Finite Group Databasebeta

← C2×C3×#6380 #13055388

Subgroups — C2×C3×#6380 #13055388

Order 1344.

420 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊D7C211yesmaximal, characteristic
2C2×C3×C7×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊Dic7C211yesmaximal, characteristic, complemented
4C3×#6380C214yesmaximal, direct product factor
5C2×#6380C311yesmaximal, characteristic, direct product factor, Hall
6C24×C3×C7C2211yescharacteristic
7C23×C3×D7C411yescharacteristic, complemented
8C22×C3×C4×C7—21no—
9C2×C3×C22⋊D7—21no—
10C2×C3×C22⋊D7—21no—
11C2×C3×C22⋊D7—21no—
12C2×C3×C22⋊D7—21no—
13C2×C3×C22⋊D7C412yescomplemented
14C2×C3×C22⋊D7C2212yes—
15C22×C3×Dic7—21no—
16C22×C3×Dic7C411yescharacteristic, complemented
17C3×C7×C22⋊C4—21no—
18C3×C7×C22⋊C4C2212yescomplemented
19C3×C22⋊Dic7—21no—
20C3×C22⋊Dic7C2212yescomplemented
21C22×C22⋊D7C2×C311yescharacteristic
22C2×C7×C22⋊C4C2×C311yescharacteristic, complemented
23C2×C22⋊Dic7C2×C311yescharacteristic, complemented
24#6380C2×C314yesdirect product factor
25C2×C3×#99—71nomaximal, Hall
26C23×C3×C7—42no—
27C23×C3×C7C2×C411yescharacteristic
28C23×C3×C7C2×C411yescharacteristic
29C23×C3×C7D411yescharacteristic
30C23×C3×C7D411yescharacteristic
31C23×C3×C7D411yescharacteristic
32C23×C3×C7D411yescharacteristic
33C23×C3×C7C2311yescharacteristic
34C22×C3×D7—21no—
35C22×C3×D7—21no—
36C22×C3×D7—42no—
37C22×C3×D7C2×C412yescomplemented
38C2×C3×C4×C7—22no—
39C2×C3×C4×C7—42no—
40C3×C22⋊D7—22no—
41C3×C22⋊D7—22no—
42C3×C22⋊D7—22no—
43C3×C22⋊D7—22no—
44C2×C3×Dic7—21no—
45C2×C3×Dic7—21no—
46C2×C3×Dic7—22no—
47C2×C3×Dic7—42no—
48C2×C3×Dic7C2×C412yescomplemented
49C24×C7C22×C311yescharacteristic
50C23×D7C3×C411yescharacteristic, complemented
51C22×C4×C7—21no—
52C2×C22⋊D7—21no—
53C2×C22⋊D7—21no—
54C2×C22⋊D7—21no—
55C2×C22⋊D7—21no—
56C2×C22⋊D7C3×C412yescomplemented
57C2×C22⋊D7C22×C312yes—
58C22×Dic7—21no—
59C22×Dic7C3×C411yescharacteristic, complemented
60C7×C22⋊C4—21no—
61C7×C22⋊C4C22×C312yescomplemented
62C22⋊Dic7—21no—
63C22⋊Dic7C22×C312yescomplemented
64C22×C3×D4—71no—
65C2×C3×C22⋊C4—71no—
66C2×C3×C22⋊C4D711yescharacteristic, complemented
67C3×#99—74no—
68C22×C3×C7—22no—
69C22×C3×C7—22no—
70C22×C3×C7—22no—
71C22×C3×C7—41no—
72C22×C3×C7—41no—
73C22×C3×C7—41no—
74C22×C3×C7—41no—
75C22×C3×C7C22⋊C411yescharacteristic
76C22×C3×C7C22⋊C411yescharacteristic
77C22×C3×C7C22⋊C411yescharacteristic
78C22×C3×C7C22⋊C411yescharacteristic
79C22×C3×C7C22×C411yescharacteristic
80C22×C3×C7C2×D411yescharacteristic
81C22×C3×C7C2×D411yescharacteristic
82C2×C3×D7—22no—
83C2×C3×D7—42no—
84C2×C3×D7—42no—
85C2×C3×D7—42no—
86C3×C4×C7—42no—
87C3×Dic7—22no—
88C3×Dic7—42no—
89C2×#99—71noSylow
90C23×C7—42no—
91C23×C7C2×C3×C411yescharacteristic
92C23×C7C2×C3×C411yescharacteristic
93C23×C7C3×D411yescharacteristic
94C23×C7C3×D411yescharacteristic
