Finite Group Databasebeta

← #27276599

Subgroups — #27276599

Order 13568.

754 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#20274798C211yesmaximal, characteristic, complemented
2#20274792C211yesmaximal, characteristic, complemented
3#20273248C212yesmaximal, complemented
4#20273244C212yesmaximal
5C2×C53×#542C211yesmaximal, characteristic
6#16810587C2212yes—
7#16810486C2212yes—
8#16810392—21no—
9#16810392—21no—
10#16810390—21no—
11#16810390—21no—
12#16810360—24no—
13#16810357—24no—
14C53×#542—24no—
15C53×#542—24no—
16C2×C4⋊Q8×C53C411yescharacteristic, complemented
17C2×#126×C53—21no—
18C2×#124×C53—21no—
19C2×#124×C53C411yescharacteristic, complemented
20C2×#123×C53—21no—
21C2×C22⋊Q8×C53—21no—
22C2×C22⋊Q8×C53C2211yescharacteristic
23C2×#121×C53C2211yescharacteristic
24C2×C4×Q8×C53C411yescharacteristic, complemented
25C2×C4×D4×C53C411yescharacteristic, complemented
26C2×#117×C53C2211yescharacteristic
27C2×#11719917—22no—
28C2×C22⋊Dic53—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124—22no—
33C22×Q8×C53—21no—
34C22×Q8×C53C2×C411yescharacteristic
35C22×D4×C53—21no—
36C22×D4×C53C2×C411yescharacteristic
37C23×C4×C53—21no—
38C23×C4×C53C2311yescharacteristic
39C4⋊Q8×C53—22no—
40C4⋊Q8×C53—22no—
41#126×C53—28no—
42#124×C53—28no—
43#124×C53—22no—
44#124×C53—22no—
45#123×C53—28no—
46C22⋊Q8×C53—28no—
47C22⋊Q8×C53—22no—
48C22⋊Q8×C53D414yes—
49#121×C53—22no—
50#121×C53D414yes—
51C4×Q8×C53—22no—
52C4×Q8×C53—22no—
53C4×D4×C53—22no—
54C4×D4×C53—22no—
55#117×C53—22no—
56#117×C53—22no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53D411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53C2×C411yescharacteristic
73C2×C42×C53C2×C411yescharacteristic
74C24×C53—21no—
75C24×C53C2×D411yescharacteristic
76C2×Q8×C53—24no—
77C2×Q8×C53—21no—
78C2×Q8×C53—42no—
79C2×Q8×C53—82no—
80C2×Q8×C53C22⋊C412yes—
81C2×D4×C53—24no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53—82no—
85C2×D4×C53C22⋊C412yes—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—24no—
90C22×C4×C53—42no—
91C22×C4×C53—82no—
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22⋊C411yescharacteristic
94C22×C4×C53C22⋊C411yescharacteristic
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D412yes—
100C22×Dic53—42no—
101C4⋊C4×C53—24no—
102C4⋊C4×C53—24no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53C22⋊C412yes—
109C4⋊C4×C53C22⋊C412yes—
110C22⋊C4×C53—24no—
111C22⋊C4×C53—24no—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—24no—
114C22⋊C4×C53—22no—
115C22⋊C4×C53—22no—
116C22⋊C4×C53—24no—
117C42×C53—24no—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53—21no—
121C42×C53—21no—
122C22⋊Dic53—44no—
123#11719917—44no—
124C2×#88322—42no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—42no—
132C23×C53C2×C22⋊C411yescharacteristic
133C23×C53#12012yes—
134Q8×C53—42no—
135Q8×C53—42no—
136Q8×C53—84no—
137D4×C53—42no—
138D4×C53—42no—
139D4×C53—84no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—84no—
159C2×C4×C53C2×C22⋊C411yescharacteristic
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53#12012yes—
162C2×Dic53—82no—
163C2×Dic53—82no—
164C2×Dic53—82no—
165#88322—82no—
166#8519—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—42no—
173C22×C53—42no—
174C22×C53—42no—
175C22×C53—82no—
176C22×C53—84no—
177C22×C53#37911yescharacteristic
178C22×C53#41012yes—
179C4×C53—22no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—42no—
183C4×C53—42no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—42no—
189Dic53—162no—
190C2×#542D5311yescharacteristic
191#1822—531no—
192#1820—531no—
193#1511—532no—
194#1510—532no—
195C2×C53—22no—
196C2×C53—42no—
197C2×C53—82no—
198C2×C53#178211yescharacteristic
199C2×C53#179511yescharacteristic
200C2×C53#179611yescharacteristic
