Finite Group Databasebeta

← #27300392

Subgroups — #27300392

Order 13568.

404 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#427×D53C211yesmaximal, characteristic
2C2×#16810402C211yesmaximal, characteristic
3#20274750C211yesmaximal, characteristic
4#20272812C211yesmaximal, characteristic, complemented
5#20272811C211yesmaximal, characteristic, complemented
6#20272810C211yesmaximal, characteristic, complemented
7C53×#1382C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9#117×D53—21no—
10C22×#11719908C2211yescharacteristic
11C2×#13133153C2211yescharacteristic
12#16810661C2211yescharacteristic
13#16810660C411yescharacteristic
14#16810660C411yescharacteristic
15#16810659—21no—
16#16810658—21no—
17#107×D53C411yescharacteristic
18#107×D53C411yescharacteristic
19#106×D53—21no—
20#16810586—21no—
21#16810402—21no—
22#16810402—21no—
23#16810387—21no—
24C22×C16×C53C2211yescharacteristic
25C2×#108×C53C2211yescharacteristic
26C53×#427C2211yescharacteristic
27C22×C4×D53D411yescharacteristic
28C2×#11719917—21no—
29C22×#88322D411yescharacteristic
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53—21no—
34C2×C8×D53C2×C411yescharacteristic
35C2×C8×D53C2×C411yescharacteristic
36C2×C8×D53D411yescharacteristic
37C2×C8×D53D411yescharacteristic
38#13133176—21no—
39#13133175—21no—
40#13133175—21no—
41#13133174—41no—
42#13133173—41no—
43C4⋊C4×D53—21no—
44C4⋊C4×D53—21no—
45C22⋊C4×D53—41no—
46C42×D53—41no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50#13133153—21no—
51#13133153—21no—
52#13133151C2×C411yescharacteristic
53#13133151C2×C411yescharacteristic
54#13133150—21no—
55#13133150—21no—
56#13133149—21no—
57#13133148—21no—
58#13133148—21no—
59#13133129—41no—
60#13133129—41no—
61C2×M16×C53—21no—
62C22×C8×C53C2311yescharacteristic
63#117×C53—21no—
64C2×C16×C53—21no—
65C2×C16×C53—21no—
66C2×C16×C53—21no—
67#108×C53—21no—
68#108×C53—21no—
69#107×C53C2×C411yescharacteristic
70#107×C53C2×C411yescharacteristic
71#106×C53—21no—
72C23×D53SD1611yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53SD1611yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53C22⋊C411yescharacteristic
82C2×C4×D53C22⋊C411yescharacteristic
83C2×C4×D53D811yescharacteristic
84C2×C4×D53D811yescharacteristic
85M16×C53—41no—
86M16×C53—41no—
87C2×C8×C53—41no—
88C2×C8×C53C22×C411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53#4111yescharacteristic
91C2×C8×C53#4111yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C22⋊C4×C53—41no—
97C42×C53—41no—
98C22⋊Dic53—41no—
99#11719921—41no—
100#11719921—41no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103#11719919—41no—
104#11719919—41no—
105C4×Dic53—41no—
106C4×Dic53—41no—
107C4×Dic53—41no—
108#11719917—41no—
109#11719917—41no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—41no—
113C2×#88322C22⋊C411yescharacteristic
114C2×#88322C22⋊C411yescharacteristic
115C8×D53—21no—
116C8×D53—21no—
117C8×D53—21no—
118C8×D53—21no—
119C8×D53—21no—
120C8×D53—21no—
121C8×D53—21no—
122C8×D53—21no—
123C16×C53—41no—
124#11719908—41no—
125C23×C53C2×SD1611yescharacteristic
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—22no—
129C22×D53—41no—
130C22×D53#10211yescharacteristic
131C22×D53#10211yescharacteristic
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53#12311yescharacteristic
136C2×C4×C53#12311yescharacteristic
137C2×C4×C53C2×D811yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C2×C4×C53#13711yescharacteristic
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53#10211yescharacteristic
145C2×Dic53#10211yescharacteristic
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—21no—
154C4×D53—82no—
155C4×D53—82no—
156C8×C53—81no—
157C8×C53#11711yescharacteristic
158C8×C53#11711yescharacteristic
159C8×C53#12311yescharacteristic
160C8×C53#12311yescharacteristic
161#88322—21no—
162#88322—21no—
163#88322—81no—
164#12869—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53C2×#10211yescharacteristic
168C22×C53#48311yescharacteristic
169C22×C53#48311yescharacteristic
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—21no—
175C2×D53—42no—
176C2×D53—42no—
177C2×D53#35912yes—
178C4×C53—81no—
179C4×C53—81no—
180C4×C53#38111yescharacteristic
181C4×C53#41911yescharacteristic
182C4×C53#48011yescharacteristic
183C4×C53#48011yescharacteristic
184Dic53—21no—
