Finite Group Databasebeta

← #27302281

Subgroups — #27302281

Order 13568.

424 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810738C211yesmaximal, characteristic
2C2×#16810443C211yesmaximal, characteristic, complemented
3#20276343C211yesmaximal, characteristic, complemented
4#20276342C211yesmaximal, characteristic, complemented
5#20276335C211yesmaximal, characteristic, complemented
6M32×Dic53C211yesmaximal, characteristic, complemented
7C53×#2115C211yesmaximal, characteristic, complemented
8C2×#13133237C411yescharacteristic
9C2×#13133223C2211yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133157C2211yescharacteristic
12#16810738—21no—
13#16810738—21no—
14#16810738—21no—
15#16810738—21no—
16C2×#13133152C411yescharacteristic
17C2×#13133144C411yescharacteristic
18C2×#13133143C411yescharacteristic
19C2×C8×Dic53C2211yescharacteristic
20#16810443—21no—
21#16810443—21no—
22#16810407—22no—
23#16810406—22no—
24#16810404—22no—
25#16810398—21no—
26C16×Dic53—21no—
27C2×#131×C53C2211yescharacteristic
28C2×M32×C53C2211yescharacteristic
29C2×M32×C53C2211yescharacteristic
30C53×#349—22no—
31C2×#11719945C2×C411yescharacteristic
32#13133237—21no—
33#13133237—21no—
34#13133237—21no—
35#13133237—21no—
36C2×#11719925C811yescharacteristic
37C2×#11719923C811yescharacteristic
38C2×#11719924C811yescharacteristic
39C2×#11719922C811yescharacteristic
40#13133223—21no—
41#13133223—21no—
42#13133223—21no—
43#13133223—21no—
44C2×#11719917C2×C411yescharacteristic
45C2×#11719912—21no—
46C22×#88322C2×C411yescharacteristic
47C2×C8×D53—21no—
48C2×C4×Dic53C2×C411yescharacteristic
49#13133157—21no—
50#13133157—21no—
51#13133157—21no—
52#13133157D412yes—
53C2×#11719908—21no—
54C2×#11719908—21no—
55#13133152—21no—
56#13133152—21no—
57#13133144—21no—
58#13133144—21no—
59#13133143—21no—
60#13133143—21no—
61C8×Dic53D411yescharacteristic
62C8×Dic53D411yescharacteristic
63C8×Dic53D411yescharacteristic
64C8×Dic53D411yescharacteristic
65C2×#41×C53C2×C411yescharacteristic
66#131×C53—21no—
67#131×C53—21no—
68#131×C53—21no—
69#131×C53—21no—
70C2×M16×C53—21no—
71C2×M16×C53C2×C411yescharacteristic
72C22×C8×C53—21no—
73C22×C8×C53C2311yescharacteristic
74M32×C53—21no—
75M32×C53—21no—
76M32×C53—21no—
77M32×C53D412yes—
78C2×C16×C53—21no—
79C2×C16×C53—21no—
80#41×C53—21no—
81#41×C53—21no—
82#41×C53—21no—
83#41×C53—21no—
84C2×Q8×C53C2×C811yescharacteristic
85C2×D4×C53—21no—
86C2×D4×C53C2×C811yescharacteristic
87C22×C4×C53—21no—
88C22×C4×C53C22×C411yescharacteristic
89C2×C22⋊D53—21no—
90C22×Dic53—21no—
91#11719945—21no—
92#11719945—21no—
93#11719945—21no—
94#11719945—21no—
95C2×D212C2×C811yescharacteristic
96C2×C4×D53—21no—
97C2×Dic106C2×C811yescharacteristic
98M16×C53—21no—
99M16×C53—21no—
100M16×C53—41no—
101M16×C53—41no—
102C2×C8×C53—21no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53#4111yescharacteristic
110C2×C8×C53#4111yescharacteristic
111#11719925—21no—
112#11719925—21no—
113#11719924—21no—
114#11719924—21no—
115#11719923—21no—
116#11719923—21no—
117#11719922—21no—
118#11719922—21no—
119C4×Dic53—21no—
120C4×Dic53C22⋊C411yescharacteristic
121C4×Dic53C22⋊C411yescharacteristic
122#11719917—21no—
123#11719917—21no—
124C2×#88322—21no—
