Group #28363069 — subgroups

Subgroups — #28363069

Order 13952.

292 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812634	C₂	1	2	yes	maximal, complemented
3	#20294989	C₂	1	2	yes	maximal
4	#20294791	C₂	1	1	yes	maximal, characteristic, complemented
5	#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×#13182013	C₂²	1	2	yes	—
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₁₀₉	—	2	4	no	—
10	C₂×C₈×D₁₀₉	C₄	1	4	yes	—
11	#16812634	—	2	4	no	—
12	#16812611	—	2	4	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂³×D₁₀₉	Q₈	1	1	yes	characteristic
16	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
17	C₂×C₄×D₁₀₉	—	2	2	no	—
18	C₂×C₄×D₁₀₉	—	2	2	no	—
19	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
21	#13182013	—	2	2	no	—
22	#13182013	—	2	2	no	—
23	C₂×#11724784	—	2	2	no	—
24	C₂×#11724784	C₂×C₄	1	2	yes	—
25	C₂×#11724784	D₄	1	4	yes	—
26	C₈×D₁₀₉	—	2	8	no	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₉	—	2	2	no	—
29	M₁₆×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	—	2	2	no	—
31	C₂×C₈×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
34	C₂²×D₁₀₉	—	2	2	no	—
35	C₂²×D₁₀₉	—	2	4	no	—
36	C₂²×D₁₀₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	#41	1	2	yes	—
41	C₂×Dic₁₀₉	—	2	2	no	—
42	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
43	C₄×D₁₀₉	—	2	2	no	—
44	C₄×D₁₀₉	—	2	2	no	—
45	C₄×D₁₀₉	—	2	4	no	—
46	C₈×C₁₀₉	—	2	2	no	—
47	C₈×C₁₀₉	—	4	2	no	—
48	#11724784	—	4	2	no	—
49	#11724784	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₁₀₉	—	2	2	no	—
51	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₁₀₉	—	2	8	no	—
54	C₂×D₁₀₉	—	2	2	no	—
55	C₂×D₁₀₉	—	2	2	no	—
56	C₂×D₁₀₉	#108	1	4	yes	complemented
57	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₁₀₉	#117	1	1	yes	characteristic
59	C₄×C₁₀₉	#121	1	2	yes	—
60	Dic₁₀₉	—	2	2	no	—
61	C₂×C₁₀₉	—	2	2	no	—
62	C₂×C₁₀₉	#380	1	1	yes	characteristic
63	C₂×C₁₀₉	C₂×#108	1	2	yes	—
64	D₁₀₉	—	2	4	no	—
65	#1810	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1810	1	1	yes	characteristic, complemented, Sylow
67	C₂³×C₈	—	109	1	no	—
68	C₂×#108	—	109	2	no	—
69	#407	—	109	2	no	—
70	#341	—	109	1	no	—
71	#341	D₁₀₉	1	1	yes	characteristic, complemented
72	C₂³×C₄	—	109	1	no	—
73	C₂×M₁₆	—	109	2	no	—
74	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
75	C₂²×C₈	—	109	1	no	—
76	C₂²×C₈	—	109	4	no	—
77	C₂²×C₈	—	218	4	no	—
78	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
79	#108	—	218	4	no	—
80	C₂²⋊C₈	—	218	4	no	—
81	C₂⁴	—	109	1	no	—
82	C₂²×C₄	—	109	1	no	—
83	C₂²×C₄	—	109	2	no	—
84	C₂²×C₄	—	109	2	no	—
85	C₂²×C₄	—	218	2	no	—
86	C₂²×C₄	—	218	2	no	—
87	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	218	2	no	—
91	M₁₆	—	218	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	109	2	no	—
95	C₂×C₈	—	109	4	no	—
96	C₂×C₈	—	218	8	no	—
97	C₂×C₈	—	218	2	no	—
98	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
99	C₂³	—	109	2	no	—
100	C₂³	—	218	4	no	—
101	C₂³	—	218	2	no	—
102	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
103	C₂×C₄	—	109	2	no	—
104	C₂×C₄	—	218	2	no	—
105	C₂×C₄	—	218	2	no	—
106	C₂×C₄	—	218	2	no	—
107	C₂×C₄	—	218	4	no	—
108	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
109	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
110	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
111	C₂×C₄	#13182029	1	2	yes	—
112	C₈	—	2	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	109	4	no	—
115	C₈	—	436	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	109	4	no	—
118	C₂²	—	218	8	no	—
119	C₂²	—	218	2	no	—
120	C₂²	—	218	2	no	—
121	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
122	C₂²	#16812677	1	2	yes	—
123	C₄	—	218	2	no	—
124	C₄	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
125	C₄	#16812656	1	1	yes	characteristic
126	C₄	#16812672	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	218	4	no	—
129	C₂	#20294885	1	1	yes	characteristic
130	C₂	#108×D₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#16812634

