Finite Group Databasebeta

← #27302406

Subgroups — #27302406

Order 13568.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810418C214yesmaximal
2#20274703C211yesmaximal, characteristic
3C4×#13133132C211yesmaximal, characteristic
4C53×#2547C211yesmaximal, characteristic
5C2×#13133132C2212yes—
6C2×#13133132C2212yes—
7#16810483—21no—
8C4×C22⋊Dic53—21no—
9#16810418—28no—
10#16810390—21no—
11#16810390—21no—
12C2×C4⋊Q8×C53C411yescharacteristic, complemented
13C2×#124×C53C2212yes—
14C2×C4×D4×C53C2211yescharacteristic
15C53×#397—21no—
16C53×#397—21no—
17C53×#391C412yescomplemented
18C53×#380—21no—
19C53×#380—21no—
20C4×C4⋊C4×C53C411yescharacteristic, complemented
21C2×C4⋊Dic53—41no—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—41no—
25#13133132—24no—
26#13133132—22no—
27#13133132D414yes—
28#13133124—41no—
29#13133124—41no—
30C22×Q8×C53C2×C412yes—
31C22×D4×C53C2311yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C4⋊Q8×C53—41no—
35C4⋊Q8×C53—41no—
36#124×C53—24no—
37#124×C53—24no—
38C4×D4×C53—41no—
39C4×D4×C53—41no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C42×C53C2×C412yes—
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—21no—
51#95×C53—22no—
52#95×C53—22no—
53C24×C53C2×D411yescharacteristic
54C24×C53C2×D411yescharacteristic
55C2×Q8×C53—24no—
56C2×Q8×C53—42no—
57C2×Q8×C53—42no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53C2×D412yes—
61C2×D4×C53#4112yes—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×Dic53—42no—
76C22×Dic53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—44no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—24no—
90C22⋊Dic53—24no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C4⋊Dic53—82no—
95C4×Dic53—82no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53#12112yes—
106C23×C53#12411yescharacteristic
107C23×C53#12711yescharacteristic
108Q8×C53—84no—
109D4×C53—42no—
110D4×C53—42no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53#9911yescharacteristic
131C2×C4×C53#9911yescharacteristic
132C2×C4×C53C4×D412yes—
133C2×Dic53—44no—
134C2×Dic53—82no—
135C2×Dic53—82no—
136C2×Dic53—82no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139#13224—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—22no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—42no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—81no—
153C22×C53—81no—
154C22×C53#39011yescharacteristic
155C22×C53C2×#9911yescharacteristic
156C22×C53#41011yescharacteristic
157C4×C53—21no—
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—82no—
163C4×C53—82no—
164C4×C53#35412yescomplemented
165Dic53—84no—
166Dic53—162no—
167#2547D5311yescharacteristic
168C2×#354—534no—
169#1799—531no—
170C4×#99—531no—
171C2×C53—21no—
172C2×C53—21no—
173C2×C53—42no—
174C2×C53—41no—
175C2×C53—41no—
176C2×C53#196411yescharacteristic
177C2×C53C2×#35411yescharacteristic
178C2×C53#212911yescharacteristic
179C2×C4⋊Q8Dic5311yescharacteristic, complemented
180C2×#124C2×D5312yes—
181C2×C4×D4C2×D5311yescharacteristic
182C2×#99—532no—
183C2×#99—532no—
184#397—21no—
185#397—21no—
186#391Dic5312yescomplemented
187#380—21no—
188#380—21no—
189#380—1061no—
190C4×C4⋊C4Dic5311yescharacteristic, complemented
191C4×C22⋊C4—1061no—
192#354—1068no—
193#342—1061no—
194#342—1061no—
195C53#1322411yescharacteristic, complemented, Sylow
196C22×Q8C2×Dic5312yes—
197C22×D4C22×D5311yescharacteristic
198C23×C4C22⋊D5311yescharacteristic
199C23×C4C22⋊D5311yescharacteristic
200C4⋊Q8—41no—
201C4⋊Q8—41no—
202#124—24no—
203#124—24no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—2121no—
210C2×C4⋊C4C2×Dic5311yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1062no—
214C2×C22⋊C4—1062no—
215C2×C22⋊C4C22⋊D5311yescharacteristic
216C2×C22⋊C4C22⋊D5311yescharacteristic
217C2×C42—2121no—
218C2×C42C2×Dic5312yes—
219C2×C42C2×Dic5311yescharacteristic
220#99—534no—
221#99—1062no—
222#99—1064no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—2121no—
227#95—2121no—
228C24C2×C22⋊D5311yescharacteristic
229C24C2×C22⋊D5311yescharacteristic
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4D4×D5312yes—
236C2×D4#1171994712yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—2122no—
246C22×C4—2122no—
247C22×C4C22⋊Dic5311yescharacteristic
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22⋊Dic5311yescharacteristic
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22×Dic5311yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—4242no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—1064no—
264C22⋊C4—2122no—
265C22⋊C4—2122no—
266C22⋊C4—2122no—
267C42—21no—
268C42—21no—
269C42—24no—
270C42—4242no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×C22⋊Dic5311yescharacteristic
280C23#1313318812yes—
281C23#1313320411yescharacteristic
282C23#1313321311yescharacteristic
283Q8—84no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—4242no—
306C2×C4—4242no—
307C2×C4—4242no—
308C2×C4—4242no—
309C2×C4—4242no—
310C2×C4—2124no—
311C2×C4#1313313211yescharacteristic
312C2×C4#1313313211yescharacteristic
313C2×C4D4×Dic5312yes—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—21no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22#1681052611yescharacteristic
329C22C2×#1313313211yescharacteristic
330C22#1681059711yescharacteristic
331C4—22no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—82no—
337C4—82no—
338C4—8482no—
339C4—4244no—
340C4#1681041812yescomplemented
341C2—21no—
342C2—21no—
343C2—41no—
344C2—42no—
345C2—41no—
346C2#2027555211yescharacteristic
347C2C2×#1681041811yescharacteristic
348C2#2027639011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2547C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#2547D53
hypocenterC53#13224
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
12. C2×C4⋊Q8×C53C4semidirect
17. C53×#391C4semidirect
20. C4×C4⋊C4×C53C4semidirect
164. C4×C53#354semidirect
179. C2×C4⋊Q8Dic53semidirect
186. #391Dic53semidirect
190. C4×C4⋊C4Dic53semidirect
195. C53#13224semidirect
340. C4#16810418semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
#202747032
C4×#131331322
C53×#25472
#1322453