Finite Group Databasebeta

← #27293451

Subgroups — #27293451

Order 13568.

706 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 95 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×#16810484C211yesmaximal, characteristic
2#20273042C211yesmaximal, characteristic
3#20272644C214yesmaximal, complemented
4C53×#1462C211yesmaximal, characteristic
5C24×Dic53C411yescharacteristic
6C22×C22⋊Dic53C411yescharacteristic
7#16810587—21no—
8#16810484—21no—
9#16810484—21no—
10#16810484C412yes—
11#16810484C2212yes—
12C2×#13133124—21no—
13C2×#13133124—21no—
14#16810347—21no—
15#16810347C2212yes—
16C22×C22⋊C4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#323—21no—
19C53×#323C2212yes—
20C23×Dic53—21no—
21C23×Dic53—21no—
22C23×Dic53—24no—
23C23×Dic53—21no—
24C23×Dic53—41no—
25C23×Dic53—41no—
26C23×Dic53—42no—
27C23×Dic53C2×C412yes—
28C2×#11719917—41no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—42no—
31C2×C22⋊Dic53C2×C412yes—
32C22⋊#88322—22no—
33C22⋊#88322—41no—
34#13133124—22no—
35#13133124—22no—
36#13133124—22no—
37#13133124—22no—
38C25×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C2×M16×C53—41no—
42C2×C22⋊C4×C53—41no—
43C2×C22⋊C4×C53—41no—
44C2×C22⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53C2311yescharacteristic
48C22⋊C8×C53—22no—
49C22⋊C8×C53—41no—
50C24×C53—21no—
51C24×C53—21no—
52C24×C53—22no—
53C24×C53—22no—
54C24×C53—42no—
55C24×C53—42no—
56C24×C53C22⋊C411yescharacteristic
57C24×C53C22⋊C411yescharacteristic
58C24×C53C22×C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—24no—
70C22×Dic53—24no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—48no—
75C22×Dic53—48no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—42no—
81C22×Dic53—42no—
82C22×Dic53—42no—
83C22×Dic53—42no—
84C22×Dic53—42no—
85C22×Dic53—42no—
86C22×Dic53M1614yescomplemented
87M16×C53—82no—
88C2×C8×C53—42no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊Dic53—42no—
94C22⋊Dic53—42no—
95#11719917—82no—
96C2×#88322—42no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—24no—
102C23×C53—24no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—48no—
112C23×C53—48no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53#9911yescharacteristic
118C23×C53#9911yescharacteristic
119C23×C53#10012yes—
120C23×C53C2×C22⋊C411yescharacteristic
121C23×C53C2×M1612yes—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×Dic53—24no—
135C2×Dic53—44no—
136C2×Dic53—44no—
137C2×Dic53—48no—
138C2×Dic53—44no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—44no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—84no—
146C8×C53—82no—
147#88322—82no—
148#11500—531nomaximal, Sylow
149C22×C53—24no—
150C22×C53—24no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—42no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53—48no—
163C22×C53—48no—
164C22×C53—42no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53#35212yes—
170C22×C53#35212yes—
171C22×C53#40711yescharacteristic
172C22×C53C2×#9911yescharacteristic
173C22×C53C2×#10011yescharacteristic
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178Dic53—44no—
179Dic53—82no—
180C2×#381—531no—
181#1462—531no—
182#1462D5311yescharacteristic
183#1317—534no—
184C2×C53—21no—
185C2×C53—22no—
186C2×C53—22no—
187C2×C53—21no—
188C2×C53—42no—
189C2×C53—42no—
190C2×C53#146211yescharacteristic
191C2×C53C2×#35211yescharacteristic
192C2×C53C2×#35211yescharacteristic
193C24×C4—531no—
194C22×C22⋊C4—531no—
195C22×C22⋊C4C2×D5311yescharacteristic
196#407—21no—
197#407—1061no—
198#381—532no—
199#381—532no—
200#381—1061no—
201#381—1061no—
202C2×#95—1061no—
203C2×#95—1061no—
204#323—21no—
205#323—532no—
206#323—1061no—
207#323C2×D5312yes—
208C53#1150011yescharacteristic, complemented, Sylow
209C25C4×D5311yescharacteristic
210C23×C4—532no—
211C23×C4—1064no—
212C23×C4—1061no—
213C23×C4—1061no—
214C23×C4—1061no—
215C23×C4—2121no—
216C23×C4—2121no—
217C23×C4—2122no—
218C23×C4D21211yescharacteristic
219C23×C4C22⋊D5311yescharacteristic
220C2×M16—41no—
221C2×M16—2121no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—532no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—2122no—
227C2×C22⋊C4C4×D5311yescharacteristic
228C2×C22⋊C4D21211yescharacteristic
229C2×C22⋊C4C22⋊D5311yescharacteristic
230C2×C22⋊C4C22×D5311yescharacteristic
231C22⋊C8—22no—
232C22⋊C8—41no—
233C22⋊C8—1062no—
234C22⋊C8—2121no—
235#95—1062no—
236#95—1062no—
237#95—1062no—
238#95—1062no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24—42no—
244C24—42no—
245C24#1171992111yescharacteristic
246C24#1171992111yescharacteristic
247C24C2×C4×D5311yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—534no—
255C22×C4—1064no—
256C22×C4—1064no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—2122no—
261C22×C4—2122no—
262C22×C4—2122no—
263C22×C4—2128no—
264C22×C4—2128no—
265C22×C4—2122no—
266C22×C4—2122no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2122no—
270C22×C4—2122no—
271C22×C4—2122no—
272C22×C4#1171992111yescharacteristic
273C22×C4#1171992111yescharacteristic
274C22×C4C2×D21211yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276M16—82no—
277M16—4242no—
278C2×C8—42no—
279C2×C8—2122no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—2122no—
285C22⋊C4—2122no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—22no—
290C23—24no—
291C23—24no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—48no—
301C23—48no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23#1313313111yescharacteristic
307C23#1313313111yescharacteristic
308C23#1313313312yes—
309C23C2×#1171992111yescharacteristic
310C23M16×D5312yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—1064no—
324C2×C4—4244no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2124no—
329C2×C4—2124no—
330C2×C4—2128no—
331C2×C4—2124no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2124no—
335C8—82no—
336C8—4242no—
337C22—24no—
338C22—24no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—22no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—48no—
352C22—48no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22#1681041212yes—
358C22#1681041412yes—
359C22#1681058611yescharacteristic
360C22C2×#1313313111yescharacteristic
361C22C2×#1313313311yescharacteristic
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—4242no—
367C4—2124no—
368C2—21no—
369C2—21no—
370C2—22no—
371C2—22no—
372C2—42no—
373C2—42no—
374C2#2027304111yescharacteristic
375C2C2×#1681041211yescharacteristic
376C2C2×#1681041411yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1462C2
socleC22×C53C2×#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1462D53
hypocenterC53#11500
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #20272644C2semidirect
86. C22×Dic53M16semidirect
208. C53#11500semidirect

Maximal subgroups

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

subgroupindex
C2×#168104842
#202730422
#202726442
C53×#14622
#1150053