Finite Group Databasebeta

← #27275780

Subgroups — #27275780

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274805C212yesmaximal, complemented
2#20274703C211yesmaximal, characteristic
3#20273287C212yesmaximal, complemented
4#20273033C211yesmaximal, characteristic
5C2×C53×#554C211yesmaximal, characteristic
6C2×#13133132—21no—
7C2×C22⋊#88322C2212yes—
8#16810483C2212yes—
9#16810390—21no—
10#16810385—24no—
11#16810360—24no—
12#16810349—21no—
13#16810347—21no—
14C53×#554—24no—
15C53×#554—24no—
16C2×C4⋊Q8×C53C411yescharacteristic, complemented
17C2×#125×C53—21no—
18C2×C22⋊Q8×C53—21no—
19C2×C22⋊Q8×C53—21no—
20C2×C22⋊Q8×C53C411yescharacteristic, complemented
21C2×C22⋊Q8×C53C2211yescharacteristic
22C2×C4×Q8×C53C411yescharacteristic, complemented
23C2×C4×D4×C53—21no—
24C2×C4×D4×C53C2211yescharacteristic
25C22×C4⋊C4×C53C411yescharacteristic, complemented
26C22×C4⋊C4×C53C2211yescharacteristic
27C22×#88322—22no—
28C2×C4⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30#13133132—42no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133124—22no—
34C22×Q8×C53—21no—
35C22×Q8×C53C2×C411yescharacteristic
36C22×D4×C53D411yescharacteristic
37C23×C4×C53—21no—
38C23×C4×C53—21no—
39C23×C4×C53C2×C411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41C4⋊Q8×C53—22no—
42C4⋊Q8×C53—22no—
43#125×C53—28no—
44C22⋊Q8×C53—28no—
45C22⋊Q8×C53—28no—
46C22⋊Q8×C53—22no—
47C22⋊Q8×C53—22no—
48C22⋊Q8×C53—22no—
49C22⋊Q8×C53D414yes—
50C4×Q8×C53—22no—
51C4×Q8×C53—22no—
52C4×D4×C53—28no—
53C4×D4×C53—22no—
54C4×D4×C53—22no—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—22no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53C2×C411yescharacteristic
65C2×C4⋊C4×C53C2×C411yescharacteristic
66C2×C4⋊C4×C53C2×C411yescharacteristic
67C2×C4⋊C4×C53D414yes—
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53D411yescharacteristic
72C2×C22⋊C4×C53D411yescharacteristic
73C2×C42×C53—21no—
74C2×C42×C53C2×C411yescharacteristic
75C24×C53C22⋊C411yescharacteristic
76C24×C53C2×D411yescharacteristic
77C2×Q8×C53—24no—
78C2×Q8×C53—21no—
79C2×Q8×C53—42no—
80C2×Q8×C53—82no—
81C2×Q8×C53C22⋊C412yes—
82C2×D4×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53SD1612yes—
85C2×D4×C53Q1612yes—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—24no—
91C22×C4×C53—24no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53—82no—
97C22×C4×C53—82no—
98C22×C4×C53C22⋊C411yescharacteristic
99C22×C4×C53C22⋊C411yescharacteristic
100C22×C4×C53C22⋊C411yescharacteristic
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C2×D412yes—
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D411yescharacteristic
105C22×Dic53—42no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53C22⋊C412yes—
117C4⋊C4×C53C22⋊C412yes—
118C4⋊C4×C53C22⋊C412yes—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—22no—
122C22⋊C4×C53—22no—
123C42×C53—24no—
124C42×C53—21no—
125C42×C53—21no—
126C22⋊Dic53—42no—
127C22⋊Dic53—42no—
128C4⋊Dic53—44no—
129C2×#88322—44no—
130C2×#88322—42no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53#10312yes—
138C23×C53C2×C22⋊C411yescharacteristic
139C23×C53#12012yes—
140C23×C53C2×SD1611yescharacteristic
141C23×C53C2×Q1611yescharacteristic
142C23×C53#13712yes—
143Q8×C53—42no—
144Q8×C53—42no—
145Q8×C53—84no—
146D4×C53—24no—
147D4×C53—24no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—22no—
151C2×C4×C53—24no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—21no—
156C2×C4×C53—22no—
157C2×C4×C53—21no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—84no—
169C2×C4×C53—84no—
170C2×C4×C53#10312yes—
171C2×C4×C53C2×C22⋊C411yescharacteristic
172C2×C4×C53C2×C22⋊C411yescharacteristic
173C2×C4×C53#12012yes—
174C2×Dic53—82no—
175C2×Dic53—82no—
176C2×Dic53—82no—
177#88322—82no—
178#8380—531nomaximal, Sylow
179C22×C53—22no—
180C22×C53—21no—
181C22×C53—22no—
182C22×C53—21no—
183C22×C53—22no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53#37911yescharacteristic
192C22×C53C2×#10311yescharacteristic
193C22×C53#41911yescharacteristic
194C22×C53#44812yes—
195C22×C53#45112yes—
196C4×C53—22no—
197C4×C53—21no—
198C4×C53—22no—
199C4×C53—42no—
200C4×C53—42no—
201C4×C53—42no—
202C4×C53—41no—
203C4×C53—41no—
204C4×C53—42no—
205C4×C53#41012yes—
206Dic53—162no—
207C2×#554D5311yescharacteristic
208#1824—532no—
209#1799—531no—
