Finite Group Databasebeta

← #27277567

Subgroups — #27277567

Order 13568.

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

normal subgroups only

This group has 656 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#20274643C211yesmaximal, characteristic
2#20274633C211yesmaximal, characteristic
3#20272568C214yesmaximal, complemented
4C53×#2297C211yesmaximal, characteristic
5C2×#13133136—21no—
6C2×#13133134—21no—
7C2×#13133132—21no—
8C2×#13133132—21no—
9#16810587C2212yes—
10#16810486C2212yes—
11C22×#41×C53C2211yescharacteristic
12C2×#117×C53C411yescharacteristic, complemented
13C22×C42×C53C411yescharacteristic, complemented
14C53×#391—21no—
15C53×#390—21no—
16C53×#386—21no—
17C53×#386—21no—
18C4×C22⋊C4×C53C412yescomplemented
19C4×C22⋊C4×C53C2212yes—
20C2×#11719917—22no—
21C2×C22⋊Dic53—22no—
22#13133136—42no—
23#13133134—42no—
24#13133132—42no—
25#13133132—42no—
26C22⋊#88322—24no—
27#13133124—24no—
28C2×#41×C53—24no—
29C2×#41×C53—42no—
30C2×#41×C53—42no—
31C22×Q8×C53D411yescharacteristic
32C22×D4×C53D411yescharacteristic
33C22×D4×C53D411yescharacteristic
34C22×D4×C53D411yescharacteristic
35C23×C4×C53—21no—
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39#117×C53—42no—
40C2×C4⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C42×C53—21no—
45C2×C42×C53—42no—
46C2×C42×C53C2×C412yes—
47C2×C42×C53C2×C412yes—
48#95×C53—22no—
49#95×C53—22no—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C22⋊C411yescharacteristic
52C24×C53C2×D411yescharacteristic
53#41×C53—416no—
54C2×Q8×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—44no—
77C22×C4×C53—42no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53#4114yes—
83C22×Dic53—42no—
84C4⋊C4×C53—42no—
85C22⋊C4×C53—44no—
86C22⋊C4×C53—44no—
87C22⋊C4×C53—42no—
88C42×C53—42no—
89C42×C53—42no—
90C42×C53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93#11719917—42no—
94#11719917—42no—
95C2×#88322—42no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53#9912yes—
107C23×C53C2×C22⋊C411yescharacteristic
108C23×C53#12312yes—
109Q8×C53—44no—
110D4×C53—44no—
111D4×C53—44no—
112D4×C53—44no—
113C2×C4×C53—28no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—24no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—44no—
125C2×C4×C53—44no—
126C2×C4×C53—42no—
127C2×C4×C53—48no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53#9912yes—
133C2×C4×C53#11712yes—
134C2×C4×C53#12312yes—
135C2×Dic53—84no—
136C2×Dic53—82no—
137#88322—82no—
138#8739—531nomaximal, Sylow
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53—24no—
143C22×C53—42no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53—41no—
147C22×C53—42no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53#38111yescharacteristic
153C22×C53C2×#9911yescharacteristic
154C22×C53C2×#9911yescharacteristic
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—22no—
158C4×C53—44no—
159C4×C53—41no—
160C4×C53—82no—
161Dic53—162no—
162#2297D5311yescharacteristic
163#1785—531no—
164#1782—531no—
165#1284—534no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—42no—
169C2×C53—42no—
170C2×C53—81no—
171C2×C53#181911yescharacteristic
172C2×C53#182712yes—
173C22×#41C2×D5311yescharacteristic
174C2×#117Dic5311yescharacteristic, complemented
175C22×C42Dic5311yescharacteristic, complemented
176C2×#101—1061no—
177C2×#100—1061no—
178C2×#99—1061no—
179C2×#99—1061no—
180#407—532no—
181#391—21no—
182#390—21no—
183#386—21no—
184#386—21no—
185#381—532no—
186C4×C22⋊C4Dic5312yescomplemented
187C4×C22⋊C4C2×D5312yes—
188C53#873911yescharacteristic, complemented, Sylow
189C2×#41—24no—
190C2×#41—42no—
191C2×#41—42no—
192C22×Q8C22⋊D5311yescharacteristic
193C22×D4C22⋊D5311yescharacteristic
194C22×D4C22⋊D5311yescharacteristic
195C22×D4C22⋊D5311yescharacteristic
196C23×C4—21no—
197C23×C4C2×Dic5311yescharacteristic
198C23×C4C2×Dic5311yescharacteristic
199C23×C4C22×D5311yescharacteristic
200C2×M16—1062no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—1062no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42C2×Dic5312yes—
210C2×C42C2×Dic5312yes—
211#101—2122no—
212#100—2122no—
213#99—2122no—
214#99—2122no—
215C22⋊C8—1064no—
216#95—22no—
217#95—22no—
218#95—1064no—
219C24C22⋊Dic5311yescharacteristic
220C24C22⋊Dic5311yescharacteristic
221C24C2×C22⋊D5311yescharacteristic
222#41—416no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—2122no—
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22⋊Dic5311yescharacteristic
250C22×C4#1171994514yes—
251C22×C4C22×Dic5311yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253M16—2122no—
254M16—2122no—
255C2×C8—2122no—
256C4⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—2122no—
261C22⋊C4—2122no—
262C42—42no—
263C42—42no—
264C42—42no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23#1313313212yes—
276C23C2×C22⋊Dic5311yescharacteristic
277C23#1313320012yes—
278Q8—44no—
279D4—44no—
280D4—44no—
281D4—44no—
282C2×C4—28no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—48no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—4242no—
302C2×C4—4244no—
303C2×C4#1313313212yes—
304C2×C4#1313317612yes—
305C2×C4#1313320012yes—
306C8—4242no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—24no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22#1681048611yescharacteristic
321C22C2×#1313313211yescharacteristic
322C22C2×#1313313211yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—44no—
327C4—41no—
328C4—82no—
329C4—8482no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2—81no—
335C2#2027478911yescharacteristic
336C2#2027481912yes—
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2297C2
socleC22×C53#381
prime radicalC22×#41×C53C22
cosocleC23×C4×C53C23
hypercenter#2297D53
hypocenterC53#8739
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
3. #20272568C2semidirect
12. C2×#117×C53C4semidirect
13. C22×C42×C53C4semidirect
18. C4×C22⋊C4×C53C4semidirect
174. C2×#117Dic53semidirect
175. C22×C42Dic53semidirect
186. C4×C22⋊C4Dic53semidirect
188. C53#8739semidirect

Maximal subgroups

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

subgroupindex
#202746432
#202746332
#202725682
C53×#22972
#873953