C₂²	#16812760	1	1	yes	characteristic
102	C₂²	#16812761	1	2	yes	—
103	C₂²	#16812762	1	2	yes	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	8	1	no	—
107	C₄	—	872	1	no	—
108	C₄	—	109	2	no	—
109	C₄	—	109	2	no	—
110	C₄	—	109	2	no	—
111	C₄	—	109	2	no	—
112	C₂	#20295884	1	1	yes	characteristic
113	C₂	#20295885	1	2	yes	—
114	C₂	#20295886	1	1	yes	characteristic
115	C₂	#20295899	1	2	yes	—
116	C₂	#20295906	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812766

C₂

yes

maximal

#20294895

C₂

yes

maximal, characteristic

C₂²×#11724786

—

#16812766

—

#16812766

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

C₄

yes

characteristic

C₂×#13182008

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

—

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

—

C₄⋊Dic₁₀₉

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

D₈

yes

—

C₂×Dic₁₀₉

SD₁₆

yes

—

C₂×Dic₁₀₉

SD₁₆

yes

—

C₂×Dic₁₀₉

Q₁₆

yes

—

#11724786

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₂²×C₁₀₉

#103

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#102

yes

—

Dic₁₀₉

#103

yes

—

Dic₁₀₉

#106

yes

—

Dic₁₀₉

#107

yes

—

C₂×C₁₀₉

#331

yes

characteristic

C₂×C₁₀₉

#332

yes

—

C₂×C₁₀₉

#333

yes

characteristic

C₂×C₁₀₉

#340

yes

—

C₂×C₁₀₉

#343

yes

characteristic

#1296

—

109

maximal, Sylow

C₁₀₉

#1296

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

#384

—

109

—

C₂²×C₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

#89137

yes

characteristic

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₂²⋊#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

436

—

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

100

C₂²

#16812759

yes

characteristic

101

C₂²

#16812760

yes

characteristic

102

C₂²

#16812761

yes

—

103

C₂²

#16812762

yes

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

872

—

108

C₄

—

109

—

109

C₄

—

109

—

110

C₄

—

109

—

111

C₄

—

109

—

112

C₂

#20295884

yes

characteristic

113

C₂

#20295885

yes

—

114

C₂

#20295886

yes

characteristic

115

C₂

#20295899

yes

—

116

C₂

#20295906

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₂²×C₄	C₂×#89137
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₀₉	C₂²
hypercenter	C₂×C₄⋊C₄	#89137
hypocenter	C₁₀₉	#1296
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#89137

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₂²×C₄

C₂×#89137

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₂²⋊C₄

cosocle

C₂×C₄×Dic₁₀₉

C₂²

C₂×C₄⋊C₄

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
5. #16812766	C₄	semidirect
59. C₁₀₉	#1296	semidirect

H ≅ G/N

type

5. #16812766

C₄

semidirect

59. C₁₀₉

#1296

semidirect

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

subgroup	index
C₂×#16812766	2
#20294895	2
#1296	109

subgroup

index

C₂×#16812766

#20294895

#1296

109

Subgroups — #28377910

Conjugacy classes of subgroups

Complements

Maximal subgroups