Finite Group Databasebeta

← #27272109

Subgroups — #27272109

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274783C211yesmaximal, characteristic
2C2×#16810378C211yesmaximal, characteristic
3#20272538C212yesmaximal, complemented
4#20272538C212yesmaximal, complemented
5C53×#2296C211yesmaximal, characteristic
6C23×#88322—21no—
7C2×C22⋊#88322—21no—
8#16810378—21no—
9#16810378—21no—
10#16810378C2212yes—
11#16810378C2212yes—
12C2×#117×C53C411yescharacteristic
13C2×#117×C53C411yescharacteristic
14C22×C42×C53C2211yescharacteristic
15C53×#389C411yescharacteristic
16C53×#389C411yescharacteristic
17C53×#389C2211yescharacteristic
18C53×#389C2211yescharacteristic
19C53×#385—21no—
20C53×#385—21no—
21C4×C4⋊C4×C53—21no—
22C4×C4⋊C4×C53—21no—
23C22×#88322—24no—
24C22×#88322—22no—
25C22×#88322—22no—
26C22×#88322—42no—
27C22×#88322—42no—
28C22⋊#88322—42no—
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C23×C4×C53D411yescharacteristic
32#117×C53—22no—
33#117×C53—42no—
34#117×C53C814yescomplemented
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C42×C53—21no—
46C2×C42×C53—21no—
47C2×C42×C53—42no—
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53C2311yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56C24×C53C22⋊C411yescharacteristic
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—22no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—41no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C2×C812yes—
74C22×C4×C53M1612yes—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53C2×C812yescomplemented
88C4⋊C4×C53C2×C812yescomplemented
89C22⋊C4×C53—24no—
90C22⋊C4×C53—42no—
91C42×C53—24no—
92C42×C53—41no—
93C42×C53—41no—
94C42×C53—41no—
95C42×C53—41no—
96C42×C53—42no—
97C2×#88322—24no—
98C2×#88322—48no—
99C2×#88322—42no—
100C2×#88322—42no—
101C2×#88322—42no—
102C2×#88322—42no—
103C2×#88322—42no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53C22⋊C812yes—
109C23×C53#9912yes—
110C23×C53#10112yes—
111C23×C53C2×C22⋊C411yescharacteristic
112C2×C4×C53—24no—
113C2×C4×C53—24no—
114C2×C4×C53—24no—
115C2×C4×C53—24no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—44no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53C22⋊C812yes—
143C2×C4×C53#10412yes—
144C2×C4×C53C22×C811yescharacteristic
145C2×C4×C53C2×M1611yescharacteristic
146C2×C4×C53#13512yes—
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53#13711yescharacteristic
149#88322—44no—
150#88322—82no—
151#7579—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53#32414yes—
159C22×C53C2×C22⋊C811yescharacteristic
160C22×C53C2×#9911yescharacteristic
161C22×C53C2×#10111yescharacteristic
162C22×C53#41611yescharacteristic
163C22×C53#41711yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C22×C53#42111yescharacteristic
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—22no—
171C4×C53—21no—
172C4×C53—22no—
173C4×C53—42no—
174C4×C53—42no—
175C4×C53—41no—
176C4×C53—81no—
177C4×C53—81no—
178#2296D5311yescharacteristic
179#1817—531no—
180C2×#336—531no—
181#1271—532no—
182#1271—532no—
183C2×C53—24no—
184C2×C53C2×#32411yescharacteristic
185C2×C53#147611yescharacteristic
186C2×C53#147711yescharacteristic
187C2×C53#150411yescharacteristic
188C2×C53#151611yescharacteristic
189C2×C53#153111yescharacteristic
190C2×C53#155211yescharacteristic
191C23×C8—1061no—
192C2×#117Dic5311yescharacteristic
193C2×#117Dic5311yescharacteristic
194C22×C42C2×D5311yescharacteristic
195C2×C22⋊C8—1061no—
196#389Dic5311yescharacteristic
197#389Dic5311yescharacteristic
198#389C2×D5311yescharacteristic
199#389C2×D5311yescharacteristic
200#385—21no—
201#385—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204#336—532no—
205#336—532no—
206#336—1061no—
207#336—1061no—
208C53#757911yescharacteristic, complemented, Sylow
209C23×C4C2×Dic5311yescharacteristic
210C23×C4C22⋊D5311yescharacteristic
211C23×C4C22⋊D5311yescharacteristic
212C22×C8—1064no—
213C22×C8—1062no—
214C22×C8—1062no—
215C22×C8—2122no—
216C22×C8—2122no—
217#117—22no—
218#117—42no—
219#117#8832214yescomplemented
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C2×Dic5311yescharacteristic
225C2×C4⋊C4C2×Dic5311yescharacteristic
226C2×C4⋊C4C2×Dic5311yescharacteristic
227C2×C4⋊C4C2×Dic5311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—42no—
233C2×C42C2×Dic5311yescharacteristic
234C2×C42C22⋊D5311yescharacteristic
235C2×C42C22⋊D5311yescharacteristic
236C2×C42C22×D5311yescharacteristic
237C22⋊C8—2122no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242C24C22⋊Dic5311yescharacteristic
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4C2×#8832212yes—
257C22×C4#1171991712yes—
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22⋊Dic5311yescharacteristic
260C22×C4C22⋊Dic5311yescharacteristic
261C22×C4C22×Dic5311yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263C22×C4C2×C22⋊D5311yescharacteristic
264C2×C8—1064no—
265C2×C8—2122no—
266C2×C8—2122no—
267C2×C8—2122no—
268C2×C8—2128no—
269C2×C8—2122no—
270C2×C8—2122no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4C2×#8832212yescomplemented
281C4⋊C4C2×#8832212yescomplemented
282C22⋊C4—24no—
283C22⋊C4—42no—
284C42—24no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—41no—
289C42—42no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23C22⋊#8832212yes—
295C23#1313313212yes—
296C23#1313313612yes—
297C23C2×C22⋊Dic5311yescharacteristic
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—44no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4C22⋊#8832212yes—
329C2×C4#1313314412yes—
330C2×C4C22×#8832211yescharacteristic
331C2×C4C2×#1171991711yescharacteristic
332C2×C4#1313323712yes—
333C2×C4#1313324211yescharacteristic
334C2×C4#1313324711yescharacteristic
335C8—4242no—
336C8—2124no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#1681034914yes—
344C22C2×C22⋊#8832211yescharacteristic
345C22C2×#1313313211yescharacteristic
346C22C2×#1313313611yescharacteristic
347C22#1681061911yescharacteristic
348C22#1681062411yescharacteristic
349C22C2×#1313314411yescharacteristic
350C22#1681064011yescharacteristic
351C4—21no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—42no—
360C4—42no—
361C4—81no—
362C4—81no—
363C2—24no—
364C2C2×#1681034911yescharacteristic
365C2#2027309711yescharacteristic
366C2#2027310211yescharacteristic
367C2#2027321911yescharacteristic
368C2#2027327111yescharacteristic
369C2#2027333011yescharacteristic
370C2#2027341511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#2296C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2296D53
hypocenterC53#7579
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
3. #20272538C2semidirect
4. #20272538C2semidirect
34. #117×C53C8semidirect
87. C4⋊C4×C53C2×C8semidirect
88. C4⋊C4×C53C2×C8semidirect
208. C53#7579semidirect
219. #117#88322semidirect
280. C4⋊C4C2×#88322semidirect
281. C4⋊C4C2×#88322semidirect

Maximal subgroups

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

subgroupindex
#202747832
C2×#168103782
#202725382
C53×#22962
#757953