Group #28371972 — subgroups

Subgroups — #28371972

Order 13952.

348 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 189 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	#20295583	C₂	1	1	yes	maximal, characteristic
2	#20295154	C₂	1	8	yes	maximal, complemented
3	#20295052	C₂	1	2	yes	maximal
4	#20295050	C₂	1	1	yes	maximal, characteristic
5	#20294981	C₂	1	2	yes	maximal
6	C₂×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#13182013	C₂²	1	2	yes	—
8	#16812699	C₄	1	2	yes	—
9	#16812691	C₄	1	2	yes	—
10	#16812678	C₄	1	4	yes	—
11	C₄×Dic₂₁₈	C₂²	1	4	yes	—
12	#16812658	C₂²	1	2	yes	—
13	C₄⋊#11724784	C₂²	1	4	yes	—
14	#16812628	C₂²	1	8	yes	—
15	#16812609	C₂²	1	4	yes	—
16	C₈×Dic₁₀₉	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
20	#97×C₁₀₉	C₂²	1	4	yes	—
21	C₂×Dic₂₁₈	C₂×C₄	1	4	yes	—
22	#13182013	—	2	4	no	—
23	C₂×#11724784	C₂³	1	4	yes	—
24	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
26	#13182008	C₂×C₄	1	8	yes	—
27	C₂²⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
28	C₄×Dic₁₀₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
33	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
34	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
35	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₀₉	C₂³	1	2	yes	—
38	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	—	4	2	no	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₀₉	#41	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₁₀₉	C₂²×C₄	1	8	yes	—
47	Dic₂₁₈	—	4	4	no	—
48	C₈×C₁₀₉	—	2	2	no	—
49	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
50	#11724784	—	2	4	no	—
51	C₂²×C₁₀₉	—	2	2	no	—
52	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₀₉	—	2	2	no	—
55	C₄×C₁₀₉	—	2	2	no	—
56	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
57	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
59	Dic₁₀₉	—	2	8	no	—
60	C₂×C₁₀₉	—	2	2	no	—
61	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₁₀₉	#568	1	2	yes	—
63	#2961	—	109	1	no	maximal, Sylow
64	C₁₀₉	#2961	1	1	yes	characteristic, complemented, Sylow
65	#533	—	109	1	no	—
66	#446	—	109	8	no	—
67	#424	—	109	2	no	—
68	#423	—	109	1	no	—
69	#405	—	109	2	no	—
70	C₂×#97	D₁₀₉	1	1	yes	characteristic
71	C₂×M₁₆	—	109	2	no	—
72	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
74	C₄⋊Q₈	—	109	2	no	—
75	#125	—	109	2	no	—
76	C₂²⋊Q₈	—	109	4	no	—
77	C₄×Q₈	—	109	4	no	—
78	#117	—	109	2	no	—
79	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
80	C₄⋊C₈	—	109	8	no	—
81	C₄⋊C₈	—	109	4	no	—
82	#97	—	109	4	no	—
83	#97	C₂×D₁₀₉	1	4	yes	—
84	C₄×C₈	—	109	4	no	—
85	C₂×Q₈	—	109	4	no	—
86	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
88	M₁₆	—	218	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	109	4	no	—
91	C₂×C₈	Dic₂₁₈	1	4	yes	complemented
92	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
93	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
94	C₄⋊C₄	—	109	8	no	—
95	C₄⋊C₄	—	109	4	no	—
96	C₄⋊C₄	—	109	4	no	—
97	C₂²⋊C₄	—	109	4	no	—
98	C₄²	—	109	4	no	—
99	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
100	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
101	C₄²	C₂²×D₁₀₉	1	2	yes	—
102	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
103	Q₈	—	436	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	109	8	no	—
106	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
110	C₂×C₄	#13182029	1	2	yes	—
111	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	218	4	no	—
114	C₈	C₂×Dic₂₁₈	1	4	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₄×Dic₂₁₈	1	2	yes	—
117	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	218	8	no	—
121	C₄	C₄×Dic₂₁₈	1	2	yes	—
122	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
123	C₄	C₂×#13182029	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×C₄×Dic₂₁₈	1	1	yes	characteristic
126	C₂	#20295792	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295583

C₂

yes

maximal, characteristic

#20295154

C₂

yes

maximal, complemented

#20295052

C₂

yes

maximal

#20295050

C₂

yes

maximal, characteristic

#20294981

C₂

yes

maximal

C₂×#97×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

#16812699

C₄

yes

—

#16812691

C₄

yes

—

#16812678

C₄

yes

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812658

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

#16812628

C₂²

yes

—

#16812609

C₂²

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

#13182013

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

C₂×C₄

yes

—

C₂²⋊Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

Q₈

yes

complemented

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×Q₈

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₀₉

#568

yes

—

#2961

—

109

maximal, Sylow

C₁₀₉

#2961

yes

characteristic, complemented, Sylow

#533

—

109

—

#446

—

109

—

#424

—

109

—

#423

—

109

—

#405

—

109

—

C₂×#97

D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

—

109

—

#125

—

109

—

C₂²⋊Q₈

—

109

—

C₄×Q₈

—

109

—

#117

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

#97

—

109

—

#97

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₂×Q₈

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

Dic₂₁₈

yes

complemented

C₂×C₈

C₂²×D₁₀₉

yes

—

C₂×C₈

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

C₄×D₁₀₉

yes

characteristic

100

C₄²

C₄×D₁₀₉

yes

characteristic

101

C₄²

C₂²×D₁₀₉

yes

—

102

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

103

Q₈

—

436

—

104

C₂×C₄

—

105

C₂×C₄

—

109

—

106

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

107

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

108

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

109

C₂×C₄

C₂×Dic₂₁₈

yes

—

110

C₂×C₄

#13182029

yes

—

111

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

112

C₈

—

113

C₈

—

218

—

114

C₈

C₂×Dic₂₁₈

yes

—

115

C₂²

—

116

C₂²

C₄×Dic₂₁₈

yes

—

117

C₂²

C₂²×C₄×D₁₀₉

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

218

—

121

C₄

C₄×Dic₂₁₈

yes

—

122

C₄

C₂²×Dic₂₁₈

yes

characteristic

123

C₄

C₂×#13182029

yes

characteristic

124

C₂

—

125

C₂

C₂×C₄×Dic₂₁₈

yes

characteristic

126

C₂

#20295792

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₂×#97×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×#97	D₁₀₉
hypocenter	C₁₀₉	#2961
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

C₂×#97×C₁₀₉

C₂

prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂×C₄×C₁₀₉

C₂⁴

hypercenter

C₂×#97

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #20295154	C₂	semidirect
32. C₂×C₈×C₁₀₉	Q₈	semidirect
64. C₁₀₉	#2961	semidirect
91. C₂×C₈	Dic₂₁₈	semidirect

H ≅ G/N

type

2. #20295154

C₂

semidirect

32. C₂×C₈×C₁₀₉

Q₈

semidirect

64. C₁₀₉

#2961

semidirect

91. C₂×C₈

Dic₂₁₈

semidirect

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

subgroup	index
#20295583	2
#20295154	2
#20295052	2
#20295050	2
#20294981	2
C₂×#97×C₁₀₉	2
#2961	109

subgroup

index

C₂×#97×C₁₀₉

109

Subgroups — #28371972

Conjugacy classes of subgroups

Complements

Maximal subgroups