Finite Group Databasebeta

← #27303245

Subgroups — #27303245

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810856C211yesmaximal, characteristic
2C2×#16810448C211yesmaximal, characteristic, complemented
3C2×#16810424C211yesmaximal, characteristic, complemented
4C2×#16810402C211yesmaximal, characteristic, complemented
5#20274741C211yesmaximal, characteristic, complemented
6#20272807C211yesmaximal, characteristic, complemented
7C2×C53×#357C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9C2×#13133188—21no—
10C22×#11719908C2211yescharacteristic
11#16810856—22no—
12#16810856—22no—
13C2×#13133151C2211yescharacteristic
14C2×#13133151C2211yescharacteristic
15C2×#13133138—21no—
16#16810480—21no—
17#16810448—22no—
18#16810424—22no—
19#16810402—22no—
20#16810385—21no—
21C22×D8×C53C2211yescharacteristic
22C22×C16×C53C2211yescharacteristic
23C2×#107×C53C2211yescharacteristic
24C53×#357—22no—
25C22×C22⋊D53—41no—
26C22×C4×D53D411yescharacteristic
27C22×#88322D411yescharacteristic
28C2×C8×D53—21no—
29C2×C8×D53—21no—
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53—21no—
34#13133188—24no—
35#13133188—42no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C22⋊Dic53—41no—
39C2×#11719921—41no—
40C2×#11719908—21no—
41C2×#11719908—21no—
42C2×#11719908D412yescomplemented
43#13133151—21no—
44#13133151—21no—
45#13133151—21no—
46#13133151D412yes—
47#13133138—24no—
48#13133124—41no—
49C22×D4×C53—21no—
50C2×D8×C53—21no—
51C2×D8×C53—21no—
52C2×D8×C53—21no—
53C2×D8×C53—21no—
54C2×D8×C53—21no—
55C2×D8×C53D412yes—
56C22×C8×C53C2311yescharacteristic
57C2×C4⋊C4×C53—21no—
58C2×C16×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53—21no—
61#107×C53—21no—
62#107×C53—21no—
63C24×C53—41no—
64C23×D53D811yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C2×D411yescharacteristic
73C2×C22⋊D53—44no—
74C2×C22⋊D53—82no—
75C2×C22⋊D53—82no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53D811yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85D8×C53—22no—
86D8×C53—22no—
87D8×C53—41no—
88D8×C53—41no—
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C2×C8×C53#4111yescharacteristic
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C22⋊Dic53—44no—
98#11719921—82no—
99C4⋊Dic53—22no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103C2×#88322—21no—
104C2×#88322—21no—
105C2×#88322—21no—
106C8×D53—41no—
107C8×D53—41no—
108C8×D53—41no—
109C8×D53—41no—
110C16×C53—41no—
111#11719908—22no—
112C23×C53—42no—
113C23×C53—81no—
114C23×C53—81no—
115C23×C53—81no—
116C23×C53—81no—
117C23×C53—81no—
118C23×C53—81no—
119C23×C53C2×D811yescharacteristic
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53D1612yes—
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—81no—
130D4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53#12011yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12311yescharacteristic
137C2×C4×C53#13511yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C53#13711yescharacteristic
140C22⋊D53—88no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—42no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150C4×D53—41no—
151C4×D53—41no—
152C4×D53—41no—
153C4×D53—41no—
154C8×C53#12111yescharacteristic
155C8×C53#12111yescharacteristic
156C8×C53#12311yescharacteristic
157C8×C53#12411yescharacteristic
158#88322—41no—
159#13380—531nomaximal, Sylow
160C22×C53—82no—
161C22×C53—82no—
162C22×C53—82no—
163C22×C53—161no—
164C22×C53—161no—
165C22×C53—161no—
166C22×C53—161no—
167C22×C53#44711yescharacteristic
168C22×C53#46611yescharacteristic
169C22×C53#48011yescharacteristic
170C22×C53C2×D1611yescharacteristic
171C22×C53#50811yescharacteristic
172C22×C53#50911yescharacteristic
173C22×C53#51011yescharacteristic
174C2×D53—22no—
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C4×C53—161no—
182C4×C53#39411yescharacteristic
183C4×C53#45211yescharacteristic
184C4×C53#46811yescharacteristic
185C4×C53#48211yescharacteristic
186Dic53—41no—
187Dic53—82no—
188Dic53—161no—
189C2×#466—531no—
190C2×#366—531no—
191C2×#357—531no—
192C2×#357D5311yescharacteristic, complemented
193C2×C22⋊C16—531no—
194#1808—531no—
195#1380—531no—
196C2×C53—162no—
197C2×C53#203311yescharacteristic
198C2×C53#218411yescharacteristic
199C2×C53#218911yescharacteristic
200C2×C53#221511yescharacteristic
