Finite Group Databasebeta

← #27283337

Subgroups — #27283337

Order 13568.

654 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×#16810999C211yesmaximal, characteristic
2#20275642C211yesmaximal, characteristic, complemented
3C4⋊C4×#88322C211yesmaximal, characteristic, complemented
4C2×#16810375C211yesmaximal, characteristic, complemented
5C2×#16810367C211yesmaximal, characteristic, complemented
6C2×#16810366C211yesmaximal, characteristic, complemented
7C53×#1982C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C22×#11719922—21no—
10C2×#13133214C2211yescharacteristic
11#16810999—22no—
12#16810999C414yescomplemented
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133137—22no—
16C2×C4×#88322—21no—
17C2×C4×#88322C2211yescharacteristic
18#16810378—21no—
19#16810375—22no—
20#16810367—22no—
21#16810366—22no—
22C2×#127×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×#102×C53—22no—
25C4×C4⋊C4×C53C2211yescharacteristic
26C22×D212—21no—
27C22×D212—21no—
28C22×D212—21no—
29C2×#11719922—24no—
30C2×#11719922—24no—
31C2×#11719922—42no—
32C2×#11719922—42no—
33C2×#11719922—42no—
34C2×#11719922—42no—
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322D411yescharacteristic
38C22×#88322D411yescharacteristic
39#13133214—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214C2×C412yes—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322D412yes—
47#13133137—44no—
48C4×#88322—42no—
49C4×#88322C2×C412yes—
50C4×#88322D412yes—
51C22×D4×C53—21no—
52C22×D4×C53—21no—
53C22×D4×C53—21no—
54C22×C8×C53—21no—
55#127×C53—21no—
56#127×C53—21no—
57#127×C53—21no—
58#127×C53C2×C412yes—
59C2×C4⋊C4×C53D412yes—
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64#102×C53—44no—
65#95×C53—21no—
66C24×C53—41no—
67C23×D53—41no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—42no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—42no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—82no—
81C22×C4×C53—22no—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C2×D212—22no—
86C2×D212—22no—
87C2×D212—22no—
88C2×D212—42no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—42no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—82no—
98C2×C8×C53—41no—
99C2×C8×C53—42no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53D814yescomplemented
102C42×C53—41no—
103C42×C53—41no—
104C42×C53C22×C411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108#11719922—48no—
109#11719922—48no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—22no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—42no—
117C2×#88322C22⋊C412yes—
118C2×#88322C22⋊C412yes—
119C2×#88322D812yes—
120C2×#88322SD1612yes—
121C23×C53—42no—
122C23×C53—81no—
123C23×C53—81no—
124C23×C53—82no—
125C23×C53—82no—
126C23×C53#12111yescharacteristic
127C22×D53—42no—
128C22×D53—81no—
129C22×D53—81no—
130C22×D53—82no—
131C22×D53—82no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—44no—
135D4×C53—82no—
136D4×C53—82no—
137D4×C53—81no—
138D4×C53—81no—
139D4×C53—81no—
140D4×C53—81no—
141C2×C4×C53—22no—
142C2×C4×C53—24no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53C2×C22⋊C411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12411yescharacteristic
150C2×C4×C53#12711yescharacteristic
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53C2×D812yes—
153C2×C4×C53C2×SD1611yescharacteristic
154C2×C4×C53#13512yes—
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157D212—42no—
158D212—42no—
159D212—44no—
160D212—82no—
161D212—82no—
162D212—81no—
163D212—81no—
164D212—81no—
165D212—81no—
166C8×C53—81no—
167#88322—22no—
168#88322—42no—
169#88322—81no—
170#88322#10214yescomplemented
171#9734—531nomaximal, Sylow
172C22×C53—82no—
173C22×C53—82no—
174C22×C53—82no—
175C22×C53—162no—
176C22×C53—162no—
177C22×C53#39011yescharacteristic
178C22×C53#45911yescharacteristic
179C22×C53#46011yescharacteristic
180C22×C53#46612yes—
181C22×C53#47212yes—
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—82no—
185C2×D53—162no—
186C2×D53—162no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—44no—
190C4×C53C2×#10211yescharacteristic
191C4×C53#41811yescharacteristic
192C4×C53C4×D812yes—
193C4×C53#45911yescharacteristic
194C4×C53#46011yescharacteristic
195C4×C53#49311yescharacteristic
196C4×C53#49511yescharacteristic
197C2×#493—531no—
198#1982—531no—
199#1982D5311yescharacteristic, complemented
200C8×C4⋊C4—531no—
201C2×#335—531no—
202C2×#331—531no—
203C2×#331—531no—
204C2×C53—162no—
205C2×C53#166712yes—
206C2×C53#166812yes—
