Finite Group Databasebeta

← C2×#20290245 #28315894

Subgroups — C2×#20290245 #28315894

Order 13568.

1182 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811646C211yesmaximal, characteristic
2C2×#16811646C211yesmaximal, characteristic
3#20290245C218yesmaximal, direct product factor
4C2×#16811482C214yesmaximal, complemented
5C2×C4⋊Q8×D53C211yesmaximal, characteristic
6#16811646—22no—
7#16811646—22no—
8#16811646C2214yes—
9#16811646C2214yes—
10C2×#13133284C2212yes—
11C2×#13133284C2212yes—
12C2×#13133274—21no—
13C2×#13133274—21no—
14C2×#13133272—21no—
15C2×#13133272—21no—
16#16811482C22116yescomplemented
17C22×Q8×D53—21no—
18C2×#13133217C411yescharacteristic
19C2×#13133216—21no—
20C2×#13133216C2212yes—
21C2×#13133215—21no—
22C4⋊Q8×D53—22no—
23C4⋊Q8×D53—22no—
24C4⋊Q8×D53C414yes—
25C4⋊Q8×D53C2214yes—
26C2×C4⋊C4×D53—21no—
27C2×C4⋊C4×D53C412yes—
28C2×C42×D53C2211yescharacteristic
29C2×C4⋊Q8×C53C411yescharacteristic
30C22×#88324—22no—
31C22×#88324—22no—
32#13133284—22no—
33#13133284—22no—
34#13133284C2314yes—
35#13133284C2314yes—
36#13133274—24no—
37#13133274—24no—
38#13133272—24no—
39#13133272—24no—
40C22×Dic106—21no—
41C22×Dic106—22no—
42C2×Q8×D53—24no—
43C2×Q8×D53—24no—
44C2×Q8×D53—48no—
45C22×C4×D53—22no—
46C22×C4×D53C2×C411yescharacteristic
47C22×C4×D53D411yescharacteristic
48C22×C4×D53D411yescharacteristic
49#13133217—22no—
50#13133217C2×C414yes—
51#13133216—24no—
52#13133216—22no—
53#13133216—22no—
54#13133216—24no—
55#13133216C2×C414yes—
56#13133216C2314yes—
57#13133215—24no—
58#13133215—24no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53C2×C412yes—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C4⋊C4×D53—24no—
64C4⋊C4×D53—24no—
65C4⋊C4×D53—24no—
66C4⋊C4×D53C2×C418yes—
67C2×C4×Dic53D411yescharacteristic
68C2×C4×Dic53D411yescharacteristic
69C2×C4×Dic53C2311yescharacteristic
70C42×D53C2×C412yes—
71C42×D53D412yes—
72C42×D53D412yes—
73C42×D53C2312yes—
74C22×Q8×C53—21no—
75C4⋊Q8×C53—22no—
76C4⋊Q8×C53C2×C414yes—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53C2×C412yes—
79C2×C42×C53C2×C411yescharacteristic
80C23×D53C22⋊C411yescharacteristic
81C2×Q8×C53—24no—
82C2×Q8×C53—44no—
83C22×C4×C53—22no—
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22×C411yescharacteristic
87C22×Dic53—22no—
88C22×Dic53C22⋊C411yescharacteristic
89C22×Dic53C2×D411yescharacteristic
90C22×Dic53C2×D411yescharacteristic
91Q8×D53—416no—
92Q8×D53—416no—
93C2×C4×D53—24no—
94C2×C4×D53—24no—
95C2×C4×D53—24no—
96C2×C4×D53—28no—
97C2×C4×D53—48no—
98C2×C4×D53C22⋊C412yes—
99C2×C4×D53C22⋊C412yes—
100C2×C4×D53C22⋊C412yes—
101C2×C4×D53C22×C412yes—
102C2×C4×D53C2×D412yes—
103C2×C4×D53C2×D412yes—
104C2×Dic106—24no—
105C2×Dic106—24no—
106C2×Dic106—24no—
107C2×Dic106—44no—
108C2×Dic106—48no—
109C2×#88324—24no—
110C2×#88324—24no—
111C2×#88324—44no—
112C2×#88324—44no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53C22×C414yes—
116C42×C53C22⋊C412yes—
117C42×C53C22×C412yes—
