Finite Group Databasebeta

← #27272391

Subgroups — #27272391

Order 13568.

490 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 101 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#20277196C211yesmaximal, characteristic
2#20274782C211yesmaximal, characteristic
3#20274782C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272537C211yesmaximal, characteristic, complemented
7C53×#1271C211yesmaximal, characteristic, complemented
8C2×#13133175C411yescharacteristic
9C2×#13133175C411yescharacteristic
10C2×C42×D53C2211yescharacteristic
11C2×#13133129—21no—
12C2×#13133129—21no—
13#16810522C411yescharacteristic
14#16810521C2211yescharacteristic
15#16810519C411yescharacteristic
16#16810497—21no—
17#16810494—21no—
18C4⋊C4×Dic53—21no—
19#16810474—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22#16810377—21no—
23#16810377—21no—
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×#88322—21no—
31C22×#88322—21no—
32#13133175—22no—
33#13133175—22no—
34#13133175—42no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C42×D53—42no—
48#13133129—42no—
49#13133129—42no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C22×C8×C53—21no—
54C22×C8×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C42×C53C2311yescharacteristic
59#95×C53—21no—
60C23×D53C22⋊C411yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C2×C4×D53—22no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53M1612yes—
94C42×C53—41no—
95#11719921—24no—
96#11719921—42no—
97C4⋊Dic53—21no—
98C4⋊Dic53—42no—
99C4⋊Dic53—41no—
100C4⋊Dic53M1612yes—
101#11719919—42no—
102#11719919—41no—
103#11719919—41no—
104C4×Dic53—24no—
105C4×Dic53—41no—
106C4×Dic53—41no—
107C4×Dic53—41no—
108C4×Dic53—42no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322—41no—
114C2×#88322—41no—
115C23×C53C2×C22⋊C411yescharacteristic
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53#9912yes—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C2×M1611yescharacteristic
136C2×C4×C53C2×M1611yescharacteristic
137C2×C4×C53#13511yescharacteristic
138C2×C4×C53#13511yescharacteristic
139C2×C4×C53#13511yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—22no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53#10112yes—
156C4×D53—24no—
157C4×D53—24no—
158C4×D53—42no—
159C4×D53—42no—
160C4×D53—42no—
161C4×D53—42no—
162C8×C53—81no—
163C8×C53—81no—
164#88322—81no—
165#88322—81no—
166#7646—531nomaximal, Sylow
167C22×C53#40711yescharacteristic
168C22×C53C2×#9911yescharacteristic
169C22×C53C2×#10111yescharacteristic
170C22×C53#41611yescharacteristic
171C22×C53#41611yescharacteristic
172C22×C53#42111yescharacteristic
173C22×C53#42111yescharacteristic
174C2×D53—22no—
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—42no—
188C4×C53—81no—
189Dic53—22no—
190Dic53—42no—
191Dic53—41no—
192Dic53—41no—
193Dic53—81no—
194#2296—531no—
195#1817—531no—
196#1817—531no—
197#1271—531no—
198#1271—531no—
199#1271—531no—
200#1271D5311yescharacteristic, complemented
201C2×C53#146311yescharacteristic
202C2×C53#148811yescharacteristic
203C2×C53#148811yescharacteristic
204C2×C53#151611yescharacteristic
205C2×C53#151611yescharacteristic
206C2×C53#153711yescharacteristic
207C2×C53#154411yescharacteristic
208D53—42no—
209C2×#117—531no—
210C2×#117—531no—
211C22×C42—531no—
212C2×C22⋊C8—1061no—
213C2×C22⋊C8—1061no—
214#389—531no—
215#389—531no—
216#389—531no—
217#389C2×D5311yescharacteristic
218#385—1061no—
219#385—1061no—
220C4×C4⋊C4—1061no—
221C4×C4⋊C4—1061no—
222#336—531no—
223#336—531no—
224#336—1061no—
225#336—1061no—
226#336C2×D5311yescharacteristic
227#336C2×D5311yescharacteristic
228C53#764611yescharacteristic, complemented, Sylow
229C23×C4—531no—
230C23×C4—531no—
231C23×C4—531no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—1061no—
235C22×C8—1061no—
236#117—1062no—
237#117—1062no—
238#117—2122no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—531no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4C4×D5311yescharacteristic
246C2×C4⋊C4C4×D5311yescharacteristic
247C2×C22⋊C4—1061no—
248C2×C22⋊C4—1061no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—1061no—
253C2×C42—1061no—
254C2×C42—2122no—
255C2×C42C22×D5311yescharacteristic
256C22⋊C8—2122no—
257C22⋊C8—2122no—
258#95—21no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262C24—531no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—1062no—
269C22×C4—1062no—
270C22×C4—1062no—
271C22×C4—1062no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4C2×C4×D5311yescharacteristic
283C22×C4D4×D5311yescharacteristic
284C22×C4D4×D5311yescharacteristic
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8—2121no—
294C2×C8—2121no—
295C2×C8—2121no—
296C2×C8—2121no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—532no—
302C4⋊C4—1061no—
303C4⋊C4—2121no—
304C4⋊C4—2122no—
305C4⋊C4—2122no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4#1171991212yes—
309C22⋊C4—1064no—
310C22⋊C4—2122no—
311C42—41no—
312C42—1064no—
313C42—2121no—
314C42—2121no—
315C42—2121no—
316C42—2122no—
317C23—532no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23—1061no—
324C23C22⋊C4×D5311yescharacteristic
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—532no—
338C2×C4—1064no—
339C2×C4—1064no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1062no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—2122no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4M16×D5311yescharacteristic
359C2×C4C2×#1171991211yescharacteristic
360C2×C4#1313323611yescharacteristic
361C2×C4#1313323811yescharacteristic
362C2×C4#1313323811yescharacteristic
363C2×C4#1313324011yescharacteristic
364C8—81no—
365C8—81no—
366C8—4241no—
367C8—4241no—
368C22—1062no—
369C22—1062no—
370C22—1062no—
371C22—2121no—
372C22—2121no—
373C22—2121no—
374C22—2121no—
375C22#1681058511yescharacteristic
376C22#99×D5311yescharacteristic
377C22#101×D5311yescharacteristic
378C22#1681061611yescharacteristic
379C22#1681061811yescharacteristic
380C22#1681063811yescharacteristic
381C22#1681063811yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—81no—
390C4—1062no—
391C4—4241no—
392C4—2121no—
393C4—2121no—
394C4—2122no—
395C2—2122no—
396C2#2027304311yescharacteristic
397C2#2027314611yescharacteristic
398C2#2027314811yescharacteristic
399C2#2027326811yescharacteristic
400C2#2027327011yescharacteristic
401C2#2027335111yescharacteristic
402C2#2027337811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7646
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #20272539C2semidirect
5. #20272538C2semidirect
6. #20272537C2semidirect
7. C53×#1271C2semidirect
200. #1271D53semidirect
228. C53#7646semidirect

Maximal subgroups

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

subgroupindex
#202771962
#202747822
#202725392
#202725382
#202725372
C53×#12712
#764653