Finite Group Databasebeta

← #27302401

Subgroups — #27302401

Order 13568.

724 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 107 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#20279661C211yesmaximal, characteristic
2C2×#16810415C214yesmaximal, complemented
3#20274703C211yesmaximal, characteristic, complemented
4C4×#99×C53C211yesmaximal, characteristic, complemented
5C2×#13133214C411yescharacteristic, complemented
6C2×#13133213C2212yes—
7C2×#13133191—21no—
8C2×#13133191—21no—
9C2×#13133132C2212yes—
10#16810532—21no—
11#16810532—21no—
12#16810524C412yescomplemented
13#16810483—21no—
14C4×C4⋊Dic53C411yescharacteristic, complemented
15#16810415—28no—
16#16810390—21no—
17C2×C4×D4×C53C2211yescharacteristic
18C2×#99×C53C2212yes—
19C4×C22⋊C4×C53—21no—
20C53×#342—21no—
21C22×D212—21no—
22C22×D212—21no—
23C22×D212C2×C412yes—
24C22×C22⋊D53—22no—
25C22×C22⋊D53—22no—
26#13133214—41no—
27#13133214—41no—
28#13133213—24no—
29#13133213—24no—
30#13133191—44no—
31#13133191—44no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—41no—
34C2×C22⋊Dic53—22no—
35C2×#11719921—22no—
36C2×#11719921—22no—
37C2×C4×Dic53C2×C412yes—
38#13133132—24no—
39#13133124—21no—
40#13133124—41no—
41C22×D4×C53C2311yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C4×D4×C53—41no—
45C4×D4×C53—41no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53—41no—
51C2×C42×C53C2×C411yescharacteristic
52#99×C53—22no—
53#99×C53D414yescomplemented
54#95×C53—41no—
55C24×C53C2×D411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C23×D53—22no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53C2×D412yes—
61C2×D4×C53#4112yes—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C2×C22⋊D53—44no—
75C2×C22⋊D53—44no—
76C2×C22⋊D53—44no—
77C2×C22⋊D53—42no—
78C2×C22⋊D53—42no—
79C2×C22⋊D53—82no—
80C2×C22⋊D53—82no—
81C22×Dic53—22no—
82C22×Dic53—42no—
83C2×D212—24no—
84C2×D212—42no—
85C2×D212—42no—
86C2×D212—42no—
87C2×D212—42no—
88C2×D212—42no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—82no—
92C4⋊C4×C53—41no—
93C22⋊C4×C53—24no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—42no—
97C42×C53—21no—
98C42×C53—21no—
99C42×C53—82no—
100C22⋊Dic53—42no—
101C22⋊Dic53—42no—
102#11719921—44no—
103#11719921—44no—
104C4⋊Dic53—24no—
105C4⋊Dic53—82no—
106C4×Dic53—24no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53#12112yes—
117C23×C53#12411yescharacteristic
118C23×C53#12711yescharacteristic
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—42no—
124C22×D53—42no—
125C22×D53—42no—
126D4×C53—42no—
127D4×C53—42no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—22no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—44no—
140C2×C4×C53—81no—
141C2×C4×C53—81no—
142C2×C4×C53—82no—
143C2×C4×C53—82no—
144C2×C4×C53#9911yescharacteristic
145C2×C4×C53#9911yescharacteristic
146C2×C4×C53C4×D412yes—
147C22⋊D53—88no—
148C22⋊D53—84no—
149C22⋊D53—84no—
150C2×Dic53—24no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—82no—
154C2×Dic53—82no—
155C2×Dic53—82no—
156D212—84no—
157D212—82no—
158D212—84no—
159D212—82no—
160#13223—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—22no—
165C22×C53—21no—
166C22×C53—42no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53#39011yescharacteristic
176C22×C53C2×#9911yescharacteristic
177C22×C53#41011yescharacteristic
178C2×D53—82no—
179C2×D53—82no—
180C2×D53—82no—
181C2×D53—82no—
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—82no—
185C4×C53—21no—
186C4×C53—22no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—41no—
190C4×C53—84no—
191C4×C53#35312yescomplemented
192Dic53—44no—
193Dic53—162no—
194#2541—531no—
195C2×#353—534no—
196#1799—531no—
197C4×#99D5311yescharacteristic, complemented
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—41no—
201C2×C53—42no—
202C2×C53—41no—
203C2×C53#196411yescharacteristic
204C2×C53C2×#35311yescharacteristic
205C2×C53#212811yescharacteristic
206D53—162no—
207C2×#127—532no—
208C2×#127—531no—
