Finite Group Databasebeta

← #27353500

Subgroups — #27353500

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#11719945C212yesmaximal, complemented
2C4×#13133239C211yesmaximal, characteristic
3#20273540C212yesmaximal, complemented
4#20273535C212yesmaximal, complemented
5#20273529C214yesmaximal, complemented
6#20273463C211yesmaximal, characteristic, complemented
7#20273098C212yesmaximal, complemented
8C2×C53×#322C211yesmaximal, characteristic, complemented
9C2×#13133239C411yescharacteristic
10C4×#11719945C2212yes—
11C8×Dic106C2214yes—
12C4×Dic212C411yescharacteristic
13C4×Dic212C2211yescharacteristic
14C4×#11719913C412yes—
15C4×#11719913C2212yes—
16C4×D424C411yescharacteristic
17C4×D424C2211yescharacteristic
18C8×D212C2214yes—
19C8×C22⋊D53—24no—
20#16810676C2212yes—
21#16810674—22no—
22#16810661C411yescharacteristic
23#16810620C412yes—
24C4×C8×D53—22no—
25#16810376C2212yes—
26#16810374C2212yes—
27#16810355C2214yes—
28#16810352C2214yes—
29C2×C4×C8×C53C2211yescharacteristic
30C2×C4×C8×C53C2211yescharacteristic
31C2×C4×C8×C53C2211yescharacteristic
32C53×#322C2212yes—
33C53×#322C2212yes—
34C2×#11719945C2×C412yes—
35#13133239C818yescomplemented
36C2×Dic212C2×C411yescharacteristic
37C2×#11719913C2×C412yes—
38C2×D424C2×C411yescharacteristic
39C2×C8×D53—24no—
40C4×Dic106C2×C412yes—
41C4×Dic106C2312yes—
42C4×D212C2×C412yes—
43C4×D212C2312yes—
44C4×C22⋊D53—22no—
45C4×C22⋊D53—22no—
46#13133176C2×C412yes—
47#13133174—22no—
48C42×D53—22no—
49#13133151C2×C411yescharacteristic
50#13133151C2×C411yescharacteristic
51#13133149C2×C412yes—
52C4⋊#88322C2314yes—
53#13133146—24no—
54#13133142C2×C412yes—
55#13133142C2×C412yes—
56#13133139C2×C412yes—
57#13133139C2×C412yes—
58C22⋊#88322—22no—
59#13133129—24no—
60C4×#88322—22no—
61C8×Dic53—24no—
62C22×C8×C53—21no—
63C22×C8×C53C2×C411yescharacteristic
64C22×C8×C53C2×C411yescharacteristic
65C22×C8×C53D412yes—
66C2×C42×C53C2311yescharacteristic
67C4×C8×C53—21no—
68C4×C8×C53C2×C411yescharacteristic
69C4×C8×C53C2×C411yescharacteristic
70C4×C8×C53D412yes—
71C4×C8×C53C2312yes—
72C4×C8×C53C2311yescharacteristic
73C4×C8×C53C2311yescharacteristic
74C4×C8×C53C2312yes—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C2×C22⋊D53—22no—
79#11719945—24no—
80#11719945C2×C818yescomplemented
81C2×D212C22×C412yes—
82C2×C4×D53—22no—
83C2×C4×D53—22no—
84C2×C4×D53—22no—
85C2×Dic106C22×C412yes—
86C2×C8×C53—22no—
87C2×C8×C53—22no—
88C2×C8×C53—42no—
89C2×C8×C53—42no—
90C2×C8×C53C2×C814yescomplemented
91C2×C8×C53C22×C411yescharacteristic
92C2×C8×C53C22×C411yescharacteristic
93C2×C8×C53C22×C411yescharacteristic
94C2×C8×C53C22×C411yescharacteristic
95C2×C8×C53C2×D414yes—
96C42×C53C22×C411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53#4111yescharacteristic
99C42×C53C2411yescharacteristic
100C22⋊Dic53—22no—
101#11719921—22no—
102#11719921—22no—
103C4⋊Dic53C22×C412yes—
104C4⋊Dic53C22×C412yes—
105#11719919—22no—
106#11719919—22no—
107C4×Dic53—22no—