207C2×C53#197011yescharacteristic
208C2×C53#197811yescharacteristic
209C2×C53#198211yescharacteristic
210D53—162no—
211C22×D8—1061no—
212C22×D8—1061no—
213C2×#127—531no—
214C2×#127C2×D5311yescharacteristic
215#493—534no—
216#493—1062no—
217C2×C4⋊C8—531no—
218C2×C4⋊C8—531no—
219C2×C4⋊C8C2×D5311yescharacteristic
220C2×#102—22no—
221C2×#102—1062no—
222C2×C4×C8—531no—
223C2×C4×C8—1061no—
224C4×C4⋊C4C2×D5311yescharacteristic
225#336—1061no—
226#335—1062no—
227#331—1062no—
228#331—1062no—
229C53#973411yescharacteristic, complemented, Sylow
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—1061no—
234C22×D4—1061no—
235C22×D4—1061no—
236C2×D8—1064no—
237C2×D8—1064no—
238C2×D8—2122no—
239C2×D8—2122no—
240C2×D8—2122no—
241C2×D8—2122no—
242C22×C8—21no—
243C22×C8—531no—
244C22×C8—531no—
245C22×C8—1061no—
246C22×C8—1061no—
247#127—21no—
248#127—21no—
249#127—21no—
250#127—532no—
251#127—1061no—
252#127—1061no—
253#127—1061no—
254#127C4×D5312yes—
255C2×C4⋊C4C22⋊D5312yes—
256C2×C42—21no—
257C2×C42C22×D5311yescharacteristic
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—532no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264#102—44no—
265#102—2124no—
266C4×C8—532no—
267C4×C8—532no—
268C4×C8—2122no—
269#95—21no—
270C24—41no—
271C24—2121no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—82no—
285C2×D4—1062no—
286C2×D4—1062no—
287C2×D4—1062no—
288C2×D4—4242no—
289C2×D4—2121no—
290C2×D4—2121no—
291C2×D4—2121no—
292C2×D4—2121no—
293C2×D4—2122no—
294C2×D4—2121no—
295C2×D4—2122no—
296C2×D4—2121no—
297C2×D4—2121no—
298C22×C4—22no—
299C22×C4D4×D5311yescharacteristic
300C22×C4#1171994711yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302D8—2128no—
303D8—2128no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—532no—
307C2×C8—532no—
308C2×C8—532no—
309C2×C8—532no—
310C2×C8—1061no—
311C2×C8—1061no—
312C2×C8—1062no—
313C2×C8—2121no—
314C2×C8—2122no—
315C2×C8—2121no—
316C2×C8—2121no—
317C4⋊C4—22no—
318C4⋊C4#1171992214yescomplemented
319C42—41no—
320C42—41no—
321C42C2×C4×D5311yescharacteristic
322C42#1171994511yescharacteristic
323C42D4×D5311yescharacteristic
324C42D4×D5311yescharacteristic
325C23—42no—
326C23—81no—
327C23—81no—
328C23—82no—
329C23—82no—
330C23—4241no—
331C23—4241no—
332C23—4242no—
333C23—4242no—
334C23—2122no—
335C23#1313318911yescharacteristic
336D4—42no—
337D4—42no—
338D4—44no—
339D4—82no—
340D4—82no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—4242no—
346D4—4242no—
347D4—4241no—
348D4—4241no—
349D4—4241no—
350D4—4241no—
351D4—2122no—
352D4—2122no—
353D4—2124no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4C22⋊C4×D5311yescharacteristic
359C2×C4#1313317711yescharacteristic
360C2×C4C4×C22⋊D5311yescharacteristic
361C2×C4#1313319011yescharacteristic
362C2×C4#1313320311yescharacteristic
363C2×C4#1313321311yescharacteristic
364C2×C4D8×D5311yescharacteristic
365C2×C4C2×#1171992212yes—
366C2×C4SD16×D5311yescharacteristic
367C2×C4#1313323712yes—
368C2×C4#1313324111yescharacteristic
369C2×C4#1313324411yescharacteristic
370C8—81no—
371C8—534no—
372C8—1062no—
373C8—4241no—
374C8—2122no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—162no—
379C22—162no—
380C22—8482no—
381C22—8482no—
382C22—4242no—
383C22—4242no—
384C22—4242no—
385C22#1681052311yescharacteristic
386C22#1681081611yescharacteristic
387C22#1681082111yescharacteristic
388C22#1681085712yes—
389C22#1681089112yes—
390C4—21no—
391C4—21no—
392C4—44no—
393C4#102×D5311yescharacteristic
394C4#1681062611yescharacteristic
395C4C4×#1171992212yes—
396C4#1681081711yescharacteristic
397C4#1681082211yescharacteristic
398C4#1681099911yescharacteristic
399C4#1681100511yescharacteristic
400C2—162no—
401C2—8482no—
402C2#2027406812yes—
403C2#2027407312yes—
404C2#2027558011yescharacteristic
405C2#2027562411yescharacteristic
406C2#2027564011yescharacteristic
subgroupG/H
centerC23#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1982C2
socleC23×C53#121
prime radicalC2×#16810999C2
cosocleC2×C42×C53C23
hypercenter#1982D53
hypocenterC53#9734
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. #20275642C2semidirect
3. C4⋊C4×#88322C2semidirect
4. C2×#16810375C2semidirect
5. C2×#16810367C2semidirect
6. C2×#16810366C2semidirect
7. C53×#1982C2semidirect
12. #16810999C4semidirect
101. C4⋊C4×C53D8semidirect
170. #88322#102semidirect
199. #1982D53semidirect
229. C53#9734semidirect
318. C4⋊C4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168109992
#202756422
C4⋊C4×#883222
C2×#168103752
C2×#168103672
C2×#168103662
C53×#19822
#973453