Group #28379268 — subgroups

Subgroups — #28379268

Order 13952.

220 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 86 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×#16812773	C₂	1	2	yes	maximal, complemented
2	#20295922	C₂	1	4	yes	maximal, complemented
3	C₂×#16812706	C₂	1	1	yes	maximal, characteristic
4	#16812773	—	2	2	no	—
5	#16812773	C₂²	1	4	yes	—
6	C₂×#13182032	—	2	2	no	—
7	#16812706	—	2	2	no	—
8	#16812706	—	2	2	no	—
9	C₂×#13182013	C₄	1	1	yes	characteristic
10	C₂×#13182013	C₂²	1	2	yes	—
11	C₂²×#11724784	C₄	1	2	yes	—
12	C₂²×#11724784	C₂²	1	1	yes	characteristic
13	C₂×#41×C₁₀₉	C₄	1	1	yes	characteristic
14	#13182032	—	2	4	no	—
15	#13182013	—	2	2	no	—
16	#13182013	—	2	4	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₂×#11724784	—	2	2	no	—
19	C₂×#11724784	C₂×C₄	1	2	yes	—
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₂×#11724784	D₄	1	2	yes	—
22	C₂×#11724784	D₄	1	2	yes	—
23	C₂×#11724784	C₂³	1	1	yes	characteristic
24	#41×C₁₀₉	—	2	2	no	—
25	#41×C₁₀₉	—	2	2	no	—
26	C₂×Q₈×C₁₀₉	C₈	1	1	yes	characteristic
27	C₂×D₄×C₁₀₉	C₈	1	2	yes	—
28	C₂×D₄×C₁₀₉	C₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₁₀₉	C₂×C₈	1	2	yes	—
32	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
33	Q₈×C₁₀₉	—	2	2	no	—
34	D₄×C₁₀₉	—	2	4	no	—
35	D₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	#11724784	—	2	2	no	—
44	#11724784	C₂²⋊C₄	1	1	yes	characteristic
45	#11724784	C₂²⋊C₄	1	1	yes	characteristic
46	#11724784	C₂×D₄	1	2	yes	—
47	C₂²×C₁₀₉	—	2	2	no	—
48	C₂²×C₁₀₉	—	4	1	no	—
49	C₂²×C₁₀₉	—	4	2	no	—
50	C₂²×C₁₀₉	C₂²⋊C₈	1	2	yes	—
51	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₁₀₉	—	2	2	no	—
53	C₄×C₁₀₉	C₂²⋊C₈	1	2	yes	—
54	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₁₀₉	—	2	1	no	—
57	C₂×C₁₀₉	—	2	2	no	—
58	C₂×C₁₀₉	—	4	2	no	—
59	C₂×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
60	#2115	—	109	1	no	maximal, Sylow
61	C₁₀₉	#2115	1	1	yes	characteristic, complemented, Sylow
62	C₂×#131	—	109	1	no	—
63	C₂×M₃₂	—	109	2	no	—
64	#349	—	109	4	no	—
65	C₂×#41	#89137	1	1	yes	characteristic
66	#131	—	218	2	no	—
67	#131	—	218	2	no	—
68	C₂×M₁₆	—	109	2	no	—
69	C₂×M₁₆	—	109	1	no	—
70	C₂²×C₈	—	109	1	no	—
71	C₂²×C₈	—	109	2	no	—
72	M₃₂	—	109	4	no	—
73	M₃₂	—	218	2	no	—
74	C₂×C₁₆	—	218	2	no	—
75	#41	—	2	2	no	—
76	#41	—	2	2	no	—
77	C₂×Q₈	#11724786	1	1	yes	characteristic
78	C₂×D₄	#11724786	1	2	yes	—
79	C₂×D₄	#11724786	1	1	yes	characteristic
80	C₂²×C₄	C₂×#89137	1	2	yes	—
81	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
82	M₁₆	—	218	4	no	—
83	M₁₆	—	218	2	no	—
84	C₂×C₈	—	109	2	no	—
85	C₂×C₈	—	109	2	no	—
86	C₂×C₈	—	109	1	no	—
87	C₂×C₈	—	109	1	no	—
88	C₂×C₈	—	109	2	no	—
89	C₂×C₈	—	218	2	no	—
90	C₂×C₈	—	218	2	no	—
91	C₁₆	—	218	4	no	—
92	C₂³	C₂×#11724786	1	2	yes	—
93	C₂³	C₂×#11724786	1	1	yes	characteristic
94	Q₈	—	2	2	no	—
95	D₄	—	2	4	no	—
96	D₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₂²⋊#89137	1	2	yes	—
100	C₂×C₄	C₂×#11724786	1	2	yes	—
101	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
102	C₂×C₄	#13182038	1	2	yes	—
103	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
104	C₈	—	109	2	no	—
105	C₈	—	109	1	no	—
106	C₈	—	109	1	no	—
107	C₈	—	218	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	4	1	no	—
110	C₂²	—	4	2	no	—
111	C₂²	C₂²⋊#11724786	1	2	yes	—
112	C₂²	C₂²×#11724786	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	C₂²⋊#11724786	1	2	yes	—
115	C₄	C₂×C₂²⋊#89137	1	1	yes	characteristic
116	C₄	C₂×#13182038	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	2	no	—
119	C₂	—	4	2	no	—
120	C₂	C₂×C₂²⋊#11724786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812773

