Finite Group Databasebeta

← #27300152

Subgroups — #27300152

Order 13568.

422 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283486C211yesmaximal, characteristic
2C2×#16810441C211yesmaximal, characteristic, complemented
3C2×#16810431C211yesmaximal, characteristic
4C2×#16810429C211yesmaximal, characteristic
5#20274885C211yesmaximal, characteristic, complemented
6#20272810C211yesmaximal, characteristic, complemented
7C53×#1382C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9#16811091—21no—
10C22×#11719908C2211yescharacteristic
11#16810717C411yescharacteristic
12#16810717C411yescharacteristic
13#16810710—21no—
14#16810710—21no—
15#16810705C411yescharacteristic
16#16810705C411yescharacteristic
17C2×#13133153C2211yescharacteristic
18C2×#13133152—21no—
19#16810646—21no—
20#16810617—21no—
21C2×C8×Dic53C2211yescharacteristic
22#16810441—21no—
23#16810441—21no—
24#16810431—22no—
25#16810429—22no—
26C22×C16×C53C2211yescharacteristic
27C2×#108×C53C2211yescharacteristic
28C53×#427C2211yescharacteristic
29C2×#11719945—21no—
30#13133239—21no—
31#13133239—21no—
32#13133239D412yes—
33#13133239Q812yes—
34C2×Dic212C2×C411yescharacteristic
35C2×#11719913—21no—
36C2×D424C2×C411yescharacteristic
37C2×#11719917—21no—
38C22×#88322D411yescharacteristic
39#13133182—21no—
40#13133182—21no—
41#13133179—21no—
42#13133179—21no—
43#13133177—41no—
44#13133177—41no—
45#13133174—41no—
46C2×C4×Dic53D411yescharacteristic
47C42×D53—41no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50C2×#11719908—21no—
51#13133153—21no—
52#13133153—21no—
53#13133152—42no—
54#13133146—41no—
55#13133146—41no—
56#13133140—21no—
57#13133140—21no—
58#13133137—21no—
59#13133137—21no—
60C8×Dic53C2×C411yescharacteristic
61C8×Dic53C2×C411yescharacteristic
62C8×Dic53D411yescharacteristic
63C8×Dic53D411yescharacteristic
64C2×M16×C53—21no—
65C22×C8×C53C2311yescharacteristic
66#117×C53—21no—
67C2×C16×C53—21no—
68C2×C16×C53—21no—
69C2×C16×C53—21no—
70#108×C53—21no—
71#108×C53—21no—
72#107×C53C2×C411yescharacteristic
73#107×C53C2×C411yescharacteristic
74#106×C53—21no—
75C22×C4×C53C2×D411yescharacteristic
76C2×C22⋊D53—41no—
77C22×Dic53SD1611yescharacteristic
78C22×Dic53SD1611yescharacteristic
79#11719945—22no—
80#11719945—22no—
81#11719945—41no—
82#11719945—41no—
83C2×D212—21no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×Dic106—21no—
88M16×C53—41no—
89M16×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53C22×C411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53C2×Q811yescharacteristic
95C2×C8×C53#4111yescharacteristic
96C2×C8×C53#4111yescharacteristic
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C22⋊C4×C53—41no—
100C42×C53—41no—
101#11719921—41no—
102#11719921—41no—
103#11719919—41no—
104#11719919—41no—
105C4×Dic53—41no—
106C4×Dic53—41no—
107C4×Dic53C22⋊C411yescharacteristic
108C4×Dic53C22⋊C411yescharacteristic
109C4×Dic53D811yescharacteristic
110C4×Dic53D811yescharacteristic
111#11719917—41no—
112#11719917—41no—
113C2×#88322—21no—
114C2×#88322—21no—
115C2×#88322—41no—
116C2×#88322C22⋊C411yescharacteristic
117C2×#88322C22⋊C411yescharacteristic
118Dic212—21no—
119Dic212C4⋊C412yes—
120D424—21no—
121D424C4⋊C412yes—
122#11719913—22no—
123#11719913—22no—
124C16×C53—41no—
125#11719908—41no—
126C23×C53C2×SD1611yescharacteristic
127C22×D53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53C2×C22⋊C411yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53C22⋊Q811yescharacteristic
133C2×C4×C53C2×D811yescharacteristic
134C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C53#13711yescharacteristic
136C22⋊D53—42no—
137C22⋊D53—42no—
138C2×Dic53—22no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53#10211yescharacteristic
142C2×Dic53#10211yescharacteristic
143C2×Dic53#10211yescharacteristic
144C2×Dic53#10211yescharacteristic
145D212—22no—
146D212—41no—
147C4×D53—42no—
148C4×D53—42no—
149C4×D53—82no—
150C4×D53—82no—
151Dic106—22no—
152Dic106—41no—
153C8×C53—81no—
154C8×C53C2×C4⋊C411yescharacteristic
155C8×C53#11711yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53C22⋊Q811yescharacteristic
158#88322—21no—
159#88322—21no—
160#88322—81no—
161#12815—531nomaximal, Sylow
162C22×C53—21no—
163C22×C53—21no—
164C22×C53C2×#10211yescharacteristic
165C22×C53#46111yescharacteristic
166C22×C53#47611yescharacteristic
167C2×D53—42no—
168C2×D53—81no—
169C2×D53—81no—
170C4×C53—81no—
171C4×C53—81no—
172C4×C53#38011yescharacteristic
173C4×C53#41711yescharacteristic
