Group #28372434 — subgroups

Subgroups — #28372434

Order 13952.

772 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 161 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	#20295590	C₂	1	3	yes	maximal, complemented
2	#20295213	C₂	1	8	yes	maximal, complemented
3	#20295000	C₂	1	3	yes	maximal, complemented
4	#585×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic, complemented
6	#16812695	—	2	3	no	—
7	#16812688	—	2	3	no	—
8	#16812673	—	2	6	no	—
9	#16812667	C₂²	1	12	yes	complemented
10	#16812658	C₂²	1	3	yes	complemented
11	#16812614	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₁₀₉	—	2	3	no	—
13	H(5,𝔽₂)×C₁₀₉	—	2	3	no	—
14	H(5,𝔽₂)×C₁₀₉	C₂²	1	4	yes	—
15	C₂×#41×C₁₀₉	—	2	6	no	—
16	C₂×#41×C₁₀₉	C₂²	1	3	yes	—
17	C₂³×D₁₀₉	C₂³	1	2	yes	—
18	C₂×D₄₃₆	—	2	3	no	—
19	C₂×D₄₃₆	—	2	3	no	—
20	C₂×C₂²⋊D₁₀₉	—	2	12	no	—
21	C₄⋊Dic₁₀₉	—	2	3	no	—
22	C₂²⋊Dic₁₀₉	C₂³	1	6	yes	—
23	#13182006	—	2	12	no	—
24	C₄×Dic₁₀₉	—	2	3	no	—
25	#41×C₁₀₉	—	2	4	no	—
26	#41×C₁₀₉	—	2	12	no	—
27	#41×C₁₀₉	—	2	6	no	—
28	#41×C₁₀₉	—	2	12	no	—
29	#41×C₁₀₉	—	2	6	no	—
30	C₂×Q₈×C₁₀₉	—	2	6	no	—
31	C₂×Q₈×C₁₀₉	D₄	1	3	yes	complemented
32	C₂×D₄×C₁₀₉	—	2	6	no	—
33	C₂×D₄×C₁₀₉	—	2	12	no	—
34	C₂×D₄×C₁₀₉	D₄	1	3	yes	complemented
35	C₂×D₄×C₁₀₉	C₂³	1	6	yes	—
36	C₂²×C₄×C₁₀₉	—	2	4	no	—
37	C₂²×C₄×C₁₀₉	D₄	1	6	yes	complemented
38	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂³×C₁₀₉	—	2	4	no	—
40	C₂³×C₁₀₉	C₂×D₄	1	6	yes	—
41	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
42	C₂²×D₁₀₉	—	2	6	no	—
43	C₂²×D₁₀₉	—	8	2	no	—
44	Q₈×C₁₀₉	—	2	4	no	—
45	Q₈×C₁₀₉	—	2	6	no	—
46	D₄×C₁₀₉	—	2	12	no	—
47	D₄×C₁₀₉	—	2	12	no	—
48	D₄×C₁₀₉	—	2	6	no	—
49	C₂×C₄×C₁₀₉	—	2	12	no	—
50	C₂×C₄×C₁₀₉	—	2	6	no	—
51	C₂×C₄×C₁₀₉	—	2	6	no	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
53	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
55	C₂²⋊D₁₀₉	—	8	12	no	—
56	C₂×Dic₁₀₉	—	2	6	no	—
57	D₄₃₆	—	4	1	no	—
58	D₄₃₆	—	4	3	no	—
59	C₂²×C₁₀₉	—	2	6	no	—
60	C₂²×C₁₀₉	—	4	1	no	—
61	C₂²×C₁₀₉	—	4	6	no	—
62	C₂²×C₁₀₉	—	8	4	no	—
63	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
64	C₂×D₁₀₉	—	4	12	no	—
65	C₂×D₁₀₉	—	4	2	no	—
66	C₄×C₁₀₉	—	2	6	no	—
67	C₄×C₁₀₉	#120	1	1	yes	characteristic
68	C₄×C₁₀₉	#120	1	3	yes	—
