Group #28379259 — subgroups

Subgroups — #28379259

Order 13952.

260 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₂²⋊#13182032	C₂	1	4	yes	maximal, complemented
3	C₂³×#11724784	C₂	1	1	yes	maximal, characteristic
4	#16812773	—	2	4	no	—
5	C₂×#13182032	C₂²	1	4	yes	complemented
6	C₂²×#11724784	—	2	4	no	—
7	C₂²×#11724784	C₄	1	2	yes	—
8	C₂²×#11724784	C₄	1	1	yes	characteristic
9	C₂²×#11724784	C₂²	1	2	yes	—
10	C₂²×#11724784	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₀₉	C₄	1	1	yes	characteristic
12	#13182032	—	2	4	no	—
13	C₂×#11724784	—	2	2	no	—
14	C₂×#11724784	—	2	4	no	—
15	C₂×#11724784	—	2	4	no	—
16	C₂×#11724784	C₂×C₄	1	2	yes	—
17	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	D₄	1	4	yes	—
19	C₂×#11724784	C₂³	1	1	yes	characteristic
20	C₂⁴×C₁₀₉	C₈	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	—	2	2	no	—
22	C₂²×C₄×C₁₀₉	—	2	2	no	—
23	C₂²×C₄×C₁₀₉	C₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₉	C₈	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₁₀₉	—	2	2	no	—
28	C₂³×C₁₀₉	—	2	4	no	—
29	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂³×C₁₀₉	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄×C₁₀₉	—	2	4	no	—
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	#11724784	—	2	2	no	—
41	#11724784	C₂²⋊C₄	1	2	yes	—
42	#11724784	C₂×D₄	1	2	yes	—
43	C₂²×C₁₀₉	—	2	8	no	—
44	C₂²×C₁₀₉	—	2	2	no	—
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	—	2	2	no	—
47	C₂²×C₁₀₉	C₂²⋊C₈	1	2	yes	—
48	C₂²×C₁₀₉	M₃₂	1	4	yes	complemented
49	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₁₀₉	—	2	2	no	—
51	C₄×C₁₀₉	C₂²⋊C₈	1	2	yes	—
52	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₁₀₉	—	2	2	no	—
55	C₂×C₁₀₉	—	2	4	no	—
56	C₂×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂×M₃₂	1	2	yes	—
58	#2112	—	109	1	no	maximal, Sylow
59	C₁₀₉	#2112	1	1	yes	characteristic, complemented, Sylow
60	C₂³×C₈	—	109	1	no	—
61	C₂×M₃₂	—	109	2	no	—
62	C₂²⋊C₁₆	—	109	4	no	—
63	C₂³×C₄	#89137	1	1	yes	characteristic
64	C₂²×C₈	—	109	1	no	—
65	C₂²×C₈	—	109	2	no	—
66	C₂²×C₈	—	109	2	no	—
67	C₂²×C₈	—	109	1	no	—
68	C₂²×C₈	—	218	4	no	—
69	M₃₂	—	218	4	no	—
70	C₂×C₁₆	—	109	4	no	—
71	C₂⁴	#11724786	1	1	yes	characteristic
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	#11724786	1	1	yes	characteristic
75	C₂²×C₄	#11724786	1	2	yes	—
76	C₂²×C₄	C₂×#89137	1	2	yes	—
77	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
78	C₂×C₈	—	109	4	no	—
79	C₂×C₈	—	109	1	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	109	2	no	—
82	C₂×C₈	—	218	4	no	—
83	C₂×C₈	—	218	4	no	—
84	C₂×C₈	—	218	2	no	—
85	C₁₆	—	218	4	no	—
86	C₂³	—	2	2	no	—
87	C₂³	—	2	4	no	—
88	C₂³	C₂×#11724786	1	1	yes	characteristic
89	C₂³	C₂×#11724786	1	2	yes	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	C₂²⋊#89137	1	2	yes	—
95	C₂×C₄	C₂×#11724786	1	2	yes	—
96	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
97	C₂×C₄	#13182038	1	2	yes	—
98	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
99	C₈	—	109	2	no	—
100	C₈	—	109	2	no	—
101	C₈	—	218	2	no	—
102	C₂²	—	2	8	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	C₂²⋊#11724786	1	2	yes	—
107	C₂²	#16812773	1	4	yes	complemented
108	C₂²	C₂²×#11724786	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	C₂²⋊#11724786	1	2	yes	—
111	C₄	C₂×C₂²⋊#89137	1	1	yes	characteristic
112	C₄	C₂×#13182038	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	—	2	4	no	—
115	C₂	C₂×C₂²⋊#11724786	1	1	yes	characteristic
116	C₂	C₂×#16812773	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812773

