Group #28371630 — subgroups

Subgroups — #28371630

Order 13952.

476 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 253 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×#16812706	C₂	1	1	yes	maximal, characteristic
2	#20295509	C₂	1	1	yes	maximal, characteristic
3	#20295069	C₂	1	3	yes	maximal, complemented
4	#20295066	C₂	1	3	yes	maximal, complemented
5	#20295062	C₂	1	3	yes	maximal, complemented
6	C₂×#16812632	C₂	1	3	yes	maximal, complemented
7	#520×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#16812706	C₄	1	8	yes	complemented
9	C₂×#13182013	C₂²	1	3	yes	—
10	C₂²×#11724784	C₂²	1	3	yes	—
11	Q₈×Dic₁₀₉	—	2	1	no	—
12	D₄×Dic₁₀₉	—	2	3	no	—
13	#16812658	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	—
15	#16812632	C₂²	1	3	yes	—
16	#16812632	C₂²	1	6	yes	—
17	#16812632	C₂²	1	1	yes	characteristic
18	#16812630	C₂²	1	3	yes	—
19	#16812630	C₂²	1	3	yes	—
20	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₀₉	—	2	1	no	—
22	C₄×D₄×C₁₀₉	—	2	3	no	—
23	#117×C₁₀₉	C₂²	1	3	yes	—
24	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
25	C₂²×Dic₁₀₉	—	2	3	no	—
26	#13182013	C₂×C₄	1	12	yes	complemented
27	C₂×#11724784	C₂×C₄	1	12	yes	complemented
28	C₂×#11724784	C₂³	1	3	yes	—
29	C₂×#11724784	C₂³	1	1	yes	characteristic
30	C₄⋊Dic₁₀₉	—	2	3	no	—
31	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
33	C₂²⋊Dic₁₀₉	—	2	3	no	—
34	C₄×Dic₁₀₉	—	2	3	no	—
35	#41×C₁₀₉	C₂×C₄	1	4	yes	—
36	#41×C₁₀₉	Q₈	1	4	yes	—
37	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂×D₄×C₁₀₉	D₄	1	3	yes	—
39	C₂²×C₄×C₁₀₉	—	2	3	no	—
40	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
41	C₄⋊C₄×C₁₀₉	—	2	3	no	—
42	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
43	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂²⋊C₄×C₁₀₉	—	2	3	no	—
45	C₄²×C₁₀₉	—	2	3	no	—
46	C₂³×C₁₀₉	C₂×D₄	1	3	yes	—
47	Q₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
48	D₄×C₁₀₉	C₄⋊C₄	1	12	yes	—
49	C₂×C₄×C₁₀₉	—	2	1	no	—
50	C₂×C₄×C₁₀₉	—	2	3	no	—
51	C₂×C₄×C₁₀₉	—	8	3	no	—
52	C₂×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
53	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₁₀₉	C₂×Q₈	1	6	yes	—
55	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₁₀₉	—	2	1	no	—
57	C₂×Dic₁₀₉	—	2	3	no	—
58	C₂×Dic₁₀₉	—	8	3	no	—
59	#11724784	C₂²×C₄	1	2	yes	—
60	#11724784	C₂²×C₄	1	6	yes	—
61	C₂²×C₁₀₉	—	2	3	no	—
62	C₂²×C₁₀₉	—	2	3	no	—
63	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
64	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₀₉	—	4	3	no	—
66	C₄×C₁₀₉	—	4	1	no	—
67	C₄×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