125C2×#88322C22⋊C411yescharacteristic
126C2×#88322C22⋊C411yescharacteristic
127C2×#88322C2×C811yescharacteristic
128C2×#88322C2×C811yescharacteristic
129C2×#88322M1611yescharacteristic
130C2×#88322M1611yescharacteristic
131#11719912—41no—
132#11719912—41no—
133C8×D53—41no—
134C8×D53—41no—
135C16×C53—22no—
136C16×C53—41no—
137#11719908—22no—
138#11719908—41no—
139C23×C53—21no—
140C23×C53#13111yescharacteristic
141C22×D53—21no—
142Q8×C53—21no—
143Q8×C53—21no—
144D4×C53—21no—
145D4×C53—21no—
146D4×C53—41no—
147D4×C53—41no—
148C2×C4×C53—21no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53C2×C22⋊C411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53C22×C811yescharacteristic
155C2×C4×C53C2×M1611yescharacteristic
156C2×C4×C53#13111yescharacteristic
157C22⋊D53—41no—
158C22⋊D53—41no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—42no—
163D212—21no—
164D212—21no—
165C4×D53—41no—
166C4×D53—41no—
167Dic106—21no—
168Dic106—21no—
169C8×C53—41no—
170C8×C53#12111yescharacteristic
171C8×C53#12111yescharacteristic
172C8×C53#12411yescharacteristic
173C8×C53#12711yescharacteristic
174#88322—41no—
175#88322C22⋊C811yescharacteristic
176#88322C22⋊C811yescharacteristic
177#88322C22⋊C811yescharacteristic
178#88322C22⋊C811yescharacteristic
179#13197—531nomaximal, Sylow
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—81no—
183C22×C53#40811yescharacteristic
184C22×C53C8×D411yescharacteristic
185C22×C53#43611yescharacteristic
186C2×D53—41no—
187C2×D53—81no—
188C4×C53—41no—
189C4×C53#39011yescharacteristic
190C4×C53C2×C22⋊C811yescharacteristic
191C4×C53C8×D411yescharacteristic
192C4×C53#43611yescharacteristic
193Dic53—22no—
194Dic53—41no—
195Dic53—41no—
196C2×#443—531no—
197C2×#364—531no—
198#2117—531no—
199#2117—531no—
200#2115—531no—
201#2115D5311yescharacteristic, complemented
202C4×M32—531no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—81no—
207C2×C53#195411yescharacteristic
208D53—81no—
209C2×#135—531no—
210C2×#131—531no—
211C2×#131C2×D5311yescharacteristic
212C2×M32—531no—
213C2×M32—531no—
214C2×M32C2×D5311yescharacteristic
215C2×M32C2×D5311yescharacteristic
216#443—1061no—
217#443—1061no—
218#443—1061no—
219#443—1061no—
220C2×#108—531no—
221C2×#104—531no—
222C2×#104—531no—
223C2×C4×C8—531no—
224#364—1061no—
225#364—1061no—
226#350—1062no—
227#350—1062no—
228#349—22no—
229#349—1062no—
230#346—1061no—
231C4×C16—1061no—
232C53#1319711yescharacteristic, complemented, Sylow
233C2×#41—531no—
234C2×#41C4×D5311yescharacteristic
235#135—1061no—
236#135—1061no—
237#135—1061no—
238#135—1061no—
239C2×Q16—531no—
240C2×SD16—531no—
241C2×SD16—531no—
242C2×D8—531no—
243#131—21no—
244#131—21no—
245#131—21no—
246#131—21no—
247#131—1061no—
248#131—1061no—
249#131—1061no—
250#131—1061no—
251C2×M16—21no—
252C2×M16—531no—
253C2×M16—1061no—
254C2×M16C4×D5311yescharacteristic
255C22×C8—21no—
256C22×C8—531no—
257C22×C8—1061no—
258C22×C8C22×D5311yescharacteristic
259C2×C42—531no—
260M32—21no—
261M32—21no—
262M32—21no—
263M32—532no—
264M32—1061no—
265M32—1061no—
266M32—1061no—
267M32C22⋊D5312yes—
268C2×C16—21no—
269C2×C16—21no—
270C2×C16—1061no—
271C2×C16—1061no—
272#108—1061no—
273#108—1061no—
274#104—1061no—