201C2×C53#222011yescharacteristic
202C2×C53#223211yescharacteristic
203C2×C53#223411yescharacteristic
204D53—42no—
205C22×D8C2×D5311yescharacteristic
206C23×C8—531no—
207C2×#121—1061no—
208C22×C16—531no—
209C22×C16C2×D5311yescharacteristic
210#466—1062no—
211#466—1062no—
212C2×#107—531no—
213C2×#107—531no—
214C2×#107C2×D5311yescharacteristic
215C2×#102—1061no—
216#380—1061no—
217#366—1062no—
218#357—22no—
219#357—1062no—
220C22⋊C16—1062no—
221#340—1061no—
222C53#1338011yescharacteristic, complemented, Sylow
223C22×D4—21no—
224C22×D4—2121no—
225C23×C4—531no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8C22⋊D5312yes—
232C22×C8—531no—
233C22×C8—1061no—
234C22×C8—1061no—
235C22×C8—1061no—
236C22×C8—1061no—
237C22×C8—1061no—
238C22×C8—1061no—
239C22×C8C22×D5311yescharacteristic
240#121—1064no—
241#121—2122no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C22⋊C4—2121no—
246C2×C22⋊C4—2121no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—532no—
251C2×C16—1061no—
252C2×C16—1061no—
253#107—21no—
254#107—21no—
255#107—532no—
256#107—1061no—
257#107—1061no—
258#107—1061no—
259#102—1064no—
260#95—2121no—
261C24—41no—
262C24—531no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—4242no—
270C2×D4—4242no—
271C2×D4—2124no—
272C22×C4—41no—
273C22×C4—531no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4D4×D5311yescharacteristic
283D8—22no—
284D8—22no—
285D8—41no—
286D8—41no—
287C2×C8—1061no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—2121no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8—2121no—
294C2×C8#1171994511yescharacteristic
295C2×C8#1171994511yescharacteristic
296C2×C8D4×D5311yescharacteristic
297C2×C8D4×D5311yescharacteristic
298C2×C8#1171994711yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—1062no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—4242no—
307C22⋊C4—2124no—
308C16—41no—
309C16—1062no—
310C23—42no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—532no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23—1061no—
324C23D8×D5311yescharacteristic
325D4—42no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—4248no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—4241no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—4241no—
340C2×C4—4241no—
341C2×C4—2122no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4#1313318511yescharacteristic
347C2×C4#1313319011yescharacteristic
348C2×C4#1313319911yescharacteristic
349C2×C4#1313323811yescharacteristic
350C2×C4#1313324111yescharacteristic
351C2×C4#1313324611yescharacteristic
352C8—2121no—
353C8#1313318911yescharacteristic
354C8#1313319011yescharacteristic
355C8#1313320011yescharacteristic
356C8#1313320311yescharacteristic
357C22—82no—
358C22—82no—
359C22—82no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22—161no—
364C22—1062no—
365C22—1062no—
366C22—1062no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22#1681075511yescharacteristic
372C22#1681085811yescharacteristic
373C22#1681093511yescharacteristic
374C22D16×D5311yescharacteristic
375C22#1681105411yescharacteristic
376C22#1681105711yescharacteristic
377C22#1681106211yescharacteristic
378C4—161no—
379C4—8481no—
380C4—4242no—
381C4—2121no—
382C4#1681053811yescharacteristic
383C4#1681078211yescharacteristic
384C4#1681086811yescharacteristic
385C4#1681094611yescharacteristic
386C2—162no—
387C2—2122no—
388C2#2027589511yescharacteristic
389C2#2027662511yescharacteristic
390C2#2027665211yescharacteristic
391C2#2027680011yescharacteristic
392C2#2027682711yescharacteristic
393C2#2027689411yescharacteristic
394C2#2027690511yescharacteristic
subgroupG/H
centerC23D8×D53
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC2×C53×#357C2
socleC23×C53C2×D8
prime radicalC2×#16810856C2
hypercenterC2×#357D53
hypocenterC53#13380
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#16810448C2semidirect
3. C2×#16810424C2semidirect
4. C2×#16810402C2semidirect
5. #20274741C2semidirect
6. #20272807C2semidirect
7. C2×C53×#357C2semidirect
42. C2×#11719908D4semidirect
192. C2×#357D53semidirect
222. C53#13380semidirect

Maximal subgroups

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

subgroupindex
C2×#168108562
C2×#168104482
C2×#168104242
C2×#168104022
#202747412
#202728072
C2×C53×#3572
#1338053