68	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
70	Dic₁₀₉	—	4	3	no	—
71	Dic₁₀₉	—	4	1	no	—
72	C₂×C₁₀₉	—	2	6	no	—
73	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₁₀₉	#576	1	1	yes	characteristic
75	C₂×C₁₀₉	#578	1	1	yes	characteristic
76	#2913	—	109	1	no	maximal, Sylow
77	C₁₀₉	#2913	1	1	yes	characteristic, complemented, Sylow
78	C₂×#131	—	109	1	no	—
79	#520	—	109	1	no	—
80	#520	D₁₀₉	1	1	yes	characteristic
81	#428	—	109	3	no	—
82	#428	—	109	3	no	—
83	#427	—	109	3	no	—
84	C₂×#107	—	109	3	no	—
85	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
86	#131	—	109	8	no	—
87	C₂×M₁₆	—	109	3	no	—
88	C₂²×C₈	—	109	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	—	218	1	no	—
91	C₄×D₄	—	2	3	no	—
92	C₄×D₄	—	218	3	no	—
93	#117	—	109	3	no	—
94	#117	C₂×D₁₀₉	1	3	yes	—
95	C₂×C₄⋊C₄	—	109	3	no	—
96	C₂×C₄⋊C₄	C₂×D₁₀₉	1	3	yes	—
97	#107	—	109	3	no	—
98	#107	—	109	6	no	—
99	#107	—	109	1	no	—
100	#106	—	109	3	no	—
101	#106	—	109	3	no	—
102	#41	Dic₂₁₈	1	4	yes	—
103	#41	C₄×D₁₀₉	1	4	yes	—
104	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
105	C₂×D₄	C₂²⋊D₁₀₉	1	3	yes	—
106	C₂²×C₄	—	2	3	no	—
107	C₂²×C₄	—	218	3	no	—
108	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
109	M₁₆	—	109	12	no	—
110	C₂×C₈	—	109	12	no	—
111	C₂×C₈	—	109	3	no	—
112	C₂×C₈	—	109	1	no	—
113	C₄⋊C₄	—	2	3	no	—
114	C₄⋊C₄	—	109	1	no	—
115	C₄⋊C₄	—	109	3	no	—
116	C₄⋊C₄	—	218	3	no	—
117	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
118	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
119	C₂²⋊C₄	—	2	3	no	—
120	C₂²⋊C₄	—	218	3	no	—
121	C₄²	—	2	3	no	—
122	C₄²	—	218	3	no	—
123	C₂³	C₂×C₂²⋊D₁₀₉	1	3	yes	—
124	Q₈	#13182008	1	4	yes	—
125	D₄	#13182008	1	12	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	8	3	no	—
129	C₂×C₄	—	872	3	no	—
130	C₂×C₄	—	218	1	no	—
131	C₂×C₄	—	218	3	no	—
132	C₂×C₄	C₂×C₄×D₁₀₉	1	6	yes	—
133	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
134	C₂×C₄	C₂×Dic₂₁₈	1	6	yes	—
135	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
136	C₈	—	109	2	no	—
137	C₈	—	109	6	no	—
138	C₂²	—	2	3	no	—
139	C₂²	—	2	3	no	—
140	C₂²	C₂×#13182008	1	6	yes	—
141	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
142	C₄	—	4	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	436	3	no	—
145	C₄	—	436	1	no	—
146	C₄	C₂×#13182008	1	6	yes	—
147	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
148	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
149	C₂	—	2	6	no	—
150	C₂	C₂²×#13182008	1	1	yes	characteristic
151	C₂	#20295829	1	1	yes	characteristic
152	C₂	#20295841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812706