210#1520—532no—
211#1459—531no—
212C2×C53—22no—
213C2×C53—21no—
214C2×C53—21no—
215C2×C53—22no—
216C2×C53—42no—
217C2×C53#178211yescharacteristic
218C2×C53#178711yescharacteristic
219C2×C53#179511yescharacteristic
220#554—24no—
221#554—24no—
222C2×C4⋊Q8Dic5311yescharacteristic, complemented
223C2×#125—21no—
224C2×C22⋊Q8—21no—
225C2×C22⋊Q8—21no—
226C2×C22⋊Q8Dic5311yescharacteristic, complemented
227C2×C22⋊Q8C2×D5311yescharacteristic
228C2×C4×Q8Dic5311yescharacteristic, complemented
229C2×C4×D4—21no—
230C2×C4×D4C2×D5311yescharacteristic
231C22×C4⋊C4Dic5311yescharacteristic, complemented
232C22×C4⋊C4C2×D5311yescharacteristic
233C2×#99—1061no—
234C2×C22⋊C8—532no—
235#380—532no—
236#342—1061no—
237#340—1064no—
238#328—1064no—
239#324—1061no—
240#323—1061no—
241C53#838011yescharacteristic, complemented, Sylow
242C22×Q8—21no—
243C22×Q8C2×Dic5311yescharacteristic
244C22×D4C22⋊D5311yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×Dic5311yescharacteristic
248C23×C4C22×D5311yescharacteristic
249C22×C8—1062no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252#125—28no—
253C22⋊Q8—28no—
254C22⋊Q8—28no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8—22no—
258C22⋊Q8C22⋊D5314yes—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×D4—28no—
262C4×D4—22no—
263C4×D4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—1062no—
274C2×C4⋊C4C2×Dic5311yescharacteristic
275C2×C4⋊C4C2×Dic5311yescharacteristic
276C2×C4⋊C4C2×Dic5311yescharacteristic
277C2×C4⋊C4C22⋊D5311yescharacteristic
278C2×C4⋊C4C22⋊D5314yes—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—1062no—
282C2×C22⋊C4C22⋊D5311yescharacteristic
283C2×C22⋊C4C22⋊D5311yescharacteristic
284C2×C42—21no—
285C2×C42C2×Dic5311yescharacteristic
286#99—2122no—
287C22⋊C8—1062no—
288C22⋊C8—1062no—
289#95—1062no—
290C24C22⋊Dic5311yescharacteristic
291C24C2×C22⋊D5311yescharacteristic
292C2×Q8—24no—
293C2×Q8—21no—
294C2×Q8—42no—
295C2×Q8—82no—
296C2×Q8C22⋊Dic5312yes—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4#1171992412yes—
300C2×D4#1171992512yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—2122no—
314C22×C4C22⋊Dic5311yescharacteristic
315C22×C4C22⋊Dic5311yescharacteristic
316C22×C4C22⋊Dic5311yescharacteristic
317C22×C4C22×Dic5311yescharacteristic
318C22×C4C2×C22⋊D5312yes—
319C22×C4C2×C22⋊D5312yes—
320C22×C4C2×C22⋊D5311yescharacteristic
321C2×C8—2124no—
322C2×C8—2122no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—2124no—
334C4⋊C4C22⋊Dic5312yes—
335C4⋊C4C22⋊Dic5312yes—
336C4⋊C4C22⋊Dic5312yes—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—2122no—
342C22⋊C4—2122no—
343C42—24no—
344C42—21no—
345C42—21no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23#1313314112yes—
353C23C2×C22⋊Dic5311yescharacteristic
354C23#1313318612yes—
355C23C2×#1171992411yescharacteristic
356C23C2×#1171992511yescharacteristic
357C23#1313324812yes—
358Q8—42no—
359Q8—42no—
360Q8—84no—
361D4—24no—
362D4—24no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—4242no—
386C2×C4—4242no—
387C2×C4—4242no—
388C2×C4#1313314112yes—
389C2×C4C2×C22⋊Dic5311yescharacteristic
390C2×C4C2×C22⋊Dic5311yescharacteristic
391C2×C4#1313318612yes—
392C8—4242no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22#1681047911yescharacteristic
406C22C2×#1313314111yescharacteristic
407C22#1681063211yescharacteristic
408C22#1681076212yes—
409C22#1681077912yes—
410C4—22no—
411C4—21no—
412C4—22no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—41no—
419C4—8482no—
420C4#1681059712yes—
421C2—22no—
422C2—21no—
423C2—21no—
424C2—22no—
425C2—42no—
426C2#2027463311yescharacteristic
427C2#2027465011yescharacteristic
428C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#554C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenterC2×#554D53
hypocenterC53#8380
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #20274805C2semidirect
3. #20273287C2semidirect
16. C2×C4⋊Q8×C53C4semidirect
20. C2×C22⋊Q8×C53C4semidirect
22. C2×C4×Q8×C53C4semidirect
25. C22×C4⋊C4×C53C4semidirect
222. C2×C4⋊Q8Dic53semidirect
226. C2×C22⋊Q8Dic53semidirect
228. C2×C4×Q8Dic53semidirect
231. C22×C4⋊C4Dic53semidirect
241. C53#8380semidirect

Maximal subgroups

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

subgroupindex
#202748052
#202747032
#202732872
#202730332
C2×C53×#5542
#838053