Finite Group Databasebeta

← #27293248

Subgroups — #27293248

Order 13568.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277118C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272660C212yesmaximal, complemented
4#20272658C212yesmaximal, complemented
5C53×#1458C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7C22×C4×Dic53C411yescharacteristic
8#16810676—21no—
9#16810492C411yescharacteristic
10#16810492C2211yescharacteristic
11#16810488C411yescharacteristic
12#16810488C2211yescharacteristic
13#16810481—21no—
14#16810481—21no—
15C4×#11719919—21no—
16C4×#11719919—21no—
17#16810349C2212yes—
18#16810347—21no—
19C2×#117×C53C2211yescharacteristic
20C53×#431—21no—
21C53×#324C2212yes—
22C53×#323—21no—
23C23×Dic53—21no—
24#13133176—41no—
25#13133176—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—22no—
36C2×C4×Dic53C814yescomplemented
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53C2×C411yescharacteristic
39C4⋊#88322—41no—
40C22⋊#88322—22no—
41C4×#88322—41no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C23×C4×C53C2×C411yescharacteristic
47#117×C53—41no—
48#117×C53—41no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C4⋊C8×C53—41no—
56C22⋊C8×C53—22no—
57C4×C8×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C2×C812yes—
66C22×C4×C53M1612yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—24no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—42no—
76C22×Dic53—42no—
77C2×C8×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C22⋊Dic53—42no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89#11719919—42no—
90#11719919—42no—
91#11719919—42no—
92#11719919—42no—
93C4×Dic53—24no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—41no—
97C4×Dic53—41no—
98C4×Dic53C2×C812yes—
99C4×Dic53C2×C812yes—
100C2×#88322—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53C22⋊C812yes—
106C23×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53C22⋊C812yes—
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#10112yes—
125C2×C4×C53C22×C811yescharacteristic
126C2×C4×C53C2×M1611yescharacteristic
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—48no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C8×C53—82no—
141#88322—82no—
142#11447—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53C2×C22⋊C811yescharacteristic
151C22×C53C2×#9911yescharacteristic
152C22×C53C2×#10111yescharacteristic
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53#32412yes—
160C4×C53#32412yes—
161Dic53—24no—
162Dic53—42no—
163Dic53—81no—
164Dic53—81no—
165#2285—531no—
166#1458—531no—
167#1458D5311yescharacteristic
168#1325—532no—
169#1324—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53C2×#32411yescharacteristic
174C2×C53#212611yescharacteristic
175C2×C53#213211yescharacteristic
176C2×#117—531no—
177C2×#117C2×D5311yescharacteristic
178C22×C42—531no—
179#431—21no—
180#431—1061no—
181#384—531no—
182#384—531no—
183#382—531no—
184#382—531no—
185#380—1061no—
186#380—1061no—
187C4×C4⋊C4—1061no—
188C4×C4⋊C4—1061no—
189#324—532no—
190#324C2×D5312yes—
191#323—21no—
192#323—1061no—
193C53#1144711yescharacteristic, complemented, Sylow
194C23×C4—1061no—
195C23×C4C4×D5311yescharacteristic
196#117—41no—
197#117—41no—
198#117—2121no—
199#117—2121no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4C4×D5311yescharacteristic
207C2×C4⋊C4C22×D5311yescharacteristic
208C2×C22⋊C4—1061no—
209C2×C22⋊C4D21211yescharacteristic
210C2×C22⋊C4C22⋊D5311yescharacteristic
211C2×C42—531no—
212C2×C42—531no—
213C2×C42—534no—
214C2×C42—1062no—
215C2×C42—1061no—
216C2×C42—1061no—
217C2×C42D21211yescharacteristic
218C2×C42C22⋊D5311yescharacteristic
219C4⋊C8—41no—
220C4⋊C8—2121no—
221C22⋊C8—22no—
222C22⋊C8—1062no—
223C4×C8—41no—
224C4×C8—2121no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229C24#1171992111yescharacteristic
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—1061no—
234C22×C4—1064no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—2122no—
239C22×C4—2122no—
240C22×C4C8×D5312yes—
241C22×C4#1171991212yes—
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4C2×C4×D5311yescharacteristic
246C22×C4C2×D21211yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248C2×C8—42no—
249C2×C8—2122no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—2121no—
253C4⋊C4—2121no—
254C4⋊C4—2122no—
255C4⋊C4—2122no—
256C4⋊C4—2122no—
257C4⋊C4—2122no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—2122no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—532no—
266C42—532no—
267C42—1064no—
268C42—1061no—
269C42—1061no—
270C42—2121no—
271C42—2121no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1313312912yes—
277C23C2×#1171992111yescharacteristic
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—2128no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4#1313312912yes—
306C2×C4#1313313111yescharacteristic
307C2×C4#1313313111yescharacteristic
308C2×C4#1313313512yes—
309C2×C4C2×C8×D5311yescharacteristic
310C2×C4C2×#1171991211yescharacteristic
311C8—82no—
312C8—4242no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22C2×#1313312911yescharacteristic
321C22C2×#1313313111yescharacteristic
322C22C2×#1313313511yescharacteristic
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—1064no—
330C4—4241no—
331C4—4241no—
332C4—2122no—
333C4#1681034812yes—
334C4#1681034812yes—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C2×#1681034811yescharacteristic
339C2#2027637811yescharacteristic
340C2#2027639811yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1458C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1458D53
hypocenterC53#11447
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20272660C2semidirect
4. #20272658C2semidirect
36. C2×C4×Dic53C8semidirect
193. C53#11447semidirect

Maximal subgroups

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

subgroupindex
#202771182
#202730292
#202726602
#202726582
C53×#14582
#1144753