Finite Group Databasebeta

← C3×#89575 #13055235

Subgroups — C3×#89575 #13055235

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×#1161C211yesmaximal, characteristic
2C2×C3×#1131C211yesmaximal, characteristic, complemented
3C3×D7×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×C7×#103C211yesmaximal, characteristic, complemented
5C3×#6409C211yesmaximal, characteristic, complemented
6C3×#6395C211yesmaximal, characteristic, complemented
7C3×#6383C211yesmaximal, characteristic, complemented
8#89575C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C3×#1161—22no—
11C3×#1161C414yescomplemented
12C2×C3×D28—21no—
13C2×C3×D28C2211yescharacteristic
14C2×C3×C4×D7—21no—
15C2×C3×C4×D7—21no—
16C2×C3×C4×D7C2211yescharacteristic
17C2×C3×C7×C8C2211yescharacteristic
18C3×C7×C4⋊C4C2211yescharacteristic
19C3×C4⋊Dic7C2211yescharacteristic
20C3×Dic7⋊C4—21no—
21C2×C3×C7⋊C8C2211yescharacteristic
22C3×#1131—22no—
23C2×#1161C2×C311yescharacteristic
24C2×#1131C2×C311yescharacteristic, complemented
25D7×C4⋊C4C2×C311yescharacteristic, complemented
26C7×#103C2×C311yescharacteristic, complemented
27#6409C2×C311yescharacteristic, complemented
28#6395C2×C311yescharacteristic, complemented
29#6383C2×C311yescharacteristic, complemented
30C3×#417—71nomaximal, Hall
31C22×C3×D7—21no—
32C22×C3×D7D411yescharacteristic
33C3×C7×Q8—21no—
34C3×C7×Q8C2×C412yescomplemented
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×C3×Dic7—21no—
39C2×C3×Dic7D411yescharacteristic
40C3×D28—24no—
41C3×D28—21no—
42C3×D28C2×C412yescomplemented
43C3×C4×D7—24no—
44C3×C4×D7—42no—
45C3×C4×D7C2×C412yes—
46C3×C4×D7D412yes—
47C3×C7×C8—21no—
48C3×C7⋊C8—21no—
49C2×C7×Q8C22×C311yescharacteristic
50#1161—22no—
51#1161C3×C414yescomplemented
52C2×D28—21no—
53C2×D28C22×C311yescharacteristic
54C2×C4×D7—21no—
55C2×C4×D7—21no—
56C2×C4×D7C22×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×#103D711yescharacteristic, complemented
68C3×#102—71no—
69C3×#102—71no—
70C22×C3×C7C2×D411yescharacteristic
71C2×C3×D7—21no—
72C2×C3×D7—21no—
73C2×C3×D7—22no—
74C2×C3×D7—41no—
75C2×C3×D7—41no—
76C2×C3×D7C22⋊C412yes—
77C3×C4×C7—22no—
78C3×C4×C7—41no—
79C3×C4×C7C22×C411yescharacteristic
80C3×C4×C7C2×D411yescharacteristic
81C3×Dic7—41no—
82C3×Dic7C22⋊C412yes—
83#417—71noSylow
84C22×D7—21no—
85C22×D7C3×D411yescharacteristic
86C7×Q8—21no—
87C7×Q8C2×C3×C412yescomplemented
88C2×C4×C7—21no—
89C2×C4×C7—21no—
90C2×C4×C7C23×C311yescharacteristic
91C2×Dic7—21no—
92C2×Dic7C3×D411yescharacteristic
93D28—24no—
94D28—21no—
95D28C2×C3×C412yescomplemented
96C4×D7—24no—
97C4×D7—42no—
98C4×D7C2×C3×C412yes—
99C4×D7C3×D412yes—
100C7×C8—21no—
101C7⋊C8—21no—
102C3×#41—74no—
103C3×#41—142no—
104C2×C3×Q8C2×D711yescharacteristic
105C2×C3×D4—71no—
106C2×C3×D4—141no—
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×C7C2×C22⋊C411yescharacteristic
117C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
118C2×C3×C7#13711yescharacteristic
119C3×D7—22no—
120C3×D7—42no—
121C2×#41—71no—
122C2×M16—71no—
123C2×C4⋊C4—71no—
124#103—71no—
125#103C3×D711yescharacteristic, complemented
126#102—71no—
