Finite Group Databasebeta

← #27299694

Subgroups — #27299694

Order 13568.

790 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 127 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#20276407C211yesmaximal, characteristic, complemented
2#20276387C211yesmaximal, characteristic, complemented
3#20276365C212yesmaximal, complemented
4#20274819C212yesmaximal, complemented
5C53×#3508C211yesmaximal, characteristic
6#16810607C2212yes—
7#16810597C2212yes—
8#16810420—22no—
9#16810416—22no—
10#16810410—24no—
11#16810383—22no—
12#16810379—22no—
13C53×#559—22no—
14C53×#550—22no—
15C53×#546—24no—
16C53×#540—22no—
17C53×#535C2211yescharacteristic
18C53×#535C2211yescharacteristic
19C53×#533C411yescharacteristic, complemented
20C53×#532—22no—
21C2×#127×C53C411yescharacteristic, complemented
22C53×#518C411yescharacteristic, complemented
23C53×#518C411yescharacteristic, complemented
24C4×#41×C53C2211yescharacteristic
25C2×#11719917—22no—
26#13133176—22no—
27#13133136—22no—
28#13133134—22no—
29#13133132—22no—
30#13133132—22no—
31C2×#41×C53—24no—
32C2×#41×C53C2311yescharacteristic
33C22×D4×C53—22no—
34C22×D4×C53C2×C411yescharacteristic
35C22×D4×C53C2×C411yescharacteristic
36C4⋊Q8×C53—21no—
37#127×C53—21no—
38#127×C53—21no—
39#127×C53—22no—
40#127×C53—21no—
41#127×C53D412yes—
42#125×C53—21no—
43#124×C53—24no—
44#124×C53—21no—
45#124×C53D412yes—
46#123×C53—24no—
47#123×C53—24no—
48C22⋊Q8×C53—22no—
49#121×C53—24no—
50#121×C53—24no—
51#121×C53—24no—
52#121×C53—22no—
53#121×C53—22no—
54#120×C53—22no—
55#120×C53D414yes—
56C4×Q8×C53—22no—
57C4×Q8×C53—21no—
58C4×D4×C53—24no—
59C4×D4×C53—22no—
60C4×D4×C53—22no—
61C4×D4×C53—22no—
62C4×D4×C53—22no—
63C4×D4×C53—22no—
64C4×D4×C53—21no—
65#117×C53C2×C411yescharacteristic
66#117×C53C2×C411yescharacteristic
67#117×C53C2×C411yescharacteristic
68#117×C53D412yes—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—22no—
71C2×C42×C53C2×C411yescharacteristic
72C2×C42×C53D412yes—
73C24×C53—21no—
74C24×C53C22⋊C412yes—
75#41×C53—42no—
76#41×C53—88no—
77C2×Q8×C53—22no—
78C2×Q8×C53C2×D411yescharacteristic
79C2×D4×C53—24no—
80C2×D4×C53—24no—
81C2×D4×C53—24no—
82C2×D4×C53—24no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—42no—
87C2×D4×C53—42no—
88C2×D4×C53—84no—
89C2×D4×C53C22⋊C412yes—
90C2×D4×C53C2×D411yescharacteristic
91C2×D4×C53C2×D412yes—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—24no—
95C22×C4×C53—24no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D412yes—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—22no—
105C22⋊C4×C53—24no—
106C22⋊C4×C53—24no—
107C22⋊C4×C53—21no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—24no—
111C22⋊C4×C53—24no—
112C22⋊C4×C53C22⋊C412yes—
113C42×C53—21no—
114C42×C53—21no—
115C42×C53—22no—
116C42×C53C22⋊C411yescharacteristic
117C42×C53C22⋊C411yescharacteristic
118C42×C53C22⋊C412yes—
119C42×C53C22⋊C412yes—
120C22⋊Dic53—42no—
121C4⋊Dic53—42no—
122C4×Dic53—42no—
123#11719917—42no—
124#11719917—42no—
125C2×#88322—42no—
126C23×C53—22no—
127C23×C53—22no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—82no—
131C23×C53—82no—
132C23×C53C2×C22⋊C411yescharacteristic
133C23×C53#12012yes—
134Q8×C53—41no—
135Q8×C53—44no—
136D4×C53—42no—
137D4×C53—41no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—48no—
141D4×C53—42no—
142D4×C53—42no—
143D4×C53—84no—
144D4×C53—84no—
145D4×C53—84no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—21no—
154C2×C4×C53—42no—
155C2×C4×C53—44no—
156C2×C4×C53—84no—
157C2×C4×C53—84no—
158C2×C4×C53C2×C22⋊C411yescharacteristic
159C2×C4×C53C2×C22⋊C411yescharacteristic
160C2×C4×C53#12012yes—
161C2×Dic53—82no—
162#88322—82no—
163#12722—531nomaximal, Sylow
164C22×C53—22no—
165C22×C53—21no—
166C22×C53—42no—
167C22×C53—41no—
168C22×C53—44no—
169C22×C53—44no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—162no—
173C22×C53—162no—
174C22×C53#37911yescharacteristic
175C4×C53—22no—
176C4×C53—22no—
177C4×C53—21no—
178C4×C53—42no—
179C4×C53—41no—
180C4×C53—42no—
181C4×C53—42no—
182C4×C53#41012yes—
