Finite Group Databasebeta

← #27286814

Subgroups — #27286814

Order 13568.

504 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 109 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#20285863C211yesmaximal, characteristic
2#20283092C211yesmaximal, characteristic
3#20275693C211yesmaximal, characteristic
4#20273400C211yesmaximal, characteristic
5#20273399C211yesmaximal, characteristic
6#20272547C211yesmaximal, characteristic
7C53×#3230C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9#16811026—21no—
10#16810973—21no—
11#16810956—22no—
12#16810918—22no—
13#16810676C2211yescharacteristic
14#16810647C2211yescharacteristic
15#16810530—22no—
16#16810528—22no—
17#16810488—22no—
18C42×Dic53C2211yescharacteristic
19#16810382—22no—
20#16810370—22no—
21#16810369—22no—
22C53×#533C2211yescharacteristic
23C2×C4⋊Q8×C53C2211yescharacteristic
24C53×#500—21no—
25C53×#487—21no—
26C53×#484—22no—
27C53×#477—22no—
28C53×#431C2211yescharacteristic
29C22×Dic106—21no—
30C2×#11719917—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D412yes—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53D412yes—
40#13133150—24no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322D412yes—
44#13133141—24no—
45C22⋊#88322—21no—
46C4×#88322—21no—
47#13133124—28no—
48C22×Q8×C53—21no—
49C4⋊Q8×C53—21no—
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53D411yescharacteristic
54C4⋊Q8×C53D411yescharacteristic
55#125×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60#117×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D412yes—
63C2×C42×C53C2311yescharacteristic
64#107×C53—24no—
65C4⋊C8×C53D412yes—
66#103×C53—24no—
67C22⋊C8×C53—21no—
68C4×C8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—44no—
72C2×Q8×C53—41no—
73C2×Q8×C53—41no—
74C22×C4×C53—22no—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—22no—
79C22×Dic53—22no—
80C2×Dic106—44no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83C2×C8×C53—22no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C42×C53—41no—
96C42×C53C2×D411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53C2×D411yescharacteristic
99C42×C53#4111yescharacteristic
100C4⋊Dic53—22no—
101C4⋊Dic53—22no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C4×Dic53—22no—
105C4×Dic53—42no—
106C4×Dic53—42no—
107C4×Dic53Q1614yes—
108#11719917—42no—
109C2×#88322—22no—
110C2×#88322—41no—
111C23×C53#12011yescharacteristic
112Q8×C53—42no—
113Q8×C53—42no—
114Q8×C53—82no—
115Q8×C53—82no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—44no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#12011yescharacteristic
123C2×C4×C53#12011yescharacteristic
124C2×C4×C53#12111yescharacteristic
125C2×C4×C53#12111yescharacteristic
126C2×C4×C53#12111yescharacteristic
127C2×C4×C53#12711yescharacteristic
128C2×C4×C53C2×Q1612yes—
129C2×C4×C53#13712yes—
130C2×Dic53—24no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—44no—
134C2×Dic53—42no—
135Dic106—82no—
136Dic106—82no—
137C8×C53—42no—
138#88322—42no—
139#88322—81no—
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—81no—
149C4×C53—81no—
150C4×C53#45412yes—
151C4×C53#45412yes—
152C4×C53#46212yes—
153C4×C53#49411yescharacteristic
154C4×C53#49711yescharacteristic
155Dic53—44no—
156Dic53—82no—
157#3230—531no—
158#3230D5311yescharacteristic
159#2862—531no—
160#1994—531no—
161#1549—531no—
162#1549—531no—
163#1276—531no—
164C2×C53—22no—
165C2×C53#200311yescharacteristic
166C2×C53#200311yescharacteristic
167C2×C53#201611yescharacteristic
168#533C2×D5311yescharacteristic
169C2×C4⋊Q8—531no—
170C2×C4⋊Q8C2×D5311yescharacteristic
171#500—21no—
172#500—1061no—
173#487—21no—
174#487—1061no—
175#484—22no—
176#484—1062no—
177#477—22no—
178#477—1062no—
179#431—531no—
180#431C2×D5311yescharacteristic
181#423—531no—
182#391—1062no—
183#391—1062no—
184#382—1062no—
185C43—531no—
186#339—1062no—
187#332—1062no—
188#332—1062no—
189C53#1029511yescharacteristic, complemented, Sylow
190C22×Q8—21no—
191C22×Q8—1061no—
192C2×M16—1061no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—1061no—
198C4⋊Q8—1061no—
199C4⋊Q8—1061no—
200C4⋊Q8—1061no—
201C4⋊Q8C22⋊D5311yescharacteristic
202C4⋊Q8C22⋊D5311yescharacteristic
203#125—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208#117—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—532no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4C22⋊D5312yes—
213C2×C42—532no—
214C2×C42—1061no—
215C2×C42—1061no—
216C2×C42C22×D5311yescharacteristic
217#107—24no—
218#107—1064no—
219C4⋊C8—532no—
220C4⋊C8—1061no—
221C4⋊C8—1061no—
222C4⋊C8C22⋊D5312yes—
223#103—24no—
224#103—1064no—
225C22⋊C8—21no—
226C22⋊C8—1061no—
227C4×C8—21no—
228C4×C8—1061no—
229#95—1068no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—44no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—2124no—
236C2×Q8—2121no—
237C2×Q8—2121no—
238C22×C4—22no—
239C22×C4—1062no—
240C22×C4—1062no—
241C22×C4D4×D5311yescharacteristic
242C22×C4D4×D5311yescharacteristic
243C22×C4C2×C22⋊D5311yescharacteristic
244M16—2122no—
245C2×C8—22no—
246C2×C8—1062no—
247C2×C8—2121no—
248C4⋊C4—21no—
249C4⋊C4—22no—
250C4⋊C4—21no—
251C4⋊C4—22no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—1062no—
259C4⋊C4—1062no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C22⋊C4—41no—
263C42—41no—
264C42—534no—
265C42—1062no—
266C42—2122no—
267C42—2122no—
268C42D4×D5311yescharacteristic
269C42#1171994711yescharacteristic
270C42C2×C22⋊D5311yescharacteristic
271C42C2×C22⋊D5311yescharacteristic
272C23#1313318511yescharacteristic
273Q8—42no—
274Q8—42no—
275Q8—82no—
276Q8—82no—
277Q8—4242no—
278Q8—4242no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—1064no—
286C2×C4—2124no—
287C2×C4—2122no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4#1313318511yescharacteristic
291C2×C4#1313318611yescharacteristic
292C2×C4#1313318811yescharacteristic
293C2×C4#1313318911yescharacteristic
294C2×C4#1313318911yescharacteristic
295C2×C4#1313321311yescharacteristic
296C2×C4Q16×D5312yes—
297C2×C4#1313324912yes—
298C8—42no—
299C8—4241no—
300C8—2122no—
301C22—21no—
302C22—21no—
303C22#1681053111yescharacteristic
304C22#1681077712yes—
305C4—21no—
306C4—21no—
307C4—82no—
308C4—81no—
309C4—81no—
310C4—4242no—
311C4—2124no—
312C4#1681079412yes—
313C4#1681079612yes—
314C4#1681083112yes—
315C4#1681100111yescharacteristic
316C4#1681101611yescharacteristic
317C2—22no—
318C2#2027573411yescharacteristic
319C2#2027573511yescharacteristic
320C2#2027581311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3230C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#3230D53
hypocenterC53#10295
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

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
#202858632
#202830922
#202756932
#202734002
#202733992
#202725472
C53×#32302
#1029553