108C4×Dic53—22no—
109C4×Dic53—22no—
110C2×#88322—24no—
111Dic212C2×C814yescomplemented
112D424C2×C814yescomplemented
113#11719913C2×C818yescomplemented
114C8×D53—48no—
115C23×C53C4×D411yescharacteristic
116C22×D53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53C4×D411yescharacteristic
120C2×C4×C53C4×D411yescharacteristic
121C2×C4×C53C4×D411yescharacteristic
122C2×C4×C53C22×C812yes—
123C2×C4×C53#13112yes—
124C2×C4×C53C23×C411yescharacteristic
125C2×C4×C53C22×D411yescharacteristic
126C2×C4×C53C2×#4111yescharacteristic
127C22⋊D53—28no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131D212—22no—
132D212C22×C814yes—
133C4×D53—28no—
134C4×D53—44no—
135C4×D53—44no—
136Dic106—22no—
137Dic106C22×C814yes—
138C8×C53—21no—
139C8×C53—21no—
140C8×C53—24no—
141C8×C53—42no—
142C8×C53C22×C812yes—
143C8×C53C22×C812yes—
144#88322—44no—
145#17849—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53C8×D412yes—
149C22×C53C2×C4×D411yescharacteristic
150C2×D53—24no—
151C2×D53—42no—
152C2×D53—42no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53C8×D412yes—
156C4×C53#44412yes—
157C4×C53C23×C811yescharacteristic
158C4×C53C2×#13111yescharacteristic
159C4×C53#57512yes—
160Dic53—24no—
161Dic53—42no—
162Dic53—42no—
163C8×#41—532no—
164C4×#135—531no—
165#1580—532no—
166#1579—532no—
167SD16⋊C8—534no—
168#1564—531no—
169#1476—532no—
170C2×#322D5311yescharacteristic, complemented
171C2×C53—22no—
172C2×C53C2×C8×D411yescharacteristic
173C2×C53#294311yescharacteristic
174C2×C53C2×#44411yescharacteristic
175D53—44no—
176C2×#135—531no—
177C4×#41—532no—
178C8×Q8—534no—
179C4×Q16—531no—
180C4×Q16—531no—
181C4×SD16—532no—
182C4×SD16—532no—
183C4×D8—531no—
184C4×D8—531no—
185C8×D4—534no—
186C8×D4—1064no—
187#431—532no—
188#431—1062no—
189#427—531no—
190#416—532no—
191C2×C4×C8—1062no—
192C2×C4×C8C2×D5311yescharacteristic
193C2×C4×C8C2×D5311yescharacteristic
194C2×C4×C8C2×D5311yescharacteristic
195#335—532no—
196#334—532no—
197Q8⋊C8—534no—
198D4⋊C8—534no—
199#322C2×D5312yes—
200#322C2×D5312yes—
201C53#1784911yescharacteristic, complemented, Sylow
202C2×#41—532no—
203#135—538no—
204C2×Q16—531no—
205C2×SD16—532no—
206C2×D8—531no—
207C22×C8—21no—
208C22×C8—1064no—
209C22×C8C4×D5311yescharacteristic
210C22×C8C4×D5311yescharacteristic
211C22×C8D21212yes—
212C4×Q8—532no—
213C4×Q8—532no—
214C4×D4—532no—
215C4×D4—532no—
216C4×D4—1062no—
217C4×D4—1062no—
218#117—532no—
219#117—1062no—
220C2×C42—1062no—
221C2×C42C22×D5311yescharacteristic
222#107—531no—
223#107—531no—
224#106—532no—
225C4⋊C8—534no—
226C4⋊C8—1064no—
227#103—532no—
228#103—532no—
229#102—532no—
230#102—532no—
231C22⋊C8—1062no—
232C22⋊C8—1064no—
233C4×C8—21no—
234C4×C8—1064no—
235C4×C8—1062no—
236C4×C8C4×D5311yescharacteristic
237C4×C8C4×D5311yescharacteristic
238C4×C8D21212yes—
239C4×C8C22×D5312yes—
240C4×C8C22×D5311yescharacteristic
241C4×C8C22×D5311yescharacteristic
242C4×C8C22×D5312yes—
243#41—538no—
244#41—1064no—
245C2×Q8—532no—
