Finite Group Databasebeta

← #27278062

Subgroups — #27278062

Order 13568.

532 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 119 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#20283744C211yesmaximal, characteristic
2#20274893C212yesmaximal, complemented
3C4×#13133132C211yesmaximal, characteristic
4#20272561C212yesmaximal, complemented
5C53×#1869C211yesmaximal, characteristic
6#16810695—22no—
7#16810695—22no—
8#16810680—21no—
9C2×#13133146—22no—
10C2×#13133132C412yes—
11#16810587—21no—
12C2×C22⋊#88322C2212yes—
13C4×C22⋊Dic53C2212yes—
14#16810390C411yescharacteristic
15#16810390C411yescharacteristic
16#16810377—22no—
17C23×C8×C53C2211yescharacteristic
18C2×C4×D4×C53C2211yescharacteristic
19C2×C22⋊C8×C53—21no—
20C2×C22⋊C8×C53C2211yescharacteristic
21C53×#336—21no—
22C2×#11719917—21no—
23C2×#11719917—21no—
24C22×#88322D412yes—
25C22×#88322D412yes—
26C2×C22⋊Dic53C2×C412yes—
27C2×C22⋊Dic53C2×C412yes—
28C2×C4×Dic53—22no—
29C4⋊#88322—21no—
30C4⋊#88322—21no—
31#13133146—44no—
32#13133132—24no—
33C22⋊#88322—22no—
34C22⋊#88322—22no—
35C22⋊#88322—22no—
36#13133128—22no—
37#13133124—22no—
38C22×D4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41C22×C8×C53—21no—
42C22×C8×C53—22no—
43C22×C8×C53—21no—
44C22×C8×C53—42no—
45C4×D4×C53—22no—
46C4×D4×C53—22no—
47C2×C4⋊C4×C53Q811yescharacteristic
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53Q811yescharacteristic
50C2×C42×C53D411yescharacteristic
51C22⋊C8×C53—22no—
52C22⋊C8×C53—42no—
53C24×C53C4⋊C411yescharacteristic
54C24×C53C22×C411yescharacteristic
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53—21no—
58C2×D4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53C4⋊C411yescharacteristic
66C22×C4×C53C4⋊C411yescharacteristic
67C22×C4×C53C4⋊C411yescharacteristic
68C22×C4×C53C2×D412yes—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×Q811yescharacteristic
71C22×C4×C53#4112yes—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C2×C8×C53—22no—
75C2×C8×C53—24no—
76C2×C8×C53—41no—
77C2×C8×C53—42no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C4⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C42×C53—21no—
86C42×C53—21no—
87C22⋊Dic53—22no—
88C22⋊Dic53—22no—
89C22⋊Dic53—22no—
90C22⋊Dic53M1614yescomplemented
91C4×Dic53—42no—
92C4×Dic53—42no—
93#11719917—81no—
94#11719917—81no—
95C2×#88322—22no—
96C2×#88322—22no—
97C2×#88322—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C2×C4⋊C411yescharacteristic
107C23×C53C4×D412yes—
108C23×C53C2×M1612yes—
109C23×C53#13111yescharacteristic
110C23×C53#13111yescharacteristic
111D4×C53—42no—
112D4×C53—42no—
113C2×C4×C53—21no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—81no—
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53C22⋊Q812yes—
131C2×C4×C53#12311yescharacteristic
132C2×Dic53—24no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C8×C53—42no—
139C8×C53—81no—
140#88322—42no—
141#88322—42no—
142#8852—531nomaximal, Sylow
143C22×C53—22no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—21no—
147C22×C53—42no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—42no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—81no—
155C22×C53#38511yescharacteristic
156C22×C53#42311yescharacteristic
157C22×C53#42411yescharacteristic
158C22×C53#43512yes—
159C22×C53#43612yes—
160C4×C53—21no—
161C4×C53—22no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53B(4,𝔽2)11yescharacteristic
