Finite Group Databasebeta

← #27294773

Subgroups — #27294773

Order 13568.

758 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#16810540C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic, complemented
3#20272717C214yesmaximal, complemented
4C53×#1821C211yesmaximal, characteristic, complemented
5C23×D212C411yescharacteristic
6C22×#11719921—21no—
7C22×#11719921—21no—
8#16810587—21no—
9#16810540—21no—
10#16810540—21no—
11#16810540C412yes—
12#16810540C2212yes—
13C2×#13133124C411yescharacteristic
14#16810349—21no—
15#16810349C2212yescomplemented
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#343—21no—
19C53×#343C2212yescomplemented
20C24×D53—21no—
21C22×D212—21no—
22C22×D212—41no—
23C22×D212—41no—
24C22×D212—44no—
25C22×D212C2×C412yes—
26C23×Dic53—21no—
27C2×#11719917—41no—
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—22no—
32C2×#11719921—44no—
33C2×#11719921—44no—
34C22⋊#88322—22no—
35C22⋊#88322—41no—
36#13133124—21no—
37#13133124C2×C412yes—
38C23×C4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C2×M16×C53—22no—
42C2×C4⋊C4×C53—41no—
43C2×C4⋊C4×C53—41no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53C2311yescharacteristic
48C22⋊C8×C53—41no—
49C22⋊C8×C53—41no—
50C24×C53C22⋊C411yescharacteristic
51C23×D53—24no—
52C23×D53—22no—
53C23×D53—42no—
54C23×D53—42no—
55C23×D53—42no—
56C23×D53—42no—
57C23×D53—42no—
58C23×D53—41no—
59C23×D53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—41no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—22no—
77C22×Dic53—42no—
78C22×Dic53—42no—
79C22×Dic53—42no—
80C2×D212—42no—
81C2×D212—42no—
82C2×D212—44no—
83C2×D212—48no—
84C2×D212—44no—
85C2×D212—44no—
86C2×D212—44no—
87M16×C53—42no—
88M16×C53—42no—
89C2×C8×C53—42no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94#11719921—44no—
95#11719921—44no—
96#11719921—44no—
97#11719921—44no—
98#11719917—82no—
99C2×#88322—42no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53#9911yescharacteristic
105C23×C53#9911yescharacteristic
106C23×C53C2×C22⋊C411yescharacteristic
107C22×D53—24no—
108C22×D53—42no—
109C22×D53—44no—
110C22×D53—48no—
111C22×D53—48no—
112C22×D53—42no—
113C22×D53—42no—
114C22×D53—44no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—84no—
119C22×D53—84no—
120C22×D53—82no—
121C22×D53—82no—
122C22×D53—82no—
123C22×D53—82no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
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—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53AGL(1,ℤ/8)12yes—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—84no—
148D212—44no—
149D212—48no—
150D212—44no—
151C8×C53—82no—
152#88322—82no—
153#11802—531nomaximal, Sylow
154C22×C53—24no—
155C22×C53—24no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53AΣL(1,𝔽16)12yes—
163C22×C53#35512yes—
164C22×C53C2×#9911yescharacteristic
165C22×C53#41811yescharacteristic
166C22×C53#42111yescharacteristic
167C2×D53—44no—
168C2×D53—44no—
169C2×D53—44no—
170C2×D53—82no—
171C2×D53—82no—
172C2×D53—88no—
173C2×D53—88no—
174C2×D53—82no—
175C2×D53—82no—
176C4×C53—22no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182Dic53—82no—
183C2×#394—531no—
184#1821D5311yescharacteristic, complemented
185#1463—531no—
186#1345—534no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—22no—
190C2×C53—22no—
191C2×C53#151011yescharacteristic
192C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
193C2×C53C2×#35511yescharacteristic
194D53—84no—
195C23×D4—531no—
196C22×C4⋊C4C2×D5311yescharacteristic
197C22×C22⋊C4—1061no—