C₂

yes

maximal, complemented

#20295922

C₂

yes

maximal, complemented

C₂×#16812706

C₂

yes

maximal, characteristic

#16812773

—

#16812773

C₂²

yes

—

—

—

—

C₄

yes

characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₄

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂×#41×C₁₀₉

C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

D₄

yes

—

C₂×#11724784

D₄

yes

—

C₂×#11724784

C₂³

yes

characteristic

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₈

yes

characteristic

C₂×D₄×C₁₀₉

C₈

yes

—

C₂×D₄×C₁₀₉

C₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₈

yes

—

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

#11724784

—

#11724784

C₂²⋊C₄

yes

characteristic

#11724784

C₂²⋊C₄

yes

characteristic

#11724784

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊C₈

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊C₈

yes

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₈

yes

characteristic

#2115

—

109

maximal, Sylow

C₁₀₉

#2115

yes

characteristic, complemented, Sylow

C₂×#131

—

109

—

C₂×M₃₂

—

109

—

#349

—

109

—

C₂×#41

#89137

yes

characteristic

#131

—

218

—

#131

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

M₃₂

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

218

—

#41

—

#41

—

C₂×Q₈

#11724786

yes

characteristic

C₂×D₄

#11724786

yes

—

C₂×D₄

#11724786

yes

characteristic

C₂²×C₄

C₂×#89137

yes

—

C₂²×C₄

C₂×#89137

yes

characteristic

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₁₆

—

218

—

C₂³

C₂×#11724786

yes

—

C₂³

C₂×#11724786

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊#89137

yes

—

100

C₂×C₄

C₂×#11724786

yes

—

101

C₂×C₄

C₂×#11724786

yes

characteristic

102

C₂×C₄

#13182038

yes

—

103

C₂×C₄

C₂²×#89137

yes

characteristic

104

C₈

—

109

—

105

C₈

—

109

—

106

C₈

—

109

—

107

C₈

—

218

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²⋊#11724786

yes

—

112

C₂²

C₂²×#11724786

yes

characteristic

113

C₄

—

114

C₄

C₂²⋊#11724786

yes

—

115

C₄

C₂×C₂²⋊#89137

yes

characteristic

116

C₄

C₂×#13182038

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

C₂×C₂²⋊#11724786

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#89137
commutator	C₂²×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting, prime radical	C₂×#41×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₂²⋊C₈
cosocle	C₂×#11724784	C₂³
hypercenter	C₂×#41	#89137
hypocenter	C₁₀₉	#2115
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₄

C₂×C₂²⋊#89137

commutator

C₂²×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting, prime radical

C₂×#41×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₂×C₂²⋊C₈

C₂³

C₂×#41

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#16812773	C₂	semidirect
2. #20295922	C₂	semidirect
61. C₁₀₉	#2115	semidirect

H ≅ G/N

type

1. C₂×#16812773

C₂

semidirect

2. #20295922

C₂

semidirect

61. C₁₀₉

#2115

semidirect

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

subgroup	index
C₂×#16812773	2
#20295922	2
C₂×#16812706	2
#2115	109

subgroup

index

109

Subgroups — #28379268

Conjugacy classes of subgroups

Complements

Maximal subgroups