183Dic53—162no—
184#3508D5311yescharacteristic
185#2134—531no—
186#2128—531no—
187#2122—532no—
188#1827—532no—
189C2×C53—21no—
190C2×C53—42no—
191C2×C53—41no—
192C2×C53—82no—
193C2×C53—82no—
194C2×C53#178211yescharacteristic
195#559—22no—
196#550—22no—
197#546—24no—
198#540—22no—
199#535C2×D5311yescharacteristic
200#535C2×D5311yescharacteristic
201#533Dic5311yescharacteristic, complemented
202#532—22no—
203C2×#127Dic5311yescharacteristic, complemented
204#518Dic5311yescharacteristic, complemented
205#518Dic5311yescharacteristic, complemented
206C4×#41C2×D5311yescharacteristic
207#413—532no—
208#410—532no—
209#355—1062no—
210#353—1062no—
211AΣL(1,𝔽16)—1064no—
212#339—1062no—
213H(3,ℤ/4)—1062no—
214C53#1272211yescharacteristic, complemented, Sylow
215C2×#41—24no—
216C2×#41C22×D5311yescharacteristic
217C22×D4—22no—
218C22×D4C2×Dic5311yescharacteristic
219C22×D4C2×Dic5311yescharacteristic
220C2×M16—1062no—
221C4⋊Q8—21no—
222#127—21no—
223#127—22no—
224#127—21no—
225#127—21no—
226#127C22⋊D5312yes—
227#125—21no—
228#124—24no—
229#124—21no—
230#124C22⋊D5312yes—
231#123—24no—
232#123—24no—
233C22⋊Q8—22no—
234#121—24no—
235#121—24no—
236#121—24no—
237#121—22no—
238#121—22no—
239#120—22no—
240#120C22⋊D5314yes—
241C4×Q8—21no—
242C4×Q8—22no—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4—21no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250#117—1062no—
251#117C2×Dic5311yescharacteristic
252#117C2×Dic5311yescharacteristic
253#117C2×Dic5311yescharacteristic
254#117C22⋊D5312yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C42C2×Dic5311yescharacteristic
258C2×C42C22⋊D5312yes—
259#101—1062no—
260#100—1062no—
261#99—1062no—
262#99—1062no—
263C24—21no—
264C24C22⋊Dic5312yes—
265#41—42no—
266#41—88no—
267C2×Q8—22no—
268C2×Q8C2×C22⋊D5311yescharacteristic
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—84no—
279C2×D4C22⋊Dic5312yes—
280C2×D4C2×C22⋊D5311yescharacteristic
281C2×D4C2×C22⋊D5312yes—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4C22×Dic5311yescharacteristic
287C22×C4C2×C22⋊D5312yes—
288M16—2122no—
289M16—2122no—
290C2×C8—2122no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—2122no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—2122no—
307C22⋊C4C22⋊Dic5312yes—
308C42—21no—
309C42—21no—
310C42—22no—
311C42—2122no—
312C42C22⋊Dic5312yes—
313C42C22⋊Dic5311yescharacteristic
314C42C22⋊Dic5311yescharacteristic
315C42C22⋊Dic5312yes—
316C23—22no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—82no—
321C23—82no—
322C23C2×C22⋊Dic5311yescharacteristic
323C23#1313318612yes—
324Q8—41no—
325Q8—44no—
326D4—42no—
327D4—41no—
328D4—42no—
329D4—42no—
330D4—48no—
331D4—42no—
332D4—42no—
333D4—84no—
334D4—84no—
335D4—84no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—4242no—
349C2×C4C2×C22⋊Dic5311yescharacteristic
350C2×C4C2×C22⋊Dic5311yescharacteristic
351C2×C4#1313318612yes—
352C8—4242no—
353C22—22no—
354C22—21no—
355C22—42no—
356C22—41no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22—162no—
362C22—162no—
363C22#1681047911yescharacteristic
364C4—22no—
365C4—22no—
366C4—21no—
367C4—42no—
368C4—41no—
369C4—42no—
370C4—42no—
371C4—8482no—
372C4#1681059712yes—
373C2—21no—
374C2—42no—
375C2—41no—
376C2—82no—
377C2—82no—
378C2#2027463311yescharacteristic
subgroupG/H
centerC2#20274633
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
Fitting, prime radicalC53×#3508C2
socleC2×C53#1782
cosocleC2×#41×C53C23
hypercenter#3508D53
hypocenterC53#12722
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #20276407C2semidirect
2. #20276387C2semidirect
3. #20276365C2semidirect
4. #20274819C2semidirect
19. C53×#533C4semidirect
21. C2×#127×C53C4semidirect
22. C53×#518C4semidirect
23. C53×#518C4semidirect
201. #533Dic53semidirect
203. C2×#127Dic53semidirect
204. #518Dic53semidirect
205. #518Dic53semidirect
214. C53#12722semidirect

Maximal subgroups

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

subgroupindex
#202764072
#202763872
#202763652
#202748192
C53×#35082
#1272253