Finite Group Databasebeta

← #28306335

Subgroups — #28306335

Order 13568.

592 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 112 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#20290570C211yesmaximal, characteristic
2#20289891C211yesmaximal, characteristic
3#20289588C212yesmaximal
4#20289588C212yesmaximal
5#20283066C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×#13133274C2212yes—
8C2×(C2×#6157)⋊C4—21no—
9C2×(C2×#6157)⋊C4—21no—
10C2×(C2×#6157)⋊C4C2212yes—
11#16811489—21no—
12C2×#13133216C411yescharacteristic, complemented
13C2×#13133195—21no—
14C2×C42×D53C2211yescharacteristic
15#16810527—21no—
16#16810521C411yescharacteristic, complemented
17#16810518—21no—
18#16810492C411yescharacteristic, complemented
19#16810482—21no—
20#16810480C2211yescharacteristic
21#16810480C2211yescharacteristic
22C53×#389C411yescharacteristic, complemented
23C23×#6157—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26#13133274—24no—
27(C2×#6157)⋊C4—22no—
28(C2×#6157)⋊C4—42no—
29(C2×#6157)⋊C4—42no—
30(C2×#6157)⋊C4Q814yescomplemented
31C22×Dic106—21no—
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D411yescharacteristic
34C22×C4×D53C2311yescharacteristic
35#13133216—42no—
36#13133195—44no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C42×D53—42no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C42×C53C2×C411yescharacteristic
56#95×C53—21no—
57C23×D53C2×D411yescharacteristic
58C22×#6157—24no—
59C22×#6157—42no—
60C22×#6157—42no—
61C22×#6157—42no—
62C22×#6157—42no—
63C22×#6157—42no—
64C22×#6157—42no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53SD1612yes—
83C2×C4×D53Q1612yes—
84C2×C4×D53C2×Q812yes—
85C2×C4×D53#4112yes—
86C2×Dic106—42no—
87C2×Dic106—42no—
88C2×Dic106—42no—
89C2×#88324—42no—
90C2×#88324—42no—
91C2×#88324—42no—
92C4⋊C4×C53—42no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C42×C53—41no—
96#11719921—42no—
97#11719921—42no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100#11719919—42no—
101#11719919—42no—
102#11719919—42no—
103#11719919—42no—
104C4×Dic53—22no—
105C4×Dic53—22no—
106C4×Dic53—41no—
107C23×C53C2×C22⋊C411yescharacteristic
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53C22⋊Q812yes—
113C22×D53C2×SD1611yescharacteristic
114C22×D53C2×Q1611yescharacteristic
115C22×D53AGL(1,ℤ/8)12yes—
116C2×#6157—44no—
117C2×#6157—44no—
118C2×#6157—44no—
119C2×#6157—82no—
120C2×#6157—82no—
121C2×#6157—82no—
122C2×#6157—82no—
123C2×C4×C53—21no—
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—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53#10312yes—
137C2×C4×C53C4×Q812yes—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53#9912yes—
151C2×Dic53#10312yes—
152C2×Dic53#12411yescharacteristic
153C2×Dic53C4⋊Q811yescharacteristic
154C4×D53—42no—
155C4×D53—42no—
156C4×D53—42no—
157C4×D53—42no—
158C4×D53—44no—
159Dic106—84no—
160#88324—82no—
161#9911—531nomaximal, Sylow
162C22×C53#39111yescharacteristic
163C22×C53C2×#9911yescharacteristic
164C22×C53#41011yescharacteristic
165C22×C53C2×#10311yescharacteristic
166C22×C53#41811yescharacteristic
167C22×C53#42111yescharacteristic
168C22×C53#42111yescharacteristic
169C2×D53—22no—
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—21no—
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
177C2×D53#47612yes—
178C2×D53#47712yes—
179#6157—84no—
180C4×C53—22no—
181C4×C53—22no—
182C4×C53—41no—
183C4×C53—81no—
184C4×C53—81no—
185Dic53—21no—
186Dic53—21no—
187Dic53—22no—
188Dic53—22no—
189Dic53—81no—
190Dic53—81no—
191#2857—531no—
192#1797—531no—
193#1434—531no—
194#1294—532no—
195#1294—532no—
196C2×C53#151011yescharacteristic
197C2×C53#151411yescharacteristic
198C2×C53#151811yescharacteristic
199C2×C53#152011yescharacteristic
