Finite Group Databasebeta

← #27271117

Subgroups — #27271117

Order 13568.

560 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 75 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#20274789C211yesmaximal, characteristic
2#20273042C211yesmaximal, characteristic
3C53×#1437C211yesmaximal, characteristic
4#16810587—21no—
5#16810479—21no—
6#16810390—21no—
7#16810390C412yescomplemented
8#16810347—21no—
9#16810347C412yescomplemented
10C22×C22⋊C4×C53—21no—
11C22×C22⋊C4×C53C2211yescharacteristic
12C22×C42×C53C411yescharacteristic, complemented
13C2×#95×C53—21no—
14C2×#95×C53C411yescharacteristic, complemented
15C2×#11719917—41no—
16C2×C22⋊Dic53—22no—
17C2×C22⋊Dic53—41no—
18C2×C22⋊Dic53—41no—
19C22⋊#88322—22no—
20C22⋊#88322—41no—
21C25×C53D411yescharacteristic
22C23×C4×C53—21no—
23C23×C4×C53—21no—
24C23×C4×C53C2×C411yescharacteristic
25C23×C4×C53D411yescharacteristic
26C2×C22⋊C4×C53—24no—
27C2×C22⋊C4×C53—41no—
28C2×C22⋊C4×C53—41no—
29C2×C22⋊C4×C53—42no—
30C2×C22⋊C4×C53—42no—
31C2×C22⋊C4×C53C2×C411yescharacteristic
32C2×C22⋊C4×C53C2×C411yescharacteristic
33C2×C22⋊C4×C53D411yescharacteristic
34C2×C22⋊C4×C53Q811yescharacteristic
35C2×C42×C53—42no—
36C2×C42×C53—41no—
37#95×C53—42no—
38#95×C53—41no—
39C24×C53—22no—
40C24×C53—21no—
41C24×C53—21no—
42C24×C53—41no—
43C24×C53—41no—
44C24×C53—41no—
45C24×C53—41no—
46C24×C53—41no—
47C24×C53C22⋊C411yescharacteristic
48C24×C53C22⋊C411yescharacteristic
49C24×C53C4⋊C411yescharacteristic
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C4211yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C4⋊C411yescharacteristic
75C22×C4×C53C4⋊C411yescharacteristic
76C22×Dic53—42no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—48no—
82C42×C53—42no—
83C42×C53—42no—
84C22⋊Dic53—42no—
85C22⋊Dic53—42no—
86C22⋊Dic53—81no—
87C22⋊Dic53—81no—
88C22⋊Dic53—81no—
89C22⋊Dic53—81no—
90#11719917—81no—
91#11719917—81no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—42no—
111C23×C53—42no—
112C23×C53—42no—
113C23×C53—42no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—81no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53—81no—
122C23×C53—81no—
123C23×C53—81no—
124C23×C53#9511yescharacteristic
125C23×C53#9911yescharacteristic
126C23×C53#9911yescharacteristic
127C23×C53#10412yes—
128C2×C4×C53—24no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53—81no—
155C2×C4×C53—81no—
156C2×Dic53—82no—
157C2×Dic53—82no—
158C2×Dic53—82no—
159#88322—82no—
160#7213—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—22no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—81no—
184C22×C53—82no—
185C22×C53—82no—
186C22×C53—82no—
187C22×C53—82no—
188C22×C53—81no—
189C22×C53H(3,ℤ/4)11yescharacteristic
190C22×C53#33911yescharacteristic
191C22×C53#34211yescharacteristic
192C22×C53AΣL(1,𝔽16)11yescharacteristic
193C22×C53#35211yescharacteristic
194C22×C53#35311yescharacteristic
195C22×C53#35411yescharacteristic
196C4×C53—44no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—82no—
202Dic53—162no—
203#1819—531no—
204#1462—531no—
205#1437D5311yescharacteristic
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—42no—
213C2×C53—81no—
214C2×C53#128411yescharacteristic
215C2×C53#139011yescharacteristic
216C2×C53#139311yescharacteristic
217C22×C22⋊C4—21no—
218C22×C22⋊C4C2×D5311yescharacteristic
219C22×C42Dic5311yescharacteristic, complemented
220#407—1061no—
221#379—1061no—
222C2×#95—21no—
223C2×#95Dic5311yescharacteristic, complemented
224#342—532no—
225#342—1061no—
226#323—532no—
227#323—1061no—
228C53#721311yescharacteristic, complemented, Sylow
229C25C22⋊D5311yescharacteristic
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C2×Dic5311yescharacteristic
233C23×C4C22⋊D5311yescharacteristic
234C2×M16—2121no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—1062no—
241C2×C22⋊C4—2121no—
242C2×C22⋊C4—2121no—
243C2×C22⋊C4Dic10611yescharacteristic
244C2×C22⋊C4C4×D5311yescharacteristic
245C2×C22⋊C4C4×D5311yescharacteristic
246C2×C22⋊C4D21211yescharacteristic
247C2×C42—41no—
248C2×C42—42no—
249C22⋊C8—1062no—
250C22⋊C8—2121no—
251#95—41no—
252#95—42no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24#1171991911yescharacteristic
262C24#1171992111yescharacteristic
263C24C22⋊Dic5311yescharacteristic
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—2122no—
287C22×C4C4×Dic5311yescharacteristic
288C22×C4#1171991911yescharacteristic
289C22×C4C4⋊Dic5311yescharacteristic
290C22×C4#1171992111yescharacteristic
291M16—4241no—
292M16—4241no—
293C2×C8—2122no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—48no—
299C22⋊C4—4241no—
300C22⋊C4—4241no—
301C22⋊C4—4241no—
302C22⋊C4—4241no—
303C22⋊C4—2122no—
304C22⋊C4—2122no—
305C42—42no—
306C42—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23#1313312411yescharacteristic
339C23#1313313211yescharacteristic
340C23#1313313211yescharacteristic
341C23#1313314512yes—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—4242no—
371C2×C4—4242no—
372C2×C4—4242no—
373C8—4242no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—81no—
402C22#1681037911yescharacteristic
403C22#1681038311yescharacteristic
404C22#1681039011yescharacteristic
405C22#1681041011yescharacteristic
406C22#1681041311yescharacteristic
407C22#1681041611yescharacteristic
408C22#1681041811yescharacteristic
409C4—44no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—8482no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—81no—
424C2#2027256811yescharacteristic
425C2#2027283311yescharacteristic
426C2#2027284011yescharacteristic
subgroupG/H
centerC22#16810390
commutatorC22×C4×C53C42
FrattiniC22×C22⋊C4C2×D53
FittingC53×#1437C2
socleC22×C53#342
prime radicalC25×C53D4
cosocleC22×C22⋊C4×C53C22
hypercenter#1437D53
hypocenterC53#7213
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1437

Complements

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

NH ≅ G/Ntype
7. #16810390C4semidirect
9. #16810347C4semidirect
12. C22×C42×C53C4semidirect
14. C2×#95×C53C4semidirect
219. C22×C42Dic53semidirect
223. C2×#95Dic53semidirect
228. C53#7213semidirect

Maximal subgroups

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

subgroupindex
#202747892
#202730422
C53×#14372
#721353