Finite Group Databasebeta

← #27303435

Subgroups — #27303435

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810998C211yesmaximal, characteristic
2C2×#16810425C211yesmaximal, characteristic, complemented
3C2×#16810424C211yesmaximal, characteristic, complemented
4C2×#16810423C211yesmaximal, characteristic, complemented
5#20275449C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C2×C53×#357C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×#11719922—21no—
10C2×#13133213—21no—
11C22×#11719908C2211yescharacteristic
12#16810998—22no—
13#16810998—22no—
14C2×#13133151C2211yescharacteristic
15C2×C8×Dic53C2211yescharacteristic
16#16810522—21no—
17#16810425—22no—
18#16810424—22no—
19#16810423—22no—
20#16810385—21no—
21C22×D8×C53C2211yescharacteristic
22C22×C16×C53C2211yescharacteristic
23C2×#107×C53C2211yescharacteristic
24C53×#357—22no—
25C22×D212—21no—
26C22×C22⋊D53—41no—
27C22×C22⋊D53—41no—
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424D412yes—
34C2×#11719922—24no—
35C2×#11719922—42no—
36C2×#11719922—42no—
37C22×#88322D411yescharacteristic
38#13133213—24no—
39#13133213—42no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—41no—
42C2×C4×Dic53D411yescharacteristic
43C2×#11719908—21no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46#13133151—21no—
47#13133151—21no—
48C8×Dic53—21no—
49C8×Dic53—21no—
50#13133124—41no—
51C22×D4×C53—21no—
52C2×D8×C53—21no—
53C2×D8×C53—21no—
54C2×D8×C53—21no—
55C2×D8×C53—21no—
56C2×D8×C53—21no—
57C2×D8×C53D412yes—
58C22×C8×C53C2311yescharacteristic
59C2×C4⋊C4×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53—21no—
63#107×C53—21no—
64#107×C53D412yescomplemented
65C24×C53—41no—
66C23×D53—41no—
67C2×D4×C53—22no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C2×C22⋊D53—44no—
76C2×C22⋊D53—44no—
77C2×C22⋊D53—82no—
78C2×C22⋊D53—82no—
79C2×C22⋊D53—82no—
80C2×C22⋊D53—82no—
81C22×Dic53—41no—
82C22×Dic53D811yescharacteristic
83C22×Dic53D811yescharacteristic
84C2×D212—22no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90D8×C53—22no—
91D8×C53—22no—
92D8×C53—41no—
93D8×C53—41no—
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53#4111yescharacteristic
98C2×C8×C53#4111yescharacteristic
99C2×C8×C53#4111yescharacteristic
100C4⋊C4×C53—22no—
101C4⋊C4×C53—41no—
102#11719922—48no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4⋊Dic53—82no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322SD1612yes—
111D424—22no—
112D424—22no—
113D424—41no—
114D424—41no—
115C16×C53—41no—
116#11719908—41no—
117C23×C53—42no—
118C23×C53—81no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53—81no—
122C23×C53—81no—
123C23×C53—81no—
124C23×C53C2×D811yescharacteristic
125C22×D53—42no—
126C22×D53—81no—
127C22×D53—81no—
128C22×D53—81no—
129C22×D53—81no—
130C22×D53—81no—
131C22×D53—81no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—81no—
135D4×C53—81no—
136C2×C4×C53—42no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12411yescharacteristic
142C2×C4×C53C2×SD1611yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C22⋊D53—88no—
146C22⋊D53—88no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—81no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154C2×Dic53D1612yes—
155C2×Dic53SD3212yes—
156D212—42no—
157D212—42no—
158D212—81no—
159D212—81no—
160C8×C53#12111yescharacteristic
161C8×C53#12111yescharacteristic
162C8×C53#12311yescharacteristic
163C8×C53#12311yescharacteristic
164#88322—22no—
165#13407—531nomaximal, Sylow
166C22×C53—82no—
167C22×C53—82no—
168C22×C53—82no—
169C22×C53—161no—
170C22×C53—161no—
171C22×C53—161no—
172C22×C53—161no—
173C22×C53#44711yescharacteristic
174C22×C53#45911yescharacteristic
175C22×C53#48611yescharacteristic
176C22×C53C2×D1611yescharacteristic
177C22×C53C2×SD3211yescharacteristic
178C22×C53#50911yescharacteristic
179C22×C53#50911yescharacteristic
180C2×D53—82no—
181C2×D53—82no—
182C2×D53—82no—
183C2×D53—161no—
184C2×D53—161no—
185C2×D53—161no—
186C2×D53—161no—
187C4×C53—82no—
188C4×C53#39411yescharacteristic
189C4×C53#45011yescharacteristic
190C4×C53#46011yescharacteristic
191C4×C53#49011yescharacteristic
192Dic53—22no—
193Dic53—22no—
194Dic53—161no—
195C2×#493—531no—
196C2×#357—531no—
197C2×#357—531no—
198C2×#357—531no—
199C2×#357D5311yescharacteristic, complemented
200#1942—531no—
201#1374—531no—
202C2×C53—162no—
203C2×C53#204111yescharacteristic
204C2×C53#218411yescharacteristic
205C2×C53#218711yescharacteristic
206C2×C53#220611yescharacteristic
