Finite Group Databasebeta

← #28319422

Subgroups — #28319422

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290646C212yesmaximal, complemented
2#20290635C212yesmaximal, complemented
3#20290624C212yesmaximal
4#20290599C211yesmaximal, characteristic
5#20290597C211yesmaximal, characteristic
6#20290521C212yesmaximal, complemented
7#20290470C212yesmaximal
8#20290437C212yesmaximal, complemented
9#20283140C211yesmaximal, characteristic
10D53⋊#108C2212yes—
11#16811633C2212yes—
12#16811609C2212yes—
13#16811602C2212yes—
14#16811586C2214yes—
15#16811505C2214yes—
16#16811494C2214yes—
17#16811492C2212yes—
18#16811491C2212yes—
19#16811490C2214yes—
20C2×#13133223C2211yescharacteristic
21C2×#13133223C2211yescharacteristic
22C4×#11719945C2211yescharacteristic
23#16810580C411yescharacteristic, complemented
24#16810580C411yescharacteristic, complemented
25#16810580C2211yescharacteristic
26#16810580C2211yescharacteristic
27#16810578C411yescharacteristic, complemented
28#16810578C411yescharacteristic, complemented
29C4×#11719917C411yescharacteristic, complemented
30#16810575C411yescharacteristic, complemented
31#16810575C2211yescharacteristic
32#97×D53C411yescharacteristic, complemented
33#97×D53C2211yescharacteristic
34C2×#97×C53C411yescharacteristic, complemented
35D53⋊M16C2314yes—
36#13133296C2314yes—
37#13133288—22no—
38#13133287—22no—
39#13133286—22no—
40#13133285—22no—
41#13133278—22no—
42#13133277—22no—
43#13133276—22no—
44#13133275—22no—
45#13133271—24no—
46C8×#6157—24no—
47C2×#11719945C2311yescharacteristic
48#13133223—24no—
49#13133223—22no—
50#13133223—22no—
51C2×#11719917C2×C411yescharacteristic
52C2×#11719917C2×C411yescharacteristic
53C2×#11719912C2×C411yescharacteristic
54C2×#11719912C2×C411yescharacteristic
55C2×#11719912C2311yescharacteristic
56C2×#11719912C2311yescharacteristic
57C2×C8×D53C2×C411yescharacteristic
58C2×C8×D53C2×C411yescharacteristic
59C2×C8×D53C2311yescharacteristic
60C2×C8×D53C2311yescharacteristic
61C4×Dic106—21no—
62C4×D212—21no—
63C4×C22⋊D53—21no—
64C4×C22⋊D53—21no—
65#13133176C2×C411yescharacteristic
66#13133174C2×C411yescharacteristic
67#13133174C2311yescharacteristic
68C42×D53C2×C411yescharacteristic
69C42×D53C2311yescharacteristic
70#13133128C2×C411yescharacteristic
71#13133128C2×C411yescharacteristic
72#13133127C2×C411yescharacteristic
73#13133127C2×C411yescharacteristic
74#13133127C2×C411yescharacteristic
75#13133127C2×C411yescharacteristic
76C4×#88322C2×C411yescharacteristic
77C4×#88322C2×C411yescharacteristic
78C8×Dic53C2×C411yescharacteristic
79C8×Dic53C2×C411yescharacteristic
80C8×Dic53C2×C411yescharacteristic
81C8×Dic53C2×C411yescharacteristic
82C22×C8×C53C2×C411yescharacteristic
83C22×C8×C53C2×C411yescharacteristic
84C2×C42×C53C2×C411yescharacteristic
85#97×C53C2×C411yescharacteristic
86#97×C53C2×C411yescharacteristic
87#97×C53C2×C411yescharacteristic
88#97×C53C2×C411yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53C22×C411yescharacteristic
91C2×C22⋊D53#4111yescharacteristic
92C2×C22⋊D53#4111yescharacteristic
93#11719945—22no—
94#11719945—22no—
95C2×D212#4111yescharacteristic
96C2×C4×D53—21no—
97C2×C4×D53—21no—
98C2×C4×D53C22×C411yescharacteristic
99C2×C4×D53C2411yescharacteristic
100C2×Dic106#4111yescharacteristic
101C22⋊#6157—24no—
102#11719940—24no—
103C4⋊#6157—24no—
104C4×#6157—24no—
105#11719936—24no—