166C4×C53#45811yescharacteristic
167Dic53—44no—
168Dic53—82no—
169#2973—531no—
170#1869D5311yescharacteristic
171#1850—532no—
172C4×#99—531no—
173#1282—532no—
174C2×C53—22no—
175C2×C53—22no—
176C2×C53—21no—
177C2×C53—21no—
178C2×C53—81no—
179C2×C53#185111yescharacteristic
180C2×C53#186711yescharacteristic
181C2×C53#186811yescharacteristic
182C23×C8C2×D5311yescharacteristic
183C2×C4×D4C2×D5311yescharacteristic
184#435—1062no—
185#435—1062no—
186#432—1061no—
187C2×C4⋊C8—1062no—
188C2×#99—532no—
189#407—1061no—
190C2×C22⋊C8—21no—
191C2×C22⋊C8—532no—
192C2×C22⋊C8C2×D5311yescharacteristic
193C4×C22⋊C4—532no—
194#342—531no—
195#342—531no—
196#336—21no—
197#336—1062no—
198C53#885211yescharacteristic, complemented, Sylow
199C22×D4C4×D5311yescharacteristic
200C23×C4C22⋊D5311yescharacteristic
201C23×C4C22×D5311yescharacteristic
202C2×M16—1061no—
203C2×M16—1061no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—42no—
208C22×C8—532no—
209C22×C8—532no—
210C4×D4—22no—
211C4×D4—22no—
212C2×C4⋊C4Dic10611yescharacteristic
213C2×C22⋊C4—532no—
214C2×C22⋊C4—532no—
215C2×C22⋊C4Dic10611yescharacteristic
216C2×C22⋊C4C4×D5311yescharacteristic
217C2×C42—1062no—
218C2×C42C22⋊D5311yescharacteristic
219C4⋊C8—1061no—
220C4⋊C8—1061no—
221C4⋊C8—2124no—
222#99—1064no—
223C22⋊C8—22no—
224C22⋊C8—42no—
225C22⋊C8—1062no—
226C22⋊C8—1062no—
227C22⋊C8—1062no—
228#97—1062no—
229#95—1062no—
230C24#1171991911yescharacteristic
231C24C2×C4×D5311yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4#1171991911yescharacteristic
245C22×C4#1171991911yescharacteristic
246C22×C4#1171991911yescharacteristic
247C22×C4C2×Dic10611yescharacteristic
248C22×C4D4×D5312yes—
249C22×C4#1171994712yes—
250C22×C4C2×C22⋊D5311yescharacteristic
251M16—4241no—
252M16—4241no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—1062no—
262C2×C8—1062no—
263C2×C8—2122no—
264C2×C8—2122no—
265C4⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—534no—
269C22⋊C4—1062no—
270C22⋊C4—1062no—
271C22⋊C4—1062no—
272C42—21no—
273C42—21no—
274C42—2122no—
275C42—2122no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×#1171991911yescharacteristic
284C23#1313317712yes—
285C23M16×D5312yes—
286C23#1313322311yescharacteristic
287C23#1313322311yescharacteristic
288D4—42no—
289D4—42no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—1064no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4#1313318511yescharacteristic
313C2×C4#1313319412yes—
314C2×C4#1313319811yescharacteristic
315C8—42no—
316C8—81no—
317C8—2122no—
318C8—2122no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1681049511yescharacteristic
332C22#1681064611yescharacteristic
333C22#1681064911yescharacteristic
334C22#1681069712yes—
335C22#1681069812yes—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—4242no—
342C4—2124no—
343C4#1681080911yescharacteristic
344C4#1681081211yescharacteristic
345C2—22no—
346C2—22no—
347C2—21no—
348C2—21no—
349C2—81no—
350C2#2027489711yescharacteristic
351C2#2027499511yescharacteristic
352C2#2027500011yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1869C2
socleC22×C53#385
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#1869D53
hypocenterC53#8852
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. #20274893C2semidirect
4. #20272561C2semidirect
90. C22⋊Dic53M16semidirect
198. C53#8852semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202748932
C4×#131331322
#202725612
C53×#18692
#885253