Finite Group Databasebeta

← #27302322

Subgroups — #27302322

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 360 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
5C2×#16810398C211yesmaximal, characteristic, complemented
6#20274878C211yesmaximal, characteristic, complemented
7C2×C53×#357C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×#11719922—21no—
10C2×#13133213—21no—
11C22×#11719908C2211yescharacteristic
12#16810998—22no—
13#16810998C414yescomplemented
14C2×#13133151C2211yescharacteristic
15C2×#13133148—21no—
16C2×C8×Dic53C2211yescharacteristic
17#16810488—21no—
18#16810425—22no—
19#16810424—22no—
20#16810423—22no—
21#16810398—22no—
22C22×D8×C53C2211yescharacteristic
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C53×#357—22no—
26C22×D212—21no—
27C22×C22⋊D53—42no—
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424C2×C412yes—
34C2×#11719922—24no—
35C2×#11719922—42no—
36C2×#11719922—42no—
37C22×#88322D411yescharacteristic
38#13133213—24no—
39#13133213—42no—
40C2×C4⋊Dic53—21no—
41C2×C4×Dic53D411yescharacteristic
42C2×#11719908—21no—
43C2×#11719908—21no—
44C2×#11719908D412yescomplemented
45#13133151—21no—
46#13133151—21no—
47#13133148—42no—
48C8×Dic53C2×C412yes—
49C8×Dic53D412yes—
50#13133124—42no—
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×C53C2×C412yes—
58C22×C8×C53C2311yescharacteristic
59C2×C4⋊C4×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53D412yescomplemented
63#107×C53—21no—
64#107×C53—21no—
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—48no—
76C2×C22⋊D53—84no—
77C2×C22⋊D53—84no—
78C22×Dic53—41no—
79C22×Dic53D812yes—
80C2×D212—22no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86D8×C53—22no—
87D8×C53—22no—
88D8×C53—41no—
89D8×C53—41no—
90C2×C8×C53C22×C411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53#4111yescharacteristic
95C2×C8×C53#4111yescharacteristic
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98#11719922—48no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4×Dic53C22⋊C412yes—
102C4×Dic53SD1612yes—
103C2×#88322C22⋊C412yes—
104C2×#88322D812yes—
105C2×#88322SD1612yes—
106D424—22no—
107D424—22no—
108D424—41no—
109D424—41no—
110C16×C53—22no—
111#11719908—22no—
112C23×C53—42no—
113C23×C53—81no—
114C23×C53—81no—
115C23×C53—82no—
116C23×C53—82no—
117C23×C53C2×D811yescharacteristic
118C22×D53—42no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53—82no—
122C22×D53—82no—
123D4×C53—42no—
124D4×C53—42no—
125D4×C53—81no—
126D4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53—82no—
129C2×C4×C53C2×C22⋊C411yescharacteristic
130C2×C4×C53#12411yescharacteristic
131C2×C4×C53#12711yescharacteristic
132C2×C4×C53C2×D811yescharacteristic
133C2×C4×C53C2×SD1611yescharacteristic
134C2×C4×C53C2×SD1611yescharacteristic
135C22⋊D53—816no—
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—81no—
139C2×Dic53—82no—
140C2×Dic53#10214yes—
141D212—42no—
142D212—42no—
143D212—81no—
144D212—81no—
145C8×C53C4×D411yescharacteristic
146C8×C53C4×D411yescharacteristic
147C8×C53#12111yescharacteristic
148C8×C53#12111yescharacteristic
149#88322#10214yes—
150#13206—531nomaximal, Sylow
151C22×C53—84no—
152C22×C53—82no—
153C22×C53—162no—
154C22×C53—161no—
155C22×C53—161no—
156C22×C53C2×#10211yescharacteristic
157C22×C53#48611yescharacteristic
158C22×C53#49311yescharacteristic
159C22×C53#50912yes—
160C22×C53#50912yes—
161C2×D53—84no—
162C2×D53—82no—
163C2×D53—162no—
164C2×D53—161no—
165C2×D53—161no—
166C4×C53—161no—
167C4×C53#39011yescharacteristic
168C4×C53C2×#10211yescharacteristic
169C4×C53#48611yescharacteristic
170C4×C53#49211yescharacteristic
171Dic53—24no—
172Dic53—161no—
173C2×#493—531no—
174C2×#357—531no—
175C2×#357—531no—
176C2×#357—531no—
177C2×#357D5311yescharacteristic, complemented
178C2×#346—531no—
179#1846—531no—
180C2×C53—162no—
181C2×C53#195711yescharacteristic
182C2×C53#214012yes—
183C2×C53#224412yes—
184C2×C53#225011yescharacteristic
185C2×C53#225011yescharacteristic
