Finite Group Databasebeta

← #28305738

Subgroups — #28305738

Order 13568.

920 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 126 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
1#20290571C211yesmaximal, characteristic, complemented
2#20290441C211yesmaximal, characteristic, complemented
3C2×#16811466C214yesmaximal, complemented
4C2×#16810526C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×C22⋊#88324C2212yes—
7C2×C22⋊#88324C2212yes—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489—21no—
11#16811466—28no—
12C22×C4⋊Dic53C411yescharacteristic
13C22×C22⋊Dic53C412yes—
14C22×C22⋊Dic53C2212yes—
15C22×C4×Dic53C2211yescharacteristic
16#16810526—21no—
17#16810526—21no—
18#16810526—21no—
19#16810526—21no—
20C23×D4×C53C411yescharacteristic
21C22×#88324—22no—
22C22×#88324—22no—
23#13133284—42no—
24C22⋊#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324D414yescomplemented
27#13133272—42no—
28C23×Dic53C2×C411yescharacteristic
29C23×Dic53C2×C412yes—
30C23×Dic53C2311yescharacteristic
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—42no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53—24no—
39C2×C22⋊Dic53—24no—
40C2×C22⋊Dic53—44no—
41C2×C4×Dic53—42no—
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C25×C53C2×C412yes—
47C22×D4×C53—24no—
48C22×D4×C53—21no—
49C22×D4×C53—21no—
50C22×D4×C53—42no—
51C22×D4×C53—42no—
52C23×C4×C53C2×C411yescharacteristic
53C24×C53—22no—
54C24×C53—22no—
55C24×C53—24no—
56C24×C53—24no—
57C24×C53—42no—
58C24×C53—42no—
59C24×C53—42no—
60C24×C53—42no—
61C24×C53M1614yes—
62C24×C53C22×C411yescharacteristic
63C2×D4×C53—24no—
64C2×D4×C53—24no—
65C2×D4×C53—416no—
66C2×D4×C53—42no—
67C2×D4×C53—42no—
68C2×D4×C53—42no—
69C2×D4×C53—42no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×Dic53—22no—
76C22×Dic53—22no—
77C22×Dic53—22no—
78C22×Dic53—24no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—42no—
83C22×Dic53—42no—
84C22×Dic53C22⋊C411yescharacteristic
85C22×Dic53C22⋊C411yescharacteristic
86C22×Dic53C2×D411yescharacteristic
87C22×Dic53C2×D411yescharacteristic
88C22×Dic53C2×D412yes—
89C22×Dic53#4112yes—
90C2×#88324—24no—
91C2×#88324—42no—
92C2×#88324—42no—
93C2×#88324—42no—
94C2×#88324—42no—
95C2×#88324—42no—
96C22⋊Dic53—44no—
97C22⋊Dic53—44no—
98C22⋊Dic53—44no—
99C22⋊Dic53—42no—
100C22⋊Dic53—42no—
101C4⋊Dic53—42no—
102C4⋊Dic53—42no—
103C4×Dic53—22no—
104C4×Dic53—22no—
105C4×Dic53—41no—
106C4×Dic53—41no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—24no—
112C23×C53—28no—
113C23×C53—44no—
114C23×C53—44no—
115C23×C53—44no—
116C23×C53—44no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—44no—
124C23×C53—84no—
125C23×C53—84no—
126C23×C53—84no—
127C23×C53—84no—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53C4×D412yes—
130C23×C53C2×M1612yes—
131C23×C53#13112yes—
132D4×C53—48no—
133D4×C53—48no—
134C2×C4×C53—28no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53#9912yes—
140C2×C4×C53#10012yes—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—44no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—84no—
157C2×Dic53#10012yes—
158C2×Dic53#12112yes—
159C2×Dic53#12411yescharacteristic
160C2×Dic53#12711yescharacteristic
161#88324—44no—
162#88324—82no—
163#9653—531nomaximal, Sylow
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—44no—
171C22×C53—44no—
172C22×C53—44no—
173C22×C53—88no—
174C22×C53—88no—
175C22×C53—82no—
176C22×C53—82no—
177C22×C53—82no—
178C22×C53—82no—
179C22×C53#39011yescharacteristic
180C22×C53#40711yescharacteristic
181C22×C53#40811yescharacteristic
182C22×C53C2×#9911yescharacteristic
183C22×C53#41011yescharacteristic
184C22×C53C2×#10011yescharacteristic
185C22×C53C2×#10011yescharacteristic
186C22×C53#43514yes—
187C4×C53—22no—
188C4×C53—22no—
189Dic53—22no—
190Dic53—21no—
191Dic53—21no—
192Dic53—22no—
193Dic53—41no—
194Dic53—82no—
195C2×#390—531no—
196#1797—531no—
197#1749—531no—
198C2×#323—534no—
199C2×C53—24no—
200C2×C53—84no—
201C2×C53#146211yescharacteristic
