Finite Group Databasebeta

← #28301903

Subgroups — #28301903

Order 13568.

528 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#20290500C211yesmaximal, characteristic
2C4×#13133284C211yesmaximal, characteristic
3C2×#16811482C212yesmaximal
4#20289593C212yesmaximal
5C4×#13133216C211yesmaximal, characteristic
6C2×#13133284C2212yes—
7C2×#13133284C2212yes—
8C2×#13133272—21no—
9C2×C4×#88324—21no—
10#16811489—21no—
11#16811489—21no—
12#16811482C418yescomplemented
13C2×#13133216C2211yescharacteristic
14C2×C4×Dic106—21no—
15#16810527—21no—
16#16810522C411yescharacteristic
17#16810520—21no—
18#16810519C411yescharacteristic
19#16810492C2211yescharacteristic
20C4×#11719919—21no—
21C4×C4⋊Dic53C411yescharacteristic
22C42×Dic53C2211yescharacteristic
23C4×C4⋊C4×C53C411yescharacteristic
24C22×#88324—22no—
25C22×#88324—22no—
26#13133284—22no—
27#13133284—22no—
28#13133284C2×C414yes—
29#13133284C2×C414yes—
30#13133272—42no—
31C4×#88324—42no—
32C22×Dic106—21no—
33#13133216—22no—
34#13133216C2×C414yes—
35C4×Dic106—44no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53C2311yescharacteristic
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C42×C53—21no—
56C2×C42×C53C2×C411yescharacteristic
57#95×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×Dic106—24no—
70C2×Dic106—42no—
71C2×Dic106—41no—
72C2×Dic106—41no—
73C2×#88324—24no—
74C2×#88324—24no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53C4212yes—
82C42×C53—22no—
83C42×C53—42no—
84C4⋊Dic53—21no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87C4⋊Dic53C4212yes—
88#11719919—24no—
89#11719919—42no—
90#11719919—42no—
91#11719919—42no—
92C4×Dic53—22no—
93C4×Dic53—22no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—41no—
97C4×Dic53—42no—
98C4×Dic53SD1612yes—
99C4×Dic53Q1612yes—
100C4×Dic53C22×C412yes—
101C4×Dic53#4112yes—
102C23×C53C2×C22⋊C411yescharacteristic
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—22no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53#10112yes—
116C2×C4×C53#10312yes—
117C2×C4×C53C2×C4211yescharacteristic
118C2×C4×C53#11711yescharacteristic
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—22no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53#10312yes—
131C2×Dic53C4×D412yes—
132C2×Dic53C4×D412yes—
133C2×Dic53C2×SD1611yescharacteristic
134C2×Dic53C2×Q1611yescharacteristic
135C2×Dic53#13512yes—
136C2×Dic53AGL(1,ℤ/8)12yes—
137C2×Dic53AGL(1,ℤ/8)11yescharacteristic
138C2×Dic53#13711yescharacteristic
139Dic106—44no—
140Dic106—44no—
141#88324—44no—
142#88324—44no—
143#8237—531nomaximal, Sylow
144C22×C53C4×C22⋊C411yescharacteristic
145C22×C53C2×#10111yescharacteristic
146C22×C53#41311yescharacteristic
147C22×C53C2×#10311yescharacteristic
148C22×C53#41611yescharacteristic
149C22×C53#41711yescharacteristic
150C22×C53#41811yescharacteristic
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—42no—
156C4×C53—81no—
157Dic53—22no—
158Dic53—41no—
159Dic53—42no—
160Dic53—81no—
161Dic53C4×SD1612yes—
162Dic53C4×Q1612yes—
163Dic53#44012yes—
164Dic53#44212yes—
165C4×C4⋊Q8—531no—
166#1767—531no—
167C4×C4⋊C8—531no—
168C2×#332—532no—
169#1296—532no—
170C2×C53#154911yescharacteristic
171C2×C53#155011yescharacteristic
172C2×C53#155211yescharacteristic
