Finite Group Databasebeta

← #27272629

Subgroups — #27272629

Order 13568.

532 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 107 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#20283046C211yesmaximal, characteristic
2#20274695C211yesmaximal, characteristic, complemented
3#20272539C212yesmaximal, complemented
4#20272535C212yesmaximal, complemented
5C53×#1749C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133191C411yescharacteristic
8C2×C4⋊#88322—21no—
9C2×C22⋊#88322—21no—
10#16810525C411yescharacteristic
11#16810525C2211yescharacteristic
12#16810521C411yescharacteristic
13#16810521C2211yescharacteristic
14#16810514—21no—
15#16810514—21no—
16#16810502—21no—
17#16810483C411yescharacteristic
18#16810378C2212yes—
19C22×C42×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C2×#97×C53—21no—
22C53×#336C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×#88322—22no—
26#13133200—44no—
27#13133191—42no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C4⋊#88322—42no—
40C22⋊#88322—42no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C22×C8×C53—22no—
48C2×C42×C53—42no—
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53C2311yescharacteristic
53C22⋊C8×C53—24no—
54#97×C53—42no—
55C24×C53C22⋊C411yescharacteristic
56C23×D53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C2×C22⋊D53—42no—
74C2×C22⋊D53—42no—
75C2×C22⋊D53—42no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C2×D212—22no—
80C2×D212—41no—
81C2×D212—41no—
82C2×C8×C53—24no—
83C2×C8×C53—42no—
84C2×C8×C53—42no—
85C42×C53—22no—
86C42×C53—22no—
87C42×C53—41no—
88C42×C53—41no—
89C22⋊Dic53—42no—
90#11719921—42no—
91#11719921—42no—
92#11719921—42no—
93#11719921—41no—
94#11719921—42no—
95#11719921—41no—
96C4⋊Dic53—22no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99#11719919—42no—
100#11719919—42no—
101C2×#88322—42no—
102C2×#88322—42no—
103C2×#88322—42no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53#10412yes—
111C23×C53C2×C22⋊C411yescharacteristic
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
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—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—44no—
134C2×C4×C53—44no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53#9912yes—
138C2×C4×C53#13112yes—
139C2×C4×C53AGL(1,ℤ/8)12yes—
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C22⋊D53—84no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152D212—82no—
153C8×C53—44no—
154#88322—82no—
155#7698—531nomaximal, Sylow
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53#40811yescharacteristic
164C22×C53C2×#9911yescharacteristic
165C22×C53#41311yescharacteristic
166C22×C53#41711yescharacteristic
167C22×C53#41811yescharacteristic
168C22×C53C2×#10411yescharacteristic
169C22×C53#42111yescharacteristic
170C2×D53—81no—
171C2×D53—81no—
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C2×D53—81no—
177C4×C53—22no—
178C4×C53—22no—
179C4×C53—22no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—41no—
183C4×C53—41no—
184Dic53—81no—
185Dic53—81no—
186Dic53—81no—
187#2854—531no—
188#1797—531no—
189#1749D5311yescharacteristic, complemented
190#1271—532no—
191#1270—532no—
192C2×C53—42no—
193C2×C53#146811yescharacteristic
194C2×C53#148311yescharacteristic
195C2×C53#149611yescharacteristic
196C2×C53#150711yescharacteristic
197C2×C53#151311yescharacteristic
198C2×C53#153211yescharacteristic
199C2×C53#155711yescharacteristic
200D53—161no—
201C2×#123—1061no—
202C2×#121—531no—
203C22×C42C2×D5311yescharacteristic
204C2×C4⋊C8—1061no—
205C2×C22⋊C8—21no—
206C2×C22⋊C8—1061no—
207C2×#97—21no—
208#390—531no—
209#390—531no—
210#389—531no—
211#389—531no—
212#388—1061no—
213#388—1061no—
214#386—1061no—
215#380—531no—
216#336—532no—
217#336C2×D5312yes—
218C53#769811yescharacteristic, complemented, Sylow
219C22×D4—531no—
220C22×D4—1061no—
221C23×C4C4×D5311yescharacteristic
222C23×C4D21211yescharacteristic
223C23×C4C22⋊D5311yescharacteristic
224C22×C8—22no—
225C22×C8—1062no—
226#123—2124no—
227#121—2122no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1061no—
239C2×C42—42no—
240C2×C42C4×D5311yescharacteristic
241C2×C42D21211yescharacteristic
242C2×C42C22⋊D5311yescharacteristic
243C2×C42C22×D5311yescharacteristic
244C4⋊C8—2122no—
245C22⋊C8—24no—
246C22⋊C8—2122no—
247#97—42no—
248#95—1061no—
249#95—1061no—
250#95—1061no—
251C24—1061no—
252C24#1171992111yescharacteristic
253C2×D4—1062no—
254C2×D4—2121no—
255C2×D4—2121no—
256C2×D4—2122no—
257C2×D4—2122no—
258C2×D4—2122no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4#1171992111yescharacteristic
273C22×C4#1171992111yescharacteristic
274C22×C4#1171992111yescharacteristic
275C22×C4C2×C4×D5311yescharacteristic
276C22×C4C2×D21211yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—2122no—
282C2×C8—2122no—
283C2×C8—2122no—
284C4⋊C4—1062no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2122no—
288C4⋊C4—2122no—
289C22⋊C4—2122no—
290C22⋊C4—2122no—
291C22⋊C4—2122no—
292C22⋊C4—2122no—
293C22⋊C4—2122no—
294C22⋊C4—2121no—
295C22⋊C4—2121no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42—41no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—2121no—
307C23—2121no—
308C23—2121no—
309C23—2121no—
310C23—2121no—
311C23—2121no—
312C23—2121no—
313C23#1313314512yes—
314C23C2×#1171992111yescharacteristic
315D4—4242no—
316D4—4244no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4#1313313112yes—
345C2×C4#1313322512yes—
346C2×C4#1313324212yes—
347C2×C4#1313324511yescharacteristic
348C2×C4#1313325011yescharacteristic
349C8—44no—
350C8—4242no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—4241no—
359C22—4241no—
360C22—4241no—
361C22—4241no—
362C22—4241no—
363C22—4241no—
364C22—4241no—
365C22#1681059111yescharacteristic
366C22C2×#1313313111yescharacteristic
367C22#1681060511yescharacteristic
368C22#1681062511yescharacteristic
369C22#1681062711yescharacteristic
370C22C2×#1313314511yescharacteristic
371C22#1681063911yescharacteristic
372C4—22no—
373C4—21no—
374C4—21no—
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—4241no—
380C4—4241no—
381C4—4241no—
382C2—42no—
383C2—8481no—
384C2#2027306411yescharacteristic
385C2#2027312911yescharacteristic
386C2#2027318311yescharacteristic
387C2#2027323011yescharacteristic
388C2#2027325911yescharacteristic
389C2#2027333311yescharacteristic
390C2#2027343811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1749C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C4
cosocleC2×C42×C53C23
hypercenter#1749D53
hypocenterC53#7698
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. #20274695C2semidirect
3. #20272539C2semidirect
4. #20272535C2semidirect
5. C53×#1749C2semidirect
189. #1749D53semidirect
218. C53#7698semidirect

Maximal subgroups

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

subgroupindex
#202830462
#202746952
#202725392
#202725352
C53×#17492
#769853