Group #28363871 — subgroups

Subgroups — #28363871

Order 13952.

396 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16812696	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812615	C₂	1	4	yes	maximal, complemented
3	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
4	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₃₆	C₄	1	4	yes	—
6	C₂²×D₄₃₆	C₂²	1	2	yes	—
7	C₂×#13182013	C₂²	1	2	yes	—
8	#16812696	—	2	1	no	—
9	#16812696	—	2	1	no	—
10	#16812696	—	2	2	no	—
11	#16812615	—	2	8	no	—
12	#16812610	—	2	1	no	—
13	C₄×#11724784	—	2	1	no	—
14	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₉	—	2	2	no	—
17	C₂³×D₁₀₉	C₂×C₄	1	4	yes	—
18	C₂×D₄₃₆	—	2	4	no	—
19	C₂×D₄₃₆	—	2	4	no	—
20	C₂×D₄₃₆	—	2	4	no	—
21	C₂×D₄₃₆	—	2	2	no	—
22	C₂×D₄₃₆	—	2	2	no	—
23	C₂×D₄₃₆	—	2	4	no	—
24	C₂×D₄₃₆	—	4	8	no	—
25	#13182013	—	2	2	no	—
26	#13182013	D₄	1	4	yes	complemented
27	C₂×#11724784	—	2	2	no	—
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₀₉	—	2	4	no	—
31	C₂×C₈×C₁₀₉	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₁₀₉	—	2	4	no	—
37	C₂²×D₁₀₉	—	2	8	no	—
38	C₂²×D₁₀₉	—	4	8	no	—
39	C₂×C₄×C₁₀₉	—	2	4	no	—
40	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	#41	1	2	yes	—
46	D₄₃₆	—	4	8	no	—
47	D₄₃₆	—	4	8	no	—
48	D₄₃₆	—	4	4	no	—
49	D₄₃₆	—	4	2	no	—
50	D₄₃₆	—	4	2	no	—
51	C₈×C₁₀₉	—	4	2	no	—
52	#11724784	—	2	4	no	—
53	C₂²×C₁₀₉	—	2	2	no	—
54	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
56	C₂×D₁₀₉	—	4	4	no	—
57	C₂×D₁₀₉	—	4	8	no	—
58	C₂×D₁₀₉	—	8	8	no	—
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	#390	1	1	yes	characteristic
66	C₂×C₁₀₉	C₂×#100	1	2	yes	—
67	D₁₀₉	—	8	4	no	—
68	#1965	—	109	1	no	maximal, Sylow
69	C₁₀₉	#1965	1	1	yes	characteristic, complemented, Sylow
70	C₂×#127	—	109	1	no	—
71	#423	D₁₀₉	1	1	yes	characteristic, complemented
72	C₂×#100	—	109	4	no	—
73	C₄×M₁₆	—	109	1	no	—
74	C₂²×D₄	—	109	4	no	—
75	C₂²×D₄	—	109	2	no	—
76	C₂×M₁₆	—	109	2	no	—
77	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
78	#127	—	218	1	no	—
79	#127	—	218	1	no	—
80	#127	—	218	2	no	—
81	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	#100	—	218	8	no	—
84	#97	—	218	1	no	—
85	C₄×C₈	—	218	1	no	—
86	C₂⁴	—	109	4	no	—
87	C₂×D₄	—	436	8	no	—
88	C₂×D₄	—	218	4	no	—
89	C₂×D₄	—	218	4	no	—
90	C₂×D₄	—	218	4	no	—
91	C₂×D₄	—	218	2	no	—
92	C₂×D₄	—	218	2	no	—
93	C₂×D₄	—	218	4	no	—
94	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
96	M₁₆	—	2	4	no	—
97	M₁₆	—	109	4	no	—
98	M₁₆	—	218	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	218	2	no	—
101	C₄²	D₄₃₆	1	1	yes	characteristic
102	C₄²	D₄₃₆	1	1	yes	characteristic
103	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
104	C₂³	—	436	8	no	—
105	C₂³	—	218	8	no	—
106	C₂³	—	218	4	no	—
107	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
108	D₄	—	436	8	no	—
109	D₄	—	436	8	no	—
110	D₄	—	436	4	no	—
111	D₄	—	436	2	no	—
112	D₄	—	436	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	#13182006	1	2	yes	—
115	C₂×C₄	#13182006	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×D₄₃₆	1	1	yes	characteristic
119	C₂×C₄	#13182030	1	2	yes	—
120	C₈	—	4	2	no	—
121	C₈	—	218	4	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	872	8	no	—
124	C₂²	—	436	4	no	—
125	C₂²	—	436	8	no	—
126	C₂²	C₂×#13182006	1	1	yes	characteristic
127	C₂²	#16812661	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	#16812615	1	4	yes	complemented
130	C₄	#16812674	1	2	yes	—
131	C₄	#16812688	1	1	yes	characteristic
132	C₄	#16812695	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	872	4	no	—
135	C₂	#20294927	1	1	yes	characteristic
136	C₂	C₂×#16812615	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812696