118C4⋊Dic53—24no—
119C4⋊Dic53—22no—
120C4⋊Dic53C22×C414yes—
121#11719919—24no—
122#11719919—28no—
123#11719919—24no—
124C4×Dic53C22⋊C412yes—
125C4×Dic53C22⋊C412yes—
126C4×Dic53C22×C411yescharacteristic
127C4×Dic53C2×D411yescharacteristic
128C4×Dic53C2×D411yescharacteristic
129C4×Dic53C2×D412yes—
130C4×Dic53C2×D412yes—
131C4×Dic53C2411yescharacteristic
132C23×C53C2×C22⋊C411yescharacteristic
133C22×D53—22no—
134C22×D53—22no—
135C22×D53#10114yes—
136C22×D53C2×C22⋊C412yes—
137Q8×C53—48no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—24no—
142C2×C4×C53—44no—
143C2×C4×C53C2×C22⋊C412yes—
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×C4×C53C2×C22⋊C412yes—
146C2×C4×C53C23×C411yescharacteristic
147C2×Dic53—22no—
148C2×Dic53—24no—
149C2×Dic53—44no—
150C2×Dic53C2×C22⋊C412yes—
151C2×Dic53C2×C22⋊C411yescharacteristic
152C2×Dic53C2×C22⋊C411yescharacteristic
153C2×Dic53AGL(1,ℤ/8)14yes—
154C2×Dic53#13714yes—
155C2×Dic53C22×D411yescharacteristic
156C2×Dic53C22×D411yescharacteristic
157C4×D53—24no—
158C4×D53—24no—
159C4×D53—24no—
160C4×D53—24no—
161C4×D53—24no—
162C4×D53—24no—
163C4×D53—416no—
164Dic106—48no—
165Dic106—48no—
166Dic106—48no—
167#88324—44no—
168#88324—44no—
169C2×#1556—531nomaximal, Sylow
170C22×C53C2×#10112yes—
171C22×C53#41812yes—
172C22×C53#41912yes—
173C22×C53C22×C22⋊C411yescharacteristic
174C2×D53—28no—
175C2×D53—22no—
176C2×D53—22no—
177C2×D53C2×#10114yes—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—22no—
181C4×C53—22no—
182C4×C53—44no—
183Dic53—22no—
184Dic53—44no—
185Dic53#41812yes—
186Dic53#41912yes—
187Dic53C2×AGL(1,ℤ/8)12yes—
188Dic53C2×#13712yes—
189C22×C4⋊Q8—531no—
190C2×#432—531no—
191C2×#432—531no—
192#1556—538no—
193C2×#332—534no—
194C2×C53#155614yescomplemented
195C2×C53C22×#10111yescharacteristic
196C2×C53C2×#41811yescharacteristic
197C2×C53C2×#41911yescharacteristic
198D53—24no—
199C23×Q8—1061no—
200C2×C4⋊Q8—534no—
201C2×C4⋊Q8—534no—
202C2×C4⋊Q8—532no—
203C2×C4⋊Q8—531no—
204C2×C4⋊Q8—1061no—
205C2×C4⋊Q8—1062no—
206C2×C4⋊Q8—1062no—
207C2×C4⋊Q8—1061no—
208C2×C4⋊Q8#615711yescharacteristic
209C22×C4⋊C4—532no—
210C22×C4⋊C4—1061no—
211C22×C42—531no—
212#432—534no—
213#432—534no—
214#432—1062no—
215#432—1062no—
216C2×C4⋊C8—532no—
217C2×C4⋊C8—532no—
218C2×C22⋊C8—1061no—
219C2×C22⋊C8—1061no—
220C2×#97—1061no—
221C2×#97—1061no—
222#332—5316no—
223C53C2×#155611yescharacteristic, complemented, Sylow
224C22×Q8—21no—
225C22×Q8—1062no—
226C22×Q8—1064no—
227C22×Q8—1064no—
228C22×Q8—1061no—
229C22×Q8—2128no—
230C23×C4—531no—
231C23×C4—531no—
232C23×C4—531no—
233C23×C4—1062no—
234C22×C8—1062no—
235C22×C8—1062no—
236C4⋊Q8—22no—
237C4⋊Q8—534no—
238C4⋊Q8—534no—
239C4⋊Q8—534no—
240C4⋊Q8—1064no—
241C4⋊Q8—1062no—
242C4⋊Q8—1062no—
243C4⋊Q8—1064no—
244C4⋊Q8—1064no—
245C4⋊Q8—1064no—
246C4⋊Q8—1062no—
247C4⋊Q8C2×#615714yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—532no—
250C2×C4⋊C4—538no—
251C2×C4⋊C4—1064no—
252C2×C4⋊C4—1064no—
253C2×C4⋊C4—1064no—
