Finite Group Databasebeta

← #27300028

Subgroups — #27300028

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283137C211yesmaximal, characteristic
2#20276470C211yesmaximal, characteristic, complemented
3C2×#16810441C211yesmaximal, characteristic, complemented
4C2×#16810408C211yesmaximal, characteristic
5C2×#16810406C211yesmaximal, characteristic
6#20272804C211yesmaximal, characteristic, complemented
7C53×#1379C211yesmaximal, characteristic, complemented
8C2×#13133225C411yescharacteristic
9C2×#13133223C2211yescharacteristic
10#16811091C411yescharacteristic
11C2×#13133157C2211yescharacteristic
12C22×#11719908C2211yescharacteristic
13#16810580—21no—
14#16810579C411yescharacteristic
15#16810577—21no—
16#16810575—21no—
17M16×Dic53C411yescharacteristic
18C2×C8×Dic53C2211yescharacteristic
19C4×C8×D53—21no—
20#16810441—21no—
21#16810441—21no—
22#16810441—21no—
23#16810441—21no—
24#16810408—22no—
25#16810406—22no—
26C2×M32×C53C2211yescharacteristic
27C22×C16×C53C2211yescharacteristic
28C53×#405C2211yescharacteristic
29C2×#11719945C2×C411yescharacteristic
30#13133225—22no—
31#13133225—22no—
32#13133223—21no—
33#13133223—21no—
34#13133223D412yes—
35#13133223Q812yes—
36C2×#11719917C2×C411yescharacteristic
37C2×#11719912—21no—
38C2×#11719912—21no—
39C22×#88322C2×C411yescharacteristic
40C2×C8×D53—21no—
41C2×C8×D53—21no—
42#13133182C811yescharacteristic
43#13133182C811yescharacteristic
44#13133179C811yescharacteristic
45#13133179C811yescharacteristic
46#13133177—21no—
47#13133177—21no—
48#13133174—21no—
49C2×C4×Dic53C2×C411yescharacteristic
50C42×D53—21no—
51#13133157—21no—
52#13133157—21no—
53C2×#11719908—21no—
54C2×#11719908—21no—
55C2×#11719908—21no—
56C2×#11719908—21no—
57#13133128—21no—
58#13133127—21no—
59C4×#88322—21no—
60C8×Dic53—21no—
61C8×Dic53D411yescharacteristic
62C8×Dic53D411yescharacteristic
63C8×Dic53D411yescharacteristic
64C8×Dic53D411yescharacteristic
65C2×M16×C53C2×C411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67#117×C53C2×C411yescharacteristic
68M32×C53—21no—
69M32×C53—21no—
70C2×C16×C53—21no—
71C2×C16×C53—21no—
72C2×C16×C53—21no—
73C2×C16×C53—21no—
74#97×C53—21no—
75C4×C8×C53—21no—
76C22×C4×C53C22×C411yescharacteristic
77C2×C22⋊D53—21no—
78C22×Dic53M1611yescharacteristic
79C22×Dic53M1611yescharacteristic
80#11719945—21no—
81#11719945—21no—
82#11719945C4⋊C412yes—
83#11719945M1612yes—
84C2×D212C2×C811yescharacteristic
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×Dic106C2×C811yescharacteristic
89M16×C53—21no—
90M16×C53M1612yes—
91C2×C8×C53—21no—
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53C2×Q811yescharacteristic
96C2×C8×C53#4111yescharacteristic
97C2×C8×C53#4111yescharacteristic
98C4⋊C4×C53C2×C811yescharacteristic
99C4⋊C4×C53C2×C811yescharacteristic
100C22⋊C4×C53—21no—
101C42×C53—21no—
102#11719921—21no—
103#11719921—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53C22⋊C411yescharacteristic
109C4×Dic53C22⋊C411yescharacteristic
110C4×Dic53C2×C811yescharacteristic
111C4×Dic53C2×C811yescharacteristic
112#11719917—21no—
113#11719917C4⋊C412yes—
