Group #28363873 — subgroups

Subgroups — #28363873

Order 13952.

396 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 105 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	#20295050	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812616	C₂	1	4	yes	maximal, complemented
3	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₂²	1	2	yes	—
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₄⋊#11724784	—	2	2	no	—
8	#16812616	—	2	8	no	—
9	#16812610	—	2	1	no	—
10	C₄×#11724784	—	2	1	no	—
11	C₂²×D₄×C₁₀₉	C₄	1	4	yes	—
12	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
13	#127×C₁₀₉	—	2	1	no	—
14	#127×C₁₀₉	—	2	1	no	—
15	#127×C₁₀₉	—	2	2	no	—
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	#13182013	—	2	2	no	—
18	#13182013	—	2	4	no	—
19	#13182013	D₄	1	4	yes	complemented
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂⁴×C₁₀₉	C₂×C₄	1	4	yes	—
23	C₂×D₄×C₁₀₉	—	2	4	no	—
24	C₂×D₄×C₁₀₉	—	2	4	no	—
25	C₂×D₄×C₁₀₉	—	2	2	no	—
26	C₂×D₄×C₁₀₉	—	2	2	no	—
27	C₂×D₄×C₁₀₉	—	2	4	no	—
28	C₂×D₄×C₁₀₉	—	2	4	no	—
29	C₂×D₄×C₁₀₉	—	4	8	no	—
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
33	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
34	C₄²×C₁₀₉	D₄	1	2	yes	—
35	C₂³×C₁₀₉	—	2	4	no	—
36	C₂³×C₁₀₉	—	2	8	no	—
37	C₂³×C₁₀₉	—	4	8	no	—
38	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	D₄×C₁₀₉	—	4	8	no	—
40	D₄×C₁₀₉	—	4	8	no	—
41	D₄×C₁₀₉	—	4	4	no	—
42	D₄×C₁₀₉	—	4	2	no	—
43	D₄×C₁₀₉	—	4	2	no	—
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	#41	1	2	yes	—
51	#11724784	—	2	4	no	—
52	#11724784	—	4	2	no	—
53	C₂²×C₁₀₉	—	2	2	no	—
54	C₂²×C₁₀₉	—	4	4	no	—
55	C₂²×C₁₀₉	—	4	8	no	—
56	C₂²×C₁₀₉	—	8	8	no	—
57	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
59	C₄×C₁₀₉	—	2	2	no	—
60	C₄×C₁₀₉	#100	1	4	yes	complemented
61	C₄×C₁₀₉	#121	1	2	yes	—
62	C₄×C₁₀₉	#124	1	1	yes	characteristic
63	C₄×C₁₀₉	#127	1	1	yes	characteristic
64	C₂×C₁₀₉	—	2	2	no	—
65	C₂×C₁₀₉	—	8	4	no	—
66	C₂×C₁₀₉	#390	1	1	yes	characteristic
67	C₂×C₁₀₉	C₂×#100	1	2	yes	—
68	#1965	—	109	1	no	maximal, Sylow
69	C₁₀₉	#1965	1	1	yes	characteristic, complemented, Sylow
70	C₂×#127	D₁₀₉	1	1	yes	characteristic
71	#423	—	109	1	no	—
72	C₂×#100	—	109	4	no	—
73	C₄×M₁₆	—	109	1	no	—
74	C₂²×D₄	Dic₁₀₉	1	4	yes	—
75	C₂²×D₄	C₂×D₁₀₉	1	2	yes	—
76	C₂×M₁₆	—	109	2	no	—
77	C₂×M₁₆	—	109	2	no	—
78	#127	—	2	1	no	—
79	#127	—	2	1	no	—
80	#127	—	2	2	no	—
81	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
82	C₄⋊C₈	—	218	2	no	—
83	#100	—	218	8	no	—
84	#97	—	218	1	no	—
85	C₄×C₈	—	218	1	no	—
86	C₂⁴	C₂×Dic₁₀₉	1	4	yes	—
87	C₂×D₄	—	2	4	no	—
88	C₂×D₄	—	2	4	no	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂×D₄	—	4	8	no	—
94	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
96	M₁₆	—	109	4	no	—
97	M₁₆	—	218	4	no	—
98	M₁₆	—	218	2	no	—
99	C₂×C₈	—	218	2	no	—
100	C₂×C₈	—	218	2	no	—
101	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
102	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
103	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
104	C₂³	—	2	4	no	—
105	C₂³	—	2	8	no	—
106	C₂³	—	4	8	no	—
107	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
108	D₄	—	4	8	no	—
109	D₄	—	4	8	no	—
110	D₄	—	4	4	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
116	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
119	C₂×C₄	#13182028	1	2	yes	—
120	C₈	—	436	2	no	—
121	C₈	—	218	4	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	4	8	no	—
125	C₂²	—	8	8	no	—
126	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
127	C₂²	D₄×Dic₁₀₉	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	#16812616	1	4	yes	complemented
130	C₄	#16812670	1	2	yes	—
131	C₄	#16812686	1	1	yes	characteristic
132	C₄	#16812695	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	8	4	no	—
135	C₂	#20294929	1	1	yes	characteristic
136	C₂	C₂×#16812616	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295050

