Finite Group Databasebeta

← C3×#89543 #13055203

Subgroups — C3×#89543 #13055203

Order 1344.

324 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1159C211yesmaximal, characteristic
2C2×C3×#1131C211yesmaximal, characteristic, complemented
3C3×D7×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×C7×#102C211yesmaximal, characteristic, complemented
5C3×#6406C211yesmaximal, characteristic, complemented
6C3×#6390C211yesmaximal, characteristic, complemented
7C3×#6384C211yesmaximal, characteristic, complemented
8#89543C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C3×C22⋊D7—21no—
11C22×C3×Dic7—21no—
12C3×#1159—22no—
13C3×#1159C414yescomplemented
14C2×C3×C4×D7—21no—
15C2×C3×C4×D7C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C2×C3×C7×C8C2211yescharacteristic
18C3×C7×C4⋊C4C2211yescharacteristic
19C3×C4⋊Dic7C2211yescharacteristic
20C3×Dic7⋊C4—21no—
21C2×C3×C7⋊C8C2211yescharacteristic
22C3×#1131—22no—
23C2×#1159C2×C311yescharacteristic
24C2×#1131C2×C311yescharacteristic, complemented
25D7×C4⋊C4C2×C311yescharacteristic, complemented
26C7×#102C2×C311yescharacteristic, complemented
27#6406C2×C311yescharacteristic, complemented
28#6390C2×C311yescharacteristic, complemented
29#6384C2×C311yescharacteristic, complemented
30C3×#417—71nomaximal, Hall
31C23×C3×C7—21no—
32C22×C3×D7D411yescharacteristic
33C3×C7×D4—21no—
34C3×C7×D4C2×C412yescomplemented
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37C3×C22⋊D7—24no—
38C2×C3×Dic7—21no—
39C2×C3×Dic7—24no—
40C2×C3×Dic7—21no—
41C2×C3×Dic7D411yescharacteristic
42C3×C4×D7—42no—
43C3×C4×D7C2×C412yes—
44C3×C4×D7D412yes—
45C3×Dic14—21no—
46C3×Dic14C2×C412yescomplemented
47C3×C7×C8—21no—
48C3×C7⋊C8—21no—
49C2×C7×D4C22×C311yescharacteristic
50C2×C22⋊D7—21no—
51C22×Dic7—21no—
52#1159—22no—
53#1159C3×C414yescomplemented
54C2×C4×D7—21no—
55C2×C4×D7C22×C311yescharacteristic
56C2×Dic14C22×C311yescharacteristic
57C2×C7×C8C22×C311yescharacteristic
58C7×C4⋊C4C22×C311yescharacteristic
59C4⋊Dic7C22×C311yescharacteristic
60Dic7⋊C4—21no—
61C2×C7⋊C8C22×C311yescharacteristic
62#1131—22no—
63C2×C3×#41—71no—
64C2×C3×M16—71no—
65C2×C3×C4⋊C4—71no—
66C3×#103—71no—
67C3×#103—71no—
68C3×#102—71no—
69C3×#102D711yescharacteristic, complemented
70C22×C3×C7—22no—
71C22×C3×C7—41no—
72C22×C3×C7—41no—
73C22×C3×C7C2×D411yescharacteristic
74C2×C3×D7—21no—
75C2×C3×D7—21no—
76C2×C3×D7C22⋊C412yes—
77C3×C4×C7—41no—
78C3×C4×C7C22×C411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80C3×Dic7—22no—
81C3×Dic7—41no—
82C3×Dic7C22⋊C412yes—
83#417—71noSylow
84C23×C7—21no—
85C22×D7C3×D411yescharacteristic
86C7×D4—21no—
87C7×D4C2×C3×C412yescomplemented
88C2×C4×C7—21no—
89C2×C4×C7C23×C311yescharacteristic
90C22⋊D7—24no—
91C2×Dic7—21no—
92C2×Dic7—24no—
93C2×Dic7—21no—
94C2×Dic7C3×D411yescharacteristic
95C4×D7—42no—
96C4×D7C2×C3×C412yes—
97C4×D7C3×D412yes—
98Dic14—21no—
99Dic14C2×C3×C412yescomplemented
100C7×C8—21no—
101C7⋊C8—21no—
102C3×#41—74no—
103C3×#41—142no—
104C2×C3×Q8—71no—
105C2×C3×D4—141no—
106C2×C3×D4C2×D711yescharacteristic
107C22×C3×C4—71no—
108C22×C3×C4—141no—
109C22×C3×C4—141no—
110C3×M16—142no—
111C2×C3×C8—71no—
112C2×C3×C8C2×D711yescharacteristic
113C3×C4⋊C4—71no—
114C3×C4⋊C4—141no—
115C3×C4⋊C4C2×D711yescharacteristic
116C2×C3×C7—42no—
117C2×C3×C7C2×C22⋊C411yescharacteristic
118C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
119C2×C3×C7#13711yescharacteristic
120C3×D7—22no—
121C2×#41—71no—
122C2×M16—71no—
123C2×C4⋊C4—71no—