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322C22⋊C411yescharacteristic
118C2×#88322C22⋊C411yescharacteristic
119#11719912—24no—
120#11719912—22no—
121#11719912—22no—
122C8×D53—24no—
123C8×D53—22no—
124C8×D53—22no—
125C16×C53—41no—
126C16×C53—41no—
127#11719908—41no—
128#11719908—41no—
129C23×C53C2×M1611yescharacteristic
130C22×D53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53C2×C4⋊C411yescharacteristic
135C2×C4×C53#11711yescharacteristic
136C2×C4×C53C22×C811yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53C2×M1611yescharacteristic
139C22⋊D53—22no—
140C22⋊D53—22no—
141C2×Dic53—21no—
142C2×Dic53—22no—
143C2×Dic53—41no—
144C2×Dic53C22⋊C811yescharacteristic
145C2×Dic53C22⋊C811yescharacteristic
146C2×Dic53C22⋊C811yescharacteristic
147C2×Dic53C22⋊C811yescharacteristic
148D212—21no—
149D212C4⋊C812yes—
150C4×D53—22no—
151C4×D53—22no—
152C4×D53—42no—
153C4×D53—42no—
154Dic106—21no—
155Dic106C4⋊C812yes—
156C8×C53—22no—
157C8×C53#12111yescharacteristic
158C8×C53#12111yescharacteristic
159C8×C53C22⋊Q811yescharacteristic
160C8×C53C22⋊Q811yescharacteristic
161#88322—22no—
162#88322—21no—
163#88322—21no—
164#12787—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53C2×C22⋊C811yescharacteristic
168C22×C53#42311yescharacteristic
169C22×C53#43211yescharacteristic
170C2×D53—22no—
171C2×D53—41no—
172C2×D53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53#38011yescharacteristic
176C4×C53#40711yescharacteristic
177C4×C53C2×C4⋊C811yescharacteristic
178C4×C53#43111yescharacteristic
179Dic53—22no—
180Dic53—21no—
181Dic53—21no—
182Dic53—21no—
183Dic53#35012yescomplemented
184C4×#131—531no—
185#2150—531no—
186C2×C4⋊C16—531no—
187C2×#350—531no—
188C2×#350—531no—
189#1379—531no—
190#1379D5311yescharacteristic, complemented
191C2×C53—21no—
192C2×C53—21no—
193C2×C53#179711yescharacteristic
194C2×C53C2×#35011yescharacteristic
195C2×C53#211711yescharacteristic
196D53—42no—
197C2×#131—531no—
198C2×#131—531no—
199C4×#41—531no—
200C2×M32—531no—
201C2×M32C2×D5311yescharacteristic
202C22×C16—531no—
203C22×C16C2×D5311yescharacteristic
204#405—531no—
205#405—1061no—
206#405—1061no—
207#405C2×D5311yescharacteristic
208C4×M16—531no—
209C4×M16—1061no—
210C2×C4×C8—531no—
211C2×C4×C8—1061no—
212C4⋊C16—1061no—
213C4⋊C16—1061no—
214C4⋊C16—1061no—
215C4⋊C16—1061no—
216#350—1062no—
217#350—1062no—
218C53#1278711yescharacteristic, complemented, Sylow
219C2×#41—531no—
220#131—532no—
221#131—532no—
222#131—1062no—
223#131—1062no—
224#131—1061no—
225#131—1061no—
226C2×M16—531no—
227C2×M16—1061no—
228C2×M16—1061no—
229C2×M16C4×D5311yescharacteristic
230C22×C8—531no—
231C22×C8—1061no—
232C22×C8—1061no—
233C22×C8C22×D5311yescharacteristic
234C4×Q8—531no—
235C4×Q8—531no—
236C4×D4—531no—
237C4×D4—531no—
238C4×D4—1061no—
239C4×D4—1061no—
240#117—1061no—
241#117C4×D5311yescharacteristic
242C2×C42—531no—
243C2×C42—1061no—
244M32—21no—
245M32—21no—
246M32—1061no—
247M32—1061no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—1061no—
253C2×C16—1061no—
254C2×C16—1061no—
255C2×C16—1061no—
256#97—21no—
257#97—1061no—
258#97—1061no—
259C4×C8—21no—
260C4×C8—531no—