202C2×C53#146611yescharacteristic
203C2×C53#146911yescharacteristic
204C2×C53#147111yescharacteristic
205C2×C53#195511yescharacteristic
206C2×C53#196411yescharacteristic
207C2×C53#196511yescharacteristic
208C23×D4#615711yescharacteristic
209C22×C4⋊C4—531no—
210C22×C22⋊C4—532no—
211C22×C22⋊C4—532no—
212C22×C42—531no—
213C2×C4⋊C8—1061no—
214C2×C22⋊C8—532no—
215C2×C22⋊C8—532no—
216C2×#97—1061no—
217#390—1061no—
218#390—1061no—
219#390—1061no—
220#390—1061no—
221#336—1061no—
222#336—1061no—
223#323—1068no—
224C53#965311yescharacteristic, complemented, Sylow
225C25C2×#615712yes—
226C22×D4—24no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—42no—
230C22×D4—42no—
231C23×C4—531no—
232C23×C4—531no—
233C23×C4—532no—
234C23×C4C2×#615711yescharacteristic
235C22×C8—1062no—
236C22×C8—1062no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4—1061no—
239C2×C4⋊C4—2122no—
240C2×C22⋊C4—1064no—
241C2×C22⋊C4—1064no—
242C2×C22⋊C4—1062no—
243C2×C22⋊C4—1062no—
244C2×C22⋊C4—1062no—
245C2×C22⋊C4—1062no—
246C2×C22⋊C4—2124no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—2122no—
252C4⋊C8—2122no—
253C22⋊C8—534no—
254C22⋊C8—1062no—
255C22⋊C8—1064no—
256#97—2122no—
257C24—22no—
258C24—22no—
259C24—24no—
260C24—24no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24#1171994014yes—
266C24C22×#615711yescharacteristic
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—416no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—532no—
281C22×C4—532no—
282C22×C4—531no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—1064no—
286C22×C4—1062no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4—2122no—
291C22×C4—2122no—
292C22×C4C22⋊#615711yescharacteristic
293C22×C4C22⋊#615711yescharacteristic
294C2×C8—1064no—
295C2×C8—2122no—
296C2×C8—2122no—
297C2×C8—2122no—
298C2×C8—2122no—
299C2×C8—2122no—
300C4⋊C4—2122no—
301C4⋊C4—2122no—
302C22⋊C4—2124no—
303C22⋊C4—2122no—
304C22⋊C4—2122no—
305C22⋊C4—2124no—
306C22⋊C4—2124no—
307C42—1062no—
308C42—1062no—
309C42—2121no—
310C42—2121no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—24no—
316C23—28no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—44no—
328C23—84no—
329C23—84no—
330C23—84no—
331C23—84no—
332C23C2×C22⋊#615711yescharacteristic
333C23D4×#615712yes—
334C23C2×#1171994012yes—
335C23#1313330412yes—
336D4—48no—
337D4—48no—
338C2×C4—28no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—532no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—532no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—4244no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2124no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4#1313327512yes—
364C2×C4#1313327712yes—
365C8—4242no—
366C8—2124no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—44no—
374C22—44no—
375C22—44no—
376C22—88no—
377C22—88no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22#1681157511yescharacteristic
383C22#1681159311yescharacteristic
384C22#1681159711yescharacteristic
385C22C2×#1313327511yescharacteristic
386C22#1681160311yescharacteristic
387C22C2×#1313327711yescharacteristic
388C22D53⋊#10011yescharacteristic
389C22#1681165414yes—
390C4—22no—
391C4—22no—
392C4—1061no—
393C4—1062no—
394C4—1061no—
395C4—1062no—
396C4—4242no—
397C4—2121no—
398C2—24no—
399C2—84no—
400C2#2028996011yescharacteristic
401C2#2028997511yescharacteristic
402C2#2028998511yescharacteristic
403C2#2028999011yescharacteristic
404C2#2029083911yescharacteristic
405C2#2029085411yescharacteristic
406C2#2029085711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
Fitting, prime radicalC23×D4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC23×D4#6157
hypocenterC53#9653
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #20290571C2semidirect
2. #20290441C2semidirect
3. C2×#16811466C2semidirect
26. C22⋊#88324D4semidirect
224. C53#9653semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202904412
C2×#168114662
C2×#168105262
#965353