C₂

yes

maximal, complemented

#20294989

C₂

yes

maximal

#20294791

C₂

yes

maximal, characteristic, complemented

#341×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

—

C₂×C₈×D₁₀₉

C₄

yes

—

#16812634

—

#16812611

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂³×D₁₀₉

Q₈

yes

characteristic

C₂²×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#11724784

D₄

yes

—

C₈×D₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

C₂²⋊C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#108

yes

complemented

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#380

yes

characteristic

C₂×C₁₀₉

C₂×#108

yes

—

D₁₀₉

—

#1810

—

109

maximal, Sylow

C₁₀₉

#1810

yes

characteristic, complemented, Sylow

C₂³×C₈

—

109

—

C₂×#108

—

109

—

#407

—

109

—

#341

—

109

—

#341

D₁₀₉

yes

characteristic, complemented

C₂³×C₄

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#108

—

218

—

C₂²⋊C₈

—

218

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂³

—

109

—

100

C₂³

—

218

—

101

C₂³

—

218

—

102

C₂³

Q₈×D₁₀₉

yes

characteristic

103

C₂×C₄

—

109

—

104

C₂×C₄

—

218

—

105

C₂×C₄

—

218

—

106

C₂×C₄

—

218

—

107

C₂×C₄

—

218

—

108

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

109

C₂×C₄

D₄×D₁₀₉

yes

—

110

C₂×C₄

D₄×D₁₀₉

yes

characteristic

111

C₂×C₄

#13182029

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

109

—

115

C₈

—

436

—

116

C₂²

—

117

C₂²

—

109

—

118

C₂²

—

218

—

119

C₂²

—

218

—

120

C₂²

—

218

—

121

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

122

C₂²

#16812677

yes

—

123

C₄

—

218

—

124

C₄

C₂²⋊C₄×D₁₀₉

yes

characteristic

125

C₄

#16812656

yes

characteristic

126

C₄

#16812672

yes

—

127

C₂

—

128

C₂

—

218

—

129

C₂

#20294885

yes

characteristic

130

C₂

#108×D₁₀₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#341×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×D₁₀₉	Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#341	D₁₀₉
hypocenter	C₁₀₉	#1810
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#341×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×D₁₀₉

Q₈

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#16812634	C₂	semidirect
4. #20294791	C₂	semidirect
5. #341×C₁₀₉	C₂	semidirect
56. C₂×D₁₀₉	#108	semidirect
66. C₁₀₉	#1810	semidirect
71. #341	D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×#16812634

C₂

semidirect

4. #20294791

C₂

semidirect

5. #341×C₁₀₉

C₂

semidirect

56. C₂×D₁₀₉

#108

semidirect

66. C₁₀₉

#1810

semidirect

71. #341

D₁₀₉

semidirect

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

subgroup	index
C₂²×C₈×D₁₀₉	2
C₂×#16812634	2
#20294989	2
#20294791	2
#341×C₁₀₉	2
#1810	109

subgroup

index

C₂²×C₈×D₁₀₉

#341×C₁₀₉

109

Subgroups — #28363069

Conjugacy classes of subgroups

Complements

Maximal subgroups