254C2×C4⋊C4—1062no—
255C2×C4⋊C4—1062no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4C2×#615712yes—
258C2×C42—531no—
259C2×C42—531no—
260C2×C42—531no—
261C2×C42—532no—
262C2×C42—532no—
263C2×C42—532no—
264C2×C42—532no—
265C2×C42C2×#615711yescharacteristic
266C4⋊C8—534no—
267C4⋊C8—534no—
268C4⋊C8—1062no—
269C4⋊C8—1062no—
270C22⋊C8—1064no—
271C22⋊C8—1064no—
272#97—1064no—
273#97—1064no—
274C24—531no—
275C2×Q8—24no—
276C2×Q8—44no—
277C2×Q8—1064no—
278C2×Q8—1064no—
279C2×Q8—1064no—
280C2×Q8—2124no—
281C2×Q8—2128no—
282C2×Q8—21216no—
283C2×Q8—21216no—
284C22×C4—22no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—532no—
289C22×C4—532no—
290C22×C4—532no—
291C22×C4—532no—
292C22×C4—532no—
293C22×C4—532no—
294C22×C4—1064no—
295C22×C4—1064no—
296C22×C4—1064no—
297C22×C4—1068no—
298C22×C4—1062no—
299C22×C4—2128no—
300C22×C4C22⋊#615711yescharacteristic
301C22×C4C22⋊#615711yescharacteristic
302C22×C4C22×#615711yescharacteristic
303C2×C8—1064no—
304C2×C8—1064no—
305C2×C8—2124no—
306C2×C8—2124no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—534no—
310C4⋊C4—1068no—
311C4⋊C4—1064no—
312C4⋊C4—1064no—
313C4⋊C4—1062no—
314C4⋊C4—1064no—
315C4⋊C4C22×#615714yes—
316C42—532no—
317C42—532no—
318C42—531no—
319C42—531no—
320C42—532no—
321C42—532no—
322C42—531no—
323C42—531no—
324C42C22⋊#615712yes—
325C42C22×#615712yes—
326C23—534no—
327C23—532no—
328C23—1062no—
329C23—1062no—
330C23C2×C22⋊#615711yescharacteristic
331Q8—48no—
332Q8—2128no—
333Q8—2128no—
334Q8—2128no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—44no—
340C2×C4—534no—
341C2×C4—534no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—532no—
347C2×C4—1064no—
348C2×C4—1064no—
349C2×C4—1064no—
350C2×C4—1064no—
351C2×C4—1064no—
352C2×C4—1064no—
353C2×C4—1062no—
354C2×C4—1064no—
355C2×C4—21216no—
356C2×C4—2124no—
357C2×C4C2×C22⋊#615711yescharacteristic
358C2×C4C2×C22⋊#615712yes—
359C2×C4C2×C22⋊#615712yes—
360C2×C4C23×#615711yescharacteristic
361C8—2124no—
362C8—2124no—
363C22—534no—
364C22—1062no—
365C22—1062no—
366C22—1068no—
367C22D53⋊#10112yes—
368C22#1681161512yes—
369C22#1681161612yes—
370C22C22×C22⋊#615711yescharacteristic
371C4—22no—
372C4—22no—
373C4—21no—
374C4—21no—
375C4—44no—
376C4—532no—
377C4—532no—
378C4—532no—
379C4—532no—
380C4—1062no—
381C4—2124no—
382C2—1064no—
383C2#2029024514yesdirect product factor
384C2C2×D53⋊#10111yescharacteristic
385C2C2×#1681161511yescharacteristic
386C2C2×#1681161611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×C4⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC4×Dic53C24
hypercenterC2×C4⋊Q8#6157
hypocenterC53C2×#1556
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20290245C2direct (N × H)
4. C2×#16811482C2semidirect
16. #16811482C22semidirect
194. C2×C53#1556semidirect
223. C53C2×#1556semidirect
383. C2#20290245direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168116462
#202902452
C2×#168114822
C2×C4⋊Q8×D532
C2×#155653