Finite Group Databasebeta

← #27300057

Subgroups — #27300057

Order 13568.

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

normal subgroups only

This group has 540 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
1C4×#13133198C211yesmaximal, characteristic
2C2×#16810412C212yesmaximal
3#20274707C211yesmaximal, characteristic
4#20272832C212yesmaximal
5C53×#1799C211yesmaximal, characteristic
6C2×#13133198C2211yescharacteristic
7C22×C4×Dic53C411yescharacteristic, complemented
8#16810647—21no—
9C2×#13133134C2212yes—
10#16810511—21no—
11#16810504—21no—
12#16810503C2211yescharacteristic
13#16810495C411yescharacteristic, complemented
14#16810484C411yescharacteristic, complemented
15C4×#11719919—21no—
16C4×C22⋊Dic53—21no—
17C4×C22⋊Dic53C411yescharacteristic, complemented
18#16810412—24no—
19#16810392—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#99×C53C2212yes—
22C53×#380—21no—
23C53×#342—21no—
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C2×#11719917—22no—
27#13133198—21no—
28#13133198—21no—
29#13133198D412yes—
30#13133198Q812yes—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53C2×C411yescharacteristic
36C2×C22⋊Dic53C2×C411yescharacteristic
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—41no—
41C2×C4×Dic53—41no—
42C4⋊#88322—41no—
43C4⋊#88322—41no—
44#13133134—22no—
45#13133134—22no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×D4×C53C2311yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53D411yescharacteristic
54C4×D4×C53—41no—
55C4×D4×C53—41no—
56C2×C4⋊C4×C53—41no—
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53C2×C411yescharacteristic
60C2×C22⋊C4×C53D411yescharacteristic
61C2×C42×C53D411yescharacteristic
62#99×C53—22no—
63#99×C53—22no—
64#95×C53—41no—
65C24×C53C22×C411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C2×D4×C53C2×D411yescharacteristic
70C2×D4×C53C2×Q811yescharacteristic
71C2×D4×C53#4111yescharacteristic
72C2×D4×C53#4111yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—42no—
91C22×Dic53—41no—
92C22×Dic53—41no—
93C22×Dic53—41no—
94C22×Dic53C4⋊C412yescomplemented
95C4⋊C4×C53—41no—
96C4⋊C4×C53—82no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—42no—
100C22⋊C4×C53—42no—
101C42×C53—21no—
102C42×C53—21no—
103C22⋊Dic53—21no—
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106C22⋊Dic53—42no—
107C22⋊Dic53—41no—
108C22⋊Dic53C4⋊C412yescomplemented
109#11719919—22no—
110#11719919—22no—
111#11719919—42no—
112C4×Dic53—42no—
113C4×Dic53—42no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116#11719917—42no—
117#11719917—42no—
118C2×#88322—42no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53C2×C4⋊C411yescharacteristic
129C23×C53#11711yescharacteristic
130C23×C53#12111yescharacteristic
131C23×C53C22⋊Q811yescharacteristic
132D4×C53—41no—
133D4×C53—41no—
134D4×C53—41no—
135D4×C53—41no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—22no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—81no—
149C2×C4×C53—82no—
150C2×C4×C53—82no—
151C2×C4×C53—82no—
152C2×C4×C53#9911yescharacteristic
153C2×C4×C53#9911yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53C22⋊Q811yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—42no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—42no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172#88322—82no—
173#12794—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—42no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—81no—
190C22×C53#38011yescharacteristic
191C22×C53C2×#9911yescharacteristic
192C22×C53#41011yescharacteristic
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—162no—
200C4×C53#35212yescomplemented
201Dic53—42no—
202Dic53—42no—
203Dic53—42no—
204Dic53—81no—
205C4×#123—531no—
206C2×#352—532no—
207#1800—531no—
208#1799D5311yescharacteristic
209#1390—532no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—41no—
213C2×C53—41no—
214C2×C53—41no—
215C2×C53—41no—
216C2×C53#179911yescharacteristic