95C23×C7C3×D411yescharacteristic
96C23×C7C3×D411yescharacteristic
97C23×C7C23×C311yescharacteristic
98C22×D7—21no—
99C22×D7—21no—
100C22×D7—42no—
101C22×D7C2×C3×C412yescomplemented
102C2×C4×C7—22no—
103C2×C4×C7—42no—
104C22⋊D7—22no—
105C22⋊D7—22no—
106C22⋊D7—22no—
107C22⋊D7—22no—
108C2×Dic7—21no—
109C2×Dic7—21no—
110C2×Dic7—22no—
111C2×Dic7—42no—
112C2×Dic7C2×C3×C412yescomplemented
113C24×C3—71no—
114C24×C3C2×D711yescharacteristic
115C2×C3×D4—72no—
116C2×C3×D4—72no—
117C2×C3×D4—141no—
118C2×C3×D4—141no—
119C2×C3×D4—141no—
120C2×C3×D4—141no—
121C22×C3×C4—21no—
122C22×C3×C4—71no—
123C22×C3×C4—141no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4—72no—
126C3×C22⋊C4—141no—
127C3×C22⋊C4C2×D712yescomplemented
128C2×C3×C7—21no—
129C2×C3×C7—21no—
130C2×C3×C7—21no—
131C2×C3×C7—21no—
132C2×C3×C7—21no—
133C2×C3×C7—21no—
134C2×C3×C7#9912yescomplemented
135C2×C3×C7C2×C22⋊C411yescharacteristic
136C3×D7—42no—
137C22×D4—71no—
138C2×C22⋊C4—71no—
139C2×C22⋊C4C3×D711yescharacteristic, complemented
140#99—74no—
141C22×C7—22no—
142C22×C7—22no—
143C22×C7—22no—
144C22×C7—41no—
145C22×C7—41no—
146C22×C7—41no—
147C22×C7—41no—
148C22×C7C3×C22⋊C411yescharacteristic
149C22×C7C3×C22⋊C411yescharacteristic
150C22×C7C3×C22⋊C411yescharacteristic
151C22×C7C3×C22⋊C411yescharacteristic
152C22×C7C22×C3×C411yescharacteristic
153C22×C7C2×C3×D411yescharacteristic
154C22×C7C2×C3×D411yescharacteristic
155C2×D7—22no—
156C2×D7—42no—
157C2×D7—42no—
158C2×D7—42no—
159C4×C7—42no—
160Dic7—22no—
161Dic7—42no—
162C23×C3—42no—
163C23×C3—72no—
164C23×C3—141no—
165C23×C3—141no—
166C23×C3—282no—
167C23×C3C4×D711yescharacteristic
168C23×C3C4×D711yescharacteristic
169C23×C3D2811yescharacteristic
170C23×C3D2811yescharacteristic
171C23×C3C22⋊D711yescharacteristic
172C23×C3C22⋊D711yescharacteristic
173C23×C3C22×D711yescharacteristic
174C3×D4—142no—
175C3×D4—142no—
176C3×D4—142no—
177C3×D4—142no—
178C2×C3×C4—22no—
179C2×C3×C4—42no—
180C2×C3×C4—72no—
181C2×C3×C4—141no—
182C2×C3×C4—141no—
183C2×C3×C4—142no—
184C2×C3×C4—282no—
185C3×C7C2×#9911yescharacteristic, complemented, Hall
186C24—71no—
187C24C2×C3×D711yescharacteristic
188C2×D4—72no—
189C2×D4—72no—
190C2×D4—141no—
191C2×D4—141no—
192C2×D4—141no—
193C2×D4—141no—
194C22×C4—21no—
195C22×C4—71no—
196C22×C4—141no—
197C22⋊C4—21no—
198C22⋊C4—72no—
199C22⋊C4—141no—
200C22⋊C4C2×C3×D712yescomplemented
201C2×C7—21no—
202C2×C7—21no—
203C2×C7—21no—
204C2×C7—21no—
205C2×C7—21no—
206C2×C7—21no—
207C2×C7C3×#9912yescomplemented
208C2×C7C2×C3×C22⋊C411yescharacteristic
209D7—42no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3—142no—
218C22×C3—282no—
219C22×C3—282no—
220C22×C3—282no—
221C22×C3(C2×D7)⋊C411yescharacteristic
222C22×C3(C2×D7)⋊C411yescharacteristic
223C22×C3(C2×D7)⋊C411yescharacteristic
224C22×C3(C2×D7)⋊C411yescharacteristic
225C22×C3C2×C4×D711yescharacteristic