246C2×D4—532no—
247C2×D4—1062no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—1062no—
251C22×C4C2×C4×D5311yescharacteristic
252C22×C4C2×D21211yescharacteristic
253C22×C4#1171994511yescharacteristic
254Q16—534no—
255SD16—538no—
256D8—534no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—1064no—
262C2×C8—2128no—
263C2×C8C8×D5314yescomplemented
264C2×C8C2×C4×D5311yescharacteristic
265C2×C8C2×C4×D5311yescharacteristic
266C2×C8C2×C4×D5311yescharacteristic
267C2×C8C2×C4×D5311yescharacteristic
268C2×C8C2×D21214yes—
269C4⋊C4—532no—
270C4⋊C4—532no—
271C4⋊C4—1062no—
272C4⋊C4—1062no—
273C22⋊C4—1062no—
274C22⋊C4—1062no—
275C22⋊C4—1062no—
276C42—1062no—
277C42—1062no—
278C42—1062no—
279C42C2×C4×D5311yescharacteristic
280C42C2×D21211yescharacteristic
281C42#1171994511yescharacteristic
282C42C23×D5311yescharacteristic
283C23—1062no—
284C23C4×D21211yescharacteristic
285Q8—534no—
286Q8—1062no—
287D4—534no—
288D4—1068no—
289D4—1062no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—1062no—
293C2×C4—1068no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—2124no—
297C2×C4—2124no—
298C2×C4C4×D21211yescharacteristic
299C2×C4C4×D21211yescharacteristic
300C2×C4C4×D21211yescharacteristic
301C2×C4C2×C8×D5312yes—
302C2×C4#1313322312yes—
303C2×C4C22×C4×D5311yescharacteristic
304C2×C4C22×D21211yescharacteristic
305C2×C4C2×#1171994511yescharacteristic
306C8—21no—
307C8—21no—
308C8—24no—
309C8—42no—
310C8—2124no—
311C8C2×C8×D5312yes—
312C8C2×C8×D5312yes—
313C22—21no—
314C22—21no—
315C22—1064no—
316C22—2122no—
317C22—2122no—
318C22C8×D21212yes—
319C22C2×C4×D21211yescharacteristic
320C4—21no—
321C4—21no—
322C4—1064no—
323C4—2122no—
324C4—2122no—
325C4C8×D21212yes—
326C4#1681074612yes—
327C4C22×C8×D5311yescharacteristic
328C4C2×#1313322311yescharacteristic
329C4#1681142312yes—
330C2—22no—
331C2—2124no—
332C2C2×C8×D21211yescharacteristic
333C2#2028351711yescharacteristic
334C2C2×#1681074611yescharacteristic
subgroupG/H
centerC42C2×D212
commutatorC4×C53C23×C8
FrattiniC42C23×D53
FittingC2×C53×#322C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133239C4
cosocleC42×C53C24
hypercenterC2×#322D53
hypocenterC53#17849
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C42 ◁ C2×C4×C8 ◁ C2×#322

Complements

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

NH ≅ G/Ntype
1. C8×#11719945C2semidirect
3. #20273540C2semidirect
4. #20273535C2semidirect
5. #20273529C2semidirect
6. #20273463C2semidirect
7. #20273098C2semidirect
8. C2×C53×#322C2semidirect
35. #13133239C8semidirect
80. #11719945C2×C8semidirect
90. C2×C8×C53C2×C8semidirect
111. Dic212C2×C8semidirect
112. D424C2×C8semidirect
113. #11719913C2×C8semidirect
170. C2×#322D53semidirect
201. C53#17849semidirect
263. C2×C8C8×D53semidirect

Maximal subgroups

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

subgroupindex
C8×#117199452
C4×#131332392
#202735402
#202735352
#202735292
#202734632
#202730982
C2×C53×#3222
#1784953