Finite Group Databasebeta

← #27295138

Subgroups — #27295138

Order 13568.

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

normal subgroups only

This group has 764 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#20274707C211yesmaximal, characteristic, complemented
2#20273033C211yesmaximal, characteristic, complemented
3#20272718C214yesmaximal, complemented
4C53×#2408C211yesmaximal, characteristic
5#16810647—21no—
6C2×#13133134—21no—
7C2×C22⋊#88322—21no—
8#16810392C2212yescomplemented
9#16810349C2212yescomplemented
10#16810347—21no—
11C23×D4×C53C411yescharacteristic
12C2×#121×C53—21no—
13C2×#120×C53—21no—
14C2×C4×D4×C53C2211yescharacteristic
15C53×#394C2212yes—
16C53×#392C412yes—
17C53×#390—21no—
18C53×#380C411yescharacteristic
19C53×#379—21no—
20C2×#11719917—22no—
21C22×#88322—41no—
22C4⋊#88322—41no—
23C4⋊#88322—41no—
24#13133134—42no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C25×C53—21no—
28C22×D4×C53—22no—
29C22×D4×C53—21no—
30C22×D4×C53—41no—
31C22×D4×C53—41no—
32C22×D4×C53—44no—
33C22×D4×C53C2×C412yes—
34C22×D4×C53D411yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53D411yescharacteristic
37#121×C53—44no—
38#120×C53—42no—
39#120×C53—42no—
40C4×D4×C53—41no—
41C4×D4×C53—41no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51#95×C53C2×C412yes—
52C24×C53—22no—
53C24×C53—42no—
54C24×C53—44no—
55C24×C53—42no—
56C24×C53—42no—
57C24×C53—82no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C22⋊C411yescharacteristic
60C2×D4×C53—24no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—42no—
64C2×D4×C53—42no—
65C2×D4×C53—48no—
66C2×D4×C53—44no—
67C2×D4×C53—44no—
68C2×D4×C53—44no—
69C2×D4×C53—44no—
70C2×D4×C53—44no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—82no—
74C2×D4×C53D812yes—
75C2×D4×C53SD1612yes—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—42no—
82C22×C4×C53—42no—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C4⋊C4×C53—41no—
89C4⋊C4×C53—42no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—44no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—42no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—82no—
97C42×C53—21no—
98C42×C53—21no—
99#11719917—42no—
100#11719917—42no—
101C2×#88322—42no—
102C2×#88322—42no—
103C2×#88322—82no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—24no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—42no—
116C23×C53—48no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—42no—
120C23×C53—84no—
121C23×C53—88no—
122C23×C53—88no—
123C23×C53—84no—
124C23×C53#10212yes—
125C23×C53#10212yes—
126C23×C53C2×C22⋊C411yescharacteristic
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—44no—
130D4×C53—44no—
131D4×C53—44no—
132D4×C53—44no—
133D4×C53—88no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—22no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—81no—
150C2×C4×C53#9911yescharacteristic
151C2×C4×C53#9911yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C2×SD1611yescharacteristic
154#88322—82no—
155#88322—82no—
156#11880—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—44no—
165C22×C53—44no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—44no—
171C22×C53—81no—
172C22×C53—82no—
173C22×C53—82no—
174C22×C53—816no—
175C22×C53—82no—
176C22×C53—82no—
177C22×C53C2×#9911yescharacteristic
178C22×C53C2×#10211yescharacteristic
179C22×C53#42111yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—82no—
187C4×C53AΣL(1,𝔽16)12yes—
188#2408D5311yescharacteristic
189#1800—531no—
190#1459—531no—
191#1345—534no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—42no—
195C2×C53—42no—
196C2×C53—84no—
197C2×C53#151911yescharacteristic
198C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
199C2×C53#213411yescharacteristic
200C23×D4Dic5311yescharacteristic
201C2×#121—21no—
202C2×#120—21no—
203C2×C4×D4C2×D5311yescharacteristic
204#423—1061no—
205C2×#100—1061no—
206C2×C22⋊C8—1061no—
207#394C2×D5312yes—
208#392Dic5312yes—
209#390—21no—
210#380Dic5311yescharacteristic
211#379—21no—
212#343—532no—
213#324—532no—
214#323—1061no—
215C53#1188011yescharacteristic, complemented, Sylow
216C25—21no—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4—41no—
220C22×D4—41no—
221C22×D4—44no—
222C22×D4C2×Dic5312yes—
223C22×D4C22⋊D5311yescharacteristic
224C23×C4C2×Dic5311yescharacteristic
225C23×C4C22⋊D5311yescharacteristic
226C2×M16—1062no—
227C22×C8—2121no—
228#121—44no—
229#120—42no—
230#120—42no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22×D5311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4C2×Dic5311yescharacteristic
240C2×C22⋊C4C22⋊D5311yescharacteristic
241C2×C42C22⋊D5311yescharacteristic
242C4⋊C8—2121no—
243C4⋊C8—2121no—
244#100—2122no—
245C22⋊C8—1062no—
246C22⋊C8—2121no—
247#95C2×Dic5312yes—
248C24—22no—
249C24—42no—
250C24—44no—
251C24—42no—
252C24—42no—
253C24—82no—
254C24C22⋊Dic5311yescharacteristic
255C24C22⋊Dic5311yescharacteristic
256C2×D4—24no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—48no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—82no—
270C2×D4#1171992212yes—
271C2×D4#1171992312yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4C22⋊Dic5311yescharacteristic
280C22×C4C22⋊Dic5311yescharacteristic
281C22×C4C22×Dic5311yescharacteristic
282C22×C4C2×C22⋊D5311yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284M16—2122no—
285M16—2122no—
286C2×C8—4242no—
287C2×C8—2122no—
288C2×C8—2122no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—44no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—82no—
298C42—21no—
299C42—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—24no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—48no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—84no—
317C23—88no—
318C23—88no—
319C23—84no—
320C23#1313313812yes—
321C23#1313313812yes—
322C23C2×C22⋊Dic5311yescharacteristic
323D4—42no—
324D4—42no—
325D4—44no—
326D4—44no—
327D4—44no—
328D4—44no—
329D4—88no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4#1313313211yescharacteristic
347C2×C4#1313313211yescharacteristic
348C2×C4C2×#1171992211yescharacteristic
349C2×C4C2×#1171992311yescharacteristic
350C8—4242no—
351C8—4242no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—44no—
360C22—44no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—44no—
366C22—81no—
367C22—82no—
368C22—82no—
369C22—816no—
370C22—82no—
371C22—82no—
372C22C2×#1313313211yescharacteristic
373C22C2×#1313313811yescharacteristic
374C22#1681064011yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—82no—
382C4#1681041012yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2—84no—
388C2#2027328311yescharacteristic
389C2C2×#1681041011yescharacteristic
390C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
Fitting, prime radicalC53×#2408C2
socleC22×C53C2×#99
cosocleC2×C4⋊C4×C53C23
hypercenter#2408D53
hypocenterC53#11880
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #20274707C2semidirect
2. #20273033C2semidirect
3. #20272718C2semidirect
8. #16810392C22semidirect
9. #16810349C22semidirect
215. C53#11880semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202730332
#202727182
C53×#24082
#1188053