Finite Group Databasebeta

← #27274236

Subgroups — #27274236

Order 13568.

874 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C2×#16810585C212yesmaximal
2#407×D53C211yesmaximal, characteristic
3#20272603C212yesmaximal, complemented
4#20272602C211yesmaximal, characteristic, complemented
5C53×#1298C211yesmaximal, characteristic, complemented
6C23×C4×D53C2211yescharacteristic
7C22×#11719912—22no—
8C2×M16×D53C412yes—
9#16810587C2211yescharacteristic
10#16810586C412yes—
11#16810585—22no—
12#16810585—22no—
13#16810585—22no—
14#16810585C414yes—
15#16810585C414yes—
16C2×C22⋊#88322C2212yes—
17C2×#13133129—22no—
18C22⋊C8×D53—22no—
19C53×#407C2211yescharacteristic
20C2×C22⋊C8×C53C2212yes—
21C24×D53Q811yescharacteristic
22C23×Dic53Q811yescharacteristic
23C22×C4×D53—44no—
24C22×C4×D53C2×C411yescharacteristic
25C22×C4×D53C2×C412yes—
26C22×C4×D53C2×C412yes—
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D412yes—
29C22×C4×D53D412yes—
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32C2×#11719917C2×C412yes—
33C2×#11719912—22no—
34C2×#11719912—24no—
35C2×#11719912—24no—
36C2×#11719912—48no—
37M16×D53—44no—
38C22×#88322—22no—
39C2×C8×D53—22no—
40C22⋊#88322—22no—
41C22⋊#88322—22no—
42C22⋊#88322C2×C414yes—
43#13133129—24no—
44#13133129—24no—
45#13133129—24no—
46C23×C4×C53C2311yescharacteristic
47C2×M16×C53C2×C412yes—
48C22×C8×C53—22no—
49C22⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C22⋊C8×C53C2×C414yes—
52C24×C53C2×Q811yescharacteristic
53C23×D53—42no—
54C23×D53—44no—
55C23×D53C4⋊C412yes—
56C23×D53C4⋊C412yes—
57C23×D53C4⋊C411yescharacteristic
58C23×D53C4⋊C411yescharacteristic
59C22×C4×C53—42no—
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C22×C412yes—
62C22×C4×C53C2×D412yes—
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—42no—
65C22×Dic53C4⋊C411yescharacteristic
66C22×Dic53C4⋊C412yes—
67C22×Dic53C4⋊C411yescharacteristic
68C22×Dic53C4⋊C412yes—
69C2×C4×D53—24no—
70C2×C4×D53—28no—
71C2×C4×D53—44no—
72C2×C4×D53—42no—
73C2×C4×D53—42no—
74C2×C4×D53—44no—
75C2×C4×D53—44no—
76C2×C4×D53—42no—
77C2×C4×D53C4211yescharacteristic
78C2×C4×D53C4211yescharacteristic
79C2×C4×D53C4212yes—
80C2×C4×D53C22⋊C412yes—
81C2×C4×D53C22⋊C412yes—
82C2×C4×D53C22⋊C411yescharacteristic
83C2×C4×D53C22⋊C412yes—
84C2×C4×D53C22⋊C412yes—
85C2×C4×D53C22⋊C412yes—
86C2×C4×D53C22⋊C411yescharacteristic
87M16×C53—42no—
88C2×C8×C53—22no—
89C2×C8×C53—24no—
90C2×C8×C53—44no—
91#11719917—42no—
92C2×#88322—22no—
93C2×#88322—24no—
94C2×#88322—44no—
95#11719912—44no—
96#11719912—48no—
97#11719912—48no—
98C8×D53—44no—
99C23×C53—24no—
100C23×C53—41no—
101C23×C53C2×C4⋊C411yescharacteristic
102C23×C53C2×C4⋊C412yes—
103C22×D53—24no—
104C22×D53—24no—
105C22×D53—24no—
106C22×D53—22no—
107C22×D53—42no—
108C22×D53—41no—
109C22×D53—42no—
110C22×D53—42no—
111C22×D53—42no—
112C22×D53—42no—
113C22×D53—416no—
114C22×D53#9511yescharacteristic
115C22×D53#9512yes—
116C22×D53#9511yescharacteristic
117C2×C4×C53—22no—
118C2×C4×C53—24no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53C2×C4211yescharacteristic
122C2×C4×C53C2×C22⋊C412yes—
123C2×C4×C53C2×C22⋊C411yescharacteristic
124C2×Dic53—24no—
125C2×Dic53—22no—
126C2×Dic53—42no—
127C2×Dic53—41no—
128C2×Dic53#9511yescharacteristic
129C2×Dic53#9512yes—
130C2×Dic53#9511yescharacteristic
131C4×D53—22no—
132C4×D53—24no—
133C4×D53—24no—
134C4×D53—22no—
135C4×D53—22no—
136C4×D53—24no—
137C4×D53—42no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53#9914yes—
141C8×C53—44no—
142C8×C53—42no—
143#88322—44no—
144#88322—42no—
145#8055—531nomaximal, Sylow
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—21no—
150C22×C53—44no—
151C22×C53C2×#9511yescharacteristic
152C22×C53#43014yes—
153C2×D53—22no—
154C2×D53—22no—
155C2×D53—22no—
156C2×D53—22no—
157C2×D53—22no—
158C2×D53—21no—
159C2×D53—48no—
160C2×D53—44no—
161C2×D53—48no—
162C2×D53—44no—
