Finite Group Databasebeta

← #27285183

Subgroups — #27285183

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3#20275642C211yesmaximal, characteristic
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2546C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133214C2211yescharacteristic
10#16810971—22no—
11#16810971—22no—
12#16810823—22no—
13#16810823—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133137—22no—
17C2×#13133137—21no—
18C2×#13133137—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810369—22no—
21Q8⋊#88322—22no—
22C2×C4⋊Q8×C53—21no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4×Q8×C53C2211yescharacteristic
25C53×#398—22no—
26C53×#391—21no—
27C53×#389—22no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719924—24no—
33C2×#11719924—42no—
34C2×#11719924—42no—
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41#13133214D412yes—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133137—24no—
47#13133137—28no—
48#13133137—44no—
49C4×#88322—21no—
50C4×#88322D412yes—
51C22×Q8×C53—21no—
52C22×Q8×C53D411yescharacteristic
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—42no—
58C4⋊Q8×C53—42no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C2×C4⋊C4×C53—24no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D412yes—
68C2×C4⋊C4×C53D412yes—
69C2×C42×C53—21no—
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72#95×C53—24no—
73#95×C53—21no—
74C23×D53—41no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×Q8×C53—41no—
84C2×Q8×C53—41no—
85C2×Q8×C53—41no—
86C2×Q8×C53D812yes—
87C22×C4×C53—22no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C2×D212—22no—
95C2×D212—22no—
96C2×D212—22no—
97C2×D212—44no—
98C2×D212—41no—
99C2×D212—42no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—42no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—48no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53SD1614yescomplemented
118C42×C53—41no—
119C42×C53—41no—
120C42×C53—41no—
121C42×C53—41no—
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53#4111yescharacteristic
126#11719924—48no—
127C2×#88322—22no—
128C2×#88322—22no—
129C2×#88322—42no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—42no—
134C23×C53#12011yescharacteristic
135C22×D53—42no—
136C22×D53—81no—
137C22×D53—82no—
138C22×D53—81no—
139C22×D53—82no—
140Q8×C53—22no—
141Q8×C53—22no—
142Q8×C53—41no—
143Q8×C53—41no—
144Q8×C53—81no—
145Q8×C53—81no—
146Q8×C53—81no—
147Q8×C53—81no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—24no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12711yescharacteristic
166C2×C4×C53C2×D811yescharacteristic
167C2×C4×C53C2×SD1612yes—
168C2×C4×C53#13511yescharacteristic
169C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53#13712yes—
172D212—44no—
173D212—42no—
174D212—42no—
175D212—44no—
176D212—82no—
177D212—81no—
178D212—81no—
179D212—81no—
180D212—81no—
181#88322—42no—
182#88322—42no—
183#88322—81no—
184#10051—531nomaximal, Sylow
185C22×C53#39211yescharacteristic
186C22×C53#44711yescharacteristic
187C22×C53#44812yes—
188C22×C53#44911yescharacteristic
189C22×C53#45412yes—
190C2×D53—82no—
191C2×D53—82no—
192C2×D53—82no—
193C2×D53—162no—
194C2×D53—162no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—44no—
198C4×C53—42no—
199C4×C53—82no—
200C4×C53—81no—
201C4×C53#44711yescharacteristic
202C4×C53#44911yescharacteristic
203C4×C53#46112yes—
204C4×C53#46611yescharacteristic
205C4×C53#46911yescharacteristic
206C4×C53#49211yescharacteristic
207C4×C53#49711yescharacteristic
208C2×#486—531no—
209C2×#460—531no—
210#2546D5311yescharacteristic, complemented
211#1982—531no—
212C2×#332—531no—
213C2×Q8⋊C8—531no—
214#1279—531no—
215C2×C53#162612yes—
216C2×C53#162712yes—
217C2×C53#199911yescharacteristic
218C2×C53#200311yescharacteristic
219C2×C53#201311yescharacteristic
220D53—162no—
221C22×SD16—1061no—
222C2×C4⋊Q8—21no—
223C2×C4⋊Q8C2×D5311yescharacteristic
224C2×#127—531no—
225C2×C4×Q8C2×D5311yescharacteristic
226#486—1062no—
227#486—1062no—
228#460—1062no—
