Finite Group Databasebeta

← #27286817

Subgroups — #27286817

Order 13568.

504 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 107 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#20285868C211yesmaximal, characteristic
2#20285866C211yesmaximal, characteristic
3#20283092C211yesmaximal, characteristic
4#20275691C211yesmaximal, characteristic
5#20273401C211yesmaximal, characteristic
6#20272547C211yesmaximal, characteristic
7C53×#1549C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9C2×#13133217C2211yescharacteristic
10#16811028—21no—
11#16811026—21no—
12#16810975—21no—
13#16810974—21no—
14#16810959—22no—
15#16810958—22no—
16#16810919—22no—
17#16810915—22no—
18#16810676C2211yescharacteristic
19#16810530—22no—
20#16810528—22no—
21#16810488—22no—
22C42×Dic53C2211yescharacteristic
23#16810384—22no—
24#16810368—22no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C53×#431C2211yescharacteristic
27C53×#423C2211yescharacteristic
28C53×#332—22no—
29C22×Dic106—21no—
30#13133217—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217D411yescharacteristic
35#13133217D411yescharacteristic
36#13133209—21no—
37#13133196—21no—
38#13133196—21no—
39C4×Dic106—21no—
40C4×Dic106—21no—
41#13133176—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53D412yes—
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53D412yes—
47#13133151—24no—
48#13133150—24no—
49C4⋊#88322D412yes—
50#13133142—24no—
51#13133140—24no—
52C22⋊#88322—21no—
53C4×#88322—21no—
54#13133124—28no—
55C22×Q8×C53—21no—
56C2×M16×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D412yes—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53D412yes—
67C22⋊C8×C53—21no—
68C4×C8×C53—21no—
69C2×Q8×C53—44no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C22×C4×C53—22no—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—22no—
77C22×Dic53—22no—
78C2×Dic106—21no—
79C2×Dic106—21no—
80C2×Dic106—44no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83M16×C53—42no—
84C2×C8×C53—22no—
85C2×C8×C53—41no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C42×C53C2×D411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53C2×D411yescharacteristic
93C42×C53#4111yescharacteristic
94C22⋊Dic53—41no—
95C4⋊Dic53—22no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—22no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103#11719919—41no—
104#11719919—41no—
105C4×Dic53—22no—
106C4×Dic53—42no—
107C4×Dic53—42no—
108C4×Dic53—41no—
109C4×Dic53Q1614yes—
110C2×#88322—22no—
111C23×C53#12011yescharacteristic
112Q8×C53—82no—
113Q8×C53—82no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—44no—
117C2×C4×C53—42no—
118C2×C4×C53#12011yescharacteristic
119C2×C4×C53#12011yescharacteristic
120C2×C4×C53#12111yescharacteristic
121C2×C4×C53#12111yescharacteristic
122C2×C4×C53#12111yescharacteristic
123C2×C4×C53#12711yescharacteristic
124C2×C4×C53C2×Q1612yes—
125C2×C4×C53#13712yes—
126C2×Dic53—24no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—44no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—42no—
133Dic106—42no—
134Dic106—42no—
135Dic106—82no—
136Dic106—82no—
137C8×C53—42no—
138C8×C53—81no—
139#88322—42no—
140#10295—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53#39211yescharacteristic
144C22×C53#45112yes—
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—82no—
148C4×C53#45412yes—
149C4×C53#45412yes—
150C4×C53#46212yes—
151C4×C53#49411yescharacteristic
152C4×C53#49711yescharacteristic
153Dic53—44no—
154Dic53—81no—
155Dic53—81no—
156Dic53—82no—
157#3230—531no—
158#3230—531no—
159#2862—531no—
160#1994—531no—
161#1549—531no—
