Finite Group Databasebeta

← #27295243

Subgroups — #27295243

Order 13568.

372 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 77 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#20280078C211yesmaximal, characteristic
2#20274710C211yesmaximal, characteristic
3#20273037C211yesmaximal, characteristic
4#20272746C211yesmaximal, characteristic
5#20272744C211yesmaximal, characteristic
6#20272742C211yesmaximal, characteristic
7C53×#1353C211yesmaximal, characteristic
8C2×#13133182C411yescharacteristic
9C2×#13133182C2211yescharacteristic
10#16810673—21no—
11#16810646—21no—
12C2×#13133135—21no—
13#16810561C411yescharacteristic
14#16810544C411yescharacteristic
15#16810543C2211yescharacteristic
16#16810527—21no—
17#16810520—21no—
18#16810506—21no—
19#16810505—21no—
20#16810487C411yescharacteristic
21#16810392C2211yescharacteristic
22#16810349C2211yescharacteristic
23#16810348—21no—
24C53×#400C2211yescharacteristic
25C53×#343C2211yescharacteristic
26C53×#324C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C22×Dic106D411yescharacteristic
29C2×#11719917—21no—
30#13133182—41no—
31#13133182—41no—
32#13133182—41no—
33#13133182—41no—
34C2×C4⋊Dic53—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919C2×C411yescharacteristic
39C2×#11719919D411yescharacteristic
40C2×C4×Dic53—21no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44#13133146—41no—
45#13133146—41no—
46#13133146—41no—
47#13133135—42no—
48C22⋊#88322—21no—
49C8×Dic53—41no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124C2×C411yescharacteristic
56C2×M16×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2311yescharacteristic
59C22⋊C8×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53C2×C411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C2×Dic106—41no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C2×Dic106—42no—
78C2×Dic106SD1612yes—
79C2×Dic106Q1612yes—
80M16×C53—41no—
81M16×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊Dic53—42no—
87#11719919—41no—
88#11719919—41no—
89#11719919—42no—
90#11719919—42no—
91#11719919—41no—
92#11719919—41no—
93C4×Dic53—21no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—42no—
100#11719917—41no—
101#11719917—41no—
102C2×#88322—41no—
103C2×#88322—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53C2×SD1611yescharacteristic
117C2×C4×C53C2×Q1611yescharacteristic
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—22no—
123C2×Dic53—21no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53#9911yescharacteristic
132C2×Dic53#9911yescharacteristic
133C2×Dic53#10312yes—
134C2×Dic53#10312yes—
135Dic106—42no—
136Dic106—42no—
137Dic106—42no—
138Dic106—42no—
139C8×C53—81no—
140C8×C53—81no—
141#88322—81no—
142#88322—81no—
143#11905—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53C2×#9911yescharacteristic
147C22×C53C2×#10311yescharacteristic
148C22×C53#42111yescharacteristic
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—81no—
154C4×C53—81no—
155Dic53—21no—
156Dic53—21no—
157Dic53—22no—
158Dic53—22no—
159Dic53—42no—
160Dic53—41no—
161Dic53—81no—
162#2581—531no—
163#1801—531no—
164#1461—531no—
165#1354—531no—
166#1354—531no—
167#1353—531no—
168#1353D5311yescharacteristic
169C2×C53—21no—
170C2×C53—21no—
171C2×C53#152011yescharacteristic
172C2×C53#212011yescharacteristic
173C2×C53#213311yescharacteristic
174C2×C4×Q8—531no—
175C2×C4×Q8—531no—
176#431—1061no—
177#423—1061no—
178C2×#101—1061no—
179#400—531no—
180#400C2×D5311yescharacteristic
181#395—531no—
182#395—531no—
183#391—1061no—
184#389—1061no—
185#387—1061no—
186#387—1061no—
187#382—531no—
188#343—531no—
189#343C2×D5311yescharacteristic
190#324—531no—
191#324—1061no—
192#324C2×D5311yescharacteristic
193C53#1190511yescharacteristic, complemented, Sylow
194C22×Q8—531no—
195C22×Q8—531no—
196C2×M16—21no—
197C2×M16—1061no—
198C4×Q8—2121no—
199C4×Q8—2121no—
200C4×Q8—2121no—
201C4×Q8—2121no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4C4×D5311yescharacteristic
209C2×C4⋊C4C22×D5311yescharacteristic
210C2×C42—531no—
211C2×C42—531no—
212C2×C42—531no—
213C2×C42—1061no—
214C4⋊C8—2121no—
215C4⋊C8—2121no—
216C4⋊C8—2121no—
217#101—2122no—
218C22⋊C8—21no—
219C22⋊C8—1061no—
220C4×C8—2121no—
221#95—21no—
222#95—21no—
223#95—531no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95C4×D5311yescharacteristic
230C2×Q8—532no—
231C2×Q8—532no—
232C2×Q8—2121no—
233C2×Q8—2121no—
234C2×Q8—2121no—
235C2×Q8—2122no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4C2×C4×D5311yescharacteristic
245C22×C4D4×D5311yescharacteristic
246C22×C4D4×D5311yescharacteristic
247M16—41no—
248M16—41no—
249M16—2121no—
250M16—2121no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—2121no—
254C2×C8—2121no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2122no—
262C4⋊C4—2122no—
263C4⋊C4—2122no—
264C42—1061no—
265C42—1061no—
266C42—1061no—
267C42—1061no—
268C42—1061no—
269C42—1061no—
270C42—2122no—
271C23C22⋊C4×D5311yescharacteristic
272Q8—2122no—
273Q8—2122no—
274Q8—2122no—
275Q8—2122no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—531no—
288C2×C4—531no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1062no—
296C2×C4—1061no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4SD16×D5311yescharacteristic
305C2×C4Q16×D5311yescharacteristic
306C8—81no—
307C8—81no—
308C8—4241no—
309C8—4241no—
310C22—21no—
311C22—21no—
312C22#99×D5311yescharacteristic
313C22#103×D5311yescharacteristic
314C22#1681063811yescharacteristic
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—81no—
320C4—81no—
321C4—1062no—
322C4—1062no—
323C4—1061no—
324C4—1061no—
325C4—4241no—
326C4—2122no—
327C4—2121no—
328C2—21no—
329C2—21no—
330C2#2027328511yescharacteristic
331C2#2027635511yescharacteristic
332C2#2027640111yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1353C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1353D53
hypocenterC53#11905
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1353

Complements

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

NH ≅ G/Ntype
193. C53#11905semidirect

Maximal subgroups

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

subgroupindex
#202800782
#202747102
#202730372
#202727462
#202727442
#202727422
C53×#13532
#1190553