Finite Group Databasebeta

← #27286801

Subgroups — #27286801

Order 13568.

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

normal subgroups only

This group has 588 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#20285660C211yesmaximal, characteristic
2#20283111C211yesmaximal, characteristic
3#20275693C211yesmaximal, characteristic
4#20273400C211yesmaximal, characteristic
5#20273395C211yesmaximal, characteristic
6#20272547C211yesmaximal, characteristic
7C53×#3230C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×#13133215—22no—
10#16811002—21no—
11#16810975—21no—
12#16810833—22no—
13#16810778—22no—
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16#16810492—22no—
17#16810492—22no—
18C42×Dic53C2211yescharacteristic
19#16810382—22no—
20#16810371—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×Dic106D412yes—
30C2×#11719925—24no—
31C2×#11719917—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133215—48no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×#11719919—28no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53D412yes—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133140—24no—
47C22⋊#88322—21no—
48C4×#88322—21no—
49C22×Q8×C53—21no—
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53D411yescharacteristic
55C4⋊Q8×C53D411yescharacteristic
56#125×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61#117×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53D412yes—
64C2×C42×C53C2311yescharacteristic
65#107×C53—24no—
66C4⋊C8×C53D412yes—
67#103×C53—24no—
68C22⋊C8×C53—21no—
69C4×C8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—44no—
73C2×Q8×C53—41no—
74C2×Q8×C53—41no—
75C22×C4×C53—22no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—22no—
80C22×Dic53—22no—
81C2×Dic106—22no—
82C2×Dic106—22no—
83C2×Dic106—44no—
84C2×C8×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C42×C53—41no—
97C42×C53C2×D411yescharacteristic
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53#4111yescharacteristic
101#11719925—84no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106#11719919—416no—
107C4×Dic53—24no—
108C4×Dic53—24no—
109C4×Dic53—22no—
110C4×Dic53Q1614yescomplemented
111#11719917—42no—
112C2×#88322—22no—
113C2×#88322—41no—
114C23×C53#12011yescharacteristic
115Q8×C53—42no—
116Q8×C53—42no—
117Q8×C53—82no—
118Q8×C53—82no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—44no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53#12011yescharacteristic
126C2×C4×C53#12011yescharacteristic
127C2×C4×C53#12111yescharacteristic
128C2×C4×C53#12111yescharacteristic
129C2×C4×C53#12111yescharacteristic
130C2×C4×C53#12711yescharacteristic
131C2×C4×C53C2×Q1612yes—
132C2×C4×C53#13712yes—
133C2×Dic53—24no—
134C2×Dic53—24no—
135C2×Dic53—24no—
136C2×Dic53—44no—
137C2×Dic53—42no—
138Dic106—44no—
139Dic106—82no—
140C8×C53—42no—
141#88322—42no—
142#88322—81no—
143#10293—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53#39211yescharacteristic
147C22×C53#45112yes—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—82no—
151C4×C53—81no—
152C4×C53—81no—
153C4×C53#45312yes—
154C4×C53#45612yes—
155C4×C53#46212yes—
156C4×C53#49411yescharacteristic
157C4×C53#49711yescharacteristic
158Dic53—28no—
159Dic53—82no—
160#3230D5311yescharacteristic
161#3209—531no—
162#2868—531no—
163#1994—531no—
164#1549—531no—
165#1548—531no—
166#1276—531no—
167C2×C53—22no—
168C2×C53#200211yescharacteristic
169C2×C53#200511yescharacteristic
170C2×C53#201611yescharacteristic
171#533C2×D5311yescharacteristic
172C2×C4⋊Q8—531no—
173C2×C4⋊Q8—1062no—
174C2×C4⋊Q8C2×D5311yescharacteristic
175#500—21no—
176#494—1061no—
177#487—21no—
178#487—1061no—
179#484—22no—
180#477—22no—
181#462—1062no—
182#451—1062no—
183#431—531no—
184#431C2×D5311yescharacteristic
185#423—531no—
186#384—1062no—
187#384—1062no—
188C43—531no—
189#339—1062no—
190#333—1062no—
191#332—1062no—
192C53#1029311yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8—532no—
195C2×Q16—1064no—
196C2×M16—1061no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—1061no—
202C4⋊Q8—1061no—
203C4⋊Q8—1061no—
204C4⋊Q8—1061no—
205C4⋊Q8—2128no—
206C4⋊Q8C22⋊D5311yescharacteristic
207C4⋊Q8C22⋊D5311yescharacteristic
208#125—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213#117—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1068no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4C22⋊D5312yes—
219C2×C42—532no—
220C2×C42—1061no—
221C2×C42—1061no—
222C2×C42C22×D5311yescharacteristic
223#107—24no—
224C4⋊C8—532no—
225C4⋊C8—1061no—
226C4⋊C8—1061no—
227C4⋊C8C22⋊D5312yes—
228#103—24no—
229#103—1064no—
230C22⋊C8—21no—
231C22⋊C8—1061no—
232C4×C8—21no—
233C4×C8—1061no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—44no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—1062no—
240C2×Q8—1062no—
241C2×Q8—2124no—
242C22×C4—22no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4D4×D5311yescharacteristic
246C22×C4D4×D5311yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248Q16—4244no—
249M16—2122no—
250C2×C8—22no—
251C2×C8—1062no—
252C2×C8—2121no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—21216no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C22⋊C4—41no—
269C42—41no—
270C42—534no—
271C42—1064no—
272C42—1064no—
273C42—1062no—
274C42D4×D5311yescharacteristic
275C42#1171994711yescharacteristic
276C42C2×C22⋊D5311yescharacteristic
277C42C2×C22⋊D5311yescharacteristic
278C23#1313318511yescharacteristic
279Q8—42no—
280Q8—42no—
281Q8—82no—
282Q8—82no—
283Q8—4242no—
284Q8—2124no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—1064no—
292C2×C4—1064no—
293C2×C4—1064no—
294C2×C4—2124no—
295C2×C4—2122no—
296C2×C4#1313318511yescharacteristic
297C2×C4#1313318611yescharacteristic
298C2×C4#1313318811yescharacteristic
299C2×C4#1313318911yescharacteristic
300C2×C4#1313318911yescharacteristic
301C2×C4#1313321311yescharacteristic
302C2×C4Q16×D5312yes—
303C2×C4#1313324912yes—
304C8—42no—
305C8—4241no—
306C8—2122no—
307C22—21no—
308C22—21no—
309C22#1681053111yescharacteristic
310C22#1681077712yes—
311C4—21no—
312C4—21no—
313C4—82no—
314C4—81no—
315C4—81no—
316C4—1068no—
317C4—4242no—
318C4#1681078912yes—
319C4#1681080612yes—
320C4#1681083112yes—
321C4#1681100111yescharacteristic
322C4#1681101611yescharacteristic
323C2—22no—
324C2#2027572711yescharacteristic
325C2#2027574911yescharacteristic
326C2#2027581311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3230C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#3230D53
hypocenterC53#10293
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
110. C4×Dic53Q16semidirect
192. C53#10293semidirect

Maximal subgroups

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

subgroupindex
#202856602
#202831112
#202756932
#202734002
#202733952
#202725472
C53×#32302
#1029353