Finite Group Databasebeta

← #27288824

Subgroups — #27288824

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275650C211yesmaximal, characteristic
2#20274871C211yesmaximal, characteristic
3#20274721C211yesmaximal, characteristic
4#20274717C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic
6#20272549C211yesmaximal, characteristic
7C53×#1296C211yesmaximal, characteristic
8C2×C4×Dic106C2211yescharacteristic
9C2×#13133148—21no—
10C2×#13133148—21no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133142—21no—
13C2×#13133140—21no—
14C2×C4×#88322C2211yescharacteristic
15#16810493C2211yescharacteristic
16#16810386—21no—
17#16810385—21no—
18#16810377—21no—
19#16810377—21no—
20C2×C4⋊C8×C53C2211yescharacteristic
21C2×C4⋊C8×C53C2211yescharacteristic
22C53×#384C2211yescharacteristic
23C22×Dic106D411yescharacteristic
24C22×#88322—21no—
25C22×#88322D411yescharacteristic
26C22×#88322D411yescharacteristic
27C4×Dic106—21no—
28C4×Dic106—21no—
29C4×Dic106—21no—
30C4×Dic106—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53D411yescharacteristic
35C2×#11719919—21no—
36C2×C4×Dic53—21no—
37#13133148—42no—
38#13133148—42no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41#13133142—42no—
42#13133140—42no—
43C4×#88322—21no—
44C4×#88322—21no—
45C22×C8×C53—21no—
46C22×C8×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C42×C53C2311yescharacteristic
51C4⋊C8×C53—21no—
52C4⋊C8×C53—21no—
53C4⋊C8×C53—21no—
54C4⋊C8×C53—21no—
55C22×C4×C53—41no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53#4111yescharacteristic
59C22×Dic53—21no—
60C22×Dic53—21no—
61C22×Dic53—41no—
62C2×Dic106—21no—
63C2×Dic106—21no—
64C2×Dic106—21no—
65C2×Dic106—21no—
66C2×Dic106—21no—
67C2×Dic106SD1612yes—
68C2×C8×C53—41no—
69C2×C8×C53—41no—
70C2×C8×C53—41no—
71C2×C8×C53—41no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—81no—
79C42×C53#4111yescharacteristic
80C42×C53#4111yescharacteristic
81C42×C53#4111yescharacteristic
82C42×C53#4111yescharacteristic
83C4⋊Dic53—21no—
84C4⋊Dic53—21no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89C4⋊Dic53—81no—
90#11719919—41no—
91#11719919—41no—
92C4×Dic53—41no—
93C4×Dic53—41no—
94C2×#88322—21no—
95C2×#88322—21no—
96C2×#88322—21no—
97C2×#88322—21no—
98C2×#88322—41no—
99C2×#88322—41no—
100C2×#88322—41no—
101C2×#88322SD1612yes—
102C2×#88322Q1612yes—
103C23×C53#12111yescharacteristic
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—81no—
107C2×C4×C53—81no—
108C2×C4×C53—81no—
109C2×C4×C53#12311yescharacteristic
110C2×C4×C53#12311yescharacteristic
111C2×C4×C53#12311yescharacteristic
112C2×C4×C53#12411yescharacteristic
113C2×C4×C53#12611yescharacteristic
114C2×C4×C53#12611yescharacteristic
115C2×C4×C53C2×SD1611yescharacteristic
116C2×C4×C53C2×SD1611yescharacteristic
117C2×C4×C53C2×Q1611yescharacteristic
118C2×C4×C53#13511yescharacteristic
119C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C53#13711yescharacteristic
122C2×C4×C53#13711yescharacteristic
123C2×Dic53—22no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—81no—
130C2×Dic53—81no—
131C2×Dic53—81no—
132Dic106—22no—
133Dic106—22no—
134Dic106—41no—
135Dic106—41no—
136C8×C53—81no—
137C8×C53—81no—
138#88322—22no—
139#88322—22no—
140#88322—81no—
141#10620—531nomaximal, Sylow
142C22×C53#39711yescharacteristic
143C22×C53#46011yescharacteristic
144C22×C53#46211yescharacteristic
145C22×C53#46511yescharacteristic
146C22×C53#46811yescharacteristic
147C22×C53#47111yescharacteristic
148C22×C53#47111yescharacteristic
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—161no—
152C4×C53#48111yescharacteristic
153C4×C53#48211yescharacteristic
154C4×C53#48311yescharacteristic
155C4×C53#48311yescharacteristic
156C4×C53#48411yescharacteristic
157C4×C53#48511yescharacteristic
158C4×C53#48711yescharacteristic
159C4×C53#48911yescharacteristic
160Dic53—42no—
161Dic53—81no—
162Dic53—161no—
163#1984—531no—
164#1844—531no—
165#1803—531no—
166#1803—531no—
167#1296—531no—
168#1296D5311yescharacteristic