106D53⋊C8—24no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53—22no—
110C2×C8×C53C22×C411yescharacteristic
111C2×C8×C53C22×C411yescharacteristic
112C2×C8×C53C22×C411yescharacteristic
113C2×C8×C53C22×C411yescharacteristic
114C42×C53—21no—
115C42×C53C22×C411yescharacteristic
116C42×C53C22×C411yescharacteristic
117C22⋊Dic53—21no—
118#11719921—21no—
119#11719921—21no—
120C4⋊Dic53—21no—
121#11719919—21no—
122#11719919—21no—
123C4×Dic53—21no—
124C4×Dic53—21no—
125C4×Dic53C22×C411yescharacteristic
126C4×Dic53C22×C411yescharacteristic
127#11719917—22no—
128#11719917—22no—
129C2×#88322C22×C411yescharacteristic
130C2×#88322C22×C411yescharacteristic
131C2×#88322C22×C411yescharacteristic
132C2×#88322C22×C411yescharacteristic
133#11719912—22no—
134#11719912—22no—
135#11719912—22no—
136#11719912—22no—
137C8×D53—21no—
138C8×D53—21no—
139C8×D53—22no—
140C8×D53#4114yes—
141C8×D53#4112yes—
142C8×D53#4112yes—
143C23×C53#11711yescharacteristic
144C22×D53#11711yescharacteristic
145C22×D53C2×#4111yescharacteristic
146C2×#6157—44no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—81no—
152C2×C4×C53#11711yescharacteristic
153C2×C4×C53C23×C411yescharacteristic
154C22⋊D53—22no—
155C22⋊D53—22no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53#11711yescharacteristic
159C2×Dic53C2×#4111yescharacteristic
160D212—22no—
161C4×D53—21no—
162C4×D53—21no—
163C4×D53—81no—
164C4×D53—81no—
165C4×D53C2×#4112yes—
166C4×D53C2×#4112yes—
167Dic106—22no—
168#88324—44no—
169C8×C53—21no—
170C8×C53—21no—
171C8×C53#11711yescharacteristic
172C8×C53#11711yescharacteristic
173C8×C53#11712yes—
174#88322—21no—
175#88322—21no—
176#88322#11712yes—
177#88322#11712yes—
178#20788—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—41no—
181C22×C53C2×#11711yescharacteristic
182C2×D53—21no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53#52912yes—
186#6157—84no—
187C4×C53—21no—
188C4×C53—41no—
189C4×C53—41no—
190C4×C53C2×#11711yescharacteristic
191C4×C53C2×#11711yescharacteristic
192Dic53—21no—
193Dic53—41no—
194Dic53—41no—
195Dic53#52912yes—
196#2875—531no—
197#1841—532no—
198#1833—532no—
199#1827—532no—
200#1813—531no—
201#1812—531no—
202#1773—532no—
203#1757—532no—
204#1747—532no—
205C2×C53—21no—
206C2×C53—41no—
207C2×C53#228311yescharacteristic
208D53—22no—
209D53—41no—
210C2×#131—531no—
211C2×#131—531no—
212C4×#41—531no—
213C2×#108—532no—
214#427—532no—
215#413—532no—
216#410—532no—
217#405—531no—
218#405—531no—
219#405—531no—
220#405—531no—
221#405—531no—
222#405—531no—
223#405—534no—
224C4×M16—531no—
225C4×M16—531no—
226C4×M16—531no—
227C2×#97—531no—
228C2×#97—531no—
229C2×#97#615711yescharacteristic, complemented
230#344—534no—
231#339—534no—
232#338—532no—
233H(3,ℤ/4)—532no—
234H(3,ℤ/4)—534no—
235C53#2078811yescharacteristic, complemented, Sylow
236C2×#41—531no—
237#131—1064no—
238#131—1062no—
239#131—1062no—
240C2×M16—531no—
241C2×M16—531no—
242C2×M16—531no—
243C2×M16—531no—
244C2×M16—531no—
245C2×M16—531no—
246C2×M16—534no—
247C22×C8—531no—
248C22×C8—531no—
249C22×C8—531no—
250C22×C8—531no—
251C22×C8C2×#615711yescharacteristic
252C22×C8C2×#615711yescharacteristic
253C4×Q8—1061no—
254C4×D4—1061no—
255C4×D4—1061no—
256C4×D4—1061no—
257#117—531no—
258#117—531no—
259#117—531no—
260#117—534no—
261C2×C42—531no—
262C2×C42—531no—
263C2×C42C2×#615711yescharacteristic
264#108—1062no—
265#108—1062no—
266#107—1062no—
