Finite Group Databasebeta

← #28303313

Subgroups — #28303313

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290569C211yesmaximal, characteristic
2C4×C22⋊#88324C211yesmaximal, characteristic
3#20289551C212yesmaximal, complemented
4#20289551C212yesmaximal, complemented
5#117×Dic53C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9C2×C4×#88324C2212yes—
10#16811489—21no—
11#16811489—21no—
12C2×#13133176C411yescharacteristic
13C22×C4×Dic53C2211yescharacteristic
14C4⋊C4×Dic53C411yescharacteristic
15C4⋊C4×Dic53C411yescharacteristic
16C4⋊C4×Dic53C2211yescharacteristic
17C4⋊C4×Dic53C2211yescharacteristic
18C22⋊C4×Dic53—21no—
19C22⋊C4×Dic53—21no—
20#16810469—21no—
21C42×Dic53—21no—
22C2×#117×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—22no—
26C4⋊#88324—22no—
27C22⋊#88324—24no—
28C22⋊#88324—42no—
29C4×#88324—22no—
30C4×#88324—22no—
31C23×Dic53D411yescharacteristic
32C23×Dic53D411yescharacteristic
33#13133176—22no—
34#13133176C814yescomplemented
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C22⋊Dic53—21no—
38C2×C4×Dic53—22no—
39C2×C4×Dic53—22no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53C2311yescharacteristic
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C23×C4×C53C2×C411yescharacteristic
54#117×C53—22no—
55#117×C53—22no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53—21no—
59C2×C42×C53—21no—
60C24×C53C22⋊C411yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C2×C812yes—
69C22×C4×C53M1612yes—
70C22×C4×C53C22×C411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—22no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—41no—
80C22×Dic53C22⋊C411yescharacteristic
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C2×D411yescharacteristic
83C22×Dic53C2×D411yescharacteristic
84C2×#88324—24no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53M1612yes—
93C4⋊C4×C53M1612yes—
94C22⋊C4×C53—24no—
95C42×C53—24no—
96C22⋊Dic53—24no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2×C812yes—
100C4⋊Dic53C2×C812yes—
101C4×Dic53—24no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—22no—
105C4×Dic53—22no—
106C4×Dic53—42no—
107C4×Dic53—42no—
108C4×Dic53—42no—
109C4×Dic53—41no—
110C4×Dic53—41no—
111C4×Dic53#4112yes—
112C4×Dic53#4112yes—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53C22⋊C812yes—
120C23×C53C2×C22⋊C411yescharacteristic
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53C22⋊C812yes—
134C2×C4×C53#11711yescharacteristic
135C2×C4×C53#11711yescharacteristic
136C2×C4×C53C22×C811yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53C2×M1611yescharacteristic
139C2×C4×C53C2×M1611yescharacteristic
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—44no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53#10412yes—
157C2×Dic53#12312yes—
158C2×Dic53#12312yes—
159C2×Dic53#13512yes—
160C2×Dic53AGL(1,ℤ/8)11yescharacteristic
161C2×Dic53#13711yescharacteristic
162#88324—44no—
163#88324—82no—
164#8708—531nomaximal, Sylow
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53#38111yescharacteristic
173C22×C53C2×C22⋊C811yescharacteristic
174C22×C53#40711yescharacteristic
175C22×C53#41611yescharacteristic
176C22×C53#41711yescharacteristic
177C22×C53C2×#10411yescharacteristic
178C22×C53#42111yescharacteristic
179C4×C53—22no—
180C4×C53—42no—
181C4×C53—42no—
182C4×C53#43111yescharacteristic
183C4×C53#43111yescharacteristic
184C4×C53#43211yescharacteristic
185C4×C53#43211yescharacteristic
186Dic53—22no—
187Dic53—22no—
188Dic53—41no—
189Dic53—42no—
190Dic53—42no—
191Dic53#48212yes—
192Dic53#48512yes—
193C4×#117—531no—
194#1797—531no—
195C4×C22⋊C8—531no—
196#1279—532no—
197#1279—532no—
198C2×C53—42no—
199C2×C53#147611yescharacteristic
200C2×C53#147711yescharacteristic
201C2×C53#148811yescharacteristic
202C2×C53#148911yescharacteristic
203C2×C53#181711yescharacteristic