69	Dic₁₀₉	—	4	6	no	—
70	C₂×C₁₀₉	—	2	3	no	—
71	C₂×C₁₀₉	—	4	6	no	—
72	C₂×C₁₀₉	C₂×#120	1	1	yes	characteristic
73	D₁₀₉	—	8	2	no	—
74	#3027	—	109	1	no	maximal, Sylow
75	C₁₀₉	#3027	1	1	yes	characteristic, complemented, Sylow
76	#585	D₁₀₉	1	1	yes	characteristic, complemented
77	#535	—	109	3	no	—
78	B(4,𝔽₂)	—	109	8	no	—
79	#410	—	109	3	no	—
80	2₋¹⁺⁴	—	2	3	no	—
81	H(5,𝔽₂)	—	2	3	no	—
82	H(5,𝔽₂)	C₂×D₁₀₉	1	4	yes	—
83	C₂×#41	—	2	6	no	—
84	C₂×#41	C₂×D₁₀₉	1	3	yes	—
85	C₂²×D₄	—	109	1	no	—
86	#127	—	218	3	no	—
87	#124	—	218	3	no	—
88	#121	—	218	6	no	—
89	#120	—	109	12	no	—
90	#117	—	109	3	no	—
91	#99	—	109	12	no	—
92	C₂⁴	—	109	2	no	—
93	#41	—	2	4	no	—
94	#41	—	2	12	no	—
95	#41	—	2	6	no	—
96	#41	—	2	12	no	—
97	#41	—	2	6	no	—
98	C₂×Q₈	—	2	6	no	—
99	C₂×Q₈	C₂²⋊D₁₀₉	1	3	yes	complemented
100	C₂×D₄	—	2	6	no	—
101	C₂×D₄	—	2	12	no	—
102	C₂×D₄	—	218	12	no	—
103	C₂×D₄	—	218	3	no	—
104	C₂×D₄	—	218	3	no	—
105	C₂×D₄	C₂²⋊D₁₀₉	1	3	yes	complemented
106	C₂×D₄	C₂²×D₁₀₉	1	6	yes	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	C₂²⋊D₁₀₉	1	6	yes	complemented
109	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
110	C₄⋊C₄	—	218	3	no	—
111	C₂²⋊C₄	—	109	6	no	—
112	C₂²⋊C₄	—	218	12	no	—
113	C₄²	—	218	3	no	—
114	C₂³	—	2	4	no	—
115	C₂³	—	872	2	no	—
116	C₂³	—	218	6	no	—
117	C₂³	C₂×C₂²⋊D₁₀₉	1	6	yes	—
118	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	6	no	—
121	D₄	—	2	12	no	—
122	D₄	—	2	12	no	—
123	D₄	—	2	6	no	—
124	D₄	—	872	12	no	—
125	D₄	—	436	1	no	—
126	D₄	—	436	3	no	—
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	6	no	—
130	C₂×C₄	—	218	6	no	—
131	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	6	yes	—
132	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
133	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
134	C₂²	—	2	6	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	6	no	—
137	C₂²	—	8	4	no	—
138	C₂²	—	436	12	no	—
139	C₂²	—	436	2	no	—
140	C₂²	C₂²×C₂²⋊D₁₀₉	1	3	yes	—
141	C₄	—	2	6	no	—
142	C₄	—	436	6	no	—
143	C₄	#16812668	1	1	yes	characteristic
144	C₄	#16812668	1	3	yes	—
145	C₂	—	2	3	no	—
146	C₂	—	4	6	no	—
147	C₂	—	872	2	no	—
148	C₂	C₂×#16812668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295590

