Finite Group Databasebeta

← C2×#20290613 #28319156

Subgroups — C2×#20290613 #28319156

Order 13568.

1566 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 280 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×#16811566C212yesmaximal
2#20290613C218yesmaximal, direct product factor
3C2×#16811501C214yesmaximal
4C22×#13133175C211yesmaximal, characteristic
5C22×C22⋊#6157C2214yes—
6#16811566—24no—
7#16811566C2218yes—
8C2×(C2×#6157)⋊C4—24no—
9#16811501C22116yes—
10C23×C4×D53C2211yescharacteristic
11C2×#13133175—24no—
12C2×#13133175—24no—
13C2×#13133175C414yescomplemented
14C2×#13133175C2214yes—
15C22×C4⋊Dic53C411yescharacteristic, complemented
16C22×#11719921C2212yes—
17C22×C4×Dic53C412yescomplemented
18C22×C4⋊C4×C53C411yescharacteristic, complemented
19C23×#6157—24no—
20C2×C22⋊#6157—216no—
21C2×C22⋊#6157—24no—
22C2×C22⋊#6157C2318yes—
23(C2×#6157)⋊C4—216no—
24C24×D53C2311yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C412yes—
27C22×C4×D53—28no—
28C22×C4×D53C2×C412yes—
29C22×C4×D53D414yes—
30C22×C4×D53D414yes—
31C22×C4×D53C2312yes—
32#13133175—216no—
33#13133175—216no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53C2×C414yescomplemented
37C2×#11719921—24no—
38C2×#11719921—24no—
39C2×#11719921C2×C414yes—
40C2×#11719921C2314yes—
41C2×C4×Dic53—24no—
42C2×C4×Dic53—24no—
43C2×C4×Dic53C2×C418yescomplemented
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53C2×C412yes—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53—22no—
48C2×C4⋊C4×C53C2×C414yescomplemented
49C24×C53C22×C411yescharacteristic
50C23×D53—42no—
51C23×D53—42no—
52C23×D53C22×C411yescharacteristic
53C23×D53C2×D412yes—
54C23×D53C2×D412yes—
55C23×D53#4118yes—
56C23×D53C2411yescharacteristic
57C22×#6157—28no—
58C22×#6157—48no—
59C22×#6157—48no—
60C22×#6157—48no—
61C22×C4×C53—24no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—44no—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22×C412yes—
68C22×C4×C53C22×C414yes—
69C22×Dic53—24no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—44no—
73C22×Dic53C22⋊C412yes—
74C22×Dic53C22⋊C412yes—
75C22×Dic53C22×C412yes—
76C22×Dic53C22×C414yes—
77C2×C4×D53—28no—
78C2×C4×D53—216no—
79C2×C4×D53—216no—
80C2×C4×D53—28no—
81C2×C4×D53C22⋊C418yes—
82C2×C4×D53C2×D418yes—
83C22⋊#6157—216no—
84C22⋊#6157—48no—
85C4⋊C4×C53—28no—
86#11719921—28no—
87#11719921—28no—
88C4⋊Dic53—28no—
89C4×Dic53—216no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53C2×C22⋊C411yescharacteristic
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53#11714yes—
95C23×C53C23×C411yescharacteristic
96C22×D53—24no—
97C22×D53—24no—
98C22×D53—41no—
99C22×D53—41no—
100C22×D53—42no—
101C22×D53—42no—
102C22×D53—416no—
103C22×D53—41no—
104C22×D53—41no—
105C22×D53C2×C22⋊C412yes—
106C22×D53#11714yes—
107C22×D53#12318yes—
108C22×D53#12318yes—
109C22×D53C22×D412yes—
110C22×D53C2×#4114yes—
111C2×#6157—48no—
112C2×#6157—48no—
113C2×#6157—48no—
114C2×#6157—816no—
115C2×C4×C53—24no—
116C2×C4×C53—24no—
117C2×C4×C53—24no—
118C2×C4×C53—24no—
119C2×C4×C53—44no—
120C2×C4×C53—44no—
121C2×C4×C53—44no—
122C2×C4×C53C2×C22⋊C414yes—
123C2×Dic53—24no—
124C2×Dic53—24no—
125C2×Dic53—24no—
126C2×Dic53—24no—
127C2×Dic53—44no—
128C2×Dic53—44no—
129C2×Dic53—44no—
130C2×Dic53C2×C22⋊C414yes—
131C4×D53—216no—
132C4×D53—216no—
133C2×#1820—531nomaximal, Sylow
134C22×C53—28no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53#38112yes—
140C22×C53#38112yes—
141C22×C53#41012yes—
142C22×C53#41012yes—
143C22×C53C22×C22⋊C411yescharacteristic
144C22×C53C2×#11712yes—
145C2×D53—216no—
146C2×D53—22no—
147C2×D53—22no—
148C2×D53—22no—
149C2×D53—22no—
150C2×D53—44no—
151C2×D53—48no—
152C2×D53—44no—
153C2×D53#38114yes—
154C2×D53C2×#12314yes—
155#6157—88no—
156C4×C53—24no—
157C4×C53—44no—
158Dic53—24no—
159Dic53—44no—
160C22×#117—531no—
161C2×#381—532no—
162#1820—538no—
163C2×#342—534no—
164C2×C53—22no—
