Finite Group Databasebeta

← C2×A4×C22⋊D7 #13062181

Subgroups — C2×A4×C22⋊D7 #13062181

Order 1344.

634 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C23×C7×A4C211yesmaximal, characteristic, complemented
2C22×A4×D7C211yesmaximal, characteristic, complemented
3A4×C22⋊D7C214yesmaximal, direct product factor
4C2×A4×Dic7C211yesmaximal, characteristic, complemented
5C23×C22⋊D7C311yesmaximal, characteristic, complemented, Hall
6C22×C7×A4—22no—
7C22×C7×A4C2211yescharacteristic
8C22×C7×A4C2212yescomplemented
9C2×A4×D7—22no—
10C2×A4×D7C2212yescomplemented
11C2×C3×C22⋊D7—41nomaximal
12A4×Dic7C2212yescomplemented
13C25×C7C2×C311yescharacteristic, complemented
14C24×D7C2×C311yescharacteristic, complemented
15C22×C22⋊D7—31no—
16C22×C22⋊D7—31no—
17C22×C22⋊D7—31no—
18C22×C22⋊D7—31no—
19C22×C22⋊D7—34no—
20C22×C22⋊D7C2×C314yescomplemented
21C23×Dic7C2×C311yescharacteristic, complemented
22C2×D4×A4—71nomaximal, Hall
23C23×C3×C7—41no—
24C22×C3×D7—41no—
25C2×C7×A4—22no—
26C2×C7×A4D412yescomplemented
27C2×C7×A4C2311yescharacteristic
28A4×D7—22no—
29C3×C22⋊D7—44no—
30C2×C3×Dic7—41no—
31C24×C7—22no—
32C24×C7—31no—
33C24×C7—32no—
34C24×C7—31no—
35C24×C7—62no—
36C24×C7C22×C311yescharacteristic
37C24×C7C22×C312yescomplemented
38C23×D7—22no—
39C23×D7—31no—
40C23×D7—31no—
41C23×D7—32no—
42C23×D7—62no—
43C23×D7C22×C312yescomplemented
44C2×C22⋊D7—34no—
45C2×C22⋊D7—34no—
46C2×C22⋊D7—32no—
47C2×C22⋊D7—34no—
48C2×C22⋊D7—31no—
49C2×C22⋊D7—31no—
50C2×C22⋊D7—34no—
51C2×C22⋊D7—34no—
52C2×C22⋊D7—34no—
53C2×C22⋊D7—34no—
54C2×C22⋊D7—34no—
55C2×C22⋊D7—31no—
56C2×C22⋊D7A411yescharacteristic, direct product factor
57C22×Dic7—31no—
58C22×Dic7—31no—
59C22×Dic7—32no—
60C22×Dic7C22×C312yescomplemented
61C23×A4—71no—
62C23×A4D711yescharacteristic, complemented
63D4×A4—74no—
64C2×C4×A4—71no—
65C22×C3×C7—41no—
66C22×C3×C7—42no—
67C22×C3×C7—82no—
68C2×C3×D7—42no—
69C2×C3×D7—82no—
70C7×A4C2×D411yescharacteristic, complemented
71C3×Dic7—42no—
72C23×D4—71noSylow
73C23×C7—22no—
74C23×C7—31no—
75C23×C7—32no—
76C23×C7—31no—
77C23×C7—32no—
78C23×C7—32no—
79C23×C7—31no—
80C23×C7—32no—
81C23×C7—32no—
82C23×C7—62no—
83C23×C7—62no—
84C23×C7—62no—
85C23×C7—64no—
86C23×C7—62no—
87C23×C7—62no—
88C23×C7—64no—
89C23×C7C3×D412yescomplemented
90C23×C7C2×A411yescharacteristic, complemented
91C23×C7C23×C311yescharacteristic
92C22×D7—22no—
93C22×D7—32no—
94C22×D7—31no—
95C22×D7—32no—
96C22×D7—32no—
97C22×D7—32no—
98C22×D7—62no—
99C22×D7—62no—
100C22×D7—64no—
101C22×D7—64no—
102C22×D7—62no—
103C22×D7—62no—
104C22×D7—62no—
105C22×D7C2×A411yescharacteristic, complemented
