Finite Group Databasebeta

← #28303363

Subgroups — #28303363

Order 13568.

630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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#20290572C211yesmaximal, characteristic
2#20290572C211yesmaximal, characteristic
3#20289593C214yesmaximal, complemented
4#384×D53C211yesmaximal, characteristic
5C2×#13133284C2212yes—
6C2×#13133284C2212yes—
7C2×#13133274—21no—
8C2×#13133274—21no—
9#16811489—21no—
10#16811489—21no—
11C2×C4⋊C4×D53—21no—
12C2×C4⋊C4×D53—21no—
13C2×C4⋊C4×D53C412yes—
14C2×C42×D53C2211yescharacteristic
15#16810493C411yescharacteristic
16#16810492—21no—
17#16810492—21no—
18#16810492C2212yes—
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—24no—
23#13133284—24no—
24#13133274—42no—
25#13133274—42no—
26C22×C4×D53—22no—
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C412yes—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C4⋊C4×D53—44no—
37C4⋊C4×D53—44no—
38C4⋊C4×D53—42no—
39C4⋊C4×D53—42no—
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43C42×D53—24no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C42×C53C2×C411yescharacteristic
48C23×D53C22⋊C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—22no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C2×D411yescharacteristic
56C22×Dic53C2×D411yescharacteristic
57C2×C4×D53—22no—
58C2×C4×D53—22no—
59C2×C4×D53—22no—
60C2×C4×D53—22no—
61C2×C4×D53—22no—
62C2×C4×D53—22no—
63C2×C4×D53—22no—
64C2×C4×D53—44no—
65C2×C4×D53—44no—
66C2×C4×D53—44no—
67C2×C4×D53D812yes—
68C2×C4×D53SD1612yes—
69C2×C4×D53SD1612yes—
70C2×C4×D53Q1612yes—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C42×C53—22no—
81C4⋊Dic53—42no—
82C4⋊Dic53—42no—
83C4⋊Dic53—42no—
84#11719919—44no—
85#11719919—42no—
86#11719919—42no—
87#11719919—44no—
88C4×Dic53—22no—
89C4×Dic53—22no—
90C4×Dic53#4114yes—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—21no—
96C22×D53#10112yes—
97C22×D53#10212yes—
98C22×D53#10312yes—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53#10212yes—
109C2×C4×C53#10312yes—
110C2×C4×C53#11712yes—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53#10012yes—
117C2×Dic53#12312yes—
118C2×Dic53#12312yes—
119C2×Dic53C2×D811yescharacteristic
120C2×Dic53C2×SD1611yescharacteristic
121C2×Dic53C2×SD1611yescharacteristic
122C2×Dic53C2×Q1611yescharacteristic
123C2×Dic53AGL(1,ℤ/8)12yes—
124C2×Dic53#13712yes—
125C4×D53—28no—
126C4×D53—24no—
127C4×D53—24no—
128C4×D53—24no—
129C4×D53—24no—
130C4×D53—88no—
131#88324—82no—
132#88324—82no—
133#8724—531nomaximal, Sylow
134C22×C53#38111yescharacteristic
135C22×C53C2×#10011yescharacteristic
136C22×C53C2×#10111yescharacteristic
137C22×C53C2×#10211yescharacteristic
138C22×C53C2×#10311yescharacteristic
139C22×C53#41811yescharacteristic
140C22×C53#41911yescharacteristic
141C2×D53—22no—
142C2×D53—22no—
143C2×D53—22no—
144C2×D53—22no—
145C2×D53—22no—
146C2×D53—22no—
147C2×D53#33214yes—
148C4×C53—22no—
149C4×C53—22no—
150C4×C53—22no—
151C4×C53—82no—
152Dic53—22no—
153Dic53—82no—
154Dic53#48012yes—
155Dic53#48112yes—
156Dic53#48312yes—
157Dic53#48412yes—
158C2×#384—531no—
159#1797—531no—
160#1797—531no—
161#1296—534no—
162C2×C53C2×#33211yescharacteristic
163C2×C53#154511yescharacteristic
164C2×C53#154811yescharacteristic
165C2×C53#155611yescharacteristic
166C2×C53#182111yescharacteristic