165C2×C53—22no—
166C2×C53#182014yescomplemented
167C2×C53C2×#38111yescharacteristic
168C2×C53C2×#41011yescharacteristic
169C2×C53C2×#41011yescharacteristic
170D53—24no—
171D53—24no—
172C24×C4—531no—
173C2×#117—534no—
174C2×#117—534no—
175C2×#117—1064no—
176C2×#117—1064no—
177C22×C4⋊C4—531no—
178C22×C4⋊C4#615711yescharacteristic, complemented
179C22×C22⋊C4—532no—
180C22×C22⋊C4—534no—
181C22×C42—532no—
182#381—538no—
183#381—1064no—
184C2×#95—1064no—
185#342—5316no—
186C53C2×#182011yescharacteristic, complemented, Sylow
187C25—531no—
188C23×C4—531no—
189C23×C4—534no—
190C23×C4—534no—
191C23×C4—532no—
192C23×C4—532no—
193C23×C4—532no—
194C23×C4—1068no—
195C23×C4—1064no—
196C23×C4C2×#615712yes—
197C23×C4C2×#615711yescharacteristic
198#117—10616no—
199#117—10616no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—534no—
203C2×C4⋊C4—1062no—
204C2×C4⋊C4—1062no—
205C2×C4⋊C4C2×#615714yescomplemented
206C2×C22⋊C4—534no—
207C2×C22⋊C4—534no—
208C2×C22⋊C4—538no—
209C2×C22⋊C4—1064no—
210C2×C22⋊C4—1064no—
211C2×C22⋊C4—10616no—
212C2×C22⋊C4—1064no—
213C2×C42—538no—
214C2×C42—1064no—
215C2×C42—1064no—
216#95—10616no—
217C24—532no—
218C24—532no—
219C24—538no—
220C24—531no—
221C24—531no—
222C24—2122no—
223C24—2122no—
224C24C22×#615711yescharacteristic
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—44no—
229C22×C4—532no—
230C22×C4—532no—
231C22×C4—532no—
232C22×C4—538no—
233C22×C4—538no—
234C22×C4—534no—
235C22×C4—10616no—
236C22×C4—10616no—
237C22×C4—1068no—
238C22×C4—1068no—
239C22×C4—1064no—
240C22×C4—1062no—
241C22×C4—1062no—
242C22×C4—1068no—
243C22×C4—2128no—
244C22×C4—2128no—
245C22×C4—2128no—
246C22×C4—2124no—
247C22×C4C22⋊#615712yes—
248C22×C4C22⋊#615712yes—
249C22×C4C22×#615712yes—
250C22×C4C22×#615714yes—
251C4⋊C4—28no—
252C4⋊C4—1068no—
253C22⋊C4—1068no—
254C22⋊C4—1068no—
255C22⋊C4—10616no—
256C22⋊C4—2128no—
257C42—10616no—
258C23—22no—
259C23—22no—
260C23—538no—
261C23—532no—
262C23—532no—
263C23—538no—
264C23—534no—
265C23—534no—
266C23—1064no—
267C23—1064no—
268C23—21216no—
269C23—2122no—
270C23—2121no—
271C23—2121no—
272C23—2122no—
273C23—2121no—
274C23—2121no—
275C23C2×C22⋊#615711yescharacteristic
276C23C2×C22⋊#615711yescharacteristic
277C23#1313329614yes—
278C23C23×#615711yescharacteristic
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—534no—
287C2×C4—10616no—
288C2×C4—10616no—
289C2×C4—1064no—
290C2×C4—1064no—
291C2×C4—1064no—
292C2×C4—1064no—
293C2×C4—42416no—
294C2×C4—2128no—
295C2×C4—2128no—
296C2×C4—2128no—
297C2×C4—2124no—
298C2×C4—2124no—
299C2×C4—2124no—
300C2×C4C2×C22⋊#615714yes—
301C22—28no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—534no—
307C22—534no—
308C22—10616no—
309C22—1062no—
310C22—1062no—
311C22—1062no—
312C22—1062no—
313C22—2128no—
314C22—2124no—
315C22—2124no—
316C22#1681156612yes—
317C22#1681156612yes—
318C22#1681160312yes—
319C22#1681160412yes—
320C22C22×C22⋊#615711yescharacteristic
321C22C2×#1313329612yes—
322C4—24no—
323C4—44no—
324C4—1064no—
325C4—4248no—
326C4—2124no—
327C2—22no—
328C2—22no—
329C2—1064no—
330C2—1064no—
331C2#2029061314yesdirect product factor
332C2C2×#1681156611yescharacteristic
333C2C2×#1681160311yescharacteristic
334C2C2×#1681160411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C23
cosocleC23×D53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53C2×#1820
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20290613C2direct (N × H)
13. C2×#13133175C4semidirect
15. C22×C4⋊Dic53C4semidirect
17. C22×C4×Dic53C4semidirect
18. C22×C4⋊C4×C53C4semidirect
36. C2×C4⋊Dic53C2×C4semidirect
43. C2×C4×Dic53C2×C4semidirect
48. C2×C4⋊C4×C53C2×C4semidirect
166. C2×C53#1820semidirect
178. C22×C4⋊C4#6157semidirect
186. C53C2×#1820semidirect
205. C2×C4⋊C4C2×#6157semidirect
331. C2#20290613direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115662
#202906132
C2×#168115012
C22×#131331752
C2×#182053