Finite Group Databasebeta

← #27294833

Subgroups — #27294833

Order 13568.

388 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 79 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#20277983C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272744C212yesmaximal, complemented
4#20272741C212yesmaximal, complemented
5C53×#1822C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic
7#16810680—21no—
8#16810562C2211yescharacteristic
9#16810559C411yescharacteristic
10#16810556—21no—
11#16810546—21no—
12#16810543C2211yescharacteristic
13#16810535C411yescharacteristic
14C4×#11719919—21no—
15C4×C4⋊Dic53C411yescharacteristic
16C4×C22⋊Dic53—21no—
17#16810349C2212yes—
18#16810347—21no—
19C2×#117×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#343—21no—
22C53×#343C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C4×Dic106—41no—
25C4×Dic106—41no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C4⋊#88322—41no—
36C22⋊#88322—22no—
37#13133128—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C2×M16×C53—22no—
46#117×C53—41no—
47#117×C53—41no—
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53C2311yescharacteristic
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C22⋊C8×C53—41no—
55C22⋊C8×C53—41no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C2×Dic106—41no—
72C2×Dic106—42no—
73M16×C53—42no—
74M16×C53—42no—
75C2×C8×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C22⋊Dic53—42no—
85C22⋊Dic53—42no—
86C4⋊Dic53—42no—
87C4⋊Dic53—41no—
88#11719919—42no—
89#11719919—42no—
90#11719919—42no—
91C4×Dic53—42no—
92C4×Dic53—41no—
93C4×Dic53—41no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53C2×C22⋊C411yescharacteristic
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53#9911yescharacteristic
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#13712yes—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130Dic106—42no—
131Dic106—42no—
132C8×C53—82no—
133#88322—82no—
134#11816—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—42no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—81no—
142C22×C53C2×#9911yescharacteristic
143C22×C53#41911yescharacteristic
144C22×C53#42111yescharacteristic
145C4×C53—22no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152Dic53—42no—
153Dic53—42no—
154Dic53—81no—
155Dic53—81no—
156#2396—531no—
157#1822D5311yescharacteristic
158#1464—531no—
159#1354—532no—
160#1353—532no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53—81no—
164C2×C53#151111yescharacteristic
165C2×C53#212011yescharacteristic
166C2×C53#213211yescharacteristic
167C2×C4×Q8—531no—
168C2×#117C2×D5311yescharacteristic
169#432—1061no—
170#407—21no—
171#400—531no—
172#399—531no—
173#398—1061no—
174#396—1061no—
175#395—531no—
176#393—531no—
177C4×C4⋊C4—531no—
178C4×C4⋊C4—1061no—
179C4×C22⋊C4—1061no—
180#343—21no—
181#343C2×D5312yes—
182#324—532no—
183#323—1061no—
184C53#1181611yescharacteristic, complemented, Sylow
185C22×Q8—531no—
186C23×C4C22⋊D5311yescharacteristic
187C2×M16—22no—
188C4×Q8—2121no—
189C4×Q8—2121no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4C22⋊D5311yescharacteristic
198C2×C4⋊C4C22×D5311yescharacteristic
199C2×C22⋊C4—1061no—
200C2×C22⋊C4—1061no—
201C2×C22⋊C4C4×D5311yescharacteristic
202C2×C22⋊C4D21211yescharacteristic
203C2×C42—1061no—
204C2×C42—1061no—
205C2×C42C4×D5311yescharacteristic
206C2×C42D21211yescharacteristic
207C4⋊C8—2121no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—1062no—
211#97—2121no—
212#95—531no—
213#95—531no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217#95—1061no—
218C24#1171992111yescharacteristic
219C2×Q8—2121no—
220C2×Q8—2122no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—1061no—
226C22×C4—1061no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4#1171992111yescharacteristic
230C22×C4#1171992111yescharacteristic
231C22×C4#1171992111yescharacteristic
232C22×C4C2×C4×D5311yescharacteristic
233C22×C4C2×D21211yescharacteristic
234C22×C4C2×C22⋊D5311yescharacteristic
235M16—42no—
236M16—42no—
237C2×C8—42no—
238C2×C8—2122no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—2121no—
242C4⋊C4—2122no—
243C4⋊C4—2122no—
244C4⋊C4—2122no—
245C4⋊C4—2122no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—2122no—
249C22⋊C4—2122no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—2122no—
255C42—2121no—
256C42—2121no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C2×#1171992111yescharacteristic
263Q8—2122no—
264Q8—2122no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—1062no—
280C2×C4—1062no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4#1313313111yescharacteristic
292C2×C4#1313313111yescharacteristic
293C2×C4#1313324712yes—
294C8—82no—
295C8—4242no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1313313111yescharacteristic
304C22#1681063111yescharacteristic
305C22#1681064111yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—4241no—
314C4—4241no—
315C4—2122no—
316C4—2122no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2027324711yescharacteristic
321C2#2027635711yescharacteristic
322C2#2027639811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1822C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1822D53
hypocenterC53#11816
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #20272744C2semidirect
4. #20272741C2semidirect
184. C53#11816semidirect

Maximal subgroups

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

subgroupindex
#202779832
#202730492
#202727442
#202727412
C53×#18222
#1181653