Finite Group Databasebeta

← #27293345

Subgroups — #27293345

Order 13568.

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

normal subgroups only

This group has 488 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
1C4×#13133198C211yesmaximal, characteristic
2#20273036C211yesmaximal, characteristic
3#20272664C212yesmaximal, complemented
4#20272644C212yesmaximal, complemented
5C53×#1459C211yesmaximal, characteristic
6C2×#13133198C411yescharacteristic
7C22×C4×Dic53C411yescharacteristic
8#16810676—21no—
9#16810511—21no—
10#16810504—21no—
11#16810503C411yescharacteristic
12#16810495C2211yescharacteristic
13#16810484C2211yescharacteristic
14C4×#11719919—21no—
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53C411yescharacteristic
17#16810347—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25#13133198—22no—
26#13133198C814yescomplemented
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53C2×C411yescharacteristic
32C2×C22⋊Dic53C2×C411yescharacteristic
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—41no—
37C2×C4×Dic53—41no—
38C4⋊#88322—41no—
39C22⋊#88322—22no—
40C4×#88322—41no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×D4×C53C2×C411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C22×C8×C53—41no—
50C4×D4×C53—41no—
51C4×D4×C53—41no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53C2311yescharacteristic
55C2×C42×C53D411yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C22×C411yescharacteristic
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53C2×C812yes—
63C2×D4×C53M1612yes—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—42no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53C2×C812yescomplemented
85C2×C8×C53—42no—
86C2×C8×C53—82no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53—21no—
91C42×C53—21no—
92C22⋊Dic53—24no—
93C22⋊Dic53—21no—
94C22⋊Dic53—42no—
95C22⋊Dic53—41no—
96C22⋊Dic53C2×C812yescomplemented
97#11719919—24no—
98#11719919—42no—
99C4×Dic53—42no—
100C4×Dic53—42no—
101C4×Dic53—41no—
102C4×Dic53—41no—
103C2×#88322—42no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53C22⋊C812yes—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C22×C811yescharacteristic
115C23×C53C2×M1611yescharacteristic
116D4×C53—42no—
117D4×C53—42no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—81no—
131C2×C4×C53C22⋊C812yes—
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53#9911yescharacteristic
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—41no—
138C2×Dic53—42no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—42no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C8×C53—82no—
151#88322—82no—
152#11472—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—22no—
158C22×C53—42no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—42no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—81no—
167C22×C53C2×C22⋊C811yescharacteristic
168C22×C53C2×#9911yescharacteristic
169C22×C53#41311yescharacteristic
170C4×C53—21no—
171C4×C53—22no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53#35212yes—
176Dic53—42no—
177Dic53—42no—
178Dic53—42no—
179Dic53—81no—
180C4×#123—531no—
181#1460—531no—
182#1459D5311yescharacteristic
183#1326—532no—
184#1317—532no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—41no—
188C2×C53—41no—
189C2×C53—42no—
190C2×C53#145911yescharacteristic
191C2×C53C2×#35211yescharacteristic
192C2×C53#212011yescharacteristic
193C2×#123—531no—
194C2×C4×D4C2×D5311yescharacteristic
195C22×C42—531no—
196#431—1061no—
197C2×C22⋊C8—21no—
198#388—1061no—
199#387—1061no—
200#386—531no—
201#385—531no—
202#381—531no—
203C4×C4⋊C4—1061no—
204C4×C22⋊C4—531no—
205C4×C22⋊C4—1061no—
206#324—21no—
207#323—532no—
208#323—1061no—
209#323C2×D5312yes—
210C53#1147211yescharacteristic, complemented, Sylow
211C22×D4C4×D5311yescharacteristic
212C23×C4—531no—
213C23×C4—531no—
214C23×C4C4×D5311yescharacteristic
215C23×C4C22⋊D5311yescharacteristic
216C22×C8—41no—
217#123—534no—
218#123—1062no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4C22⋊D5311yescharacteristic
224C2×C22⋊C4—531no—
225C2×C22⋊C4—531no—
226C2×C22⋊C4—1061no—
227C2×C22⋊C4—1061no—
228C2×C22⋊C4D21211yescharacteristic
229C2×C22⋊C4C22×D5311yescharacteristic
230C2×C42—1061no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42—2121no—
234C2×C42—2121no—
235C2×C42D21211yescharacteristic
236C4⋊C8—2121no—
237C22⋊C8—22no—
238C22⋊C8—41no—
239C22⋊C8—1062no—
240C4×C8—2121no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246C24#1171992111yescharacteristic
247C24C2×C4×D5311yescharacteristic
248C2×D4—41no—
249C2×D4—41no—
250C2×D4C8×D5312yes—
251C2×D4#1171991212yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—532no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—2122no—
265C22×C4—2121no—
266C22×C4—2121no—
267C22×C4—2121no—
268C22×C4#1171992111yescharacteristic
269C22×C4#1171992111yescharacteristic
270C22×C4#1171992111yescharacteristic
271C22×C4C2×D21211yescharacteristic
272C22×C4C2×C22⋊D5311yescharacteristic
273C2×C8—42no—
274C2×C8—82no—
275C2×C8—2122no—
276C4⋊C4—41no—
277C4⋊C4—1064no—
278C4⋊C4—2122no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—532no—
282C22⋊C4—1064no—
283C22⋊C4—1061no—
284C22⋊C4—2122no—
285C22⋊C4—2121no—
286C42—21no—
287C42—21no—
288C42—2122no—
289C42—2122no—
290C42—2121no—
291C42—2121no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23#1313312912yes—
301C23C2×#1171992111yescharacteristic
302C23C2×C8×D5311yescharacteristic
303C23C2×#1171991211yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4#1313312912yes—
336C2×C4#1313313111yescharacteristic
337C2×C4#1313313111yescharacteristic
338C8—82no—
339C8—4242no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22C2×#1313312911yescharacteristic
355C22C2×#1313313111yescharacteristic
356C22#1681060511yescharacteristic
357C4—21no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—4241no—
363C4—2122no—
364C4—2122no—
365C4—2122no—
366C4#1681041212yes—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—42no—
372C2#2027303111yescharacteristic
373C2C2×#1681041211yescharacteristic
374C2#2027635711yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC23×C53C22×C8
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1459C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1459D53
hypocenterC53#11472
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20272664C2semidirect
4. #20272644C2semidirect
26. #13133198C8semidirect
84. C22×Dic53C2×C8semidirect
96. C22⋊Dic53C2×C8semidirect
210. C53#11472semidirect

Maximal subgroups

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

subgroupindex
C4×#131331982
#202730362
#202726642
#202726442
C53×#14592
#1147253