C₂

yes

maximal, complemented

C₂²⋊#13182032

C₂

yes

maximal, complemented

C₂³×#11724784

C₂

yes

maximal, characteristic

#16812773

—

C₂×#13182032

C₂²

yes

complemented

C₂²×#11724784

—

C₂²×#11724784

C₄

yes

—

C₂²×#11724784

C₄

yes

characteristic

C₂²×#11724784

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂³×C₄×C₁₀₉

C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

D₄

yes

—

C₂×#11724784

C₂³

yes

characteristic

C₂⁴×C₁₀₉

C₈

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂³×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×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

—

#11724784

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊C₈

yes

—

C₂²×C₁₀₉

M₃₂

yes

complemented

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×C₂²⋊C₈

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

—

#2112

—

109

maximal, Sylow

C₁₀₉

#2112

yes

characteristic, complemented, Sylow

C₂³×C₈

—

109

—

C₂×M₃₂

—

109

—

C₂²⋊C₁₆

—

109

—

C₂³×C₄

#89137

yes

characteristic

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

218

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂⁴

#11724786

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#11724786

yes

characteristic

C₂²×C₄

#11724786

yes

—

C₂²×C₄

C₂×#89137

yes

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₁₆

—

218

—

C₂³

—

C₂³

—

C₂³

C₂×#11724786

yes

characteristic

C₂³

C₂×#11724786

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊#89137

yes

—

C₂×C₄

C₂×#11724786

yes

—

C₂×C₄

C₂×#11724786

yes

characteristic

C₂×C₄

#13182038

yes

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

100

C₈

—

109

—

101

C₈

—

218

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²⋊#11724786

yes

—

107

C₂²

#16812773

yes

complemented

108

C₂²

C₂²×#11724786

yes

characteristic

109

C₄

—

110

C₄

C₂²⋊#11724786

yes

—

111

C₄

C₂×C₂²⋊#89137

yes

characteristic

112

C₄

C₂×#13182038

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×C₂²⋊#11724786

yes

characteristic

116

C₂

C₂×#16812773

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₂³×C₄×C₁₀₉	C₄
prime radical	C₂⁴×C₁₀₉	C₈
cosocle	C₂×#11724784	C₂³
hypercenter	C₂³×C₄	#89137
hypocenter	C₁₀₉	#2112
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₂³×C₄×C₁₀₉

C₄

prime radical

C₂⁴×C₁₀₉

C₈

C₂³

C₂³×C₄

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₂³×C₄

N	H ≅ G/N	type
1. C₂×#16812773	C₂	semidirect
2. C₂²⋊#13182032	C₂	semidirect
5. C₂×#13182032	C₂²	semidirect
48. C₂²×C₁₀₉	M₃₂	semidirect
59. C₁₀₉	#2112	semidirect
107. C₂²	#16812773	semidirect

H ≅ G/N

type

1. C₂×#16812773

C₂

semidirect

2. C₂²⋊#13182032

C₂

semidirect

5. C₂×#13182032

C₂²

semidirect

48. C₂²×C₁₀₉

M₃₂

semidirect

59. C₁₀₉

#2112

semidirect

107. C₂²

#16812773

semidirect

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

subgroup	index
C₂×#16812773	2
C₂²⋊#13182032	2
C₂³×#11724784	2
#2112	109

subgroup

index

109

Subgroups — #28379259

Conjugacy classes of subgroups

Complements

Maximal subgroups