198C22×C22⋊C4—1061no—
199#407—21no—
200#407—1061no—
201#394—532no—
202#394—532no—
203#394—1061no—
204#394—1061no—
205C2×#95—531no—
206#343—21no—
207#343C2×D5312yescomplemented
208#324—532no—
209#324—1061no—
210C53#1180211yescharacteristic, complemented, Sylow
211C25—1061no—
212C22×D4—532no—
213C22×D4—1061no—
214C22×D4—2121no—
215C22×D4—2121no—
216C22×D4—2124no—
217C23×C4—1061no—
218C23×C4C4×D5311yescharacteristic
219C23×C4D21211yescharacteristic
220C23×C4C22⋊D5311yescharacteristic
221C2×M16—22no—
222C2×M16—2121no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C4×D5311yescharacteristic
226C2×C4⋊C4D21211yescharacteristic
227C2×C4⋊C4C22⋊D5311yescharacteristic
228C2×C4⋊C4C22×D5311yescharacteristic
229C2×C22⋊C4—1062no—
230C2×C22⋊C4—1062no—
231C2×C22⋊C4—1062no—
232C2×C22⋊C4—1062no—
233C2×C22⋊C4—2124no—
234C2×C22⋊C4—2124no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8—1062no—
238C22⋊C8—2121no—
239#95—532no—
240#95—1061no—
241C24—1064no—
242C24—1062no—
243C24—2122no—
244C24—2122no—
245C24—2122no—
246C24—2122no—
247C24—2122no—
248C24—2121no—
249C24—2121no—
250C24#1171992111yescharacteristic
251C2×D4—2122no—
252C2×D4—2122no—
253C2×D4—2124no—
254C2×D4—2124no—
255C2×D4—2128no—
256C2×D4—2124no—
257C2×D4—2124no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—1062no—
269C22×C4—2122no—
270C22×C4—2122no—
271C22×C4—2122no—
272C22×C4#1171992111yescharacteristic
273C22×C4#1171992111yescharacteristic
274C22×C4#1171992111yescharacteristic
275C22×C4C2×C4×D5311yescharacteristic
276C22×C4C2×D21211yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278M16—42no—
279M16—42no—
280M16—4242no—
281C2×C8—42no—
282C2×C8—2122no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—2124no—
288C22⋊C4—2124no—
289C22⋊C4—2124no—
290C22⋊C4—2124no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—1064no—
296C23—4244no—
297C23—4244no—
298C23—4242no—
299C23—4242no—
300C23—4242no—
301C23—4242no—
302C23—2122no—
303C23—2124no—
304C23—2128no—
305C23—2128no—
306C23—2122no—
307C23—2122no—
308C23—2122no—
309C23—2122no—
310C23—2122no—
311C23—2124no—
312C23#1313313111yescharacteristic
313C23#1313313111yescharacteristic
314C23C2×#1171992111yescharacteristic
315D4—2124no—
316D4—2124no—
317D4—2128no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—4244no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4#1313324212yes—
342C8—82no—
343C8—4242no—
344C22—24no—
345C22—24no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—4242no—
353C22—4248no—
354C22—4248no—
355C22—4242no—
356C22—4242no—
357C22—4242no—
358C22—2124no—
359C22—2124no—
360C22—2124no—
361C22#1681041112yes—
362C22#1681041912yes—
363C22C2×#1313313111yescharacteristic
364C22#1681062711yescharacteristic
365C22#1681064111yescharacteristic
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—4242no—
373C2—21no—
374C2—22no—
375C2—22no—
376C2—21no—
377C2—4244no—
378C2#2027324311yescharacteristic
379C2C2×#1681041111yescharacteristic
380C2C2×#1681041911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1821C2
socleC22×C53C2×#99
prime radicalC23×D212C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1821D53
hypocenterC53#11802
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #20273045C2semidirect
3. #20272717C2semidirect
4. C53×#1821C2semidirect
15. #16810349C22semidirect
19. C53×#343C22semidirect
184. #1821D53semidirect
207. #343C2×D53semidirect
210. C53#11802semidirect

Maximal subgroups

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

subgroupindex
C2×#168105402
#202730452
#202727172
C53×#18212
#1180253