Finite Group Databasebeta

← #27299585

Subgroups — #27299585

Order 13568.

1098 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 139 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×#16810418C212yesmaximal, complemented
2C2×#16810416C212yesmaximal, complemented
3#20274633C212yesmaximal, complemented
4C2×C53×#532C211yesmaximal, characteristic
5C2×#13133132—22no—
6C2×#13133132C2214yes—
7#16810486—22no—
8#16810418—24no—
9#16810416—24no—
10C22×#41×C53C411yescharacteristic, complemented
11C22×#41×C53C2211yescharacteristic
12C53×#532—24no—
13C53×#532—24no—
14C2×C4⋊Q8×C53C411yescharacteristic, complemented
15C2×#127×C53C2211yescharacteristic
16C2×#124×C53C411yescharacteristic, complemented
17C2×#124×C53C2211yescharacteristic
18C2×#121×C53—22no—
19C2×C4×D4×C53—22no—
20C22×C42×C53C411yescharacteristic, complemented
21C2×C22⋊Dic53—24no—
22#13133132—24no—
23#13133132—24no—
24#13133132—44no—
25#13133124—44no—
26C2×#41×C53—22no—
27C2×#41×C53—22no—
28C2×#41×C53—22no—
29C2×#41×C53—22no—
30C2×#41×C53—42no—
31C2×#41×C53—42no—
32C2×#41×C53—42no—
33C2×#41×C53—42no—
34C22×Q8×C53C2×C411yescharacteristic
35C22×Q8×C53C2×C411yescharacteristic
36C22×D4×C53—21no—
37C22×D4×C53—22no—
38C22×D4×C53C2×C411yescharacteristic
39C22×D4×C53D412yes—
40C22×D4×C53C2311yescharacteristic
41C23×C4×C53—21no—
42C23×C4×C53—21no—
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53D412yes—
45C4⋊Q8×C53—22no—
46C4⋊Q8×C53—22no—
47#127×C53—22no—
48#127×C53D414yes—
49#124×C53—22no—
50#124×C53—22no—
51#124×C53—22no—
52#124×C53D414yes—
53#121×C53—216no—
54C4×D4×C53—216no—
55C2×C4⋊C4×C53—22no—
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53—22no—
58C2×C42×C53—22no—
59C2×C42×C53—22no—
60C2×C42×C53—21no—
61C2×C42×C53C2×C411yescharacteristic
62C2×C42×C53C2×C411yescharacteristic
63C24×C53—21no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53C2×D412yes—
66#41×C53—48no—
67#41×C53—48no—
68#41×C53—48no—
69#41×C53—48no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—41no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53C22⋊C412yes—
78C2×D4×C53—24no—
79C2×D4×C53—24no—
80C2×D4×C53—28no—
81C2×D4×C53—24no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—48no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53C22⋊C412yes—
92C2×D4×C53C2×D412yes—
93C2×D4×C53C2×D412yes—
94C22×C4×C53—22no—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—24no—
100C22×C4×C53—24no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—24no—
104C22×C4×C53—24no—
105C22×C4×C53—42no—
106C22×C4×C53—44no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53C22⋊C412yes—
110C22×C4×C53C22⋊C411yescharacteristic
111C22×C4×C53C22×C411yescharacteristic
112C22×Dic53—44no—
113C4⋊C4×C53—28no—
114C22⋊C4×C53—28no—
115C22⋊C4×C53—28no—
116C42×C53—24no—
117C42×C53—21no—
118C42×C53—21no—
119C42×C53C22⋊C412yes—
120C42×C53C22⋊C412yes—
121C22⋊Dic53—44no—
122C22⋊Dic53—44no—
123C23×C53—21no—
124C23×C53—22no—
125C23×C53—21no—
126C23×C53—22no—
127C23×C53—22no—
128C23×C53—42no—
129C23×C53—44no—
130C23×C53—44no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53#9912yes—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53#12014yes—
136Q8×C53—42no—
137Q8×C53—42no—
138Q8×C53—42no—
139Q8×C53—42no—
140D4×C53—42no—
141D4×C53—42no—
142D4×C53—42no—
143D4×C53—416no—
144D4×C53—42no—
145D4×C53—48no—
146D4×C53—48no—
147C2×C4×C53—24no—
148C2×C4×C53—24no—
149C2×C4×C53—24no—
150C2×C4×C53—21no—
151C2×C4×C53—22no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—48no—
161C2×C4×C53—48no—
162C2×C4×C53—48no—
163C2×C4×C53—42no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53#9912yes—
167C2×C4×C53C2×C22⋊C411yescharacteristic
168C2×C4×C53C2×C22⋊C411yescharacteristic
169C2×Dic53—84no—
170C2×Dic53—88no—
171#12695—531nomaximal, Sylow
172C22×C53—22no—
173C22×C53—24no—
174C22×C53—22no—
175C22×C53—44no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53—44no—
179C22×C53—44no—
180C22×C53—42no—
181C22×C53—42no—
182C22×C53—82no—
183C22×C53—82no—
184C22×C53—82no—
185C22×C53#37911yescharacteristic
186C22×C53C2×#9911yescharacteristic