124#103—71no—
125#103—71no—
126#102—71no—
127#102C3×D711yescharacteristic, complemented
128C22×C7—22no—
129C22×C7—41no—
130C22×C7—41no—
131C22×C7C2×C3×D411yescharacteristic
132C2×D7—21no—
133C2×D7—21no—
134C2×D7C3×C22⋊C412yes—
135C4×C7—41no—
136C4×C7C22×C3×C411yescharacteristic
137C4×C7C2×C3×D411yescharacteristic
138Dic7—22no—
139Dic7—41no—
140Dic7C3×C22⋊C412yes—
141C23×C3—21no—
142C23×C3—71no—
143C3×Q8—72no—
144C3×Q8—141no—
145C3×D4—21no—
146C3×D4—144no—
147C3×D4C4×D712yescomplemented
148C2×C3×C4—21no—
149C2×C3×C4—71no—
150C2×C3×C4—72no—
151C2×C3×C4—72no—
152C2×C3×C4—144no—
153C2×C3×C4—141no—
154C2×C3×C4—141no—
155C2×C3×C4—282no—
156C2×C3×C4C22×D711yescharacteristic
157C3×C8—21no—
158C3×C8—141no—
159C3×C7#41711yescharacteristic, complemented, Hall
160#41—74no—
161#41—142no—
162C2×Q8—71no—
163C2×D4—141no—
164C2×D4C2×C3×D711yescharacteristic
165C22×C4—71no—
166C22×C4—141no—
167C22×C4—141no—
168M16—142no—
169C2×C8—71no—
170C2×C8C2×C3×D711yescharacteristic
171C4⋊C4—71no—
172C4⋊C4—141no—
173C4⋊C4C2×C3×D711yescharacteristic
174C2×C7—42no—
175C2×C7C2×C3×C22⋊C411yescharacteristic
176C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
177C2×C7C3×#13711yescharacteristic
178D7—22no—
179C22×C3—22no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—72no—
183C22×C3—141no—
184C22×C3—141no—
185C22×C3D4×D711yescharacteristic
186C3×C4—41no—
187C3×C4—72no—
188C3×C4—142no—
189C3×C4—281no—
190C3×C4C2×C4×D711yescharacteristic
191C3×C4D4×D711yescharacteristic
192C23—21no—
193C23—71no—
194Q8—72no—
195Q8—141no—
196D4—21no—
197D4—144no—
198D4C3×C4×D712yescomplemented
199C2×C4—21no—
200C2×C4—71no—
201C2×C4—72no—
202C2×C4—72no—
203C2×C4—144no—
204C2×C4—141no—
205C2×C4—141no—
206C2×C4—282no—
207C2×C4C22×C3×D711yescharacteristic
208C8—21no—
209C8—141no—
210C7C3×#41711yescharacteristic, complemented, Sylow
211C2×C3—42no—
212C2×C3—142no—
213C2×C3D7×C22⋊C411yescharacteristic
214C2×C3#647411yescharacteristic
215C2×C3#647811yescharacteristic
216C22—22no—
217C22—41no—
218C22—41no—
219C22—72no—
220C22—141no—
221C22—141no—
222C22C3×D4×D711yescharacteristic
223C4—41no—
224C4—72no—
225C4—142no—
226C4—281no—
227C4C2×C3×C4×D711yescharacteristic
228C4C3×D4×D711yescharacteristic
229C3#8954311yescharacteristic, direct product factor, Sylow
230C2—42no—
231C2—142no—
232C2C3×D7×C22⋊C411yescharacteristic
233C2C3×#647411yescharacteristic
234C2C3×#647811yescharacteristic
subgroupG/H
centerC22×C3D4×D7
commutatorC4×C7C22×C3×C4
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1159C2
cosocleC2×C4×C7C23×C3
hypercenterC3×#102D7
hypocenterC7C3×#417
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#1131C2semidirect
3. C3×D7×C4⋊C4C2semidirect
4. C3×C7×#102C2semidirect
5. C3×#6406C2semidirect
6. C3×#6390C2semidirect
7. C3×#6384C2semidirect
8. #89543C3direct (N × H)
13. C3×#1159C4semidirect
24. C2×#1131C2×C3semidirect
25. D7×C4⋊C4C2×C3semidirect
26. C7×#102C2×C3semidirect
27. #6406C2×C3semidirect
28. #6390C2×C3semidirect
29. #6384C2×C3semidirect
34. C3×C7×D4C2×C4semidirect
46. C3×Dic14C2×C4semidirect
53. #1159C3×C4semidirect
69. C3×#102D7semidirect
87. C7×D4C2×C3×C4semidirect
99. Dic14C2×C3×C4semidirect
127. #102C3×D7semidirect
147. C3×D4C4×D7semidirect
159. C3×C7#417semidirect
198. D4C3×C4×D7semidirect
210. C7C3×#417semidirect
229. C3#89543direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#11592
C2×C3×#11312
C3×D7×C4⋊C42
C3×C7×#1022
C3×#64062
C3×#63902
C3×#63842
#895433
C3×#4177