Finite Group Databasebeta

← #27296929

Subgroups — #27296929

Order 13568.

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

normal subgroups only

This group has 394 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#20275871C211yesmaximal, characteristic
2#20273756C211yesmaximal, characteristic, complemented
3#20273688C211yesmaximal, characteristic, complemented
4#20272745C211yesmaximal, characteristic, complemented
5#20272742C211yesmaximal, characteristic, complemented
6#20272647C211yesmaximal, characteristic
7C53×#2784C211yesmaximal, characteristic
8#16811254C2211yescharacteristic
9#16810958—21no—
10#16810779—21no—
11#16810628—21no—
12#16810392C2211yescharacteristic
13#16810383—21no—
14#16810359—21no—
15#16810359—21no—
16#16810349C2211yescharacteristic
17#16810349C2211yescharacteristic
18C53×#400C2211yescharacteristic
19C53×#396—21no—
20C53×#395C2211yescharacteristic
21C53×#393—21no—
22C53×#390C2211yescharacteristic
23C53×#387—21no—
24C53×#386—21no—
25C53×#385—21no—
26C53×#380—21no—
27C2×#11719925—41no—
28C2×#11719917—21no—
29#13133216—41no—
30#13133208—41no—
31#13133196—21no—
32#13133196—21no—
33#13133176—21no—
34#13133150—41no—
35#13133141—41no—
36#13133140—41no—
37#13133138—41no—
38#13133138—41no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22×Q8×C53D411yescharacteristic
42C22×D4×C53D411yescharacteristic
43C23×C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53C2311yescharacteristic
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C42×C53D411yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53D411yescharacteristic
61#95×C53D411yescharacteristic
62C24×C53—21no—
63C2×Q8×C53—41no—
64C2×Q8×C53—81no—
65C2×D4×C53—21no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C2×D4×C53Q1612yescomplemented
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—42no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C2×Dic106—21no—
81C2×Dic106—21no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—81no—
86C4⋊C4×C53—81no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—81no—
89C22⋊C4×C53—81no—
90C22⋊C4×C53—81no—
91C22⋊C4×C53—81no—
92C42×C53—41no—
93C22⋊Dic53—41no—
94#11719925—82no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99#11719919—41no—
100#11719919—41no—
101C4×Dic53—41no—
102#11719917—41no—
103#11719917—41no—
104C2×#88322—41no—
105C2×#88322—41no—
106C2×#88322—41no—
107C23×C53—22no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53#12011yescharacteristic
115Q8×C53—42no—
116Q8×C53—81no—
117D4×C53—42no—
118D4×C53—81no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—41no—
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—84no—
141C2×C4×C53C2×Q1611yescharacteristic
142C2×C4×C53#13711yescharacteristic
143C2×Dic53—41no—
144C2×Dic53—41no—
145Dic106—81no—
146Dic106—81no—
147#88322—81no—
148#88322—81no—
149#88322—81no—
150#12223—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53—42no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53#45111yescharacteristic
161C22×C53#45511yescharacteristic
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#2784D5311yescharacteristic
175#2027—531no—
176#1617—531no—
177#1605—531no—
178#1354—531no—
179#1353—531no—
180#1318—531no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—82no—
184C2×C53#160211yescharacteristic
185C2×C53#220011yescharacteristic
186C2×C53#220411yescharacteristic
187#544—531no—
188#484—1061no—
189#451—1061no—
190#418—1061no—
191#400C2×D5311yescharacteristic
192#396—21no—
193#395C2×D5311yescharacteristic
194#393—21no—
195#390C2×D5311yescharacteristic
196#387—21no—
197#386—21no—
198#385—21no—
199#380—21no—
200#343—531no—
201#339—1061no—
202#328—1061no—
203#328—1061no—
204#324—531no—
205#324—531no—
206C53#1222311yescharacteristic, complemented, Sylow
207C22×Q8C22⋊D5311yescharacteristic
208C22×D4C22⋊D5311yescharacteristic
209C23×C4—21no—
210C2×Q16—2121no—
211C2×M16—1061no—
212C4⋊Q8—2121no—
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⋊C4C22⋊D5311yescharacteristic
221C2×C4⋊C4C22×D5311yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42C22⋊D5311yescharacteristic
228#107—2121no—
229#103—2121no—
230#103—2121no—
231#102—2121no—
232#102—2121no—
233C22⋊C8—1061no—
234C22⋊C8—1061no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C22⋊D5311yescharacteristic
241#95C22⋊D5311yescharacteristic
242C24—21no—
243C2×Q8—41no—
244C2×Q8—81no—
245C2×Q8—1061no—
246C2×Q8—1061no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4#1171992512yescomplemented
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4C2×C22⋊D5311yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262Q16—4242no—
263M16—2121no—
264M16—2121no—
265C2×C8—2121no—
266C2×C8—2121no—
267C2×C8—2121no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
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—81no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—2121no—
285C42—41no—
286C42—2121no—
287C23—22no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1313318611yescharacteristic
295Q8—42no—
296Q8—81no—
297Q8—4241no—
298Q8—4241no—
299D4—42no—
300D4—81no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
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—84no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4C2×#1171992511yescharacteristic
326C2×C4#1313324811yescharacteristic
327C8—4241no—
328C8—4241no—
329C8—4241no—
330C22—21no—
331C22—21no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#1681077911yescharacteristic
340C22#1681080111yescharacteristic
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—82no—
356C2#2027367211yescharacteristic
357C2#2027672211yescharacteristic
358C2#2027674111yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2784C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
hypercenter#2784D53
hypocenterC53#12223
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2784

Complements

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

NH ≅ G/Ntype
2. #20273756C2semidirect
3. #20273688C2semidirect
4. #20272745C2semidirect
5. #20272742C2semidirect
68. C2×D4×C53Q16semidirect
206. C53#12223semidirect
250. C2×D4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202758712
#202737562
#202736882
#202727452
#202727422
#202726472
C53×#27842
#1222353