169#1277—531no—
170C2×C53#166611yescharacteristic
171C2×C53#167211yescharacteristic
172C2×C53#168611yescharacteristic
173C2×C53#169211yescharacteristic
174C2×C53#207211yescharacteristic
175C2×C53#207711yescharacteristic
176C2×C53#207911yescharacteristic
177C2×C4×Q8—531no—
178C2×#106—1061no—
179C2×#106—1061no—
180C2×C4⋊C8—531no—
181C2×C4⋊C8C2×D5311yescharacteristic
182C2×C4⋊C8C2×D5311yescharacteristic
183C2×#103—1061no—
184C2×#103—1061no—
185C2×C4×C8—531no—
186#384—531no—
187#384C2×D5311yescharacteristic
188#340—1061no—
189#340—1061no—
190#336—1061no—
191#336—1061no—
192C53#1062011yescharacteristic, complemented, Sylow
193C22×Q8—531no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—531no—
197C22×C8—531no—
198C22×C8—1061no—
199C4×Q8—1061no—
200C4×Q8—1061no—
201C4×Q8—1061no—
202C4×Q8—1061no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C42—1061no—
212C2×C42C22×D5311yescharacteristic
213#106—2122no—
214#106—2122no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—1061no—
220C4⋊C8—1061no—
221#103—2122no—
222#103—2122no—
223C4×C8—1061no—
224C4×C8—1061no—
225C2×Q8—532no—
226C2×Q8—1061no—
227C2×Q8—1061no—
228C2×Q8—1061no—
229C2×Q8—1061no—
230C2×Q8—1061no—
231C22×C4—41no—
232C22×C4—1061no—
233C22×C4—1061no—
234C22×C4—2121no—
235C22×C4#1171994511yescharacteristic
236C22×C4D4×D5311yescharacteristic
237C22×C4D4×D5311yescharacteristic
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—532no—
245C2×C8—532no—
246C2×C8—1061no—
247C2×C8—1061no—
248C2×C8—1061no—
249C2×C8—1061no—
250C2×C8—2121no—
251C2×C8—2121no—
252C2×C8—2121no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—81no—
258C4⋊C4—1061no—
259C4⋊C4—1061no—
260C4⋊C4—4241no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C42—2121no—
268C42—2121no—
269C42#1171994511yescharacteristic
270C42#1171994511yescharacteristic
271C42#1171994711yescharacteristic
272C42#1171994911yescharacteristic
273C23#1313319011yescharacteristic
274Q8—1062no—
275Q8—1062no—
276Q8—2121no—
277Q8—2121no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—1062no—
284C2×C4—4241no—
285C2×C4—4241no—
286C2×C4—4241no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4#1313319911yescharacteristic
293C2×C4#1313320211yescharacteristic
294C2×C4#1313320211yescharacteristic
295C2×C4#1313320311yescharacteristic
296C2×C4#1313321111yescharacteristic
297C2×C4#1313321211yescharacteristic
298C2×C4SD16×D5311yescharacteristic
299C2×C4SD16×D5311yescharacteristic
300C2×C4Q16×D5311yescharacteristic
301C2×C4#1313323611yescharacteristic
302C2×C4#1313324111yescharacteristic
303C2×C4#1313324411yescharacteristic
304C2×C4#1313324611yescharacteristic
305C2×C4#1313324911yescharacteristic
306C8—81no—
307C8—81no—
308C8—1062no—
309C8—1062no—
310C8—4241no—
311C22#1681055211yescharacteristic
312C22#1681082211yescharacteristic
313C22#1681083211yescharacteristic
314C22#1681085111yescharacteristic
315C22#1681087211yescharacteristic
316C22#1681088511yescharacteristic
317C22#1681088911yescharacteristic
318C4—21no—
319C4—21no—
320C4—161no—
321C4—8481no—
322C4—4241no—
323C4—2122no—
324C4#1681094311yescharacteristic
325C4#1681094511yescharacteristic
326C4#1681095211yescharacteristic
327C4#1681095511yescharacteristic
328C4#1681096011yescharacteristic
329C4#1681096611yescharacteristic
330C4#1681097211yescharacteristic
331C4#1681098111yescharacteristic
332C2#2027406211yescharacteristic
333C2#2027409611yescharacteristic
334C2#2027419011yescharacteristic
335C2#2027422411yescharacteristic
336C2#2027612711yescharacteristic
337C2#2027615311yescharacteristic
338C2#2027616411yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socle, prime radicalC23×C53#121
hypercenter#1296D53
hypocenterC53#10620
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
192. C53#10620semidirect

Maximal subgroups

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

subgroupindex
#202756502
#202748712
#202747212
#202747172
#202725992
#202725492
C53×#12962
#1062053