261C4×C8—531no—
262C4×C8—531no—
263C4×C8—531no—
264C4×C8—1061no—
265C4×C8—1061no—
266#41—532no—
267#41—532no—
268#41—1061no—
269#41—1061no—
270C2×Q8—531no—
271C2×D4—531no—
272C2×D4—1061no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4C2×C4×D5311yescharacteristic
279M16—21no—
280M16—532no—
281M16—1064no—
282M16—1061no—
283M16—1062no—
284M16—1062no—
285M16#1171991212yes—
286C2×C8—21no—
287C2×C8—531no—
288C2×C8—531no—
289C2×C8—1064no—
290C2×C8—1061no—
291C2×C8—1061no—
292C2×C8—1061no—
293C2×C8—1062no—
294C2×C8—1062no—
295C2×C8#1171994511yescharacteristic
296C2×C8#1171994511yescharacteristic
297C2×C8D4×D5311yescharacteristic
298C2×C8D4×D5311yescharacteristic
299C2×C8D4×D5311yescharacteristic
300C2×C8Q8×D5311yescharacteristic
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4C8×D5311yescharacteristic
304C4⋊C4C8×D5311yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C42—21no—
309C42—531no—
310C42—531no—
311C42—531no—
312C42—531no—
313C42—1061no—
314C42—1061no—
315C16—41no—
316C16—41no—
317C16—2121no—
318C16—2121no—
319C23—1061no—
320C23M16×D5311yescharacteristic
321Q8—532no—
322Q8—1061no—
323D4—532no—
324D4—1062no—
325D4—1062no—
326D4—1061no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—531no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—1062no—
334C2×C4—1061no—
335C2×C4—1062no—
336C2×C4—1062no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2121no—
340C2×C4C22⋊C4×D5311yescharacteristic
341C2×C4C4⋊C4×D5311yescharacteristic
342C2×C4#1313317411yescharacteristic
343C2×C4C2×C8×D5311yescharacteristic
344C2×C4M16×D5311yescharacteristic
345C2×C4C2×#1171991211yescharacteristic
346C8—22no—
347C8—1061no—
348C8—1061no—
349C8—1062no—
350C8#1313319011yescharacteristic
351C8#1313319011yescharacteristic
352C8#1313319511yescharacteristic
353C8#1313319511yescharacteristic
354C22—21no—
355C22—21no—
356C22—1062no—
357C22—2121no—
358C22—2121no—
359C22C22⋊C8×D5311yescharacteristic
360C22#1681064511yescharacteristic
361C22#1681067811yescharacteristic
362C4—41no—
363C4—41no—
364C4—532no—
365C4—1062no—
366C4—1061no—
367C4—1061no—
368C4—1061no—
369C4#1681048211yescharacteristic
370C4#1681058511yescharacteristic
371C4C4⋊C8×D5311yescharacteristic
372C4#1681067411yescharacteristic
373C2—21no—
374C2—21no—
375C2—2122no—
376C2#2027469411yescharacteristic
377C2#350×D5311yescharacteristic
378C2#2027634111yescharacteristic
subgroupG/H
centerC2×C8D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1379C2
socleC22×C53C2×C22⋊C8
prime radicalC2×#11719945C2×C4
cosocleC22×C8×C53C23
hypercenter#1379D53
hypocenterC53#12787
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ #1379

Complements

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

NH ≅ G/Ntype
2. #20276470C2semidirect
3. C2×#16810441C2semidirect
6. #20272804C2semidirect
7. C53×#1379C2semidirect
183. Dic53#350semidirect
190. #1379D53semidirect
218. C53#12787semidirect

Maximal subgroups

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

subgroupindex
#202831372
#202764702
C2×#168104412
C2×#168104082
C2×#168104062
#202728042
C53×#13792
#1278753