Finite Group Databasebeta

← #27301458

Subgroups — #27301458

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C16×D53C211yesmaximal, characteristic
2C2×#16810865C211yesmaximal, characteristic
3C2×#16810428C211yesmaximal, characteristic, complemented
4C2×#16810426C211yesmaximal, characteristic, complemented
5#20274741C211yesmaximal, characteristic
6#20272806C211yesmaximal, characteristic, complemented
7C53×#1380C211yesmaximal, characteristic, complemented
8C22×Dic212C2211yescharacteristic
9C22×C8×D53C2211yescharacteristic
10C2×#13133197—21no—
11C22×#11719908C2211yescharacteristic
12C2×C16×D53—22no—
13C2×C16×D53—22no—
14C2×C16×D53—22no—
15#16810865—22no—
16#16810865C414yescomplemented
17C2×#13133151C2211yescharacteristic
18C2×#13133140—21no—
19#16810480—21no—
20#16810428—22no—
21#16810426—22no—
22#16810385—21no—
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C2×#107×C53C2211yescharacteristic
26C22×Dic106—21no—
27C22×C4×D53D411yescharacteristic
28C2×Dic212—21no—
29C2×Dic212—21no—
30C2×Dic212—21no—
31C2×Dic212—21no—
32C2×Dic212—21no—
33C2×Dic212C2×C412yes—
34C22×#88322D411yescharacteristic
35C2×C8×D53—21no—
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38C2×C8×D53—21no—
39C2×C8×D53C2×C412yes—
40C2×C8×D53D412yes—
41#13133197—24no—
42#13133197—42no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—41no—
45C2×#11719921—41no—
46C2×#11719921—41no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49C2×#11719908D412yes—
50C16×D53—28no—
51#13133151—21no—
52#13133151—21no—
53#13133140—24no—
54#13133124—41no—
55C22×C8×C53C2311yescharacteristic
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C16×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53D412yes—
61#107×C53—21no—
62#107×C53—21no—
63#107×C53—21no—
64#107×C53C2×C412yes—
65C23×D53D811yescharacteristic
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—41no—
70C22×Dic53—41no—
71C22×Dic53D811yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53C22⋊C412yes—
77C2×C4×D53SD1612yes—
78C2×Dic106—22no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C2×C8×C53C22×C411yescharacteristic
85C2×C8×C53C2×D411yescharacteristic
86C2×C8×C53C2×D411yescharacteristic
87C2×C8×C53C2×D411yescharacteristic
88C2×C8×C53#4111yescharacteristic
89C2×C8×C53#4111yescharacteristic
90C4⋊C4×C53—22no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94#11719921—44no—
95#11719921—82no—
96C4⋊Dic53—44no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99C2×#88322—21no—
100C2×#88322—21no—
101C2×#88322C22⋊C412yes—
102Dic212—22no—
103Dic212—22no—
104Dic212—41no—
105Dic212—41no—
106C8×D53—22no—
107C8×D53—22no—
108C8×D53—22no—
109C8×D53—22no—
110C16×C53—22no—
111#11719908—22no—
112C23×C53C2×D811yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53#10212yes—
118C22×D53SD3212yes—
119C22×D53Q3212yes—
120C2×C4×C53—42no—
121C2×C4×C53—81no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53C2×C22⋊C411yescharacteristic
127C2×C4×C53#12111yescharacteristic
128C2×C4×C53#12111yescharacteristic
129C2×C4×C53C2×SD1611yescharacteristic
130C2×C4×C53#13511yescharacteristic
131C2×C4×C53#13511yescharacteristic
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—42no—
135C2×Dic53—81no—
136C2×Dic53—81no—
137C2×Dic53—81no—
138C2×Dic53—81no—
139C2×Dic53—81no—
140C2×Dic53#10212yes—
141C4×D53—22no—
142C4×D53—22no—
143C4×D53—22no—
144C4×D53—22no—
145Dic106—42no—
146Dic106—42no—
147Dic106—81no—
148Dic106—81no—
149C8×C53C4×D411yescharacteristic
150C8×C53C4×D411yescharacteristic
151C8×C53#12011yescharacteristic
152C8×C53#12311yescharacteristic
153#88322—22no—
154#13060—531nomaximal, Sylow
155C22×C53C2×#10211yescharacteristic
156C22×C53#46611yescharacteristic
157C22×C53#46611yescharacteristic
158C22×C53C2×SD3211yescharacteristic
159C22×C53C2×Q3211yescharacteristic
160C22×C53#50811yescharacteristic
161C22×C53#50811yescharacteristic
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—22no—
167C2×D53—22no—
168C2×D53#35814yescomplemented
169C4×C53—82no—
170C4×C53—161no—
171C4×C53#38611yescharacteristic
172C4×C53#41611yescharacteristic
173C4×C53#46511yescharacteristic
174C4×C53#46511yescharacteristic
175Dic53—22no—
176Dic53—82no—
177Dic53—161no—
178C23×C16—531no—
179C2×#467—531no—
180C2×#358—531no—
181C2×#358—531no—
182#1808—531no—
183#1380—531no—
184#1380D5311yescharacteristic, complemented
185C2×C53#189311yescharacteristic