C₂

yes

maximal, complemented

#20295213

C₂

yes

maximal, complemented

#20295000

C₂

yes

maximal, complemented

#585×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

C₂²

yes

characteristic, complemented

—

—

—

C₂²

yes

complemented

#16812658

C₂²

yes

complemented

#16812614

C₂²

yes

complemented

2₋¹⁺⁴×C₁₀₉

—

H(5,𝔽₂)×C₁₀₉

—

H(5,𝔽₂)×C₁₀₉

C₂²

yes

—

C₂×#41×C₁₀₉

—

C₂×#41×C₁₀₉

C₂²

yes

—

C₂³×D₁₀₉

C₂³

yes

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×C₂²⋊D₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

C₂³

yes

—

#13182006

—

C₄×Dic₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

complemented

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

D₄

yes

complemented

C₂×D₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

complemented

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

#120

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#120

yes

characteristic

D₁₀₉

—

#3027

—

109

maximal, Sylow

C₁₀₉

#3027

yes

characteristic, complemented, Sylow

#585

D₁₀₉

yes

characteristic, complemented

#535

—

109

—

B(4,𝔽₂)

—

109

—

#410

—

109

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₁₀₉

yes

—

C₂×#41

—

C₂×#41

C₂×D₁₀₉

yes

—

C₂²×D₄

—

109

—

#127

—

218

—

#124

—

218

—

#121

—

218

—

#120

—

109

—

#117

—

109

—

#99

—

109

—

C₂⁴

—

109

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

complemented

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

218

—

103

C₂×D₄

—

218

—

104

C₂×D₄

—

218

—

105

C₂×D₄

C₂²⋊D₁₀₉

yes

complemented

106

C₂×D₄

C₂²×D₁₀₉

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₁₀₉

yes

complemented

109

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

110

C₄⋊C₄

—

218

—

111

C₂²⋊C₄

—

109

—

112

C₂²⋊C₄

—

218

—

113

C₄²

—

218

—

114

C₂³

—

115

C₂³

—

872

—

116

C₂³

—

218

—

117

C₂³

C₂×C₂²⋊D₁₀₉

yes

—

118

C₂³

C₂³×D₁₀₉

yes

characteristic

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

D₄

—

124

D₄

—

872

—

125

D₄

—

436

—

126

D₄

—

436

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

218

—

131

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

132

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

133

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

436

—

139

C₂²

—

436

—

140

C₂²

C₂²×C₂²⋊D₁₀₉

yes

—

141

C₄

—

142

C₄

—

436

—

143

C₄

#16812668

yes

characteristic

144

C₄

#16812668

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

872

—

148

C₂

C₂×#16812668

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#16812668
commutator, cosocle	C₂³×C₁₀₉	C₂⁴
Frattini	C₂³	C₂³×D₁₀₉
Fitting	#585×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×#120
hypercenter	#585	D₁₀₉
hypocenter	C₁₀₉	#3027
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#16812668

commutator, cosocle

C₂³×C₁₀₉

C₂⁴

Frattini

C₂³

C₂³×D₁₀₉

Fitting

#585×C₁₀₉

C₂

C₂×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #20295590	C₂	semidirect
2. #20295213	C₂	semidirect
3. #20295000	C₂	semidirect
4. #585×C₁₀₉	C₂	semidirect
5. C₂²×D₄₃₆	C₂²	semidirect
9. #16812667	C₂²	semidirect
10. #16812658	C₂²	semidirect
11. #16812614	C₂²	semidirect
31. C₂×Q₈×C₁₀₉	D₄	semidirect
34. C₂×D₄×C₁₀₉	D₄	semidirect
37. C₂²×C₄×C₁₀₉	D₄	semidirect
75. C₁₀₉	#3027	semidirect
76. #585	D₁₀₉	semidirect
99. C₂×Q₈	C₂²⋊D₁₀₉	semidirect
105. C₂×D₄	C₂²⋊D₁₀₉	semidirect
108. C₂²×C₄	C₂²⋊D₁₀₉	semidirect

H ≅ G/N

type

1. #20295590

C₂

semidirect

2. #20295213

C₂

semidirect

3. #20295000

C₂

semidirect

4. #585×C₁₀₉

C₂

semidirect

5. C₂²×D₄₃₆

C₂²

semidirect

9. #16812667

C₂²

semidirect

10. #16812658

C₂²

semidirect

11. #16812614

C₂²

semidirect

31. C₂×Q₈×C₁₀₉

D₄

semidirect

34. C₂×D₄×C₁₀₉

D₄

semidirect

37. C₂²×C₄×C₁₀₉

D₄

semidirect

75. C₁₀₉

#3027

semidirect

76. #585

D₁₀₉

semidirect

99. C₂×Q₈

C₂²⋊D₁₀₉

semidirect

105. C₂×D₄

C₂²⋊D₁₀₉

semidirect

108. C₂²×C₄

C₂²⋊D₁₀₉

semidirect

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

subgroup	index
#20295590	2
#20295213	2
#20295000	2
#585×C₁₀₉	2
#3027	109

subgroup

index

#585×C₁₀₉

109

Subgroups — #28372434

Conjugacy classes of subgroups

Complements

Maximal subgroups