217C2×C53C2×#35211yescharacteristic
218C2×C53#212011yescharacteristic
219C2×#123—531no—
220C2×C4×D4C2×D5311yescharacteristic
221C22×C42—531no—
222#423—1061no—
223C2×#100—532no—
224C2×#99C2×D5312yes—
225#388—1061no—
226#387—1061no—
227#386—531no—
228#385—531no—
229#381—531no—
230#380—21no—
231C4×C4⋊C4—1061no—
232C4×C22⋊C4—531no—
233C4×C22⋊C4—1061no—
234#352—1064no—
235#343—1061no—
236#342—21no—
237C53#1279411yescharacteristic, complemented, Sylow
238C22×D4C22×D5311yescharacteristic
239C23×C4—531no—
240C23×C4—531no—
241C23×C4C4×D5311yescharacteristic
242C23×C4C22⋊D5311yescharacteristic
243C2×M16—1062no—
244#123—532no—
245#123—532no—
246#123—1061no—
247#123—1061no—
248C4×D4—41no—
249C4×D4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4D21211yescharacteristic
254C2×C22⋊C4—22no—
255C2×C22⋊C4—531no—
256C2×C22⋊C4—531no—
257C2×C22⋊C4—1061no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4C4×D5311yescharacteristic
260C2×C22⋊C4C22⋊D5311yescharacteristic
261C2×C42—1061no—
262C2×C42—1061no—
263C2×C42—1061no—
264C2×C42—2121no—
265C2×C42—2121no—
266C2×C42D21211yescharacteristic
267C4⋊C8—2121no—
268C4⋊C8—2121no—
269#100—1062no—
270#100—1062no—
271#99—22no—
272#99—22no—
273#95—41no—
274#95—1061no—
275#95—1061no—
276#95—1061no—
277#95—1061no—
278#95—1061no—
279C24C2×C4×D5311yescharacteristic
280C24C2×C22⋊D5311yescharacteristic
281C2×D4—41no—
282C2×D4—41no—
283C2×D4D4×D5311yescharacteristic
284C2×D4#1171994711yescharacteristic
285C2×D4Q8×D5311yescharacteristic
286C2×D4#1171994911yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—532no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4—1061no—
300C22×C4—2122no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4—2121no—
304C22×C4#1171992111yescharacteristic
305C22×C4#1171992111yescharacteristic
306C22×C4#1171992111yescharacteristic
307C22×C4#1171992111yescharacteristic
308C22×C4C2×D21211yescharacteristic
309M16—2122no—
310M16—2122no—
311C2×C8—2122no—
312C4⋊C4—41no—
313C4⋊C4—82no—
314C4⋊C4—1062no—
315C4⋊C4—1062no—
316C4⋊C4—2122no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—532no—
322C22⋊C4—1061no—
323C22⋊C4—1062no—
324C22⋊C4—1062no—
325C22⋊C4—2122no—
326C22⋊C4—2121no—
327C42—21no—
328C42—21no—
329C42—2122no—
330C42—2122no—
331C42—2121no—
332C42—2121no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C2×#1171992111yescharacteristic
342C23C4⋊C4×D5311yescharacteristic
343C23#1313317511yescharacteristic
344C23#1313318811yescharacteristic
345C23#1313319311yescharacteristic
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—1062no—
367C2×C4—1062no—
368C2×C4—1062no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2122no—
375C2×C4—2121no—
376C2×C4—2122no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4#1313313111yescharacteristic
383C2×C4#1313313111yescharacteristic
384C2×C4#1313319211yescharacteristic
385C2×C4#1313319711yescharacteristic
386C8—4242no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22#1681048011yescharacteristic
404C22C2×#1313313111yescharacteristic
405C22#1681059611yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—162no—
413C4—4241no—
414C4—2122no—
415C4—2122no—
416C4—2122no—
417C4#1681041212yescomplemented
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—41no—
424C2#2027470011yescharacteristic
425C2C2×#1681041211yescharacteristic
426C2#2027635411yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1799C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1799D53
hypocenterC53#12794
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
7. C22×C4×Dic53C4semidirect
13. #16810495C4semidirect
14. #16810484C4semidirect
17. C4×C22⋊Dic53C4semidirect
94. C22×Dic53C4⋊C4semidirect
108. C22⋊Dic53C4⋊C4semidirect
200. C4×C53#352semidirect
237. C53#12794semidirect
417. C4#16810412semidirect

Maximal subgroups

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

subgroupindex
C4×#131331982
C2×#168104122
#202747072
#202728322
C53×#17992
#1279453