Finite Group Databasebeta

← #27296922

Subgroups — #27296922

Order 13568.

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

normal subgroups only

This group has 346 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
2#20273756C211yesmaximal, characteristic
3#20273688C211yesmaximal, characteristic
4#20272745C211yesmaximal, characteristic
5#20272742C211yesmaximal, characteristic
6#20272649C211yesmaximal, characteristic
7C53×#2799C211yesmaximal, characteristic
8#16811254C2211yescharacteristic
9#16810958—21no—
10#16810779—21no—
11#16810632—21no—
12#16810392C2211yescharacteristic
13#16810383—21no—
14#16810359—21no—
15#16810359—21no—
16#16810349C2211yescharacteristic
17#16810349C2211yescharacteristic
18C53×#400—21no—
19C53×#400C2211yescharacteristic
20C53×#398—21no—
21C53×#395C2211yescharacteristic
22C53×#391—21no—
23C53×#391—21no—
24C53×#391C2211yescharacteristic
25C53×#387—21no—
26C53×#387—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#13133141—41no—
37#13133141—41no—
38#13133140—41no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22×Q8×C53D411yescharacteristic
42C22×Q8×C53D411yescharacteristic
43C2×C4⋊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×C42×C53—21no—
50C2×C42×C53D411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
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
62C2×Q8×C53—21no—
63C2×Q8×C53—42no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53Q1612yescomplemented
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C2×Dic106—21no—
78C2×Dic106—21no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—81no—
84C4⋊C4×C53—81no—
85C42×C53—41no—
86C42×C53—81no—
87C22⋊Dic53—41no—
88#11719925—82no—
89C4⋊Dic53—21no—
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93#11719919—41no—
94#11719919—41no—
95C4×Dic53—41no—
96#11719917—41no—
97#11719917—41no—
98C2×#88322—41no—
99C2×#88322—41no—
100C2×#88322—41no—
101C23×C53#12011yescharacteristic
102Q8×C53—42no—
103Q8×C53—42no—
104Q8×C53—81no—
105Q8×C53—81no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53C2×Q1611yescharacteristic
130C2×C4×C53#13711yescharacteristic
131C2×Dic53—41no—
132C2×Dic53—41no—
133Dic106—81no—
134Dic106—81no—
135#88322—81no—
136#88322—81no—
137#88322—81no—
138#12222—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—21no—
141C22×C53#45111yescharacteristic
142C22×C53#45511yescharacteristic
143C22×C53B(4,𝔽2)11yescharacteristic
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—42no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53—42no—
150C4×C53—81no—
151C4×C53—81no—
152C4×C53—81no—
153C4×C53—81no—
154Dic53—81no—
155Dic53—81no—
156#2799D5311yescharacteristic
157#2028—531no—
158#1617—531no—
159#1605—531no—
160#1354—531no—
161#1353—531no—
162#1319—531no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53#160211yescharacteristic
166C2×C53#219711yescharacteristic
167C2×C53#220311yescharacteristic
168#544—531no—
169#484—1061no—
170#451—1061no—
171#419—1061no—
172#400—21no—
173#400C2×D5311yescharacteristic
174#398—21no—
175#395C2×D5311yescharacteristic
176#391—21no—
177#391—21no—
178#391C2×D5311yescharacteristic
179#387—21no—
180#387—21no—
181#343—531no—
182#339—1061no—
183#328—1061no—
184#328—1061no—
185#324—531no—
186#324—531no—
187C53#1222211yescharacteristic, complemented, Sylow
188C22×Q8C22⋊D5311yescharacteristic
189C22×Q8C22⋊D5311yescharacteristic
190C2×Q16—2121no—
191C2×M16—1061no—
192C4⋊Q8—2121no—
193#125—2121no—
194C22⋊Q8—1061no—
195C22⋊Q8—1061no—
196#117—1061no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C22⋊D5311yescharacteristic
202C2×C4⋊C4C22×D5311yescharacteristic
203C2×C42—21no—
204C2×C42C22⋊D5311yescharacteristic
205#107—2121no—
206#103—2121no—
207#103—2121no—
208#103—2121no—
209#103—2121no—
210C22⋊C8—1061no—
211C22⋊C8—1061no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C22⋊D5311yescharacteristic
222#95C22⋊D5311yescharacteristic
223C2×Q8—21no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—1061no—
229C2×Q8—1061no—
230C2×Q8#1171992512yescomplemented
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4C2×C22⋊D5311yescharacteristic
238C22×C4C2×C22⋊D5311yescharacteristic
239C22×C4C2×C22⋊D5311yescharacteristic
240Q16—4242no—
241M16—2121no—
242M16—2121no—
243C2×C8—2121no—
244C2×C8—2121no—
245C2×C8—2121no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—1061no—
253C4⋊C4—1061no—
254C4⋊C4—1061no—
255C4⋊C4—1061no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C22⋊C4—2121no—
259C42—41no—
260C42—81no—
261C42—2121no—
262C23#1313318611yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—81no—
266Q8—81no—
267Q8—4241no—
268Q8—4241no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4C2×#1171992511yescharacteristic
295C2×C4#1313324811yescharacteristic
296C8—4241no—
297C8—4241no—
298C8—4241no—
299C22—21no—
300C22—21no—
301C22#1681077911yescharacteristic
302C22#1681080111yescharacteristic
303C22#1681081011yescharacteristic
304C4—21no—
305C4—21no—
306C4—42no—
307C4—41no—
308C4—41no—
309C4—42no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—4241no—
315C4—4241no—
316C2—21no—
317C2—21no—
318C2#2027367211yescharacteristic
319C2#2027670511yescharacteristic
320C2#2027673611yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2799C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2799D53
hypocenterC53#12222
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2799

Complements

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

NH ≅ G/Ntype
67. C2×Q8×C53Q16semidirect
187. C53#12222semidirect
230. C2×Q8#11719925semidirect

Maximal subgroups

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

subgroupindex
#202758762
#202737562
#202736882
#202727452
#202727422
#202726492
C53×#27992
#1222253