275#104—1061no—
276#104—1061no—
277#104—1061no—
278C4×C8—531no—
279C4×C8—531no—
280C4×C8—531no—
281C4×C8—531no—
282#41—21no—
283#41—21no—
284#41—21no—
285#41—21no—
286#41—1061no—
287#41—1061no—
288#41—1061no—
289#41—1061no—
290C2×Q8—531no—
291C2×Q8C8×D5311yescharacteristic
292C2×D4—21no—
293C2×D4—531no—
294C2×D4—1061no—
295C2×D4C8×D5311yescharacteristic
296C22×C4—21no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4C2×C4×D5311yescharacteristic
300Q16—1061no—
301Q16—1061no—
302SD16—1061no—
303SD16—1061no—
304SD16—1061no—
305SD16—1061no—
306D8—1061no—
307D8—1061no—
308M16—21no—
309M16—21no—
310M16—41no—
311M16—41no—
312M16—1061no—
313M16—1061no—
314M16—2121no—
315M16—2121no—
316C2×C8—21no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—531no—
320C2×C8—531no—
321C2×C8—531no—
322C2×C8—531no—
323C2×C8—531no—
324C2×C8—531no—
325C2×C8—1061no—
326C2×C8—2121no—
327C2×C8—2121no—
328C2×C8#1171994511yescharacteristic
329C2×C8D4×D5311yescharacteristic
330C2×C8D4×D5311yescharacteristic
331C2×C8D4×D5311yescharacteristic
332C2×C8#1171994711yescharacteristic
333C2×C8C2×C22⋊D5311yescharacteristic
334C42—531no—
335C42—531no—
336C42—1061no—
337C16—22no—
338C16—41no—
339C16—1062no—
340C16—2121no—
341C23—21no—
342C23—1061no—
343C23#1313322311yescharacteristic
344Q8—21no—
345Q8—21no—
346Q8—1061no—
347Q8—1061no—
348D4—21no—
349D4—21no—
350D4—41no—
351D4—41no—
352D4—1061no—
353D4—1061no—
354D4—2121no—
355D4—2121no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—2122no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4C22⋊C4×D5311yescharacteristic
366C2×C4#1313317711yescharacteristic
367C2×C4C4×C22⋊D5311yescharacteristic
368C2×C4C2×C8×D5311yescharacteristic
369C2×C4M16×D5311yescharacteristic
370C2×C4#1313322511yescharacteristic
371C8—41no—
372C8—531no—
373C8—531no—
374C8—531no—
375C8—531no—
376C8—2121no—
377C8#1313318911yescharacteristic
378C8#1313319011yescharacteristic
379C8#1313320311yescharacteristic
380C8#1313321311yescharacteristic
381C22—41no—
382C22—41no—
383C22—81no—
384C22—4241no—
385C22—2121no—
386C22#1681058811yescharacteristic
387C22C8×C22⋊D5311yescharacteristic
388C22#1681069811yescharacteristic
389C4—41no—
390C4—1062no—
391C4—2121no—
392C4—2121no—
393C4#1681052311yescharacteristic
394C4C22⋊C8×D5311yescharacteristic
395C4#1681068611yescharacteristic
396C4#1681069911yescharacteristic
397C2—21no—
398C2—21no—
399C2—21no—
400C2—81no—
401C2—4241no—
402C2#2027550711yescharacteristic
subgroupG/H
centerC8#13133190
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#2115C2
socleC2×C53#1954
prime radicalC2×#13133237C4
cosocleC22×C8×C53C23
hypercenter#2115D53
hypocenterC53#13197
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2115

Complements

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

NH ≅ G/Ntype
2. C2×#16810443C2semidirect
3. #20276343C2semidirect
4. #20276342C2semidirect
5. #20276335C2semidirect
6. M32×Dic53C2semidirect
7. C53×#2115C2semidirect
201. #2115D53semidirect
232. C53#13197semidirect

Maximal subgroups

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

subgroupindex
C2×#168107382
C2×#168104432
#202763432
#202763422
#202763352
M32×Dic532
C53×#21152
#1319753