200C2×C53#198711yescharacteristic
201C2×C53#199511yescharacteristic
202C2×C53#199611yescharacteristic
203D53—22no—
204D53—22no—
205C2×C4⋊Q8—531no—
206C2×C22⋊Q8—1061no—
207C22×C42—531no—
208C2×C4⋊C8—1061no—
209C2×C22⋊C8—532no—
210#391—1061no—
211#389—531no—
212#389—1061no—
213#389#615711yescharacteristic, complemented
214#384—531no—
215#380—531no—
216#380—531no—
217#380—1061no—
218C2×#95—532no—
219C2×#95—1061no—
220C2×#95—1061no—
221#336—1061no—
222C53#991111yescharacteristic, complemented, Sylow
223C22×Q8—1061no—
224C23×C4—531no—
225C23×C4—531no—
226C23×C4—531no—
227C23×C4—1062no—
228C22×C8—1062no—
229C4⋊Q8—2122no—
230C22⋊Q8—2124no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4C2×#615711yescharacteristic
239C2×C4⋊C4C2×#615711yescharacteristic
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C42—531no—
243C2×C42—531no—
244C2×C42—531no—
245C2×C42—2122no—
246C2×C42C2×#615711yescharacteristic
247C4⋊C8—2122no—
248C22⋊C8—1064no—
249#95—21no—
250#95—534no—
251#95—1061no—
252#95—1061no—
253#95—1062no—
254#95—1061no—
255#95—2122no—
256#95—2122no—
257C24—531no—
258C2×Q8—2122no—
259C2×Q8—2122no—
260C2×Q8—2122no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—532no—
267C22×C4—532no—
268C22×C4—532no—
269C22×C4—532no—
270C22×C4—1062no—
271C22×C4—1062no—
272C22×C4—1064no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4—2122no—
281C22×C4—2122no—
282C22×C4—2122no—
283C22×C4—2122no—
284C22×C4—2122no—
285C22×C4—2122no—
286C22×C4C22⋊#615711yescharacteristic
287C22×C4C22⋊#615711yescharacteristic
288C22×C4C22×#615711yescharacteristic
289C2×C8—2122no—
290C2×C8—2122no—
291C2×C8—2122no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—2121no—
296C4⋊C4—2121no—
297C4⋊C4—2122no—
298C4⋊C4—2122no—
299C4⋊C4—2122no—
300C4⋊C4—2122no—
301C22⋊C4—2122no—
302C22⋊C4—2122no—
303C42—41no—
304C42—1062no—
305C42—1062no—
306C42—2121no—
307C23—532no—
308C23—531no—
309C23—531no—
310C23—532no—
311C23—1061no—
312C23—1061no—
313C23—1061no—
314C23—1061no—
315C23C2×C22⋊#615711yescharacteristic
316Q8—4244no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—532no—
333C2×C4—532no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—4242no—
341C2×C4—4242no—
342C2×C4—4242no—
343C2×C4—4242no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4—2124no—
349C2×C4—2124no—
350C2×C4—2124no—
351C2×C4—2124no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4#1313328012yes—
359C2×C4Q8×#615712yes—
360C8—4242no—
361C22—532no—
362C22—532no—
363C22—1062no—
364C22—1062no—
365C22—1062no—
366C22—1062no—
367C22—1061no—
368C22—1061no—
369C22—1061no—
370C22—1061no—
371C22(C2×#6157)⋊Q811yescharacteristic
372C22D53⋊#9911yescharacteristic
373C22#1681160211yescharacteristic
374C22C2×#1313328011yescharacteristic
375C22#1681161511yescharacteristic
376C22#1681162011yescharacteristic
377C22#1681162011yescharacteristic
378C4—22no—
379C4—22no—
380C4—41no—
381C4—81no—
382C4—81no—
383C4—1062no—
384C4—1061no—
385C4—1061no—
386C4—1062no—
387C4—4244no—
388C4—4241no—
389C4—4241no—
390C2—1062no—
391C2—1062no—
392C2#2029012111yescharacteristic
393C2#2029013411yescharacteristic
394C2#2029014811yescharacteristic
395C2#2029015311yescharacteristic
396C2#2029089311yescharacteristic
397C2#2029090211yescharacteristic
398C2#2029090611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenter#389#6157
hypocenterC53#9911
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
12. C2×#13133216C4semidirect
16. #16810521C4semidirect
18. #16810492C4semidirect
22. C53×#389C4semidirect
30. (C2×#6157)⋊C4Q8semidirect
213. #389#6157semidirect
222. C53#9911semidirect

Maximal subgroups

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

subgroupindex
#202905702
#202898912
#202895882
#202830662
#991153