Finite Group Databasebeta

← #27294675

Subgroups — #27294675

Order 13568.

768 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13133134C211yesmaximal, characteristic, complemented
2#20273053C211yesmaximal, characteristic, complemented
3#20272718C214yesmaximal, complemented
4C53×#2408C211yesmaximal, characteristic
5C2×#13133134—21no—
6#16810587—21no—
7C4×#11719917—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—
21C2×#11719917—41no—
22#13133134—24no—
23C22⋊#88322—22no—
24C22⋊#88322—41no—
25#13133128—41no—
26C4×#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—44no—
39C4×D4×C53—41no—
40C4×D4×C53—41no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53C2×C412yes—
51C24×C53—22no—
52C24×C53—42no—
53C24×C53—44no—
54C24×C53—42no—
55C24×C53—42no—
56C24×C53—82no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C2×D4×C53—24no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
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—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—22no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C4⋊C4×C53—41no—
87C4⋊C4×C53—42no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—44no—
92C22⋊C4×C53—44no—
93C22⋊C4×C53—82no—
94C42×C53—21no—
95C42×C53—21no—
96#11719917—24no—
97#11719917—42no—
98#11719917—82no—
99C2×#88322—42no—
100C2×#88322—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—24no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—42no—
112C23×C53—42no—
113C23×C53—42no—
114C23×C53—48no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—84no—
119C23×C53—88no—
120C23×C53—88no—
121C23×C53—84no—
122C23×C53#10412yes—
123C23×C53C2×C22⋊C411yescharacteristic
124D4×C53—44no—
125D4×C53—44no—
126D4×C53—44no—
127D4×C53—44no—
128D4×C53—44no—
129D4×C53—88no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—81no—
146C2×C4×C53#9911yescharacteristic
147C2×C4×C53#9911yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)12yes—
149#88322—44no—
150#88322—82no—
151#11779—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—42no—
156C22×C53—42no—
157C22×C53—42no—
158C22×C53—44no—
159C22×C53—44no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—44no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—81no—
167C22×C53—82no—
168C22×C53—82no—
169C22×C53—816no—
170C22×C53—82no—
171C22×C53—82no—
172C22×C53C2×#9911yescharacteristic
173C22×C53#41811yescharacteristic
174C22×C53C2×#10411yescharacteristic
175C4×C53—21no—
176C4×C53—22no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—82no—
181C4×C53AΣL(1,𝔽16)12yes—
182#2408D5311yescharacteristic
183C4×#100—531no—
184#1465—531no—
185#1345—534no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—42no—
189C2×C53—81no—
190C2×C53—84no—
191C2×C53#150711yescharacteristic
192C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
193C2×C53#213411yescharacteristic
194C23×D4Dic5311yescharacteristic
195C2×#121—21no—
196C2×#120—21no—
197C2×C4×D4C2×D5311yescharacteristic
198C2×#100—1061no—
199#407—1061no—
200C4×M16—1061no—
201#394C2×D5312yes—
202#392Dic5312yes—
203#390—21no—
204#380Dic5311yescharacteristic
205#379—21no—
206#343—532no—
207#324—532no—
208#323—1061no—
209C53#1177911yescharacteristic, complemented, Sylow
210C25—21no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4—44no—
216C22×D4C2×Dic5312yes—
217C22×D4C22⋊D5311yescharacteristic
218C23×C4C2×Dic5311yescharacteristic
219C23×C4C22⋊D5311yescharacteristic
220C2×M16—1062no—
221C2×M16—2121no—
222#121—44no—
223#120—44no—
224C4×D4—41no—
225C4×D4—41no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C22×D5311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4C2×Dic5311yescharacteristic
233C2×C22⋊C4C22⋊D5311yescharacteristic
234C2×C42C22⋊D5311yescharacteristic
235#100—1064no—
236C22⋊C8—1062no—
237C22⋊C8—2121no—
238#97—2121no—
239C4×C8—2121no—
240#95C2×Dic5312yes—
241C24—22no—
242C24—42no—
243C24—44no—
244C24—42no—
245C24—42no—
246C24—82no—
247C24C22⋊Dic5311yescharacteristic
248C24C22⋊Dic5311yescharacteristic
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—48no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—82no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C22⋊Dic5311yescharacteristic
272C22×C4C22⋊Dic5311yescharacteristic
273C22×C4C22×Dic5311yescharacteristic
274C22×C4C2×C22⋊D5311yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276M16—1064no—
277M16—4242no—
278M16—2122no—
279C2×C8—2122no—
280C2×C8—2122no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—82no—
289C42—21no—
290C42—21no—
291C23—21no—
292C23—21no—
293C23—24no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—48no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—84no—
309C23—88no—
310C23—88no—
311C23—84no—
312C23#1313314412yes—
313C23C2×C22⋊Dic5311yescharacteristic
314D4—44no—
315D4—44no—
316D4—44no—
317D4—44no—
318D4—44no—
319D4—88no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4#1313313211yescharacteristic
337C2×C4#1313313211yescharacteristic
338C2×C4#1313324312yes—
339C8—4242no—
340C8—2124no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—44no—
348C22—44no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—41no—
355C22—81no—
356C22—82no—
357C22—82no—
358C22—816no—
359C22—82no—
360C22—82no—
361C22C2×#1313313211yescharacteristic
362C22#1681062811yescharacteristic
363C22C2×#1313314411yescharacteristic
364C4—22no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—82no—
370C4#1681041012yes—
371C2—21no—
372C2—21no—
373C2—42no—
374C2—81no—
375C2—84no—
376C2#2027323111yescharacteristic
377C2C2×#1681041011yescharacteristic
378C2#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#11779
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. C4×#13133134C2semidirect
2. #20273053C2semidirect
3. #20272718C2semidirect
8. #16810392C22semidirect
9. #16810349C22semidirect
209. C53#11779semidirect

Maximal subgroups

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

subgroupindex
C4×#131331342
#202730532
#202727182
C53×#24082
#1177953