187C22×C53C2×#9911yescharacteristic
188C4×C53—24no—
189C4×C53—22no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—42no—
193C4×C53—42no—
194C4×C53—42no—
195Dic53—164no—
196C2×#532D5311yescharacteristic
197C2×#354—532no—
198C2×#353—532no—
199#1782—532no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—44no—
203C2×C53—42no—
204C2×C53—82no—
205C2×C53#178111yescharacteristic
206C2×C53#212612yes—
207C22×#41Dic5311yescharacteristic, complemented
208C22×#41C2×D5311yescharacteristic
209#532—24no—
210#532—24no—
211C2×C4⋊Q8Dic5311yescharacteristic, complemented
212C2×#127C2×D5311yescharacteristic
213C2×#124Dic5311yescharacteristic, complemented
214C2×#124C2×D5311yescharacteristic
215C2×#121—22no—
216C2×C4×D4—22no—
217C22×C42Dic5311yescharacteristic, complemented
218C2×#99—534no—
219C2×#99—1062no—
220#381—1062no—
221#354—1064no—
222#353—1064no—
223C53#1269511yescharacteristic, complemented, Sylow
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—42no—
229C2×#41—42no—
230C2×#41—42no—
231C2×#41—42no—
232C22×Q8C2×Dic5311yescharacteristic
233C22×Q8C2×Dic5311yescharacteristic
234C22×D4—22no—
235C22×D4—21no—
236C22×D4C2×Dic5311yescharacteristic
237C22×D4C22⋊D5312yes—
238C22×D4C22×D5311yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C2×Dic5311yescharacteristic
242C23×C4C22⋊D5312yes—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245#127—22no—
246#127C22⋊D5314yes—
247#124—22no—
248#124—22no—
249#124—22no—
250#124C22⋊D5314yes—
251#121—216no—
252C4×D4—216no—
253C2×C4⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—1064no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42C2×Dic5311yescharacteristic
261C2×C42C2×Dic5311yescharacteristic
262#99—1064no—
263#99—1064no—
264#99—2124no—
265#95—2124no—
266C24—21no—
267C24C22⋊Dic5311yescharacteristic
268C24C2×C22⋊D5312yes—
269#41—48no—
270#41—48no—
271#41—48no—
272#41—48no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8C22⋊Dic5312yes—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—24no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—48no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4C22⋊Dic5312yes—
295C2×D4C2×C22⋊D5312yes—
296C2×D4C2×C22⋊D5312yes—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—2124no—
313C22×C4C22⋊Dic5312yes—
314C22×C4C22⋊Dic5311yescharacteristic
315C22×C4C22×Dic5311yescharacteristic
316C4⋊C4—28no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—2124no—
320C22⋊C4—2124no—
321C42—24no—
322C42—21no—
323C42—21no—
324C42C22⋊Dic5312yes—
325C42C22⋊Dic5312yes—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—42no—
332C23—44no—
333C23—44no—
334C23—41no—
335C23—41no—
336C23#1313313212yes—
337C23C2×C22⋊Dic5311yescharacteristic
338C23#1313318614yes—
339Q8—42no—
340Q8—42no—
341Q8—42no—
342Q8—42no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4—416no—
347D4—42no—
348D4—48no—
349D4—48no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—4244no—
370C2×C4—4248no—
371C2×C4#1313313212yes—
372C2×C4C2×C22⋊Dic5311yescharacteristic
373C2×C4C2×C22⋊Dic5311yescharacteristic
374C22—22no—
375C22—22no—
376C22—24no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22#1681047911yescharacteristic
388C22C2×#1313313211yescharacteristic
389C22C2×#1313313211yescharacteristic
390C4—24no—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—8484no—
398C2—21no—
399C2—21no—
400C2—44no—
401C2—42no—
402C2—82no—
403C2#2027463011yescharacteristic
404C2#2027638012yes—
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC2×C53×#532C2
socleC22×C53C2×#99
cosocleC22×D4×C53C23
hypercenterC2×#532D53
hypocenterC53#12695
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#16810418C2semidirect
2. C2×#16810416C2semidirect
3. #20274633C2semidirect
10. C22×#41×C53C4semidirect
14. C2×C4⋊Q8×C53C4semidirect
16. C2×#124×C53C4semidirect
20. C22×C42×C53C4semidirect
207. C22×#41Dic53semidirect
211. C2×C4⋊Q8Dic53semidirect
213. C2×#124Dic53semidirect
217. C22×C42Dic53semidirect
223. C53#12695semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
C2×#168104162
#202746332
C2×C53×#5322
#1269553