Finite Group Databasebeta

← #27303264

Subgroups — #27303264

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810865C211yesmaximal, characteristic
2C2×#16810450C211yesmaximal, characteristic, complemented
3C2×#16810428C211yesmaximal, characteristic, complemented
4C2×#16810426C211yesmaximal, characteristic, complemented
5C2×#16810402C211yesmaximal, characteristic
6#20274741C211yesmaximal, characteristic
7C53×#1380C211yesmaximal, characteristic, complemented
8C22×Dic212C2211yescharacteristic
9C22×C8×D53C2211yescharacteristic
10C2×#13133197—21no—
11C22×#11719908C2211yescharacteristic
12#16810865—22no—
13#16810865—22no—
14C2×#13133151C2211yescharacteristic
15C2×#13133140—21no—
16#16810480—21no—
17#16810450—22no—
18#16810428—22no—
19#16810426—22no—
20#16810402—22no—
21#16810385—21no—
22C22×C16×C53C2211yescharacteristic
23C2×#107×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C22×Dic106—21no—
26C22×C4×D53D411yescharacteristic
27C2×Dic212—21no—
28C2×Dic212—21no—
29C2×Dic212—21no—
30C2×Dic212—21no—
31C2×Dic212—21no—
32C2×Dic212D412yes—
33C22×#88322D411yescharacteristic
34C2×C8×D53—21no—
35C2×C8×D53—21no—
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38C2×C8×D53—21no—
39C2×C8×D53—21no—
40#13133197—24no—
41#13133197—42no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—41no—
44C2×#11719921—41no—
45C2×#11719921—41no—
46C2×#11719908—21no—
47C2×#11719908—21no—
48C2×#11719908D412yes—
49#13133151—21no—
50#13133151—21no—
51#13133140—24no—
52#13133124—41no—
53C22×C8×C53C2311yescharacteristic
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C16×C53—21no—
57C2×C16×C53—21no—
58C2×C16×C53—21no—
59#107×C53—21no—
60#107×C53—21no—
61#107×C53—21no—
62#107×C53D412yes—
63C23×D53D811yescharacteristic
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53D811yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×Dic106—22no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×C8×C53C2×D411yescharacteristic
83C2×C8×C53C2×D411yescharacteristic
84C2×C8×C53C2×D411yescharacteristic
85C2×C8×C53#4111yescharacteristic
86C2×C8×C53#4111yescharacteristic
87C2×C8×C53#4111yescharacteristic
88C4⋊C4×C53—22no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92#11719921—44no—
93#11719921—82no—
94C4⋊Dic53—44no—
95C4⋊Dic53—41no—
96C4⋊Dic53—41no—
97C2×#88322—21no—
98C2×#88322—21no—
99C2×#88322—21no—
100Dic212—22no—
101Dic212—22no—
102Dic212—41no—
103Dic212—41no—
104C8×D53—41no—
105C8×D53—41no—
106C8×D53—41no—
107C8×D53—41no—
108C16×C53—41no—
109#11719908—22no—
110C23×C53C2×D811yescharacteristic
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53SD3212yes—
118C2×C4×C53—42no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53—81no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53#12111yescharacteristic
126C2×C4×C53#12311yescharacteristic
127C2×C4×C53#13511yescharacteristic
128C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C53#13711yescharacteristic
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—42no—
134C2×Dic53—81no—
135C2×Dic53—81no—
136C2×Dic53—81no—
137C2×Dic53—81no—
138C2×Dic53—81no—
139C4×D53—41no—
140C4×D53—41no—
141C4×D53—41no—
142C4×D53—41no—
143Dic106—42no—
144Dic106—42no—
145Dic106—81no—
146Dic106—81no—
147C8×C53#12111yescharacteristic
148C8×C53#12111yescharacteristic
149C8×C53#12311yescharacteristic
150C8×C53#12411yescharacteristic
151#88322—41no—
152#13382—531nomaximal, Sylow
153C22×C53#44711yescharacteristic
154C22×C53#46611yescharacteristic
155C22×C53#48011yescharacteristic
156C22×C53C2×SD3211yescharacteristic
157C22×C53#50811yescharacteristic
158C22×C53#51011yescharacteristic
159C22×C53#51011yescharacteristic
160C2×D53—22no—
161C2×D53—22no—
162C2×D53—22no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C4×C53—82no—
168C4×C53—161no—
169C4×C53#39411yescharacteristic
170C4×C53#45211yescharacteristic
171C4×C53#46811yescharacteristic
172C4×C53#48211yescharacteristic
173Dic53—41no—
174Dic53—82no—
175Dic53—161no—
176C2×#467—531no—
177C2×#367—531no—
178C2×#358—531no—
179C2×#358—531no—
180C2×C22⋊C16—531no—
181#1808—531no—
182#1380D5311yescharacteristic, complemented
183C2×C53#203311yescharacteristic
184C2×C53#218511yescharacteristic
185C2×C53#218911yescharacteristic