204C2×C53#182511yescharacteristic
205C2×C53#182611yescharacteristic
206C2×#117—531no—
207C2×#117#615711yescharacteristic
208C22×C42—531no—
209C2×C4⋊C8—532no—
210C2×C22⋊C8—1061no—
211C2×C22⋊C8—1061no—
212C2×C4×C8—532no—
213C4×C4⋊C4—531no—
214C4×C4⋊C4—531no—
215C4×C4⋊C4—531no—
216C4×C4⋊C4—531no—
217C4×C22⋊C4—1061no—
218C4×C22⋊C4—1061no—
219#376—1061no—
220C43—1061no—
221#336—1061no—
222#336—1061no—
223C53#870811yescharacteristic, complemented, Sylow
224C23×C4—531no—
225C23×C4—531no—
226C23×C4C2×#615711yescharacteristic
227C22×C8—1062no—
228C22×C8—1062no—
229#117—22no—
230#117—22no—
231#117—534no—
232#117—1062no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4C2×#615711yescharacteristic
236C2×C4⋊C4C2×#615711yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—1061no—
239C2×C42—21no—
240C2×C42—531no—
241C2×C42—531no—
242C2×C42—531no—
243C2×C42—531no—
244C2×C42—1062no—
245C2×C42—1062no—
246C2×C42—1061no—
247C2×C42—1061no—
248C2×C42—1061no—
249C2×C42—1061no—
250C2×C42—1061no—
251C4⋊C8—1062no—
252C4⋊C8—1062no—
253C22⋊C8—1064no—
254C22⋊C8—2122no—
255C4×C8—1062no—
256C4×C8—1062no—
257#95—1061no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261C24C22⋊#615711yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—1062no—
273C22×C4—1061no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—2121no—
281C22×C4D53⋊C812yes—
282C22×C4#1171993612yes—
283C22×C4C22⋊#615711yescharacteristic
284C22×C4C22×#615711yescharacteristic
285C2×C8—1064no—
286C2×C8—2122no—
287C2×C8—2122no—
288C2×C8—2122no—
289C2×C8—2122no—
290C2×C8—2122no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—532no—
294C4⋊C4—532no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C4⋊C4#1171994012yes—
298C4⋊C4#1171994012yes—
299C22⋊C4—24no—
300C22⋊C4—1064no—
301C42—24no—
302C42—532no—
303C42—532no—
304C42—1062no—
305C42—1062no—
306C42—1064no—
307C42—1061no—
308C42—1061no—
309C42—2122no—
310C42—2121no—
311C42—2121no—
312C42—2122no—
313C42—2122no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23#1313327412yes—
321C23C2×C22⋊#615711yescharacteristic
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—532no—
335C2×C4—532no—
336C2×C4—531no—
337C2×C4—531no—
338C2×C4—532no—
339C2×C4—532no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—2122no—
350C2×C4—2122no—
351C2×C4—2124no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4#1313327412yes—
357C2×C4#1313329611yescharacteristic
358C2×C4#1313329611yescharacteristic
359C2×C4C2×D53⋊C811yescharacteristic
360C2×C4C2×#1171994011yescharacteristic
361C2×C4C2×#1171994011yescharacteristic
362C2×C4C2×#1171993611yescharacteristic
363C8—4242no—
364C8—2124no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22#1681156611yescharacteristic
373C22C2×#1313327411yescharacteristic
374C22#1681159311yescharacteristic
375C22#1681161311yescharacteristic
376C22#1681161411yescharacteristic
377C22D53⋊#10411yescharacteristic
378C22#1681162011yescharacteristic
379C4—22no—
380C4—42no—
381C4—42no—
382C4—532no—
383C4—532no—
384C4—1062no—
385C4—1062no—
386C4—2121no—
387C4—2122no—
388C4—2122no—
389C4#1681164011yescharacteristic
390C4#1681164011yescharacteristic
391C4#1681164411yescharacteristic
392C4#1681164411yescharacteristic
393C2—42no—
394C2#2029000811yescharacteristic
395C2#2029001211yescharacteristic
396C2#2029004611yescharacteristic
397C2#2029005011yescharacteristic
398C2#2029060611yescharacteristic
399C2#2029062011yescharacteristic
400C2#2029062311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#8708
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #20289551C2semidirect
4. #20289551C2semidirect
34. #13133176C8semidirect
223. C53#8708semidirect

Maximal subgroups

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

subgroupindex
#202905692
C4×C22⋊#883242
#202895512
#117×Dic532
#870853