229#460—1062no—
230C2×C4⋊C8—531no—
231C2×C4⋊C8—531no—
232C2×#102—1062no—
233C2×#102—1061no—
234C2×#102—1062no—
235C2×C4×C8—531no—
236#398—22no—
237#391—21no—
238#389—22no—
239C4×C4⋊C4C2×D5311yescharacteristic
240#332—1062no—
241Q8⋊C8—1062no—
242C53#1005111yescharacteristic, complemented, Sylow
243C22×Q8—21no—
244C22×Q8C22⋊D5311yescharacteristic
245C22×D4—1061no—
246C22×D4—1061no—
247C22×D4—1061no—
248C2×SD16—1064no—
249C2×SD16—2122no—
250C2×SD16—2122no—
251C22×C8—1061no—
252C22×C8—1061no—
253C22×C8—1061no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260#127—532no—
261#127—1061no—
262#127—1061no—
263#127—1061no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C22⋊D5311yescharacteristic
272C2×C4⋊C4C22⋊D5312yes—
273C2×C4⋊C4C22⋊D5312yes—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42C22×D5311yescharacteristic
277C4⋊C8—532no—
278C4⋊C8—1061no—
279C4⋊C8—1061no—
280C4⋊C8—1061no—
281#102—1064no—
282#102—1068no—
283#102—2124no—
284C4×C8—532no—
285C4×C8—1061no—
286#95—24no—
287#95—21no—
288C24—2121no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8#1171992212yes—
301C2×D4—1062no—
302C2×D4—1062no—
303C2×D4—1062no—
304C2×D4—2124no—
305C2×D4—2121no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4—2121no—
309C2×D4—2122no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2122no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4C2×C22⋊D5311yescharacteristic
319C22×C4C2×C22⋊D5311yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321SD16—2128no—
322C2×C8—1062no—
323C2×C8—1062no—
324C2×C8—2121no—
325C2×C8—2122no—
326C2×C8—2122no—
327C2×C8—2121no—
328C2×C8—2121no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—48no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4#1171992414yescomplemented
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42D4×D5311yescharacteristic
345C42D4×D5311yescharacteristic
346C42D4×D5311yescharacteristic
347C42#1171994711yescharacteristic
348C23—4241no—
349C23—4242no—
350C23—4241no—
351C23—4242no—
352C23—2122no—
353C23#1313318611yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362D4—4242no—
363D4—4241no—
364D4—4241no—
365D4—4241no—
366D4—4241no—
367D4—2124no—
368D4—2122no—
369D4—2122no—
370D4—2124no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4#1313318511yescharacteristic
384C2×C4#1313318511yescharacteristic
385C2×C4#1313318811yescharacteristic
386C2×C4#1313318911yescharacteristic
387C2×C4#1313318911yescharacteristic
388C2×C4#1313321311yescharacteristic
389C2×C4C2×#1171992211yescharacteristic
390C2×C4C2×#1171992412yes—
391C2×C4#1313323711yescharacteristic
392C2×C4#1313324211yescharacteristic
393C2×C4#1313324311yescharacteristic
394C2×C4#1313324812yes—
395C8—4241no—
396C8—2122no—
397C8—2122no—
398C22—8482no—
399C22—8482no—
400C22—4242no—
401C22—4242no—
402C22—4242no—
403C22#1681053111yescharacteristic
404C22#1681075711yescharacteristic
405C22#1681076212yes—
406C22#1681076811yescharacteristic
407C22#1681079612yes—
408C4—21no—
409C4—21no—
410C4—44no—
411C4—42no—
412C4—81no—
413C4—82no—
414C4#1681075611yescharacteristic
415C4#1681076611yescharacteristic
416C4#1681082512yes—
417C4#1681085711yescharacteristic
418C4#1681087411yescharacteristic
419C4#1681099711yescharacteristic
420C4#1681101511yescharacteristic
421C2—8482no—
422C2#2027381112yes—
423C2#2027381812yes—
424C2#2027571111yescharacteristic
425C2#2027573511yescharacteristic
426C2#2027579811yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2546C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#2546D53
hypocenterC53#10051
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2546

Complements

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

NH ≅ G/Ntype
4. C2×#16810369C2semidirect
5. C2×Q8⋊#88322C2semidirect
6. #20272556C2semidirect
7. C53×#2546C2semidirect
117. C4⋊C4×C53SD16semidirect
210. #2546D53semidirect
242. C53#10051semidirect
339. C4⋊C4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
C2×#168108232
#202756422
C2×#168103692
C2×Q8⋊#883222
#202725562
C53×#25462
#1005153