209C2×#121—1061no—
210C2×#121—1061no—
211C2×C4×D4C2×D5311yescharacteristic
212C2×#99—532no—
213C2×#99C2×D5312yes—
214#392—1061no—
215#392—1061no—
216#390—532no—
217#380—1061no—
218C4×C4⋊C4—531no—
219C4×C22⋊C4—21no—
220#353—1068no—
221#342—21no—
222#342—1061no—
223C53#1322311yescharacteristic, complemented, Sylow
224C22×D4—532no—
225C22×D4—1062no—
226C22×D4—1062no—
227C22×D4—1061no—
228C22×D4—1061no—
229C22×D4C22×D5311yescharacteristic
230C23×C4D21211yescharacteristic
231C23×C4C22⋊D5311yescharacteristic
232#127—1064no—
233#127—1064no—
234#127—2121no—
235#127—2121no—
236#121—2124no—
237#121—2124no—
238C4×D4—41no—
239C4×D4—41no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—2121no—
242C2×C4⋊C4C4×D5311yescharacteristic
243C2×C22⋊C4—22no—
244C2×C22⋊C4—1062no—
245C2×C22⋊C4—1062no—
246C2×C22⋊C4—1062no—
247C2×C22⋊C4D21211yescharacteristic
248C2×C22⋊C4C22⋊D5311yescharacteristic
249C2×C42—41no—
250C2×C42—532no—
251C2×C42C4×D5311yescharacteristic
252#99—22no—
253#99—1064no—
254#99D21214yescomplemented
255#95—41no—
256#95—1061no—
257#95—2121no—
258C24—1062no—
259C24C2×D21211yescharacteristic
260C24C2×C22⋊D5311yescharacteristic
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—1064no—
264C2×D4—4242no—
265C2×D4—4242no—
266C2×D4—4242no—
267C2×D4—2122no—
268C2×D4—2122no—
269C2×D4—2122no—
270C2×D4—2122no—
271C2×D4—2122no—
272C2×D4—2124no—
273C2×D4—2124no—
274C2×D4—2124no—
275C2×D4—2122no—
276C2×D4—2122no—
277C2×D4—2121no—
278C2×D4—2121no—
279C2×D4C2×D21212yes—
280C2×D4#1171994512yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—1062no—
289C22×C4—2122no—
290C22×C4#1171992111yescharacteristic
291C22×C4#1171992111yescharacteristic
292C22×C4#1171992111yescharacteristic
293C22×C4#1171992111yescharacteristic
294C22×C4C2×C4×D5311yescharacteristic
295C4⋊C4—41no—
296C4⋊C4—1064no—
297C4⋊C4—4242no—
298C22⋊C4—24no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—2124no—
303C22⋊C4—2124no—
304C22⋊C4—2122no—
305C22⋊C4—2122no—
306C42—21no—
307C42—21no—
308C42—82no—
309C42—1064no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—2122no—
319C23—2122no—
320C23—2122no—
321C23—2122no—
322C23—2122no—
323C23—2122no—
324C23—2122no—
325C23C2×#1171992111yescharacteristic
326C23#1313319112yes—
327C23#1313320511yescharacteristic
328C23#1313321411yescharacteristic
329D4—42no—
330D4—42no—
331D4—4244no—
332D4—4242no—
333D4—4244no—
334D4—4248no—
335D4—4244no—
336D4—4244no—
337D4—4242no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—1064no—
355C2×C4—4242no—
356C2×C4—4242no—
357C2×C4—4242no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4#1313313111yescharacteristic
361C2×C4#1313313111yescharacteristic
362C2×C4C4×D21212yes—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—21no—
368C22—42no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—4242no—
378C22—4242no—
379C22—4242no—
380C22—4242no—
381C22—4242no—
382C22—4242no—
383C22—4242no—
384C22#1681052511yescharacteristic
385C22C2×#1313313111yescharacteristic
386C22#1681059611yescharacteristic
387C4—22no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—84no—
393C4—8482no—
394C4—2124no—
395C4#1681041512yescomplemented
396C2—21no—
397C2—21no—
398C2—42no—
399C2—41no—
400C2—41no—
401C2—8482no—
402C2#2027555111yescharacteristic
403C2C2×#1681041511yescharacteristic
404C2#2027638611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radical#20279661C2
cosocleC22×D4×C53C23
hypercenterC4×#99D53
hypocenterC53#13223
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16810415C2semidirect
3. #20274703C2semidirect
4. C4×#99×C53C2semidirect
5. C2×#13133214C4semidirect
12. #16810524C4semidirect
14. C4×C4⋊Dic53C4semidirect
53. #99×C53D4semidirect
191. C4×C53#353semidirect
197. C4×#99D53semidirect
223. C53#13223semidirect
254. #99D212semidirect
395. C4#16810415semidirect

Maximal subgroups

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

subgroupindex
#202796612
C2×#168104152
#202747032
C4×#99×C532
#1322353