163C2×D53#34212yes—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53C2×#9912yes—
167Dic53—21no—
168Dic53—22no—
169Dic53#34212yes—
170C2×#407—531no—
171C2×#407—532no—
172#1298—531no—
173#1298—532no—
174#1298D5311yescharacteristic, complemented
175C2×C53—24no—
176C2×C53—41no—
177C2×C53#173812yes—
178C2×C53C2×#34211yescharacteristic
179D53—22no—
180D53—42no—
181D53—41no—
182C24×C4—531no—
183C22×M16—532no—
184C22×M16—1062no—
185#407—534no—
186#407—534no—
187#407—531no—
188#407—532no—
189#407—1062no—
190#407—1062no—
191#407—1062no—
192#407C2×D5311yescharacteristic
193C2×C22⋊C8—532no—
194C2×C22⋊C8—1062no—
195C2×C22⋊C8—1062no—
196C2×C22⋊C8C2×D5312yes—
197C53#805511yescharacteristic, complemented, Sylow
198C25—531no—
199C23×C4—531no—
200C23×C4—531no—
201C23×C4—532no—
202C23×C4—532no—
203C23×C4—532no—
204C23×C4—531no—
205C23×C4—532no—
206C23×C4—531no—
207C23×C4—531no—
208C23×C4—2124no—
209C23×C4C22×D5311yescharacteristic
210C2×M16—532no—
211C2×M16—1062no—
212C2×M16—1064no—
213C2×M16—1064no—
214C2×M16—2128no—
215C2×M16—2124no—
216C2×M16C4×D5312yes—
217C22×C8—22no—
218C22×C8—1062no—
219C22×C8—1062no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—534no—
223C22⋊C8—1064no—
224C22⋊C8—1064no—
225C22⋊C8—1064no—
226C22⋊C8—1062no—
227C22⋊C8—1062no—
228C22⋊C8C4×D5314yes—
229C24—532no—
230C24—531no—
231C24—532no—
232C24—531no—
233C24—2124no—
234C24—2122no—
235C24Q8×D5311yescharacteristic
236C22×C4—42no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—532no—
240C22×C4—531no—
241C22×C4—532no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—532no—
245C22×C4—532no—
246C22×C4—532no—
247C22×C4—531no—
248C22×C4—532no—
249C22×C4—532no—
250C22×C4—532no—
251C22×C4—1068no—
252C22×C4—1064no—
253C22×C4—2124no—
254C22×C4—2122no—
255C22×C4—2124no—
256C22×C4—2122no—
257C22×C4—2124no—
258C22×C4—2122no—
259C22×C4—2122no—
260C22×C4C2×C4×D5311yescharacteristic
261C22×C4C2×C4×D5312yes—
262C22×C4D4×D5312yes—
263C22×C4D4×D5311yescharacteristic
264M16—42no—
265M16—2128no—
266M16—2128no—
267M16—2124no—
268M16—2122no—
269C2×C8—24no—
270C2×C8—22no—
271C2×C8—44no—
272C2×C8—1064no—
273C2×C8—1062no—
274C2×C8—2124no—
275C2×C8—2124no—
276C23—24no—
277C23—41no—
278C23—531no—
279C23—531no—
280C23—532no—
281C23—1064no—
282C23—1062no—
283C23—1064no—
284C23—1064no—
285C23—2122no—
286C23—2121no—
287C23—2122no—
288C23—2122no—
289C23—2122no—
290C23—2122no—
291C23—21216no—
292C23C4⋊C4×D5311yescharacteristic
293C23C4⋊C4×D5312yes—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—531no—
299C2×C4—534no—
300C2×C4—531no—
301C2×C4—532no—
302C2×C4—1064no—
303C2×C4—1062no—
304C2×C4—1064no—
305C2×C4—1062no—
306C2×C4—1064no—
307C2×C4—1062no—
308C2×C4—1062no—
309C2×C4—1064no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4—2121no—
315C2×C4C42×D5311yescharacteristic
316C2×C4C22⋊C4×D5312yes—
317C2×C4C22⋊C4×D5311yescharacteristic
318C8—44no—
319C8—42no—
320C8—2124no—
321C8—2122no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—44no—
327C22—532no—
328C22—1062no—
329C22—1062no—
330C22—1062no—
331C22—1062no—
332C22—1062no—
333C22—1061no—
334C22—2128no—
335C22—2128no—
336C22—2124no—
337C22—2124no—
338C22#95×D5311yescharacteristic
339C22#1681067014yes—
340C4—22no—
341C4—21no—
342C4—532no—
343C4—1062no—
344C4—1061no—
345C4#99×D5312yes—
346C2—24no—
347C2—41no—
348C2—1062no—
349C2—2122no—
350C2—2121no—
351C2#2027444012yes—
352C2#342×D5311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1298C2
socleC22×C53C2×#95
prime radicalC24×D53Q8
cosocleC23×C4×C53C23
hypercenter#1298D53
hypocenterC53#8055
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
3. #20272603C2semidirect
4. #20272602C2semidirect
5. C53×#1298C2semidirect
174. #1298D53semidirect
197. C53#8055semidirect

Maximal subgroups

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

subgroupindex
C2×#168105852
#407×D532
#202726032
#202726022
C53×#12982
#805553