267#106—1062no—
268#101—1062no—
269#100—1062no—
270#99—1062no—
271#99—1062no—
272#97—531no—
273#97—531no—
274#97—531no—
275#97—531no—
276#97—531no—
277#97—531no—
278#97—1064no—
279#97C2×#615711yescharacteristic
280#97C2×#615711yescharacteristic
281#97C2×#615711yescharacteristic
282#97C2×#615711yescharacteristic
283C4×C8—531no—
284C4×C8—531no—
285C4×C8—531no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—531no—
289C4×C8—1064no—
290#41—1062no—
291#41—1062no—
292C2×Q8—531no—
293C2×D4—531no—
294C2×D4—531no—
295C2×D4—531no—
296C22×C4—21no—
297C22×C4—531no—
298C22×C4—531no—
299C22×C4—1061no—
300C22×C4—1061no—
301C22×C4C22×#615711yescharacteristic
302M16—1062no—
303M16—1062no—
304M16—1062no—
305M16—1062no—
306M16—1062no—
307M16—1062no—
308M16—1064no—
309M16—1064no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—531no—
314C2×C8—531no—
315C2×C8—534no—
316C2×C8—532no—
317C2×C8—532no—
318C2×C8—531no—
319C2×C8—531no—
320C2×C8—1061no—
321C2×C8—1061no—
322C2×C8—1062no—
323C2×C8—1064no—
324C2×C8C22×#615711yescharacteristic
325C2×C8C22×#615711yescharacteristic
326C2×C8C22×#615711yescharacteristic
327C2×C8C22×#615711yescharacteristic
328C4⋊C4—1061no—
329C4⋊C4—1061no—
330C4⋊C4—1064no—
331C4⋊C4—1061no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1064no—
336C42—21no—
337C42—531no—
338C42—531no—
339C42—1061no—
340C42—1061no—
341C42—1064no—
342C42C22×#615711yescharacteristic
343C42C22×#615711yescharacteristic
344C23—531no—
345C23—531no—
346C23#1313329611yescharacteristic
347Q8—1062no—
348D4—1062no—
349D4—1062no—
350D4—1062no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—81no—
356C2×C4—531no—
357C2×C4—532no—
358C2×C4—532no—
359C2×C4—531no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—4241no—
365C2×C4—4241no—
366C2×C4—2124no—
367C2×C4#1313329611yescharacteristic
368C2×C4C23×#615711yescharacteristic
369C8—21no—
370C8—21no—
371C8—532no—
372C8—532no—
373C8—1061no—
374C8—1061no—
375C8—2124no—
376C8#1313329611yescharacteristic
377C8#1313329611yescharacteristic
378C8#1313329612yes—
379C22—21no—
380C22—41no—
381C22—532no—
382C22—1061no—
383C22—2121no—
384C22—2121no—
385C22C2×#1313329611yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—532no—
390C4—1061no—
391C4—4244no—
392C4—2121no—
393C4—2121no—
394C4C2×#1313329611yescharacteristic
395C4C2×#1313329611yescharacteristic
396C2—21no—
397C2—41no—
398C2—1062no—
399C2—2121no—
400C2#2029116211yescharacteristic
subgroupG/H
centerC4C2×#13133296
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×#97×C53C4
socleC2×C53#2283
prime radicalC2×#11719945C23
cosocleC2×C4×D53C24
hypercenterC2×#97#6157
hypocenterC53#20788
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ #97 ◁ C2×#97

Complements

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

NH ≅ G/Ntype
1. #20290646C2semidirect
2. #20290635C2semidirect
6. #20290521C2semidirect
8. #20290437C2semidirect
23. #16810580C4semidirect
24. #16810580C4semidirect
27. #16810578C4semidirect
28. #16810578C4semidirect
29. C4×#11719917C4semidirect
30. #16810575C4semidirect
32. #97×D53C4semidirect
34. C2×#97×C53C4semidirect
229. C2×#97#6157semidirect
235. C53#20788semidirect

Maximal subgroups

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

subgroupindex
#202906462
#202906352
#202906242
#202905992
#202905972
#202905212
#202904702
#202904372
#202831402
#2078853