207C2×C53#220711yescharacteristic
208C2×C53#224011yescharacteristic
209C2×C53#224411yescharacteristic
210D53—162no—
211C22×D8—531no—
212C22×D8—1061no—
213C22×D8C2×D5311yescharacteristic
214C2×#127—1061no—
215C22×C16—531no—
216C22×C16C2×D5311yescharacteristic
217#493—1062no—
218#493—1062no—
219C2×#107—531no—
220C2×#107C2×D5311yescharacteristic
221C2×C4×C8—531no—
222#389—1061no—
223#357—22no—
224#357—1062no—
225#357—1062no—
226#357—1062no—
227#340—1061no—
228C53#1340711yescharacteristic, complemented, Sylow
229C22×D4—21no—
230C22×D4—1061no—
231C22×D4—2121no—
232C22×D4—2121no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8—532no—
239C2×D8—1064no—
240C2×D8—1061no—
241C2×D8—1061no—
242C2×D8—1061no—
243C2×D8—1061no—
244C2×D8—1061no—
245C2×D8—2122no—
246C2×D8—2122no—
247C2×D8D21212yes—
248C22×C8—531no—
249C22×C8C22×D5311yescharacteristic
250#127—1064no—
251#127—2122no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—2121no—
255C2×C42—531no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—1061no—
260C2×C16—1061no—
261C2×C16—1061no—
262#107—21no—
263#107—1061no—
264#107—1061no—
265#107C22⋊D5312yescomplemented
266C4×C8—1061no—
267C4×C8—1061no—
268#95—2121no—
269C24—41no—
270C24—2121no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—1062no—
278C2×D4—4242no—
279C2×D4—4242no—
280C2×D4—4242no—
281C2×D4—4242no—
282C2×D4—2124no—
283C2×D4—2124no—
284C2×D4—2121no—
285C2×D4—2121no—
286C2×D4—2121no—
287C2×D4—2121no—
288C2×D4—2121no—
289C22×C4—41no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—2121no—
293C22×C4D4×D5311yescharacteristic
294D8—22no—
295D8—22no—
296D8—41no—
297D8—41no—
298D8—1062no—
299D8—1062no—
300D8—2128no—
301D8—2121no—
302D8—2121no—
303C2×C8—532no—
304C2×C8—1061no—
305C2×C8—1061no—
306C2×C8C2×D21211yescharacteristic
307C2×C8#1171994511yescharacteristic
308C2×C8D4×D5311yescharacteristic
309C2×C8#1171994711yescharacteristic
310C2×C8#1171994711yescharacteristic
311C2×C8C2×C22⋊D5311yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—4242no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C42—1061no—
318C42—1061no—
319C16—41no—
320C16—2121no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—4241no—
329C23—4241no—
330C23—4241no—
331C23—4241no—
332C23—4241no—
333C23—4241no—
334C23—2122no—
335C23D8×D5311yescharacteristic
336D4—42no—
337D4—42no—
338D4—81no—
339D4—81no—
340D4—4248no—
341D4—4248no—
342D4—4241no—
343D4—4241no—
344D4—2122no—
345D4—2122no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—532no—
350C2×C4—532no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4#1313318711yescharacteristic
359C2×C4#1313318911yescharacteristic
360C2×C4#1313320311yescharacteristic
361C2×C4SD16×D5311yescharacteristic
362C2×C4#1313324111yescharacteristic
363C2×C4#1313324411yescharacteristic
364C8—1062no—
365C8#1313318811yescharacteristic
366C8#1313319111yescharacteristic
367C8#1313319911yescharacteristic
368C8#1313320111yescharacteristic
369C22—82no—
370C22—82no—
371C22—82no—
372C22—161no—
373C22—161no—
374C22—161no—
375C22—161no—
376C22—8481no—
377C22—8481no—
378C22—8481no—
379C22—8481no—
380C22—4242no—
381C22—4242no—
382C22—4242no—
383C22#1681075811yescharacteristic
384C22#1681081611yescharacteristic
385C22#1681096811yescharacteristic
386C22D16×D5311yescharacteristic
387C22SD32×D5311yescharacteristic
388C22#1681105711yescharacteristic
389C22#1681106011yescharacteristic
390C4—82no—
391C4—8481no—
392C4—1062no—
393C4—1062no—
394C4#1681053911yescharacteristic
395C4#1681077511yescharacteristic
396C4#1681082111yescharacteristic
397C4#1681098611yescharacteristic
398C2—162no—
399C2—8482no—
400C2#2027594811yescharacteristic
401C2#2027662811yescharacteristic
402C2#2027664511yescharacteristic
403C2#2027674811yescharacteristic
404C2#2027675311yescharacteristic
405C2#2027693711yescharacteristic
406C2#2027695511yescharacteristic
subgroupG/H
centerC23D8×D53
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC2×C53×#357C2
socleC23×C53C2×D8
prime radicalC2×#16810998C2
hypercenterC2×#357D53
hypocenterC53#13407
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×#16810425C2semidirect
3. C2×#16810424C2semidirect
4. C2×#16810423C2semidirect
5. #20275449C2semidirect
6. #20272792C2semidirect
7. C2×C53×#357C2semidirect
64. #107×C53D4semidirect
199. C2×#357D53semidirect
228. C53#13407semidirect
265. #107C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109982
C2×#168104252
C2×#168104242
C2×#168104232
#202754492
#202727922
C2×C53×#3572
#1340753