C₂

yes

maximal, characteristic, complemented

C₂×#16812616

C₂

yes

maximal, complemented

C₄×#13182013

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

—

—

—

—

C₂²×D₄×C₁₀₉

C₄

yes

—

C₂²×D₄×C₁₀₉

C₂²

yes

—

#127×C₁₀₉

—

#127×C₁₀₉

—

#127×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

—

D₄

yes

complemented

C₂×#11724784

—

C₂×#11724784

—

C₂⁴×C₁₀₉

C₂×C₄

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#100

yes

complemented

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

C₂×#100

yes

—

#1965

—

109

maximal, Sylow

C₁₀₉

#1965

yes

characteristic, complemented, Sylow

C₂×#127

D₁₀₉

yes

characteristic

#423

—

109

—

C₂×#100

—

109

—

C₄×M₁₆

—

109

—

C₂²×D₄

Dic₁₀₉

yes

—

C₂²×D₄

C₂×D₁₀₉

yes

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

#127

—

#127

—

#127

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

218

—

#100

—

218

—

#97

—

218

—

C₄×C₈

—

218

—

C₂⁴

C₂×Dic₁₀₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

218

—

100

C₂×C₈

—

218

—

101

C₄²

C₂²⋊D₁₀₉

yes

characteristic

102

C₄²

C₂²⋊D₁₀₉

yes

characteristic

103

C₄²

C₂²⋊D₁₀₉

yes

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₂²×Dic₁₀₉

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

115

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

116

C₂×C₄

D₄×D₁₀₉

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

119

C₂×C₄

#13182028

yes

—

120

C₈

—

436

—

121

C₈

—

218

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

127

C₂²

D₄×Dic₁₀₉

yes

—

128

C₄

—

129

C₄

#16812616

yes

complemented

130

C₄

#16812670

yes

—

131

C₄

#16812686

yes

characteristic

132

C₄

#16812695

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

#20294929

yes

characteristic

136

C₂

C₂×#16812616

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₉
commutator	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting, prime radical	C₂×#127×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#127	D₁₀₉
hypocenter	C₁₀₉	#1965
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₉

commutator

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting, prime radical

C₂×#127×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #20295050	C₂	semidirect
2. C₂×#16812616	C₂	semidirect
3. C₄×#13182013	C₂	semidirect
19. #13182013	D₄	semidirect
60. C₄×C₁₀₉	#100	semidirect
69. C₁₀₉	#1965	semidirect
129. C₄	#16812616	semidirect

H ≅ G/N

type

1. #20295050

C₂

semidirect

2. C₂×#16812616

C₂

semidirect

3. C₄×#13182013

C₂

semidirect

19. #13182013

D₄

semidirect

60. C₄×C₁₀₉

#100

semidirect

69. C₁₀₉

#1965

semidirect

129. C₄

#16812616

semidirect

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

subgroup	index
#20295050	2
C₂×#16812616	2
C₄×#13182013	2
C₂×#127×C₁₀₉	2
#1965	109

subgroup

index

C₂×#127×C₁₀₉

109

Subgroups — #28363873

Conjugacy classes of subgroups

Complements

Maximal subgroups