185Dic53—21no—
186Dic53—81no—
187Dic53—81no—
188C2×#427—531no—
189C2×C22⋊C16—531no—
190#1810—531no—
191#1382—531no—
192#1382—531no—
193#1382—531no—
194#1382D5311yescharacteristic, complemented
195C2×C53—21no—
196C2×C53—21no—
197C2×C53#182311yescharacteristic
198C2×C53C2×#35911yescharacteristic
199C2×C53#214311yescharacteristic
200D53—22no—
201D53—41no—
202C23×C8—531no—
203C2×#117—1061no—
204C22×C16—531no—
205C22×C16C2×D5311yescharacteristic
206C2×#108—531no—
207C2×#108C2×D5311yescharacteristic
208#427—531no—
209#427—531no—
210#427—531no—
211#427—1061no—
212#427—1061no—
213#427C2×D5311yescharacteristic
214C2×#107—531no—
215C2×#107—531no—
216C2×#106—1061no—
217#407—1061no—
218C22⋊C16—1061no—
219C22⋊C16—1061no—
220#341—1061no—
221C53#1286911yescharacteristic, complemented, Sylow
222C23×C4—531no—
223C2×M16—21no—
224C2×M16—1061no—
225C22×C8—531no—
226C22×C8—531no—
227C22×C8—531no—
228C22×C8—531no—
229C22×C8—531no—
230C22×C8—1061no—
231C22×C8—1061no—
232C22×C8—1061no—
233C22×C8—1061no—
234C22×C8C22×D5311yescharacteristic
235#117—21no—
236#117—1061no—
237#117—1061no—
238#117—1061no—
239#117—2121no—
240#117—2121no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C22⋊C4—2121no—
244C2×C42—2121no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—1061no—
249C2×C16—1061no—
250C2×C16—1061no—
251#108—21no—
252#108—21no—
253#108—1061no—
254#108—1061no—
255#107—531no—
256#107—531no—
257#107—1061no—
258#107—1061no—
259#107C4×D5311yescharacteristic
260#107C4×D5311yescharacteristic
261#106—21no—
262#106—1061no—
263#106—1061no—
264#106—1061no—
265C22⋊C8—2121no—
266C22⋊C8—2121no—
267C24—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4D4×D5311yescharacteristic
280M16—41no—
281M16—41no—
282M16—2121no—
283M16—2121no—
284C2×C8—41no—
285C2×C8—531no—
286C2×C8—531no—
287C2×C8—1061no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—1061no—
292C2×C8—1061no—
293C2×C8—1061no—
294C2×C8—1061no—
295C2×C8—1061no—
296C2×C8—1061no—
297C2×C8—2121no—
298C2×C8C2×C4×D5311yescharacteristic
299C2×C8#1171994511yescharacteristic
300C2×C8#1171994511yescharacteristic
301C2×C8D4×D5311yescharacteristic
302C2×C8#1171994711yescharacteristic
303C2×C8#1171994911yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C22⋊C4—41no—
311C22⋊C4—2121no—
312C22⋊C4—2121no—
313C22⋊C4—2121no—
314C42—41no—
315C42—2121no—
316C42—2121no—
317C42—2121no—
318C16—41no—
319C16—2121no—
320C23—531no—
321C23—531no—
322C23—1062no—
323C23—1061no—
324C23—1061no—
325C23—2121no—
326C23SD16×D5311yescharacteristic
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—531no—
330C2×C4—531no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—4242no—
342C2×C4—4242no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4C22⋊C4×D5311yescharacteristic
346C2×C4#1313319911yescharacteristic
347C2×C4#1313320211yescharacteristic
348C2×C4D8×D5311yescharacteristic
349C2×C4#1313324611yescharacteristic
350C2×C4#1313324911yescharacteristic
351C8—81no—
352C8—1061no—
353C8—1061no—
354C8—4241no—
355C8#1313317411yescharacteristic
356C8#1313317511yescharacteristic
357C8#1313319911yescharacteristic
358C8#1313320211yescharacteristic
359C22—21no—
360C22—21no—
361C22—532no—
362C22—1061no—
363C22—1061no—
364C22—1061no—
365C22—1061no—
366C22—1061no—
367C22—2122no—
368C22—2122no—
369C22#102×D5311yescharacteristic
370C22#1681095211yescharacteristic
371C22#1681095511yescharacteristic
372C4—81no—
373C4—81no—
374C4—1061no—
375C4—1061no—
376C4—4241no—
377C4—4241no—
378C4#1681048511yescharacteristic
379C4#1681063011yescharacteristic
380C4#1681093511yescharacteristic
381C4#1681093811yescharacteristic
382C2—21no—
383C2—21no—
384C2—1062no—
385C2—2121no—
386C2#2027480011yescharacteristic
387C2#359×D5311yescharacteristic
388C2#2027644111yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1382C2
socleC22×C53C2×#102
prime radicalC23×D53SD16
cosocleC22×C8×C53C23
hypercenter#1382D53
hypocenterC53#12869
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
4. #20272812C2semidirect
5. #20272811C2semidirect
6. #20272810C2semidirect
7. C53×#1382C2semidirect
194. #1382D53semidirect
221. C53#12869semidirect

Maximal subgroups

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

subgroupindex
#427×D532
C2×#168104022
#202747502
#202728122
#202728112
#202728102
C53×#13822
#1286953