162#1549D5311yescharacteristic
163#1276—531no—
164C2×C53—22no—
165C2×C53#200311yescharacteristic
166C2×C53#200311yescharacteristic
167C2×C53#201611yescharacteristic
168#533—531no—
169C2×C4⋊Q8—531no—
170C2×C4⋊Q8C2×D5311yescharacteristic
171#500—1061no—
172#500—1061no—
173#487—1061no—
174#487—1061no—
175#484—1062no—
176#484—1062no—
177#477—1062no—
178#477—1062no—
179#431—531no—
180#431C2×D5311yescharacteristic
181#423C2×D5311yescharacteristic
182#391—1062no—
183#391—1062no—
184#382—1062no—
185C43—531no—
186#339—1062no—
187#332—22no—
188#332—1062no—
189C53#1029511yescharacteristic, complemented, Sylow
190C22×Q8—21no—
191C22×Q8—1061no—
192C2×M16—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—531no—
198C4⋊Q8—531no—
199C4⋊Q8—1061no—
200C4⋊Q8—1061no—
201C4⋊Q8—1061no—
202C4⋊Q8—1061no—
203#125—1061no—
204C22⋊Q8—1061no—
205C22⋊Q8—1061no—
206C4×Q8—1061no—
207C4×Q8—1061no—
208#117—1061no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—532no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4D21212yes—
213C2×C42—532no—
214C2×C42—1061no—
215C2×C42—1061no—
216C2×C42C22×D5311yescharacteristic
217#107—1064no—
218#107—1064no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—532no—
222C4⋊C8C22⋊D5312yes—
223#103—1064no—
224#103—1064no—
225C22⋊C8—21no—
226C22⋊C8—1061no—
227C4×C8—21no—
228C4×C8—1061no—
229#95—1068no—
230C2×Q8—44no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—1061no—
234C2×Q8—1061no—
235C2×Q8—2124no—
236C2×Q8—2121no—
237C2×Q8—2121no—
238C22×C4—22no—
239C22×C4—1062no—
240C22×C4—1062no—
241C22×C4C2×D21211yescharacteristic
242C22×C4D4×D5311yescharacteristic
243C22×C4D4×D5311yescharacteristic
244M16—42no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—1062no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—1062no—
253C4⋊C4—1061no—
254C4⋊C4—1061no—
255C4⋊C4—1062no—
256C4⋊C4—1061no—
257C4⋊C4—1061no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C22⋊C4—2121no—
263C42—534no—
264C42—1062no—
265C42—2122no—
266C42—2122no—
267C42—2121no—
268C42#1171994511yescharacteristic
269C42D4×D5311yescharacteristic
270C42D4×D5311yescharacteristic
271C42C2×C22⋊D5311yescharacteristic
272C23#1313318711yescharacteristic
273Q8—82no—
274Q8—82no—
275Q8—4242no—
276Q8—4242no—
277Q8—2122no—
278Q8—2122no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—1064no—
284C2×C4—2124no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2122no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4#1313318511yescharacteristic
291C2×C4#1313318711yescharacteristic
292C2×C4#1313319011yescharacteristic
293C2×C4#1313319011yescharacteristic
294C2×C4#1313319111yescharacteristic
295C2×C4#1313321311yescharacteristic
296C2×C4Q16×D5312yes—
297C2×C4#1313324612yes—
298C8—42no—
299C8—81no—
300C8—2122no—
301C22—21no—
302C22—21no—
303C22#1681053211yescharacteristic
304C22#1681078012yes—
305C4—21no—
306C4—21no—
307C4—82no—
308C4—4241no—
309C4—4242no—
310C4—4241no—
311C4—2124no—
312C4#1681079412yes—
313C4#1681079812yes—
314C4#1681083212yes—
315C4#1681100111yescharacteristic
316C4#1681101411yescharacteristic
317C2—22no—
318C2#2027573811yescharacteristic
319C2#2027573911yescharacteristic
320C2#2027581611yescharacteristic
subgroupG/H
centerC22#16810532
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1549C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#1549D53
hypocenterC53#10295
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
189. C53#10295semidirect

Maximal subgroups

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

subgroupindex
#202858682
#202858662
#202830922
#202756912
#202734012
#202725472
C53×#15492
#1029553