Finite Group Databasebeta

← #28299669

Subgroups — #28299669

Order 13568.

640 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290441C211yesmaximal, characteristic, complemented
2#20290378C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289531C212yesmaximal, complemented
5#121×Dic53C211yesmaximal, characteristic
6C22×#11719940—21no—
7C2×C22⋊#88324—21no—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×D4×Dic53—21no—
13C2×D4×Dic53C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810526C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810516—21no—
18#16810503—21no—
19#16810483C411yescharacteristic
20C4⋊C4×Dic53C411yescharacteristic
21C22⋊C4×Dic53—21no—
22C2×#121×C53C411yescharacteristic
23C2×#11719940—24no—
24C2×#11719940—42no—
25C2×#11719940—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28C22⋊#88324—24no—
29#13133272—42no—
30C23×Dic53—21no—
31C23×Dic53D411yescharacteristic
32C23×Dic53D411yescharacteristic
33D4×Dic53—44no—
34D4×Dic53—42no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—42no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53C2311yescharacteristic
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×D4×C53—21no—
52C22×D4×C53C2×C411yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54#121×C53—22no—
55#121×C53—22no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53—21no—
58C24×C53—21no—
59C24×C53C22⋊C411yescharacteristic
60C2×D4×C53—24no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—42no—
65C2×D4×C53—42no—
66C2×D4×C53M1612yescomplemented
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—22no—
79C22×Dic53—22no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—41no—
83C22×Dic53—42no—
84C22×Dic53—42no—
85C22×Dic53—42no—
86C22×Dic53C22⋊C411yescharacteristic
87C22×Dic53C22⋊C411yescharacteristic
88C22×Dic53C2×D411yescharacteristic
89C22×Dic53C2×D411yescharacteristic
90#11719940—48no—
91C2×#88324—24no—
92C2×#88324—24no—
93C2×#88324—42no—
94C2×#88324—42no—
95C2×#88324—42no—
96C2×#88324—42no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53M1612yescomplemented
99C22⋊C4×C53—24no—
100C22⋊Dic53—42no—
101C22⋊Dic53—42no—
102C22⋊Dic53—41no—
103C22⋊Dic53—42no—
104C22⋊Dic53—41no—
105C4⋊Dic53—42no—
106C4⋊Dic53—41no—
107C4⋊Dic53—41no—
108C4×Dic53—22no—
109C4×Dic53—22no—
110C4×Dic53—41no—
111C4×Dic53—41no—
112C4×Dic53—42no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—22no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53#10412yes—
127C23×C53C2×C22⋊C411yescharacteristic
128D4×C53—42no—
129D4×C53—42no—
130D4×C53—48no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—22no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53C2×M1611yescharacteristic
142C2×C4×C53C2×M1611yescharacteristic
143C2×Dic53—24no—
144C2×Dic53—24no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—44no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—84no—
161C2×Dic53#9912yes—
162C2×Dic53#10412yes—
163C2×Dic53AGL(1,ℤ/8)11yescharacteristic
164C2×Dic53AGL(1,ℤ/8)12yes—
165C2×Dic53#13711yescharacteristic
166#88324—44no—
167#88324—44no—
168#7633—531nomaximal, Sylow
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53#40711yescharacteristic
184C22×C53C2×#9911yescharacteristic
185C22×C53#41311yescharacteristic
186C22×C53#41711yescharacteristic
187C22×C53#41811yescharacteristic
188C22×C53C2×#10411yescharacteristic
189C22×C53C2×#10411yescharacteristic
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—41no—
193C4×C53—42no—
194Dic53—21no—
195Dic53—22no—
196Dic53—21no—
197Dic53—22no—
198Dic53—22no—
199Dic53—81no—
200Dic53—81no—
201C4×#121—531no—
202#1749—531no—
203#1728—531no—
204#1271—532no—
205#1270—532no—
206C2×C53—42no—
207C2×C53—82no—
208C2×C53#146511yescharacteristic
209C2×C53#148911yescharacteristic
210C2×C53#149011yescharacteristic