174C4×C53#45911yescharacteristic
175C4×C53#47411yescharacteristic
176Dic53—21no—
177Dic53—21no—
178Dic53—21no—
179Dic53—42no—
180Dic53#35912yescomplemented
181C4×#135—531no—
182C2×C4⋊C16—531no—
183C2×#359—531no—
184C2×#359—531no—
185#1848—531no—
186#1382—531no—
187#1382D5311yescharacteristic, complemented
188C2×C53—21no—
189C2×C53—21no—
190C2×C53#180211yescharacteristic
191C2×C53C2×#35911yescharacteristic
192C2×C53#214311yescharacteristic
193D53—82no—
194C2×#135—531no—
195C4×#41—1061no—
196C22×C16—531no—
197C22×C16C2×D5311yescharacteristic
198C4×Q16—531no—
199C4×Q16—531no—
200C4×SD16—1061no—
201C4×SD16—1061no—
202C4×D8—531no—
203C4×D8—531no—
204C2×#108—531no—
205C2×#108—1061no—
206C2×#108C2×D5311yescharacteristic
207#427C2×D5311yescharacteristic
208#423—1061no—
209#416—1061no—
210C2×C4×C8—531no—
211C4⋊C16—1061no—
212C4⋊C16—1061no—
213#359—1062no—
214#359—1062no—
215C53#1281511yescharacteristic, complemented, Sylow
216C2×#41—1061no—
217#135—532no—
218#135—532no—
219#135—1061no—
220#135—1061no—
221C2×Q16—531no—
222C2×SD16—1061no—
223C2×D8—531no—
224C2×M16—21no—
225C2×M16—1061no—
226C22×C8—531no—
227C22×C8C22×D5311yescharacteristic
228C4×Q8—1061no—
229C4×Q8—1061no—
230C4×D4—1061no—
231C4×D4—1061no—
232C4×D4—2121no—
233C4×D4—2121no—
234#117—21no—
235#117—2121no—
236C2×C42—531no—
237C2×C42—2121no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—1061no—
242C2×C16—1061no—
243C2×C16—1061no—
244#108—21no—
245#108—21no—
246#108—1061no—
247#108—1061no—
248#108—2122no—
249#107C4×D5311yescharacteristic
250#107C4×D5311yescharacteristic
251#106—21no—
252C4⋊C8—2121no—
253C4⋊C8—2121no—
254#103—1061no—
255#103—1061no—
256#102—1061no—
257#102—1061no—
258C4×C8—531no—
259C4×C8—531no—
260C4×C8—531no—
261C4×C8—531no—
262#41—1062no—
263#41—1062no—
264#41—2121no—
265#41—2121no—
266C2×Q8—1061no—
267C2×D4—1061no—
268C2×D4—2121no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4D4×D5311yescharacteristic
275Q16—532no—
276Q16—1061no—
277SD16—1062no—
278SD16—1062no—
279D8—532no—
280D8—1061no—
281M16—41no—
282M16—41no—
283M16—2121no—
284M16—2121no—
285C2×C8—41no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—2121no—
291C2×C8C2×C4×D5311yescharacteristic
292C2×C8#1171994511yescharacteristic
293C2×C8#1171994511yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8Q8×D5311yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—2121no—
300C4⋊C4—2121no—
301C22⋊C4—41no—
302C22⋊C4—2121no—
303C22⋊C4—2121no—
304C42—41no—
305C42—531no—
306C42—531no—
307C42—531no—
308C42—531no—
309C42—2121no—
310C42—2121no—
311C16—41no—
312C16—2121no—
313C23—2121no—
314C23SD16×D5311yescharacteristic
315Q8—1062no—
316Q8—2121no—
317D4—1062no—
318D4—2122no—
319D4—2122no—
320D4—2121no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—1062no—
328C2×C4—4242no—
329C2×C4—4242no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4C22⋊C4×D5311yescharacteristic
335C2×C4#1313319011yescharacteristic
336C2×C4#1313319511yescharacteristic
337C2×C4D8×D5311yescharacteristic
338C2×C4#1313324411yescharacteristic
339C2×C4#1313324911yescharacteristic
340C8—81no—
341C8—1061no—
342C8—1061no—
343C8—4241no—
344C8C4⋊C4×D5311yescharacteristic
345C8#1313317411yescharacteristic
346C8#1313319011yescharacteristic
347C8#1313319511yescharacteristic
348C22—21no—
349C22—21no—
350C22—4241no—
351C22—4241no—
352C22—2122no—
353C22#102×D5311yescharacteristic
354C22#1681082711yescharacteristic
355C22#1681091211yescharacteristic
356C4—81no—
357C4—81no—
358C4—532no—
359C4—1061no—
360C4—1061no—
361C4—1061no—
362C4—2122no—
363C4#1681048211yescharacteristic
364C4#1681062311yescharacteristic
365C4#1681081711yescharacteristic
366C4#1681090211yescharacteristic
367C2—21no—
368C2—21no—
369C2—4242no—
370C2#2027471411yescharacteristic
371C2#359×D5311yescharacteristic
372C2#2027644111yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1382C2
socleC22×C53C2×#102
prime radicalC2×#13133239C22
cosocleC22×C8×C53C23
hypercenter#1382D53
hypocenterC53#12815
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #1382

Complements

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

NH ≅ G/Ntype
2. C2×#16810441C2semidirect
5. #20274885C2semidirect
6. #20272810C2semidirect
7. C53×#1382C2semidirect
180. Dic53#359semidirect
187. #1382D53semidirect
215. C53#12815semidirect

Maximal subgroups

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

subgroupindex
#202834862
C2×#168104412
C2×#168104312
C2×#168104292
#202748852
#202728102
C53×#13822
#1281553