Finite Group Databasebeta

← #27294297

Subgroups — #27294297

Order 13568.

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

normal subgroups only

This group has 514 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#20280457C211yesmaximal, characteristic
2C2×#16810349C211yesmaximal, characteristic, complemented
3#20272720C212yesmaximal, complemented
4#20272717C212yesmaximal, complemented
5C2×C53×#343C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133191C411yescharacteristic
8C2×C22⋊#88322—21no—
9#16810553C411yescharacteristic
10#16810553C2211yescharacteristic
11#16810545—21no—
12#16810545—21no—
13#16810540C411yescharacteristic
14#16810540C2211yescharacteristic
15#16810502—21no—
16#16810483C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C22×M16×C53—21no—
20C22×C4⋊C4×C53C2211yescharacteristic
21C53×#343—21no—
22C53×#343C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×#88322—41no—
26#13133200—42no—
27#13133200—42no—
28#13133191—41no—
29#13133191—41no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C22⋊#88322—22no—
41C22⋊#88322—41no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C2×M16×C53—24no—
50C2×M16×C53—22no—
51C2×M16×C53—41no—
52C2×M16×C53—42no—
53C22×C8×C53—41no—
54C2×C4⋊C4×C53—41no—
55C2×C4⋊C4×C53—41no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C4⋊C4×C53C2311yescharacteristic
60C24×C53C22⋊C411yescharacteristic
61C23×D53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—81no—
70C22×C4×C53—81no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53D812yes—
75C22×C4×C53SD1612yes—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C2×C22⋊D53—42no—
82C2×C22⋊D53—42no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C2×D212—41no—
87C2×D212—81no—
88M16×C53—24no—
89M16×C53—44no—
90M16×C53—42no—
91C2×C8×C53—44no—
92C2×C8×C53—42no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C22⋊Dic53—42no—
98#11719921—42no—
99#11719921—42no—
100#11719921—41no—
101#11719921—41no—
102#11719921—81no—
103#11719921—81no—
104C4⋊Dic53—41no—
105C4⋊Dic53—81no—
106#11719919—42no—
107#11719919—41no—
108#11719919—41no—
109C2×#88322—42no—
110C2×#88322—82no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53#9911yescharacteristic
114C23×C53#9911yescharacteristic
115C23×C53#10212yes—
116C23×C53#10412yes—
117C23×C53C2×C22⋊C411yescharacteristic
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
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×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53#10212yes—
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53C2×SD1611yescharacteristic
148C22⋊D53—82no—
149C22⋊D53—82no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159D212—82no—
160C8×C53—44no—
161#88322—82no—
162#11698—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53#32712yes—
172C22×C53#32712yes—
173C22×C53C2×#9911yescharacteristic
174C22×C53C2×#10211yescharacteristic
175C22×C53C2×#10411yescharacteristic
176C2×D53—81no—
177C2×D53—81no—
178C2×D53—81no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C2×D53—81no—
183C4×C53—21no—
184C4×C53—22no—
185C4×C53—21no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—41no—
190Dic53—81no—
191Dic53—81no—
192Dic53—81no—
193#2618—531no—
194C2×#343D5311yescharacteristic, complemented
195C2×#324—531no—
196#1346—532no—
197#1345—532no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—22no—
201C2×C53—22no—
202C2×C53C2×#32711yescharacteristic
203C2×C53#212111yescharacteristic
204C2×C53#213311yescharacteristic
205D53—161no—
206C22×M16—21no—
207C2×#123—1061no—
208C2×#121—531no—
209C22×C4⋊C4C2×D5311yescharacteristic
210C2×C22⋊C8—1061no—
211#398—531no—
212#398—531no—
213#396—1061no—
214#396—1061no—
215#394—531no—
216#394—531no—
217#386—1061no—
218#380—531no—
219#343—21no—
220#343C2×D5312yes—
221#324—532no—
222#324—1061no—
223C53#1169811yescharacteristic, complemented, Sylow
224C22×D4—531no—
225C22×D4—1061no—
226C23×C4C4×D5311yescharacteristic
227C23×C4D21211yescharacteristic
228C23×C4C22⋊D5311yescharacteristic
229C2×M16—24no—
230C2×M16—22no—
231C2×M16—41no—
232C2×M16—42no—
233C22×C8—41no—
234C22×C8—2121no—
235#123—2122no—
236#123—2122no—
237#121—2121no—
238#121—2121no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—531no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4C4×D5311yescharacteristic
246C2×C4⋊C4D21211yescharacteristic
247C2×C4⋊C4C22⋊D5311yescharacteristic
248C2×C4⋊C4C22×D5311yescharacteristic
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C22⋊C4—1061no—
252C2×C22⋊C4—1061no—
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C22⋊C8—1062no—
256C22⋊C8—2121no—
257#95—531no—
258#95—531no—
259#95—1061no—
260#95—1061no—
261C24—1061no—
262C24#1171992111yescharacteristic
263C2×D4—4241no—
264C2×D4—2121no—
265C2×D4—2121no—
266C2×D4—2121no—
267C2×D4—2122no—
268C2×D4—2122no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4#1171992111yescharacteristic
282C22×C4#1171992111yescharacteristic
283C22×C4#1171992111yescharacteristic
284C22×C4#1171992212yes—
285C22×C4#1171992412yes—
286C22×C4C2×C4×D5311yescharacteristic
287C22×C4C2×D21211yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289M16—24no—
290M16—44no—
291M16—42no—
292C2×C8—42no—
293C2×C8—44no—
294C2×C8—4242no—
295C2×C8—2122no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—4241no—
301C4⋊C4—2121no—
302C4⋊C4—2122no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C22⋊C4—4241no—
306C22⋊C4—4241no—
307C22⋊C4—2122no—
308C22⋊C4—2122no—
309C22⋊C4—2122no—
310C22⋊C4—2121no—
311C22⋊C4—2121no—
312C23—41no—
313C23—41no—
314C23—2121no—
315C23—2121no—
316C23—2121no—
317C23—2121no—
318C23—2121no—
319C23—2121no—
320C23—2121no—
321C23#1313313111yescharacteristic
322C23#1313313111yescharacteristic
323C23#1313313712yes—
324C23#1313314512yes—
325C23C2×#1171992111yescharacteristic
326D4—4242no—
327D4—4242no—
328D4—4242no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4#1313313712yes—
359C2×C4C2×#1171992211yescharacteristic
360C2×C4C2×#1171992411yescharacteristic
361C8—44no—
362C8—4242no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—4241no—
372C22—4241no—
373C22—4241no—
374C22—4241no—
375C22—4241no—
376C22—4241no—
377C22—4241no—
378C22#1681035612yes—
379C22#1681035612yes—
380C22C2×#1313313111yescharacteristic
381C22C2×#1313313711yescharacteristic
382C22C2×#1313314511yescharacteristic
383C4—21no—
384C4—22no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—4241no—
391C4—4241no—
392C4—4241no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2—22no—
397C2—8481no—
398C2C2×#1681035611yescharacteristic
399C2#2027636211yescharacteristic
400C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#343C2
socleC22×C53C2×#99
prime radicalC2×#13133191C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#343D53
hypocenterC53#11698
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#16810349C2semidirect
3. #20272720C2semidirect
4. #20272717C2semidirect
5. C2×C53×#343C2semidirect
194. C2×#343D53semidirect
223. C53#11698semidirect

Maximal subgroups

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

subgroupindex
#202804572
C2×#168103492
#202727202
#202727172
C2×C53×#3432
#1169853