Finite Group Databasebeta

← #27277458

Subgroups — #27277458

Order 13568.

496 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277113C211yesmaximal, characteristic
2C4×C22⋊#88322C211yesmaximal, characteristic
3#20272554C212yesmaximal, complemented
4#20272554C212yesmaximal, complemented
5C53×#1797C211yesmaximal, characteristic
6C2×C22⋊#88322—21no—
7C2×C4×#88322C2212yes—
8#16810491—21no—
9#16810489C411yescharacteristic
10#16810489C411yescharacteristic
11#16810486—21no—
12#16810483C411yescharacteristic
13#16810483C411yescharacteristic
14C4×C4⋊Dic53C2211yescharacteristic
15C4×C4⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53—21no—
17#16810469—21no—
18#16810378—21no—
19C22×C42×C53C2211yescharacteristic
20C2×C4⋊C8×C53C2212yes—
21C2×C22⋊C8×C53—21no—
22C53×#336—21no—
23C22×#88322—22no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C22⋊Dic53—21no—
29C2×C22⋊Dic53—21no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C22⋊#88322—24no—
33C4×#88322—24no—
34#13133124—21no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C22×C8×C53—22no—
46C2×C42×C53—24no—
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C2×C42×C53C2311yescharacteristic
51C4⋊C8×C53—24no—
52C22⋊C8×C53—42no—
53C24×C53C22⋊C411yescharacteristic
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—41no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C2×C8×C53—42no—
75C2×C8×C53—42no—
76C2×C8×C53—42no—
77C42×C53—22no—
78C42×C53—22no—
79C42×C53—22no—
80C42×C53#4114yes—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C4⋊Dic53—22no—
84C4⋊Dic53—22no—
85C4⋊Dic53—22no—
86C4⋊Dic53—22no—
87C4×Dic53—42no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C2×#88322—24no—
91C2×#88322—42no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53#10412yes—
100C23×C53C2×C22⋊C411yescharacteristic
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—44no—
113C2×C4×C53—44no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53#11712yes—
117C2×C4×C53#12312yes—
118C2×C4×C53#12312yes—
119C2×C4×C53#13112yes—
120C2×C4×C53#13112yes—
121C2×C4×C53#13512yes—
122C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C53#13711yescharacteristic
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C8×C53—82no—
137#88322—44no—
138#8711—531nomaximal, Sylow
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53#38111yescharacteristic
147C22×C53#40811yescharacteristic
148C22×C53#40811yescharacteristic
149C22×C53#41611yescharacteristic
150C22×C53#41711yescharacteristic
151C22×C53C2×#10411yescharacteristic
152C22×C53#42111yescharacteristic
153C4×C53—22no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53#43314yes—
158C4×C53#48212yes—
159C4×C53#48512yes—
160Dic53—81no—
161Dic53—81no—
162Dic53—81no—
163Dic53—81no—
164#2283—531no—
165#1797D5311yescharacteristic
166C4×C22⋊C8—531no—
167#1279—532no—
168#1279—532no—
169C2×C53—42no—
170C2×C53#148211yescharacteristic
171C2×C53#148311yescharacteristic
172C2×C53#149411yescharacteristic
173C2×C53#149511yescharacteristic
174C2×C53#181811yescharacteristic
175C2×C53#182511yescharacteristic
176C2×C53#182611yescharacteristic
177C22×C42C2×D5311yescharacteristic
178C2×C4⋊C8C2×D5312yes—
179C2×C22⋊C8—21no—
180C2×C22⋊C8—1061no—
181C2×C4×C8—532no—
182#383—1061no—
183#382—531no—
184#382—531no—
185#381—1061no—
186#380—531no—
187#380—531no—
188C4×C4⋊C4—531no—
189C4×C4⋊C4—531no—
190C4×C22⋊C4—1061no—
191#376—1061no—
192#336—21no—
193#336—1061no—
194C53#871111yescharacteristic, complemented, Sylow
195C23×C4C4×D5311yescharacteristic
196C23×C4D21211yescharacteristic
197C23×C4C22⋊D5311yescharacteristic
198C22×C8—22no—
199C22×C8—1062no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C22⋊C4—1061no—
205C2×C22⋊C4—1061no—
206C2×C42—24no—
207C2×C42—1061no—
208C2×C42—1061no—
209C2×C42C4×D5311yescharacteristic
210C2×C42D21211yescharacteristic
211C2×C42C22⋊D5311yescharacteristic
212C2×C42C22×D5311yescharacteristic
213C4⋊C8—24no—
214C22⋊C8—42no—
215C22⋊C8—1064no—
216C4×C8—1064no—
217#95—1061no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225C24#1171992111yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—41no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992111yescharacteristic
243C22×C4C2×C4×D5311yescharacteristic
244C22×C4C2×D21211yescharacteristic
245C22×C4C2×C22⋊D5311yescharacteristic
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—1064no—
250C2×C8—2122no—
251C2×C8—2122no—
252C4⋊C4—1062no—
253C4⋊C4—1062no—
254C4⋊C4—1062no—
255C4⋊C4—1062no—
256C22⋊C4—2122no—
257C22⋊C4—2122no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—2122no—
262C42—2121no—
263C42—2121no—
264C42#1171994714yes—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23#1313314312yes—
272C23C2×#1171992111yescharacteristic
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4#1313317312yes—
301C2×C4#1313319812yes—
302C2×C4#1313320112yes—
303C2×C4#1313322312yes—
304C2×C4#1313322512yes—
305C2×C4#1313323712yes—
306C2×C4#1313324311yescharacteristic
307C2×C4#1313324811yescharacteristic
308C8—82no—
309C8—2124no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22#1681048411yescharacteristic
318C22#1681058911yescharacteristic
319C22#1681059111yescharacteristic
320C22#1681061711yescharacteristic
321C22#1681062211yescharacteristic
322C22C2×#1313314311yescharacteristic
323C22#1681063911yescharacteristic
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—4241no—
329C4—4241no—
330C4—4241no—
331C4—4241no—
332C4#1681068114yes—
333C4#1681094412yes—
334C4#1681096112yes—
335C2—42no—
336C2#2027312111yescharacteristic
337C2#2027312611yescharacteristic
338C2#2027317311yescharacteristic
339C2#2027317811yescharacteristic
340C2#2027478411yescharacteristic
341C2#2027480711yescharacteristic
342C2#2027481211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1797C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1797D53
hypocenterC53#8711
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20272554C2semidirect
4. #20272554C2semidirect
194. C53#8711semidirect

Maximal subgroups

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

subgroupindex
#202771132
C4×C22⋊#883222
#202725542
C53×#17972
#871153