186D53—162no—
187C22×D8—531no—
188C22×D8—1061no—
189C22×D8C2×D5311yescharacteristic
190C2×#127—1061no—
191C22×C16—531no—
192C22×C16C2×D5311yescharacteristic
193#493—534no—
194#493—1062no—
195C2×#107—531no—
196C2×#107C2×D5311yescharacteristic
197C2×#106—1061no—
198C2×C4×C8—531no—
199#382—1061no—
200#357—22no—
201#357—1062no—
202#357—1062no—
203#357—1062no—
204#346—1062no—
205C53#1320611yescharacteristic, complemented, Sylow
206C22×D4—21no—
207C22×D4—1061no—
208C22×D4—2122no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—532no—
215C2×D8—1064no—
216C2×D8—1061no—
217C2×D8—1061no—
218C2×D8—1061no—
219C2×D8—1061no—
220C2×D8—1061no—
221C2×D8—2122no—
222C2×D8—2122no—
223C2×D8C4×D5312yes—
224C22×C8—531no—
225C22×C8C22×D5311yescharacteristic
226#127—1064no—
227#127—2122no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—1061no—
230C2×C42—531no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—532no—
234C2×C16—1061no—
235C2×C16—1061no—
236C2×C16C22⋊D5312yescomplemented
237#107—21no—
238#107—21no—
239#107—1061no—
240#107—1061no—
241#106—2122no—
242C4×C8—532no—
243C4×C8—532no—
244#95—2122no—
245C24—41no—
246C24—2121no—
247C2×D4—22no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—1062no—
254C2×D4—4244no—
255C2×D4—4244no—
256C2×D4—2128no—
257C2×D4—2121no—
258C2×D4—2121no—
259C2×D4—2121no—
260C2×D4—2121no—
261C2×D4—2121no—
262C22×C4—41no—
263C22×C4—532no—
264C22×C4—2121no—
265C22×C4D4×D5311yescharacteristic
266D8—22no—
267D8—22no—
268D8—41no—
269D8—41no—
270D8—1062no—
271D8—1062no—
272D8—2128no—
273D8—2121no—
274D8—2121no—
275C2×C8—532no—
276C2×C8—532no—
277C2×C8—532no—
278C2×C8C2×C4×D5311yescharacteristic
279C2×C8#1171994511yescharacteristic
280C2×C8D4×D5311yescharacteristic
281C2×C8D4×D5311yescharacteristic
282C2×C8#1171994711yescharacteristic
283C2×C8C2×C22⋊D5311yescharacteristic
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C42—532no—
289C42—532no—
290C16—22no—
291C16—1062no—
292C23—42no—
293C23—82no—
294C23—82no—
295C23—81no—
296C23—81no—
297C23—4242no—
298C23—4242no—
299C23—4241no—
300C23—4241no—
301C23—2122no—
302C23D8×D5311yescharacteristic
303D4—42no—
304D4—42no—
305D4—81no—
306D4—81no—
307D4—42416no—
308D4—4241no—
309D4—4241no—
310D4—2122no—
311D4—2122no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—534no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—4242no—
318C2×C4—4241no—
319C2×C4C22⋊C4×D5311yescharacteristic
320C2×C4#1313320311yescharacteristic
321C2×C4#1313321311yescharacteristic
322C2×C4D8×D5311yescharacteristic
323C2×C4SD16×D5311yescharacteristic
324C2×C4SD16×D5311yescharacteristic
325C8—534no—
326C8#1313317711yescharacteristic
327C8C4×C22⋊D5311yescharacteristic
328C8#1313318911yescharacteristic
329C8#1313319011yescharacteristic
330C22—82no—
331C22—84no—
332C22—162no—
333C22—161no—
334C22—161no—
335C22—8482no—
336C22—8481no—
337C22—8481no—
338C22—4242no—
339C22—4244no—
340C22#102×D5311yescharacteristic
341C22#1681096811yescharacteristic
342C22#1681099811yescharacteristic
343C22#1681105712yes—
344C22#1681106012yes—
345C4—161no—
346C4—8481no—
347C4—1064no—
348C4#1681052311yescharacteristic
349C4#102×D5311yescharacteristic
350C4#1681096811yescharacteristic
351C4#1681099411yescharacteristic
352C2—162no—
353C2—8482no—
354C2#2027551911yescharacteristic
355C2#2027642912yes—
356C2#2027695512yes—
357C2#2027697811yescharacteristic
358C2#2027698011yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#357C2
socleC23×C53C2×D8
prime radicalC2×#16810998C2
cosocleC22×C8×C53C23
hypercenterC2×#357D53
hypocenterC53#13206
LCSG ▷ C2×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. C2×#16810398C2semidirect
6. #20274878C2semidirect
7. C2×C53×#357C2semidirect
13. #16810998C4semidirect
44. C2×#11719908D4semidirect
62. C2×C16×C53D4semidirect
177. C2×#357D53semidirect
205. C53#13206semidirect
236. C2×C16C22⋊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
C2×#168103982
#202748782
C2×C53×#3572
#1320653