127#102—71no—
128C22×C7C2×C3×D411yescharacteristic
129C2×D7—21no—
130C2×D7—21no—
131C2×D7—22no—
132C2×D7—41no—
133C2×D7—41no—
134C2×D7C3×C22⋊C412yes—
135C4×C7—22no—
136C4×C7—41no—
137C4×C7C22×C3×C411yescharacteristic
138C4×C7C2×C3×D411yescharacteristic
139Dic7—41no—
140Dic7C3×C22⋊C412yes—
141C23×C3—71no—
142C23×C3—141no—
143C3×Q8—21no—
144C3×Q8C4×D712yescomplemented
145C3×D4—72no—
146C3×D4—144no—
147C3×D4—141no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—71no—
151C2×C3×C4—72no—
152C2×C3×C4—72no—
153C2×C3×C4—144no—
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×Q8C2×C3×D711yescharacteristic
163C2×D4—71no—
164C2×D4—141no—
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×C7C2×C3×C22⋊C411yescharacteristic
175C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
176C2×C7C3×#13711yescharacteristic
177D7—22no—
178D7—42no—
179C22×C3—72no—
180C22×C3—141no—
181C22×C3—141no—
182C22×C3—142no—
183C22×C3—281no—
184C22×C3—281no—
185C22×C3D4×D711yescharacteristic
186C3×C4—22no—
187C3×C4—41no—
188C3×C4—72no—
189C3×C4—281no—
190C3×C4C2×C4×D711yescharacteristic
191C3×C4D4×D711yescharacteristic
192C23—71no—
193C23—141no—
194Q8—21no—
195Q8C3×C4×D712yescomplemented
196D4—72no—
197D4—144no—
198D4—141no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—71no—
202C2×C4—72no—
203C2×C4—72no—
204C2×C4—144no—
205C2×C4—141no—
206C2×C4—282no—
207C2×C4C22×C3×D711yescharacteristic
208C8—21no—
209C8—141no—
210C7C3×#41711yescharacteristic, complemented, Sylow
211C2×C3—142no—
212C2×C3—282no—
213C2×C3D7×C22⋊C411yescharacteristic
214C2×C3#647711yescharacteristic
215C2×C3#648111yescharacteristic
216C22—72no—
217C22—141no—
218C22—141no—
219C22—142no—
220C22—281no—
221C22—281no—
222C22C3×D4×D711yescharacteristic
223C4—22no—
224C4—41no—
225C4—72no—
226C4—281no—
227C4C2×C3×C4×D711yescharacteristic
228C4C3×D4×D711yescharacteristic
229C3#8957511yescharacteristic, direct product factor, Sylow
230C2—142no—
231C2—282no—
232C2C3×D7×C22⋊C411yescharacteristic
233C2C3×#647711yescharacteristic
234C2C3×#648111yescharacteristic
subgroupG/H
centerC22×C3D4×D7
commutatorC4×C7C22×C3×C4
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1161C2
cosocleC2×C4×C7C23×C3
hypercenterC3×#103D7
hypocenterC7C3×#417
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

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×#103C2semidirect
5. C3×#6409C2semidirect
6. C3×#6395C2semidirect
7. C3×#6383C2semidirect
8. #89575C3direct (N × H)
11. C3×#1161C4semidirect
24. C2×#1131C2×C3semidirect
25. D7×C4⋊C4C2×C3semidirect
26. C7×#103C2×C3semidirect
27. #6409C2×C3semidirect
28. #6395C2×C3semidirect
29. #6383C2×C3semidirect
34. C3×C7×Q8C2×C4semidirect
42. C3×D28C2×C4semidirect
51. #1161C3×C4semidirect
67. C3×#103D7semidirect
87. C7×Q8C2×C3×C4semidirect
95. D28C2×C3×C4semidirect
125. #103C3×D7semidirect
144. C3×Q8C4×D7semidirect
159. C3×C7#417semidirect
195. Q8C3×C4×D7semidirect
210. C7C3×#417semidirect
229. C3#89575direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#11612
C2×C3×#11312
C3×D7×C4⋊C42
C3×C7×#1032
C3×#64092
C3×#63952
C3×#63832
#895753
C3×#4177