167C2×C53#182311yescharacteristic
168C2×C53#182411yescharacteristic
169D53—24no—
170C22×C4⋊C4—532no—
171C22×C4⋊C4—1061no—
172C22×C4⋊C4—1061no—
173C22×C42—531no—
174C2×C4⋊C8—532no—
175C2×C4⋊C8—532no—
176C2×C22⋊C8—1061no—
177C2×C22⋊C8—1061no—
178#384—532no—
179#384—531no—
180#384—1061no—
181#384—1061no—
182#384#615711yescharacteristic
183#336—1061no—
184#336—1061no—
185C53#872411yescharacteristic, complemented, Sylow
186C23×C4—531no—
187C23×C4—531no—
188C23×C4—531no—
189C23×C4—1062no—
190C22×C8—1062no—
191C22×C8—1062no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—532no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4—1062no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—2122no—
201C2×C4⋊C4—2122no—
202C2×C4⋊C4—2124no—
203C2×C4⋊C4—2124no—
204C2×C4⋊C4C2×#615712yes—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—1064no—
209C2×C42C2×#615711yescharacteristic
210C4⋊C8—1064no—
211C4⋊C8—1064no—
212C22⋊C8—2122no—
213C22⋊C8—2122no—
214C24—531no—
215C22×C4—22no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—532no—
220C22×C4—532no—
221C22×C4—532no—
222C22×C4—532no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—1062no—
226C22×C4—1062no—
227C22×C4—1062no—
228C22×C4—1062no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4—2124no—
232C22×C4—2124no—
233C22×C4—2124no—
234C22×C4C22⋊#615711yescharacteristic
235C22×C4C22⋊#615711yescharacteristic
236C22×C4C22×#615711yescharacteristic
237C2×C8—2122no—
238C2×C8—2122no—
239C2×C8—2122no—
240C2×C8—2122no—
241C2×C8—2122no—
242C2×C8—2122no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—2122no—
247C4⋊C4—2122no—
248C4⋊C4—2122no—
249C4⋊C4—2122no—
250C4⋊C4—2124no—
251C4⋊C4—2124no—
252C4⋊C4—2122no—
253C42—22no—
254C42—534no—
255C42—1062no—
256C42—1062no—
257C23—532no—
258C23—532no—
259C23—532no—
260C23—1061no—
261C23—1061no—
262C23—1061no—
263C23—1061no—
264C23C2×C22⋊#615711yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—532no—
275C2×C4—532no—
276C2×C4—531no—
277C2×C4—531no—
278C2×C4—531no—
279C2×C4—531no—
280C2×C4—532no—
281C2×C4—532no—
282C2×C4—532no—
283C2×C4—1064no—
284C2×C4—1064no—
285C2×C4—1064no—
286C2×C4—1064no—
287C2×C4—1068no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—4248no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4#1313327912yes—
295C2×C4#1313328012yes—
296C2×C4#1313329612yes—
297C8—4242no—
298C8—4242no—
299C22—534no—
300C22—1062no—
301C22—1062no—
302C22—1062no—
303C22—1062no—
304C22—1062no—
305C22—1062no—
306C22#1681156611yescharacteristic
307C22D53⋊#10011yescharacteristic
308C22D53⋊#10111yescharacteristic
309C22C2×#1313327911yescharacteristic
310C22C2×#1313328011yescharacteristic
311C22#1681161511yescharacteristic
312C22#1681161611yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—82no—
317C4—532no—
318C4—532no—
319C4—532no—
320C4—532no—
321C4—1062no—
322C4—4242no—
323C2—1064no—
324C2D53⋊#33211yescharacteristic
325C2#2029022311yescharacteristic
326C2#2029022911yescharacteristic
327C2#2029024511yescharacteristic
328C2#2029061511yescharacteristic
329C2#2029061811yescharacteristic
330C2#2029061911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#8724
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #20289593C2semidirect
185. C53#8724semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202895932
#384×D532
#872453