173C2×C53#155311yescharacteristic
174C2×C53C4×#10111yescharacteristic
175C2×C53C4×#10311yescharacteristic
176C2×C53#176211yescharacteristic
177C2×C4⋊Q8—531no—
178C2×C4×Q8—1061no—
179C2×C4⋊C8—532no—
180C2×C4⋊C8—532no—
181C2×#97—1061no—
182C2×C4×C8—1061no—
183#391—1061no—
184#389—531no—
185#389—531no—
186#389—1061no—
187#384—531no—
188C4×C4⋊C4—531no—
189C4×C4⋊C4—1061no—
190C4×C4⋊C4#615711yescharacteristic
191C43—531no—
192#336—1061no—
193#336—1061no—
194#332—538no—
195C53#823711yescharacteristic, complemented, Sylow
196C22×Q8—1061no—
197C22×C8—1062no—
198C22×C8—1062no—
199C4⋊Q8—534no—
200C4⋊Q8—1062no—
201C4×Q8—2124no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4C2×#615711yescharacteristic
210C2×C4⋊C4C2×#615711yescharacteristic
211C2×C42—21no—
212C2×C42—531no—
213C2×C42—531no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—1061no—
219C2×C42C2×#615711yescharacteristic
220C4⋊C8—534no—
221C4⋊C8—534no—
222C4⋊C8—1062no—
223C4⋊C8—1062no—
224#97—2122no—
225C4×C8—2122no—
226#95—21no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230C2×Q8—1064no—
231C2×Q8—2122no—
232C2×Q8—2121no—
233C2×Q8—2121no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4C22⋊#615711yescharacteristic
243C22×C4C22⋊#615711yescharacteristic
244C22×C4C22×#615711yescharacteristic
245C2×C8—1064no—
246C2×C8—1064no—
247C2×C8—2122no—
248C2×C8—2122no—
249C2×C8—2122no—
250C2×C8—2122no—
251C4⋊C4—21no—
252C4⋊C4—41no—
253C4⋊C4—532no—
254C4⋊C4—1064no—
255C4⋊C4—1061no—
256C4⋊C4—2121no—
257C4⋊C4—2122no—
258C4⋊C4—2122no—
259C4⋊C4—2122no—
260C4⋊C4—2122no—
261C4⋊C4C4×#615712yes—
262C42—22no—
263C42—42no—
264C42—532no—
265C42—532no—
266C42—532no—
267C42—532no—
268C42—1061no—
269C42—1062no—
270C42—1062no—
271C42—1061no—
272C42—2121no—
273C42—2122no—
274C23C2×C22⋊#615711yescharacteristic
275Q8—2124no—
276Q8—2124no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—531no—
292C2×C4—531no—
293C2×C4—532no—
294C2×C4—531no—
295C2×C4—531no—
296C2×C4—532no—
297C2×C4—532no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1062no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4#1313327812yes—
310C2×C4#1313328012yes—
311C2×C4C2×C4×#615711yescharacteristic
312C2×C4#1313329611yescharacteristic
313C8—2124no—
314C8—2124no—
315C22C4×C22⋊#615711yescharacteristic
316C22C2×#1313327811yescharacteristic
317C22#1681160911yescharacteristic
318C22C2×#1313328011yescharacteristic
319C22#1681161311yescharacteristic
320C22#1681161411yescharacteristic
321C22#1681161511yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—81no—
328C4—532no—
329C4—532no—
330C4—532no—
331C4—532no—
332C4—1062no—
333C4—4241no—
334C4—2122no—
335C4—2121no—
336C2#2029023011yescharacteristic
337C2#2029023211yescharacteristic
338C2#2029023611yescharacteristic
339C2#2029023811yescharacteristic
340C2C4×#1313327811yescharacteristic
341C2C4×#1313328011yescharacteristic
342C2#2029048511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#8237
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. #16811482C4semidirect
195. C53#8237semidirect

Maximal subgroups

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

subgroupindex
#202905002
C4×#131332842
C2×#168114822
#202895932
C4×#131332162
#823753