106C22⋊D7—38no—
107C22⋊D7—34no—
108C22⋊D7—34no—
109C22⋊D7—34no—
110C22⋊D7C2×A414yesdirect product factor
111C2×Dic7—32no—
112C2×Dic7—31no—
113C2×Dic7—32no—
114C2×Dic7—32no—
115C2×Dic7—32no—
116C2×Dic7C2×A411yescharacteristic, complemented
117C22×A4—22no—
118C22×A4—72no—
119C22×A4—142no—
120C22×A4C2×D711yescharacteristic
121C22×A4C2×D712yescomplemented
122C2×C3×D4—281no—
123C4×A4—72no—
124C2×C3×C7—42no—
125C2×C3×C7—41no—
126C2×C3×C7—82no—
127C3×D7—82no—
128C25—71no—
129C25C3×D711yescharacteristic, complemented
130C22×D4—74no—
131C22×D4—211no—
132C22×D4—211no—
133C22×D4—214no—
134C22×D4—211no—
135C22×D4—211no—
136C23×C4—71no—
137C22×C7—22no—
138C22×C7—32no—
139C22×C7—31no—
140C22×C7—32no—
141C22×C7—32no—
142C22×C7—32no—
143C22×C7—31no—
144C22×C7—31no—
145C22×C7—32no—
146C22×C7—62no—
147C22×C7—64no—
148C22×C7—62no—
149C22×C7—62no—
150C22×C7—62no—
151C22×C7—64no—
152C22×C7—62no—
153C22×C7C2×C3×D411yescharacteristic, complemented
154C22×C7C22×A411yescharacteristic
155C22×C7C22×A412yescomplemented
156C2×D7—22no—
157C2×D7—32no—
158C2×D7—62no—
159C2×D7—64no—
160C2×D7—62no—
161C2×D7—64no—
162C2×D7—62no—
163C2×D7—62no—
164C2×D7—62no—
165C2×D7C22×A412yescomplemented
166Dic7—32no—
167Dic7C22×A412yescomplemented
168C23×C3—41no—
169C23×C3—281no—
170C2×A4—22no—
171C2×A4—142no—
172C2×A4C22⋊D712yesdirect product factor
173C2×A4C22×D711yescharacteristic
174C3×D4—284no—
175C2×C3×C4—281no—
176C3×C7—41noHall
177C24—22no—
178C24—32no—
179C24—31no—
180C24—31no—
181C24—62no—
182C24—72no—
183C24—142no—
184C24—211no—
185C24—212no—
186C24—211no—
187C24—422no—
188C24C2×C3×D711yescharacteristic
189C24C2×C3×D712yescomplemented
190C2×D4—71no—
191C2×D4—211no—
192C2×D4—214no—
193C2×D4—214no—
194C2×D4—211no—
195C2×D4—214no—
196C2×D4—214no—
197C2×D4—214no—
198C2×D4—212no—
199C2×D4—211no—
200C2×D4—214no—
201C2×D4—214no—
202C2×D4—214no—
203C22×C4—72no—
204C22×C4—211no—
205C22×C4—212no—
206C22×C4—211no—
207C2×C7—22no—
208C2×C7—32no—
209C2×C7—31no—
210C2×C7—31no—
211C2×C7—62no—
212C2×C7D4×A412yescomplemented
213C2×C7C23×A411yescharacteristic
214D7—22no—
215D7—62no—
216C22×C3—41no—
217C22×C3—42no—
218C22×C3—82no—
219C22×C3—282no—
220C22×C3—562no—
221A4C2×C22⋊D711yescharacteristic, direct product factor
222C3×C4—282no—
223C23—22no—
224C23—31no—
225C23—32no—
226C23—31no—
227C23—32no—
228C23—32no—
229C23—31no—
230C23—32no—
231C23—32no—
232C23—62no—
233C23—62no—
234C23—62no—
235C23—64no—
236C23—62no—
237C23—62no—
238C23—64no—
239C23—71no—
240C23—142no—
241C23—212no—
242C23—212no—
243C23—212no—
244C23—212no—
245C23—211no—
246C23—422no—
247C23—422no—
