Finite Group Databasebeta

← #27300071

Subgroups — #27300071

Order 13568.

512 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277380C211yesmaximal, characteristic
2C2×#16810414C212yesmaximal
3#20274703C211yesmaximal, characteristic
4#20272834C212yesmaximal
5C53×#1800C211yesmaximal, characteristic
6C2×#13133198C2211yescharacteristic
7C22×#11719919C411yescharacteristic, complemented
8C2×#13133132C2212yes—
9#16810518—21no—
10#16810517—21no—
11#16810507—21no—
12#16810503C2211yescharacteristic
13#16810496—21no—
14#16810495C411yescharacteristic, complemented
15#16810484C411yescharacteristic, complemented
16#16810483—21no—
17C4×C22⋊Dic53C411yescharacteristic, complemented
18#16810414—24no—
19#16810390—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#423—21no—
22C2×#100×C53C2212yes—
23C53×#343—21no—
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26#13133198—21no—
27#13133198—21no—
28#13133198D412yes—
29#13133198Q812yes—
30C2×C4⋊Dic53—41no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—41no—
36C2×#11719919—41no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53C2×C411yescharacteristic
41C2×C22⋊Dic53C2×C411yescharacteristic
42C2×C4×Dic53—21no—
43#13133132—22no—
44#13133132—22no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—41no—
51C22×D4×C53C2311yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53D411yescharacteristic
54C2×M16×C53—22no—
55C4×D4×C53—41no—
56C4×D4×C53—41no—
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C22⋊C4×C53C2×C411yescharacteristic
59C2×C22⋊C4×C53D411yescharacteristic
60C2×C42×C53D411yescharacteristic
61C4⋊C8×C53—41no—
62C4⋊C8×C53—41no—
63#100×C53—22no—
64#100×C53—22no—
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—81no—
78C22×C4×C53—81no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53—42no—
93C22×Dic53—81no—
94C22×Dic53C4⋊C412yescomplemented
95M16×C53—42no—
96M16×C53—42no—
97C2×C8×C53—42no—
98C4⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C42×C53—21no—
102C42×C53—21no—
103C22⋊Dic53—21no—
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106C22⋊Dic53—41no—
107C22⋊Dic53—42no—
108C22⋊Dic53—42no—
109C22⋊Dic53—81no—
110C22⋊Dic53C4⋊C412yescomplemented
111C4⋊Dic53—82no—
112#11719919—22no—
113#11719919—22no—
114#11719919—42no—
115#11719919—41no—
116#11719919—41no—
117#11719919—81no—
118C4×Dic53—81no—
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—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—81no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×C4×C53#9911yescharacteristic
150C2×C4×C53#9911yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53C22⋊Q811yescharacteristic
153C2×Dic53—22no—
154C2×Dic53—22no—
155C2×Dic53—41no—
156C2×Dic53—42no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—42no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—82no—
168C2×Dic53—82no—
169C2×Dic53—82no—
170C2×Dic53—82no—
171C8×C53—82no—
172#12796—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—42no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—81no—
189C22×C53#38011yescharacteristic
190C22×C53C2×#9911yescharacteristic
191C22×C53#41011yescharacteristic
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—41no—
199Dic53—42no—
200Dic53—42no—
201Dic53—81no—
202Dic53—81no—
203Dic53—162no—
204#2322—531no—
205C2×#352—532no—
206#1800D5311yescharacteristic
207#1799—531no—
208#1390—532no—
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—41no—
212C2×C53—41no—
213C2×C53—41no—
214C2×C53—41no—
215C2×C53#179911yescharacteristic
216C2×C53#212111yescharacteristic
217C2×C53#212311yescharacteristic
218C2×#123—531no—
219C2×C4×D4C2×D5311yescharacteristic
220C22×C4⋊C4—531no—
221#423—21no—
222C2×#100C2×D5312yes—