186C2×C53#221711yescharacteristic
187C2×C53#221911yescharacteristic
188C2×C53#223511yescharacteristic
189C2×C53#223711yescharacteristic
190D53—42no—
191C22×Q16—531no—
192C23×C8—531no—
193C2×C22⋊Q8—1061no—
194C22×C16—531no—
195C22×C16C2×D5311yescharacteristic
196#467—1062no—
197#467—1062no—
198C2×#107—531no—
199C2×#107C2×D5311yescharacteristic
200C2×#107C2×D5311yescharacteristic
201C2×#103—1061no—
202#380—1061no—
203#367—1062no—
204#358—1062no—
205#358—1062no—
206C22⋊C16—1062no—
207#340—1061no—
208C53#1338211yescharacteristic, complemented, Sylow
209C22×Q8—1061no—
210C23×C4—531no—
211C2×Q16—532no—
212C2×Q16—1061no—
213C2×Q16—1061no—
214C2×Q16—1061no—
215C2×Q16—1061no—
216C2×Q16—1061no—
217C22×C8—531no—
218C22×C8—1061no—
219C22×C8—1061no—
220C22×C8—1061no—
221C22×C8—1061no—
222C22×C8—1061no—
223C22×C8—1061no—
224C22×C8C22×D5311yescharacteristic
225C22⋊Q8—1064no—
226C22⋊Q8—2122no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—2121no—
231C2×C22⋊C4—2121no—
232C2×C22⋊C4—2121no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—532no—
237C2×C16—1061no—
238C2×C16—1061no—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—1061no—
243#107—1061no—
244#107D21212yes—
245#103—1064no—
246#95—2121no—
247C24—531no—
248C2×Q8—1062no—
249C2×Q8—2121no—
250C2×Q8—2121no—
251C2×Q8—2121no—
252C2×Q8—2121no—
253C2×Q8—2121no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—531no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4D4×D5311yescharacteristic
266Q16—1062no—
267Q16—1062no—
268Q16—2121no—
269Q16—2121no—
270C2×C8—1061no—
271C2×C8—1061no—
272C2×C8—1061no—
273C2×C8—2121no—
274C2×C8—2121no—
275C2×C8—2121no—
276C2×C8—2121no—
277C2×C8C2×D21211yescharacteristic
278C2×C8#1171994511yescharacteristic
279C2×C8#1171994511yescharacteristic
280C2×C8D4×D5311yescharacteristic
281C2×C8D4×D5311yescharacteristic
282C2×C8#1171994911yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—2124no—
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C22⋊C4—4242no—
291C22⋊C4—2124no—
292C16—41no—
293C16—1062no—
294C23—532no—
295C23—1061no—
296C23—1061no—
297C23—1061no—
298C23—1061no—
299C23—1061no—
300C23—1061no—
301C23D8×D5311yescharacteristic
302Q8—4241no—
303Q8—4241no—
304Q8—2122no—
305Q8—2122no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—4241no—
316C2×C4—4241no—
317C2×C4—4241no—
318C2×C4—4241no—
319C2×C4—4241no—
320C2×C4—2122no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4#1313318711yescharacteristic
326C2×C4#1313319011yescharacteristic
327C2×C4#1313320211yescharacteristic
328C2×C4#1313323811yescharacteristic
329C2×C4#1313324411yescharacteristic
330C2×C4#1313324911yescharacteristic
331C8—2121no—
332C8#1313319011yescharacteristic
333C8#1313319211yescharacteristic
334C8#1313320211yescharacteristic
335C8#1313320511yescharacteristic
336C22—1062no—
337C22—1062no—
338C22—1062no—
339C22—2121no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22#1681075811yescharacteristic
344C22#1681085811yescharacteristic
345C22#1681093811yescharacteristic
346C22SD32×D5311yescharacteristic
347C22#1681105211yescharacteristic
348C22#1681106211yescharacteristic
349C22#1681106511yescharacteristic
350C4—82no—
351C4—161no—
352C4—8481no—
353C4—4242no—
354C4—2121no—
355C4#1681054111yescharacteristic
356C4#1681078811yescharacteristic
357C4#1681087211yescharacteristic
358C4#1681094911yescharacteristic
359C2—2122no—
360C2#2027590011yescharacteristic
361C2#2027663311yescharacteristic
362C2#2027665611yescharacteristic
363C2#2027681011yescharacteristic
364C2#2027682211yescharacteristic
365C2#2027691411yescharacteristic
366C2#2027692511yescharacteristic
subgroupG/H
centerC23D8×D53
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#1380C2
socleC23×C53C2×D8
prime radicalC23×D53D8
hypercenter#1380D53
hypocenterC53#13382
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#16810450C2semidirect
3. C2×#16810428C2semidirect
4. C2×#16810426C2semidirect
7. C53×#1380C2semidirect
182. #1380D53semidirect
208. C53#13382semidirect

Maximal subgroups

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

subgroupindex
C2×#168108652
C2×#168104502
C2×#168104282
C2×#168104262
C2×#168104022
#202747412
C53×#13802
#1338253