248C23—422no—
249C23—424no—
250C23—422no—
251C23—422no—
252C23—424no—
253C23C3×C22⋊D712yescomplemented
254C23A4×D711yescharacteristic, complemented
255C23C22×C3×D711yescharacteristic
256D4—74no—
257D4—214no—
258D4—214no—
259D4—218no—
260D4—214no—
261C2×C4—71no—
262C2×C4—212no—
263C2×C4—212no—
264C2×C4—212no—
265C2×C4—211no—
266C2×C4—212no—
267C7C2×D4×A411yescharacteristic, complemented, Sylow
268C2×C3—42no—
269C2×C3—41no—
270C2×C3—82no—
271C2×C3—562no—
272C22—22no—
273C22—32no—
274C22—31no—
275C22—32no—
276C22—32no—
277C22—32no—
278C22—31no—
279C22—31no—
280C22—32no—
281C22—62no—
282C22—64no—
283C22—62no—
284C22—62no—
285C22—62no—
286C22—64no—
287C22—62no—
288C22—72no—
289C22—142no—
290C22—212no—
291C22—422no—
292C22—422no—
293C22—424no—
294C22—422no—
295C22—422no—
296C22—424no—
297C22—422no—
298C22C2×C3×C22⋊D711yescharacteristic, complemented
299C22C2×A4×D711yescharacteristic
300C22C2×A4×D712yescomplemented
301C4—72no—
302C4—212no—
303C3—41noSylow
304C2—22no—
305C2—31no—
306C2—32no—
307C2—31no—
308C2—62no—
309C2—142no—
310C2—422no—
311C2A4×C22⋊D712yesdirect product factor
312C2C22×A4×D711yescharacteristic
subgroupG/H
centerC22C2×A4×D7
commutator, cosocleC23×C7C23×C3
FrattiniC2C22×A4×D7
FittingC25×C7C2×C3
socleC24×C7C22×C3
hypercenterC23A4×D7
hypocenterC22×C7C2×C3×D4
LCSG ▷ C23×C7 ▷ C22×C7
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C7×A4C2semidirect
2. C22×A4×D7C2semidirect
3. A4×C22⋊D7C2direct (N × H)
4. C2×A4×Dic7C2semidirect
5. C23×C22⋊D7C3semidirect
8. C22×C7×A4C22semidirect
10. C2×A4×D7C22semidirect
12. A4×Dic7C22semidirect
13. C25×C7C2×C3semidirect
14. C24×D7C2×C3semidirect
20. C22×C22⋊D7C2×C3semidirect
21. C23×Dic7C2×C3semidirect
26. C2×C7×A4D4semidirect
37. C24×C7C22×C3semidirect
43. C23×D7C22×C3semidirect
56. C2×C22⋊D7A4direct (N × H)
60. C22×Dic7C22×C3semidirect
62. C23×A4D7semidirect
70. C7×A4C2×D4semidirect
89. C23×C7C3×D4semidirect
90. C23×C7C2×A4semidirect
105. C22×D7C2×A4semidirect
110. C22⋊D7C2×A4direct (N × H)
116. C2×Dic7C2×A4semidirect
121. C22×A4C2×D7semidirect
129. C25C3×D7semidirect
153. C22×C7C2×C3×D4semidirect
155. C22×C7C22×A4semidirect
165. C2×D7C22×A4semidirect
167. Dic7C22×A4semidirect
172. C2×A4C22⋊D7direct (N × H)
189. C24C2×C3×D7semidirect
212. C2×C7D4×A4semidirect
221. A4C2×C22⋊D7direct (N × H)
253. C23C3×C22⋊D7semidirect
254. C23A4×D7semidirect
267. C7C2×D4×A4semidirect
298. C22C2×C3×C22⋊D7semidirect
300. C22C2×A4×D7semidirect
311. C2A4×C22⋊D7direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C7×A42
C22×A4×D72
A4×C22⋊D72
C2×A4×Dic72
C23×C22⋊D73
C2×C3×C22⋊D74
C2×D4×A47