Finite Group Databasebeta

← #27296854

Subgroups — #27296854

Order 13568.

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

normal subgroups only

This group has 414 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#20275876C211yesmaximal, characteristic, complemented
2#20273722C211yesmaximal, characteristic, complemented
3#20273654C211yesmaximal, characteristic
4#20272721C211yesmaximal, characteristic, complemented
5#20272718C211yesmaximal, characteristic
6#20272649C211yesmaximal, characteristic, complemented
7C53×#2728C211yesmaximal, characteristic
8#16811254C2211yescharacteristic
9#16810953—21no—
10#16810774—21no—
11#16810632—21no—
12#16810392C2211yescharacteristic
13#16810383—21no—
14#16810359—21no—
15#16810359—21no—
16#16810349C2211yescharacteristic
17#16810349C2211yescharacteristic
18C2×C22⋊Q8×C53—21no—
19C53×#400—21no—
20C53×#398C2211yescharacteristic
21C53×#397—21no—
22C53×#394C2211yescharacteristic
23C53×#391C2211yescharacteristic
24C53×#388—21no—
25C53×#386—21no—
26C53×#385—21no—
27C2×#11719923—41no—
28C2×#11719917—21no—
29#13133216—41no—
30#13133208—41no—
31#13133196—21no—
32#13133196—21no—
33#13133176—21no—
34#13133148—41no—
35#13133141—41no—
36#13133141—41no—
37#13133140—41no—
38#13133138—41no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22×Q8×C53D411yescharacteristic
42C22×D4×C53D411yescharacteristic
43C23×C4×C53—21no—
44C22⋊Q8×C53—42no—
45C22⋊Q8×C53—42no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C42×C53D411yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53D411yescharacteristic
63#95×C53D411yescharacteristic
64C24×C53—21no—
65C2×Q8×C53—21no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53SD1612yescomplemented
69C2×D4×C53—41no—
70C2×D4×C53—42no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—42no—
77C22×C4×C53—81no—
78C22×C4×C53—81no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C2×Dic106—21no—
83C2×Dic106—21no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—81no—
96C42×C53—41no—
97C22⋊Dic53—41no—
98#11719923—82no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103#11719919—41no—
104#11719919—41no—
105C4×Dic53—41no—
106#11719917—41no—
107#11719917—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C23×C53—22no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53#12011yescharacteristic
119Q8×C53—42no—
120Q8×C53—81no—
121D4×C53—42no—
122D4×C53—81no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
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×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—84no—
145C2×C4×C53C2×SD1611yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×Dic53—41no—
148C2×Dic53—41no—
149Dic106—81no—
150Dic106—81no—
151#88322—81no—
152#88322—81no—
153#88322—81no—
154#12210—531nomaximal, Sylow
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53#45011yescharacteristic
165C22×C53#45511yescharacteristic
166C22×C53B(4,𝔽2)11yescharacteristic
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—42no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—81no—
173C4×C53—81no—
174C4×C53—81no—
175C4×C53—81no—
176Dic53—81no—
177Dic53—81no—
178#2728D5311yescharacteristic
179#2028—531no—
180#1611—531no—
181#1599—531no—
182#1346—531no—
183#1345—531no—
184#1319—531no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—82no—
188C2×C53#160111yescharacteristic
189C2×C53#219711yescharacteristic
190C2×C53#220311yescharacteristic
191#544—531no—
192C2×C22⋊Q8—21no—
193#483—1061no—
194#450—1061no—
195#419—1061no—
196#400—21no—
197#398C2×D5311yescharacteristic
198#397—21no—
199#394C2×D5311yescharacteristic
200#391C2×D5311yescharacteristic
201#388—21no—
202#386—21no—
203#385—21no—
204#343—531no—
205#339—1061no—
206#328—1061no—
207#328—1061no—
208#324—531no—
209#324—531no—
210C53#1221011yescharacteristic, complemented, Sylow
211C22×Q8C22⋊D5311yescharacteristic
212C22×D4C22⋊D5311yescharacteristic
213C23×C4—21no—
214C2×SD16—2121no—
215C2×M16—1061no—
216C4⋊Q8—2121no—
217#125—2121no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220C22⋊Q8—1061no—
221C22⋊Q8—1061no—
222#117—1061no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22⋊D5311yescharacteristic
227C2×C4⋊C4C22×D5311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42C22⋊D5311yescharacteristic
234#106—2121no—
235#103—2121no—
236#103—2121no—
237#103—2121no—
238#102—2121no—
239C22⋊C8—1061no—
240C22⋊C8—1061no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95C22⋊D5311yescharacteristic
247#95C22⋊D5311yescharacteristic
248C24—21no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—1061no—
253C2×Q8—1061no—
254C2×Q8#1171992312yescomplemented
255C2×D4—41no—
256C2×D4—42no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4C2×C22⋊D5311yescharacteristic
266C22×C4C2×C22⋊D5311yescharacteristic
267C22×C4C2×C22⋊D5311yescharacteristic
268SD16—4242no—
269M16—2121no—
270M16—2121no—
271C2×C8—2121no—
272C2×C8—2121no—
273C2×C8—2121no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—2121no—
285C4⋊C4—2121no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—81no—
292C22⋊C4—2121no—
293C42—41no—
294C42—2121no—
295C23—22no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23#1313318611yescharacteristic
303Q8—42no—
304Q8—81no—
305Q8—4241no—
306Q8—4241no—
307D4—42no—
308D4—81no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
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—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—84no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4C2×#1171992311yescharacteristic
334C2×C4#1313324311yescharacteristic
335C8—4241no—
336C8—4241no—
337C8—4241no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22#1681077411yescharacteristic
348C22#1681080111yescharacteristic
349C22#1681081011yescharacteristic
350C4—21no—
351C4—21no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—4241no—
360C4—4241no—
361C2—21no—
362C2—21no—
363C2—82no—
364C2#2027366711yescharacteristic
365C2#2027670511yescharacteristic
366C2#2027673611yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2728C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
hypercenter#2728D53
hypocenterC53#12210
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2728

Complements

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

NH ≅ G/Ntype
1. #20275876C2semidirect
2. #20273722C2semidirect
4. #20272721C2semidirect
6. #20272649C2semidirect
68. C2×Q8×C53SD16semidirect
210. C53#12210semidirect
254. C2×Q8#11719923semidirect

Maximal subgroups

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

subgroupindex
#202758762
#202737222
#202736542
#202727212
#202727182
#202726492
C53×#27282
#1221053