C₂

yes

maximal, characteristic

#20295509

C₂

yes

maximal, characteristic

#20295069

C₂

yes

maximal, complemented

#20295066

C₂

yes

maximal, complemented

#20295062

C₂

yes

maximal, complemented

C₂×#16812632

C₂

yes

maximal, complemented

#520×C₁₀₉

C₂

yes

maximal, characteristic

#16812706

C₄

yes

complemented

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

Q₈×Dic₁₀₉

—

D₄×Dic₁₀₉

—

#16812658

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812632

C₂²

yes

—

#16812632

C₂²

yes

—

#16812632

C₂²

yes

characteristic

#16812630

C₂²

yes

—

#16812630

C₂²

yes

—

C₂×#41×C₁₀₉

C₂²

yes

characteristic

C₄×Q₈×C₁₀₉

—

C₄×D₄×C₁₀₉

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

—

#13182013

C₂×C₄

yes

complemented

C₂×#11724784

C₂×C₄

yes

complemented

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

C₂³

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

#41×C₁₀₉

C₂×C₄

yes

—

#41×C₁₀₉

Q₈

yes

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×D₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

—

Q₈×C₁₀₉

C₄⋊C₄

yes

—

D₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

C₂²×C₄

yes

—

#11724784

C₂²×C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×C₄⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2913

yes

characteristic, complemented, Sylow

C₂×#131

—

109

—

#520

—

109

—

#520

D₁₀₉

yes

characteristic

#428

—

109

—

#428

—

109

—

#427

—

109

—

C₂×#107

—

109

—

C₂×#41

C₂×D₁₀₉

yes

characteristic

#131

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₄×Q₈

—

C₄×Q₈

—

218

—

C₄×D₄

—

C₄×D₄

—

218

—

#117

—

109

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

#107

—

109

—

#107

—

109

—

#107

—

109

—

100

#106

—

109

—

101

#106

—

109

—

102

#41

Dic₂₁₈

yes

—

103

#41

C₄×D₁₀₉

yes

—

104

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

105

C₂×D₄

C₂²⋊D₁₀₉

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

218

—

108

C₂²×C₄

C₂²×D₁₀₉

yes

—

109

M₁₆

—

109

—

110

C₂×C₈

—

109

—

111

C₂×C₈

—

109

—

112

C₂×C₈

—

109

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

109

—

115

C₄⋊C₄

—

109

—

116

C₄⋊C₄

—

218

—

117

C₄⋊C₄

C₂²×D₁₀₉

yes

characteristic

118

C₄⋊C₄

C₂²×D₁₀₉

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

218

—

121

C₄²

—

122

C₄²

—

218

—

123

C₂³

C₂×C₂²⋊D₁₀₉

yes

—

124

Q₈

#13182008

yes

—

125

D₄

#13182008

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

872

—

130

C₂×C₄

—

218

—

131

C₂×C₄

—

218

—

132

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

133

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

134

C₂×C₄

C₂×Dic₂₁₈

yes

—

135

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

136

C₈

—

109

—

137

C₈

—

109

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×#13182008

yes

—

141

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

436

—

145

C₄

—

436

—

146

C₄

C₂×#13182008

yes

—

147

C₄

C₂²×C₄×D₁₀₉

yes

characteristic

148

C₄

C₂²×Dic₂₁₈

yes

characteristic

149

C₂

—

150

C₂

C₂²×#13182008

yes

characteristic

151

C₂

#20295829

yes

characteristic

152

C₂

#20295841

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#520×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#41×C₁₀₉	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#520	D₁₀₉
hypocenter	C₁₀₉	#2913
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#520×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂×#41×C₁₀₉

C₂²

cosocle

C₂×C₄×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

N	H ≅ G/N	type
3. #20295069	C₂	semidirect
4. #20295066	C₂	semidirect
5. #20295062	C₂	semidirect
6. C₂×#16812632	C₂	semidirect
8. #16812706	C₄	semidirect
26. #13182013	C₂×C₄	semidirect
27. C₂×#11724784	C₂×C₄	semidirect
77. C₁₀₉	#2913	semidirect

H ≅ G/N

type

3. #20295069

C₂

semidirect

4. #20295066

C₂

semidirect

5. #20295062

C₂

semidirect

6. C₂×#16812632

C₂

semidirect

8. #16812706

C₄

semidirect

26. #13182013

C₂×C₄

semidirect

27. C₂×#11724784

C₂×C₄

semidirect

77. C₁₀₉

#2913

semidirect

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

subgroup	index
C₂×#16812706	2
#20295509	2
#20295069	2
#20295066	2
#20295062	2
C₂×#16812632	2
#520×C₁₀₉	2
#2913	109

subgroup

index

#520×C₁₀₉

109

Subgroups — #28371630

Conjugacy classes of subgroups

Complements

Maximal subgroups