226C22×C3C2×D2811yescharacteristic
227C22×C3C2×C22⋊D711yescharacteristic
228C3×C4—42no—
229C3×C4—142no—
230C3×C4—282no—
231C23—42no—
232C23—72no—
233C23—141no—
234C23—141no—
235C23—282no—
236C23C3×C4×D711yescharacteristic
237C23C3×C4×D711yescharacteristic
238C23C3×D2811yescharacteristic
239C23C3×D2811yescharacteristic
240C23C3×C22⋊D711yescharacteristic
241C23C3×C22⋊D711yescharacteristic
242C23C22×C3×D711yescharacteristic
243D4—142no—
244D4—142no—
245D4—142no—
246D4—142no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—72no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—142no—
253C2×C4—282no—
254C7C2×C3×#9911yescharacteristic, complemented, Sylow
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—21no—
259C2×C3—21no—
260C2×C3—21no—
261C2×C3—282no—
262C2×C3#638012yesdirect product factor
263C2×C3C2×(C2×D7)⋊C411yescharacteristic
264C22—22no—
265C22—22no—
266C22—22no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—142no—
272C22—282no—
273C22—282no—
274C22—282no—
275C22C3×(C2×D7)⋊C411yescharacteristic
276C22C3×(C2×D7)⋊C411yescharacteristic
277C22C3×(C2×D7)⋊C411yescharacteristic
278C22C3×(C2×D7)⋊C411yescharacteristic
279C22C2×C3×C4×D711yescharacteristic
280C22C2×C3×D2811yescharacteristic
281C22C2×C3×C22⋊D711yescharacteristic
282C4—42no—
283C4—142no—
284C4—282no—
285C3C2×#638011yescharacteristic, direct product factor, Sylow
286C2—21no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—282no—
293C2C3×#638012yesdirect product factor
294C2C2×C3×(C2×D7)⋊C411yescharacteristic
subgroupG/H
centerC22×C3(C2×D7)⋊C4
commutatorC22×C7C22×C3×C4
FrattiniC23C22×C3×D7
FittingC2×C3×C7×C22⋊C4C2
socleC22×C3×C7C22⋊C4
prime radicalC22×C3×C22⋊D7C2
cosocleC23×C7C23×C3
hypercenterC2×C3×C22⋊C4D7
hypocenterC7C2×C3×#99
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C7×C22⋊C4C2semidirect
3. C2×C3×C22⋊Dic7C2semidirect
4. C3×#6380C2direct (N × H)
5. C2×#6380C3direct (N × H)
7. C23×C3×D7C4semidirect
13. C2×C3×C22⋊D7C4semidirect
16. C22×C3×Dic7C4semidirect
18. C3×C7×C22⋊C4C22semidirect
20. C3×C22⋊Dic7C22semidirect
22. C2×C7×C22⋊C4C2×C3semidirect
23. C2×C22⋊Dic7C2×C3semidirect
24. #6380C2×C3direct (N × H)
37. C22×C3×D7C2×C4semidirect
48. C2×C3×Dic7C2×C4semidirect
50. C23×D7C3×C4semidirect
56. C2×C22⋊D7C3×C4semidirect
59. C22×Dic7C3×C4semidirect
61. C7×C22⋊C4C22×C3semidirect
63. C22⋊Dic7C22×C3semidirect
66. C2×C3×C22⋊C4D7semidirect
101. C22×D7C2×C3×C4semidirect
112. C2×Dic7C2×C3×C4semidirect
127. C3×C22⋊C4C2×D7semidirect
134. C2×C3×C7#99semidirect
139. C2×C22⋊C4C3×D7semidirect
185. C3×C7C2×#99semidirect
200. C22⋊C4C2×C3×D7semidirect
207. C2×C7C3×#99semidirect
254. C7C2×C3×#99semidirect
262. C2×C3#6380direct (N × H)
285. C3C2×#6380direct (N × H)
293. C2C3×#6380direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊D72
C2×C3×C7×C22⋊C42
C2×C3×C22⋊Dic72
C3×#63802
C2×#63803
C2×C3×#997