Group #28377871 — subgroups

Subgroups — #28377871

Order 13952.

152 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 56 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	#20296043	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812754	C₂	1	1	yes	maximal, characteristic
3	#20295076	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11724786	C₄	1	2	yes	—
5	#16812766	—	2	1	no	—
6	C₂²⋊#11724786	C₄	1	2	yes	—
7	#16812754	—	2	1	no	—
8	#16812754	—	2	1	no	—
9	C₄×#11724786	—	2	1	no	—
10	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	#16812628	—	2	1	no	—
12	C₂²⋊#11724784	C₄	1	1	yes	characteristic
13	C₈×Dic₁₀₉	—	2	1	no	—
14	C₂²⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	C₂×#11724786	—	2	2	no	—
16	C₂×#11724786	—	2	2	no	—
17	C₂×#11724786	—	2	2	no	—
18	C₂×#11724786	C₈	1	4	yes	complemented
19	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×#11724784	—	2	1	no	—
22	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
23	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
24	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
25	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	—	2	1	no	—
28	C₂³×C₁₀₉	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	4	1	no	—
30	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	—	4	1	no	—
33	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
38	C₂×Dic₁₀₉	M₁₆	1	2	yes	—
39	#11724786	—	2	4	no	—
40	#11724786	—	4	2	no	—
41	C₈×C₁₀₉	—	4	1	no	—
42	#11724784	—	4	1	no	—
43	C₂²×C₁₀₉	—	2	1	no	—
44	C₂²×C₁₀₉	—	2	1	no	—
45	C₂²×C₁₀₉	#95	1	1	yes	characteristic
46	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
47	C₂²×C₁₀₉	#97	1	1	yes	characteristic
48	C₄×C₁₀₉	—	2	1	no	—
49	C₄×C₁₀₉	—	2	1	no	—
50	Dic₁₀₉	—	2	1	no	—
51	Dic₁₀₉	—	2	1	no	—
52	Dic₁₀₉	C₂²⋊C₈	1	2	yes	—
53	Dic₁₀₉	C₄⋊C₈	1	2	yes	—
54	C₂×C₁₀₉	—	2	1	no	—
55	C₂×C₁₀₉	—	2	1	no	—
56	C₂×C₁₀₉	#336	1	1	yes	characteristic
57	C₂×C₁₀₉	#344	1	1	yes	characteristic
58	C₂×C₁₀₉	#344	1	1	yes	characteristic
59	#1281	—	109	1	no	maximal, Sylow
60	C₁₀₉	#1281	1	1	yes	characteristic, complemented, Sylow
61	#431	—	109	1	no	—
62	#431	—	109	1	no	—
63	C₂×#97	—	109	1	no	—
64	C₂²×C₈	—	109	2	no	—
65	C₂×C₄²	—	109	1	no	—
66	C₄⋊C₈	—	218	1	no	—
67	C₄⋊C₈	—	218	1	no	—
68	C₂²⋊C₈	—	109	2	no	—
69	C₂²⋊C₈	—	109	1	no	—
70	C₂²⋊C₈	#89137	1	1	yes	characteristic
71	#97	—	218	1	no	—
72	#97	—	218	1	no	—
73	C₄×C₈	—	218	1	no	—
74	C₄×C₈	—	218	1	no	—
75	C₂²×C₄	—	109	1	no	—
76	C₂²×C₄	—	109	1	no	—
77	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	109	4	no	—
80	C₂×C₈	—	218	2	no	—
81	C₂×C₈	—	218	1	no	—
82	C₂×C₈	—	218	2	no	—
83	C₂×C₈	—	218	2	no	—
84	C₄²	—	109	1	no	—
85	C₄²	—	109	1	no	—
86	C₄²	—	109	1	no	—
87	C₄²	—	109	1	no	—
88	C₂³	C₄×#89137	1	1	yes	characteristic
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	109	1	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	109	2	no	—
93	C₂×C₄	—	109	2	no	—
94	C₂×C₄	—	109	1	no	—
95	C₂×C₄	—	109	1	no	—
96	C₂×C₄	—	436	1	no	—
97	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
98	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
99	C₈	—	4	1	no	—
100	C₈	—	436	1	no	—
101	C₈	—	436	2	no	—
102	C₈	—	218	4	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
106	C₂²	C₈×#89137	1	1	yes	characteristic
107	C₂²	#16812753	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	109	2	no	—
111	C₄	—	109	2	no	—
112	C₄	—	218	1	no	—
113	C₄	—	218	1	no	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	(C₂×#89137)⋊C₈	1	1	yes	characteristic
117	C₂	#20295907	1	1	yes	characteristic
118	C₂	#20295909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296043

C₂

yes

maximal, characteristic

C₂×#16812754

C₂

yes

maximal, characteristic

#20295076

C₂

yes

maximal, characteristic

C₂²×#11724786

C₄

yes

—

#16812766

—

C₂²⋊#11724786

C₄

yes

—

#16812754

—

#16812754

—

C₄×#11724786

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812628

—

C₂²⋊#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

—

C₂²⋊C₈×C₁₀₉

C₄

yes

—

—

—

C₈

yes

complemented

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂³×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

M₁₆

yes

—

#11724786

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂²×C₁₀₉

C₄×C₈

yes

characteristic

C₂²×C₁₀₉

#97

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₈

yes

—

Dic₁₀₉

C₄⋊C₈

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1281

yes

characteristic, complemented, Sylow

#431

—

109

—

#431

—

109

—

C₂×#97

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

#89137

yes

characteristic

#97

—

218

—

#97

—

218

—

C₄×C₈

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₄×#89137

yes

characteristic

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

yes

C₂×C₄

yes

C₈

—

100

C₈

—

436

—

101

C₈

—

436

—

102

C₈

—

218

—

103

C₂²

—

104

C₂²

—

105

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

106

C₂²

C₈×#89137

yes

characteristic

107

C₂²

#16812753

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

109

—

111

C₄

—

109

—

112

C₄

—

218

—

113

C₄

—

218

—

114

C₂

—

115

C₂

—

116

C₂

(C₂×#89137)⋊C₈

yes

characteristic

117

C₂

#20295907

yes

characteristic

118

C₂

#20295909

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#89137)⋊C₄
commutator	C₂²×C₁₀₉	C₄×C₈
Frattini	C₂²×C₄	C₂×#89137
Fitting	C₂²⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂³×C₁₀₉	C₄²
cosocle	C₂×C₄×Dic₁₀₉	C₂²
hypercenter	C₂²⋊C₈	#89137
hypocenter	C₁₀₉	#1281
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center

C₂²

(C₂×#89137)⋊C₄

commutator

C₂²×C₁₀₉

C₄×C₈

Frattini

C₂²×C₄

C₂×#89137

Fitting

C₂²⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂³×C₁₀₉

C₄²

cosocle

C₂×C₄×Dic₁₀₉

C₂²

C₂²⋊C₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
18. C₂×#11724786	C₈	semidirect
60. C₁₀₉	#1281	semidirect

H ≅ G/N

type

18. C₂×#11724786

C₈

semidirect

60. C₁₀₉

#1281

semidirect

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

subgroup	index
#20296043	2
C₂×#16812754	2
#20295076	2
#1281	109

subgroup

index

109

Subgroups — #28377871

Conjugacy classes of subgroups

Complements

Maximal subgroups