Finite Group Databasebeta

← #27297103

Subgroups — #27297103

Order 13568.

386 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 63 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#20276060C211yesmaximal, characteristic, complemented
2#20273722C211yesmaximal, characteristic, complemented
3#20273654C211yesmaximal, characteristic
4#20272721C211yesmaximal, characteristic, complemented
5#20272718C211yesmaximal, characteristic
6#20272661C211yesmaximal, characteristic, complemented
7C53×#2823C211yesmaximal, characteristic
8#16811254C2211yescharacteristic
9#16810953—21no—
10#16810774—21no—
11#16810663—21no—
12#16810392C2211yescharacteristic
13#16810383—21no—
14#16810359—21no—
15#16810359—21no—
16#16810349C2211yescharacteristic
17#16810349C2211yescharacteristic
18C2×#126×C53—21no—
19C53×#398—21no—
20C53×#398C2211yescharacteristic
21C53×#396—21no—
22C53×#396—21no—
23C53×#396—21no—
24C53×#394—21no—
25C53×#394C2211yescharacteristic
26C53×#384C2211yescharacteristic
27C2×#11719923—41no—
28C2×#11719917—21no—
29#13133216—41no—
30#13133208—41no—
31#13133196—21no—
32#13133196—21no—
33#13133176—21no—
34#13133150—41no—
35#13133148—41no—
36#13133148—41no—
37#13133140—41no—
38#13133138—41no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22×D4×C53D411yescharacteristic
42#126×C53—42no—
43#126×C53—42no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C4⋊C4×C53C2311yescharacteristic
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C42×C53D411yescharacteristic
60#95×C53—21no—
61#95×C53—21no—
62#95×C53D411yescharacteristic
63#95×C53D411yescharacteristic
64C24×C53—21no—
65C2×D4×C53—41no—
66C2×D4×C53—81no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C2×Dic106—21no—
76C2×Dic106—21no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—81no—
85C4⋊C4×C53—81no—
86C4⋊C4×C53—81no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—81no—
92C22⋊C4×C53—81no—
93C22⋊C4×C53—81no—
94C42×C53—21no—
95C42×C53SD1612yescomplemented
96C22⋊Dic53—41no—
97#11719923—82no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102#11719919—41no—
103#11719919—41no—
104C4×Dic53—41no—
105#11719917—22no—
106#11719917—41no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53#12011yescharacteristic
118D4×C53—42no—
119D4×C53—81no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53C2×SD1611yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×Dic53—41no—
144C2×Dic53—41no—
145Dic106—81no—
146Dic106—81no—
147#88322—42no—
148#88322—81no—
149#88322—81no—
150#12249—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53#45011yescharacteristic
161C22×C53#45611yescharacteristic
162C22×C53B(4,𝔽2)11yescharacteristic
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—42no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—81no—
169C4×C53—81no—
170C4×C53—81no—
171C4×C53—81no—
172Dic53—81no—
173Dic53—81no—
174#2823D5311yescharacteristic
175#2060—531no—
176#1611—531no—
177#1599—531no—
178#1346—531no—
179#1345—531no—
180#1325—531no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—161no—
184C2×C53#160411yescharacteristic
185C2×C53#220011yescharacteristic
186C2×C53#220311yescharacteristic
187#544—531no—
188C2×#126—21no—
189#483—1061no—
190#450—1061no—
191#428—1061no—
192#398—21no—
193#398C2×D5311yescharacteristic
194#396—21no—
195#396—21no—
196#396—21no—
197#394—21no—
198#394C2×D5311yescharacteristic
199#384C2×D5311yescharacteristic
200#343—531no—
201#339—1061no—
202#328—1061no—
203#328—1061no—
204#324—531no—
205#324—531no—
206C53#1224911yescharacteristic, complemented, Sylow
207C22×D4C22⋊D5311yescharacteristic
208C2×SD16—2121no—
209C2×M16—1061no—
210C4⋊Q8—2121no—
211#126—42no—
212#126—42no—
213#125—2121no—
214C22⋊Q8—1061no—
215C22⋊Q8—1061no—
216#117—1061no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22⋊D5311yescharacteristic
224C2×C4⋊C4C22⋊D5311yescharacteristic
225C2×C4⋊C4C22×D5311yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42C22⋊D5311yescharacteristic
233#107—2121no—
234#106—2121no—
235#106—2121no—
236#103—2121no—
237#102—2121no—
238C22⋊C8—1061no—
239C22⋊C8—1061no—
240#95—21no—
241#95—21no—
242#95C22⋊D5311yescharacteristic
243#95C22⋊D5311yescharacteristic
244C24—21no—
245C2×Q8—1061no—
246C2×Q8—1061no—
247C2×D4—41no—
248C2×D4—81no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C2×C22⋊D5311yescharacteristic
255C22×C4C2×C22⋊D5311yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257SD16—4242no—
258M16—1062no—
259M16—2121no—
260C2×C8—2121no—
261C2×C8—2121no—
262C2×C8—2121no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C22⋊C4—2121no—
287C42—21no—
288C42—2121no—
289C42#1171992312yescomplemented
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1313318611yescharacteristic
298Q8—4241no—
299Q8—4241no—
300D4—42no—
301D4—81no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4C2×#1171992311yescharacteristic
326C2×C4#1313324311yescharacteristic
327C8—4241no—
328C8—4241no—
329C8—2122no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#1681077411yescharacteristic
340C22#1681080611yescharacteristic
341C22#1681081011yescharacteristic
342C4—21no—
343C4—21no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—4241no—
352C4—4241no—
353C2—21no—
354C2—21no—
355C2—161no—
356C2#2027368411yescharacteristic
357C2#2027672211yescharacteristic
358C2#2027673611yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2823C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
hypercenter#2823D53
hypocenterC53#12249
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2823

Complements

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

NH ≅ G/Ntype
1. #20276060C2semidirect
2. #20273722C2semidirect
4. #20272721C2semidirect
6. #20272661C2semidirect
95. C42×C53SD16semidirect
206. C53#12249semidirect
289. C42#11719923semidirect

Maximal subgroups

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

subgroupindex
#202760602
#202737222
#202736542
#202727212
#202727182
#202726612
C53×#28232
#1224953