201#542—24no—
202#542—24no—
203C2×C4⋊Q8Dic5311yescharacteristic, complemented
204C2×#126—21no—
205C2×#124—21no—
206C2×#124Dic5311yescharacteristic, complemented
207C2×#123—21no—
208C2×C22⋊Q8—21no—
209C2×C22⋊Q8C2×D5311yescharacteristic
210C2×#121C2×D5311yescharacteristic
211C2×C4×Q8Dic5311yescharacteristic, complemented
212C2×C4×D4Dic5311yescharacteristic, complemented
213C2×#117C2×D5311yescharacteristic
214#407—532no—
215#381—532no—
216#343—1061no—
217#343—1061no—
218#342—1061no—
219#342—1061no—
220#328—1064no—
221#327—1064no—
222C53#851911yescharacteristic, complemented, Sylow
223C22×Q8—21no—
224C22×Q8C2×Dic5311yescharacteristic
225C22×D4—21no—
226C22×D4C2×Dic5311yescharacteristic
227C23×C4—21no—
228C23×C4C22×D5311yescharacteristic
229C2×M16—1062no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232#126—28no—
233#124—28no—
234#124—22no—
235#124—22no—
236#123—28no—
237C22⋊Q8—28no—
238C22⋊Q8—22no—
239C22⋊Q8C22⋊D5314yes—
240#121—22no—
241#121C22⋊D5314yes—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×D4—22no—
245C4×D4—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×Dic5311yescharacteristic
252C2×C4⋊C4C2×Dic5311yescharacteristic
253C2×C4⋊C4C22⋊D5311yescharacteristic
254C2×C4⋊C4C22⋊D5311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—1062no—
261C2×C22⋊C4C22⋊D5311yescharacteristic
262C2×C22⋊C4C22⋊D5311yescharacteristic
263C2×C42—21no—
264C2×C42C2×Dic5311yescharacteristic
265C2×C42C2×Dic5311yescharacteristic
266C22⋊C8—1062no—
267C22⋊C8—1062no—
268#95—1062no—
269#95—1062no—
270C24—21no—
271C24C2×C22⋊D5311yescharacteristic
272C2×Q8—24no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—82no—
276C2×Q8C22⋊Dic5312yes—
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—42no—
280C2×D4—82no—
281C2×D4C22⋊Dic5312yes—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—82no—
288C22×C4—2122no—
289C22×C4C22⋊Dic5311yescharacteristic
290C22×C4C22⋊Dic5311yescharacteristic
291C22×C4C22⋊Dic5311yescharacteristic
292C22×C4C22⋊Dic5311yescharacteristic
293C22×C4C22×Dic5311yescharacteristic
294C22×C4C2×C22⋊D5311yescharacteristic
295C22×C4C2×C22⋊D5312yes—
296C22×C4C2×C22⋊D5312yes—
297M16—2124no—
298C2×C8—2122no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4C22⋊Dic5312yes—
307C4⋊C4C22⋊Dic5312yes—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—24no—
315C22⋊C4—2124no—
316C42—24no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23C2×C22⋊Dic5311yescharacteristic
329C23#1313318612yes—
330Q8—42no—
331Q8—42no—
332Q8—84no—
333D4—42no—
334D4—42no—
335D4—84no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—84no—
355C2×C4—4242no—
356C2×C4—4242no—
357C2×C4—4242no—
358C2×C4C2×C22⋊Dic5311yescharacteristic
359C2×C4C2×C22⋊Dic5311yescharacteristic
360C2×C4#1313318612yes—
361C8—4242no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—82no—
371C22—84no—
372C22#1681047911yescharacteristic
373C22#1681059712yes—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—41no—
383C4—41no—
384C4—8482no—
385C2—22no—
386C2—42no—
387C2—82no—
388C2#2027463311yescharacteristic
389C2#2027468511yescharacteristic
390C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#542C2
socleC22×C53#379
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenterC2×#542D53
hypocenterC53#8519
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. #20274798C2semidirect
2. #20274792C2semidirect
3. #20273248C2semidirect
16. C2×C4⋊Q8×C53C4semidirect
19. C2×#124×C53C4semidirect
24. C2×C4×Q8×C53C4semidirect
25. C2×C4×D4×C53C4semidirect
203. C2×C4⋊Q8Dic53semidirect
206. C2×#124Dic53semidirect
211. C2×C4×Q8Dic53semidirect
212. C2×C4×D4Dic53semidirect
222. C53#8519semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202747922
#202732482
#202732442
C2×C53×#5422
#851953