223C2×#99—532no—
224#389—1061no—
225#388—1061no—
226#387—1061no—
227#386—531no—
228#385—531no—
229#385—1061no—
230#381—531no—
231#380—1061no—
232C4×C22⋊C4—531no—
233#352—1064no—
234#343—21no—
235#342—1061no—
236C53#1279611yescharacteristic, complemented, Sylow
237C22×D4C22×D5311yescharacteristic
238C23×C4—531no—
239C23×C4—531no—
240C23×C4C4×D5311yescharacteristic
241C23×C4D21211yescharacteristic
242C2×M16—22no—
243#123—532no—
244#123—532no—
245#123—1061no—
246#123—1061no—
247C4×D4—41no—
248C4×D4—41no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—2121no—
254C2×C4⋊C4—2121no—
255C2×C4⋊C4—2121no—
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C22⋊C4—531no—
258C2×C22⋊C4—531no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4—1061no—
261C2×C22⋊C4—1062no—
262C2×C22⋊C4C4×D5311yescharacteristic
263C2×C22⋊C4D21211yescharacteristic
264C2×C42—1061no—
265C2×C42C22⋊D5311yescharacteristic
266C4⋊C8—41no—
267C4⋊C8—41no—
268#100—22no—
269#100—22no—
270#99—1062no—
271#99—1062no—
272#95—1061no—
273#95—1061no—
274#95—1061no—
275#95—1061no—
276#95—1061no—
277#95—2121no—
278C24C2×C4×D5311yescharacteristic
279C24C2×D21211yescharacteristic
280C2×D4—41no—
281C2×D4—41no—
282C2×D4D4×D5311yescharacteristic
283C2×D4#1171994711yescharacteristic
284C2×D4Q8×D5311yescharacteristic
285C2×D4#1171994911yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4—532no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4—4241no—
299C22×C4—2122no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4#1171992111yescharacteristic
304C22×C4#1171992111yescharacteristic
305C22×C4#1171992111yescharacteristic
306C22×C4#1171992111yescharacteristic
307C22×C4C2×C22⋊D5311yescharacteristic
308M16—42no—
309M16—42no—
310C2×C8—42no—
311C4⋊C4—41no—
312C4⋊C4—1062no—
313C4⋊C4—1062no—
314C4⋊C4—4242no—
315C4⋊C4—4241no—
316C4⋊C4—2122no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—532no—
322C22⋊C4—1061no—
323C22⋊C4—1062no—
324C22⋊C4—1062no—
325C22⋊C4—4241no—
326C22⋊C4—2122no—
327C22⋊C4—2122no—
328C22⋊C4—2121no—
329C42—21no—
330C42—21no—
331C42—4241no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23C2×#1171992111yescharacteristic
341C23C4⋊C4×D5311yescharacteristic
342C23#1313317511yescharacteristic
343C23#1313319211yescharacteristic
344C23#1313319711yescharacteristic
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—1062no—
363C2×C4—1062no—
364C2×C4—4242no—
365C2×C4—4242no—
366C2×C4—4242no—
367C2×C4—4242no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2122no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2122no—
380C2×C4#1313313111yescharacteristic
381C2×C4#1313313111yescharacteristic
382C2×C4#1313318811yescharacteristic
383C2×C4#1313319311yescharacteristic
384C8—82no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22#1681048011yescharacteristic
402C22C2×#1313313111yescharacteristic
403C22#1681059611yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—8482no—
412C4—4241no—
413C4—4241no—
414C4—2122no—
415C4—2122no—
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2—41no—
421C2—41no—
422C2#2027470011yescharacteristic
423C2#2027636211yescharacteristic
424C2#2027637011yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1800C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1800D53
hypocenterC53#12796
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
7. C22×#11719919C4semidirect
14. #16810495C4semidirect
15. #16810484C4semidirect
17. C4×C22⋊Dic53C4semidirect
94. C22×Dic53C4⋊C4semidirect
110. C22⋊Dic53C4⋊C4semidirect
236. C53#12796semidirect

Maximal subgroups

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

subgroupindex
#202773802
C2×#168104142
#202747032
#202728342
C53×#18002
#1279653