211C2×C53#150411yescharacteristic
212C2×C53#150711yescharacteristic
213C2×C53#152411yescharacteristic
214C2×C53#155711yescharacteristic
215C22×M16—1061no—
216C2×#121#615711yescharacteristic
217C2×C4×D4—531no—
218C2×C4×D4—1061no—
219C22×C42—531no—
220C2×C22⋊C8—1061no—
221C2×#97—1061no—
222#390—531no—
223#389—531no—
224#388—1061no—
225#386—1061no—
226#380—531no—
227C4×C4⋊C4—531no—
228C4×C22⋊C4—1061no—
229#336—532no—
230#336—532no—
231#336—1061no—
232C53#763311yescharacteristic, complemented, Sylow
233C22×D4—21no—
234C22×D4C2×#615711yescharacteristic
235C23×C4—531no—
236C23×C4—531no—
237C23×C4—1061no—
238C23×C4C2×#615711yescharacteristic
239C2×M16—1064no—
240C2×M16—2122no—
241C2×M16—2122no—
242C22×C8—1062no—
243C22×C8—1062no—
244#121—22no—
245#121—22no—
246C4×D4—2122no—
247C4×D4—2124no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4C2×#615711yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C22⋊C4—1061no—
257C2×C42—531no—
258C2×C42—531no—
259C2×C42—531no—
260C2×C42—531no—
261C2×C42—1061no—
262C2×C42—2122no—
263C22⋊C8—1064no—
264#97—2122no—
265#95—1061no—
266#95—1061no—
267#95—1061no—
268C24—21no—
269C24C22⋊#615711yescharacteristic
270C2×D4—24no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4#1171994012yescomplemented
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—1062no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1062no—
290C22×C4—1062no—
291C22×C4—1062no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—2122no—
295C22×C4—2121no—
296C22×C4—2122no—
297C22×C4—2122no—
298C22×C4C22⋊#615711yescharacteristic
299C22×C4C22×#615711yescharacteristic
300M16—2128no—
301C2×C8—1064no—
302C2×C8—1064no—
303C2×C8—2122no—
304C2×C8—2122no—
305C2×C8—2122no—
306C2×C8—2122no—
307C4⋊C4—21no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C4⋊C4—2122no—
311C4⋊C4#1171994012yescomplemented
312C22⋊C4—24no—
313C22⋊C4—2122no—
314C22⋊C4—2122no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C22⋊C4—2122no—
318C42—1062no—
319C42—1062no—
320C42—2121no—
321C42—2121no—
322C42—2122no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#1313328112yes—
337C23C2×C22⋊#615711yescharacteristic
338D4—42no—
339D4—42no—
340D4—48no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—532no—
352C2×C4—531no—
353C2×C4—531no—
354C2×C4—532no—
355C2×C4—532no—
356C2×C4—1064no—
357C2×C4—1064no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—4244no—
367C2×C4—2124no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4C2×#1171994011yescharacteristic
375C2×C4C2×#1171994011yescharacteristic
376C8—2124no—
377C8—2124no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22#1681159311yescharacteristic
393C22D53⋊#9911yescharacteristic
394C22#1681161011yescharacteristic
395C22#1681161411yescharacteristic
396C22#1681161511yescharacteristic
397C22C2×#1313328111yescharacteristic
398C22D53⋊#10411yescharacteristic
399C4—21no—
400C4—21no—
401C4—42no—
402C4—41no—
403C4—1061no—
404C4—1062no—
405C4—1061no—
406C4—1062no—
407C4—1062no—
408C4—4241no—
409C4—4241no—
410C2—42no—
411C2—82no—
412C2#2028997111yescharacteristic
413C2#2029005011yescharacteristic
414C2#2029005411yescharacteristic
415C2#2029010011yescharacteristic
416C2#2029011011yescharacteristic
417C2#2029016611yescharacteristic
418C2#2029024711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#121×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#121#6157
hypocenterC53#7633
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20290441C2semidirect
2. #20290378C2semidirect
3. #20289533C2semidirect
4. #20289531C2semidirect
66. C2×D4×C53M16semidirect
98. C4⋊C4×C53M16semidirect
232. C53#7633semidirect
276. C2×D4#11719940semidirect
311. C4⋊C4#11719940semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202903782
#202895332
#202895312
#121×Dic532
#763353