C₂

yes

maximal, characteristic

C₂×#16812615

C₂

yes

maximal, complemented

C₄×#13182013

C₂

yes

maximal, characteristic, complemented

#423×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

C₄

yes

—

C₂²×D₄₃₆

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

—

—

—

—

—

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂³×D₁₀₉

C₂×C₄

yes

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

#13182013

—

#13182013

D₄

yes

complemented

C₂×#11724784

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

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

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#100

yes

complemented

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

C₂×#100

yes

—

D₁₀₉

—

#1965

—

109

maximal, Sylow

C₁₀₉

#1965

yes

characteristic, complemented, Sylow

C₂×#127

—

109

—

#423

D₁₀₉

yes

characteristic, complemented

C₂×#100

—

109

—

C₄×M₁₆

—

109

—

C₂²×D₄

—

109

—

C₂²×D₄

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

#127

—

218

—

#127

—

218

—

#127

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

#100

—

218

—

#97

—

218

—

C₄×C₈

—

218

—

C₂⁴

—

109

—

C₂×D₄

—

436

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

100

C₂×C₈

—

218

—

101

C₄²

D₄₃₆

yes

characteristic

102

C₄²

D₄₃₆

yes

characteristic

103

C₄²

C₂²⋊D₁₀₉

yes

—

104

C₂³

—

436

—

105

C₂³

—

218

—

106

C₂³

—

218

—

107

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

108

D₄

—

436

—

109

D₄

—

436

—

110

D₄

—

436

—

111

D₄

—

436

—

112

D₄

—

436

—

113

C₂×C₄

—

114

C₂×C₄

#13182006

yes

—

115

C₂×C₄

#13182006

yes

—

116

C₂×C₄

D₄×D₁₀₉

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

118

C₂×C₄

C₂×D₄₃₆

yes

characteristic

119

C₂×C₄

#13182030

yes

—

120

C₈

—

121

C₈

—

218

—

122

C₂²

—

123

C₂²

—

872

—

124

C₂²

—

436

—

125

C₂²

—

436

—

126

C₂²

C₂×#13182006

yes

characteristic

127

C₂²

#16812661

yes

—

128

C₄

—

129

C₄

#16812615

yes

complemented

130

C₄

#16812674

yes

—

131

C₄

#16812688

yes

characteristic

132

C₄

#16812695

yes

characteristic

133

C₂

—

134

C₂

—

872

—

135

C₂

#20294927

yes

characteristic

136

C₂

C₂×#16812615

yes

—

	subgroup	G/H
center	C₂²	C₂×#13182006
commutator	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#423×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂×#16812696	C₂
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#423	D₁₀₉
hypocenter	C₁₀₉	#1965
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#13182006

commutator

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#423×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂×#16812696

C₂

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#16812615	C₂	semidirect
3. C₄×#13182013	C₂	semidirect
4. #423×C₁₀₉	C₂	semidirect
26. #13182013	D₄	semidirect
60. C₄×C₁₀₉	#100	semidirect
69. C₁₀₉	#1965	semidirect
71. #423	D₁₀₉	semidirect
129. C₄	#16812615	semidirect

H ≅ G/N

type

2. C₂×#16812615

C₂

semidirect

3. C₄×#13182013

C₂

semidirect

4. #423×C₁₀₉

C₂

semidirect

26. #13182013

D₄

semidirect

60. C₄×C₁₀₉

#100

semidirect

69. C₁₀₉

#1965

semidirect

71. #423

D₁₀₉

semidirect

129. C₄

#16812615

semidirect

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

subgroup	index
C₂×#16812696	2
C₂×#16812615	2
C₄×#13182013	2
#423×C₁₀₉	2
#1965	109

subgroup

index

#423×C₁₀₉

109

Subgroups — #28363871

Conjugacy classes of subgroups

Complements

Maximal subgroups