186C2×C53C2×#35811yescharacteristic
187C2×C53#213811yescharacteristic
188C2×C53#221611yescharacteristic
189C2×C53#221611yescharacteristic
190C2×C53#221711yescharacteristic
191C2×C53#221711yescharacteristic
192D53—24no—
193C22×Q16—531no—
194C23×C8—531no—
195C2×C22⋊Q8—1061no—
196C22×C16—531no—
197C22×C16—1062no—
198C22×C16—1062no—
199C22×C16—1062no—
200C22×C16C2×D5311yescharacteristic
201#467—534no—
202#467—1062no—
203C2×#107—531no—
204C2×#107C2×D5311yescharacteristic
205C2×#107C2×D5311yescharacteristic
206C2×#103—1061no—
207#380—1061no—
208#358—1062no—
209#358—1062no—
210#340—1061no—
211C53#1306011yescharacteristic, complemented, Sylow
212C22×Q8—1061no—
213C23×C4—531no—
214C2×Q16—532no—
215C2×Q16—1061no—
216C2×Q16—1061no—
217C2×Q16—1061no—
218C2×Q16—1061no—
219C2×Q16—1061no—
220C22×C8—531no—
221C22×C8—532no—
222C22×C8—532no—
223C22×C8—1061no—
224C22×C8—1061no—
225C22×C8—1061no—
226C22×C8—1061no—
227C22×C8C22×D5311yescharacteristic
228C22⋊Q8—1064no—
229C22⋊Q8—2122no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—2121no—
234C2×C22⋊C4—2121no—
235C2×C22⋊C4—2121no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—532no—
239C2×C16—1068no—
240C2×C16—1061no—
241C2×C16—1061no—
242C2×C16D21212yes—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—1061no—
247#107—1061no—
248#107C4×D5312yes—
249#103—1064no—
250#95—2121no—
251C24—531no—
252C2×Q8—1062no—
253C2×Q8—2121no—
254C2×Q8—2121no—
255C2×Q8—2121no—
256C2×Q8—2121no—
257C2×Q8—2121no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—531no—
261C22×C4—532no—
262C22×C4—532no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4D4×D5311yescharacteristic
270Q16—1062no—
271Q16—1062no—
272Q16—2121no—
273Q16—2121no—
274C2×C8—532no—
275C2×C8—1061no—
276C2×C8—1061no—
277C2×C8—1062no—
278C2×C8—1062no—
279C2×C8—1062no—
280C2×C8—1062no—
281C2×C8C2×C4×D5311yescharacteristic
282C2×C8C2×D21211yescharacteristic
283C2×C8#1171994511yescharacteristic
284C2×C8D4×D5311yescharacteristic
285C2×C8D4×D5311yescharacteristic
286C2×C8#1171994911yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—2124no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C22⋊C4—4242no—
295C22⋊C4—2124no—
296C16—22no—
297C16—1062no—
298C23—532no—
299C23—532no—
300C23—532no—
301C23—1061no—
302C23—1061no—
303C23—1061no—
304C23—1061no—
305C23D8×D5311yescharacteristic
306Q8—4241no—
307Q8—4241no—
308Q8—2122no—
309Q8—2122no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—532no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—4241no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—2122no—
329C2×C4C22⋊C4×D5311yescharacteristic
330C2×C4#1313319011yescharacteristic
331C2×C4#1313319211yescharacteristic
332C2×C4SD16×D5311yescharacteristic
333C2×C4#1313323811yescharacteristic
334C2×C4#1313324011yescharacteristic
335C8—1062no—
336C8#1313317911yescharacteristic
337C8C4×D21211yescharacteristic
338C8#1313318711yescharacteristic
339C8#1313320211yescharacteristic
340C22—534no—
341C22—1062no—
342C22—1062no—
343C22—1062no—
344C22—1062no—
345C22—1062no—
346C22—1062no—
347C22#102×D5311yescharacteristic
348C22#1681085811yescharacteristic
349C22#1681086011yescharacteristic
350C22SD32×D5311yescharacteristic
351C22Q32×D5311yescharacteristic
352C22#1681105211yescharacteristic
353C22#1681105411yescharacteristic
354C4—82no—
355C4—161no—
356C4—8481no—
357C4—1062no—
358C4—4242no—
359C4#1681050111yescharacteristic
360C4#1681061811yescharacteristic
361C4#1681085311yescharacteristic
362C4#1681085511yescharacteristic
363C2—1064no—
364C2#2027514211yescharacteristic
365C2#358×D5311yescharacteristic
366C2#2027641911yescharacteristic
367C2#2027680511yescharacteristic
368C2#2027680711yescharacteristic
369C2#2027681011yescharacteristic
370C2#2027681211yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1380C2
socleC23×C53C2×D8
prime radicalC23×D53D8
cosocleC22×C8×C53C23
hypercenter#1380D53
hypocenterC53#13060
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
3. C2×#16810428C2semidirect
4. C2×#16810426C2semidirect
6. #20272806C2semidirect
7. C53×#1380C2semidirect
16. #16810865C4semidirect
168. C2×D53#358semidirect
184. #1380D53semidirect
211. C53#13060semidirect

Maximal subgroups

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

subgroupindex
C22×C16×D532
C2×#168108652
C2×#168104282
C2×#168104262
#202747412
#202728062
C53×#13802
#1306053