Finite Group Databasebeta

← #27293566

Subgroups — #27293566

Order 13568.

460 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 87 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×#13133204C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272658C212yesmaximal, complemented
4#20272648C212yesmaximal, complemented
5C53×#1465C211yesmaximal, characteristic
6C2×#13133204C411yescharacteristic
7C2×C4×Dic106C411yescharacteristic
8#16810587—21no—
9#16810528C2211yescharacteristic
10#16810526C411yescharacteristic
11#16810517—21no—
12#16810517—21no—
13#16810488C2211yescharacteristic
14C4×C22⋊Dic53—21no—
15C4×C22⋊Dic53—21no—
16C42×Dic53C411yescharacteristic
17#16810349C2212yes—
18#16810347—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#324C2212yes—
22C53×#323—21no—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—41no—
25#13133204—22no—
26#13133204—22no—
27C4×Dic106—41no—
28C4×Dic106—41no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40C22⋊#88322—22no—
41C22⋊#88322—41no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×D4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C2×M16×C53—41no—
50C4×D4×C53—41no—
51C4×D4×C53—41no—
52C2×C4⋊C4×C53C2311yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C42×C53C2×C411yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C22⋊C411yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
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—
79C2×Dic106—21no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106M1612yescomplemented
83M16×C53—82no—
84C2×C8×C53—42no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C42×C53—21no—
89C42×C53—21no—
90C22⋊Dic53—24no—
91C22⋊Dic53—24no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C4⋊Dic53—41no—
95#11719919—42no—
96C4×Dic53—21no—
97C4×Dic53—42no—
98C4×Dic53—42no—
99C4×Dic53—41no—
100C4×Dic53—41no—
101C4×Dic53—41no—
102C4×Dic53M1612yescomplemented
103#11719917—82no—
104C2×#88322—42no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53C2×C22⋊C411yescharacteristic
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—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53#9911yescharacteristic
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53C2×M1611yescharacteristic
134C2×C4×C53C2×M1611yescharacteristic
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147Dic106—42no—
148Dic106—42no—
149C8×C53—82no—
150#88322—82no—
151#11531—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—42no—
155C22×C53—42no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53#40711yescharacteristic
165C22×C53C2×#9911yescharacteristic
166C22×C53#41311yescharacteristic
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53#35412yes—
174Dic53—42no—
175Dic53—42no—
176Dic53—42no—
177Dic53—81no—
178C4×#124—531no—
179#1465—531no—
180#1465D5311yescharacteristic
181#1324—532no—
182#1319—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—81no—
186C2×C53—81no—
187C2×C53#146511yescharacteristic
188C2×C53C2×#35411yescharacteristic
189C2×C53#213211yescharacteristic
190C2×#124—531no—
191C2×C4×Q8—531no—
192C2×C4×D4C2×D5311yescharacteristic
193#407—21no—
194#407—1061no—
195#391—531no—
196#390—531no—
197#388—1061no—
198#388—1061no—
199#382—531no—
200C4×C22⋊C4—1061no—
201C4×C22⋊C4—1061no—
202C43—531no—
203#324—532no—
204#324C2×D5312yes—
205#323—21no—
206#323—1061no—
207C53#1153111yescharacteristic, complemented, Sylow
208C22×Q8—531no—
209C22×D4C4×D5311yescharacteristic
210C23×C4D21211yescharacteristic
211C23×C4C22⋊D5311yescharacteristic
212C2×M16—41no—
213C2×M16—2121no—
214#124—1062no—
215#124—1062no—
216C4×Q8—2121no—
217C4×Q8—2121no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4C22×D5311yescharacteristic
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1061no—
227C2×C22⋊C4D21211yescharacteristic
228C2×C22⋊C4C22⋊D5311yescharacteristic
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42—1061no—
234C2×C42C4×D5311yescharacteristic
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—1062no—
238C22⋊C8—2121no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242#95—1061no—
243C24#1171992111yescharacteristic
244C24#1171992111yescharacteristic
245C2×Q8—532no—
246C2×Q8—1061no—
247C2×Q8—2121no—
248C2×Q8—2121no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4#1171992111yescharacteristic
264C22×C4#1171992111yescharacteristic
265C22×C4C2×C4×D5311yescharacteristic
266C22×C4C2×D21211yescharacteristic
267C22×C4C2×C22⋊D5311yescharacteristic
268M16—82no—
269M16—4242no—
270C2×C8—42no—
271C2×C8—2122no—
272C4⋊C4—41no—
273C4⋊C4—2121no—
274C4⋊C4—2122no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—1064no—
278C22⋊C4—1064no—
279C22⋊C4—2122no—
280C22⋊C4—2122no—
281C42—21no—
282C42—21no—
283C42—532no—
284C42—1061no—
285C42—2121no—
286C42—2122no—
287C42—2122no—
288C42—2121no—
289C42—2121no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C2×#1171992111yescharacteristic
301Q8—2122no—
302Q8—2122no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4#1313313111yescharacteristic
331C2×C4#1313313111yescharacteristic
332C2×C4M16×D5311yescharacteristic
333C2×C4M16×D5311yescharacteristic
334C8—82no—
335C8—4242no—
336C22—21no—
337C22—21no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22#1681058611yescharacteristic
349C22C2×#1313313111yescharacteristic
350C22#1681060511yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—4241no—
358C4—2122no—
359C4—2122no—
360C4—2122no—
361C4#1681041712yes—
362C2—21no—
363C2—21no—
364C2—81no—
365C2—81no—
366C2#2027305111yescharacteristic
367C2C2×#1681041711yescharacteristic
368C2#2027639811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1465C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1465D53
hypocenterC53#11531
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20272658C2semidirect
4. #20272648C2semidirect
82. C2×Dic106M16semidirect
102. C4×Dic53M16semidirect
207. C53#11531semidirect

Maximal subgroups

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

subgroupindex
C4×#131332042
#202730532
#202726582
#202726482
C53×#14652
#1153153