Finite Group Databasebeta

← #27279186

Subgroups — #27279186

Order 13568.

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

normal subgroups only

This group has 482 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#20277983C211yesmaximal, characteristic
2#20273143C212yesmaximal, complemented
3#20273029C211yesmaximal, characteristic
4#20272616C212yesmaximal, complemented
5C53×#2921C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7#16810676C2212yes—
8#16810562C411yescharacteristic
9#16810559C411yescharacteristic
10#16810556—21no—
11#16810546—21no—
12#16810543C411yescharacteristic
13#16810535C411yescharacteristic
14C4×#11719919—21no—
15C4×C4⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53—21no—
17#16810353—24no—
18#16810349—21no—
19#16810347—21no—
20C2×#117×C53C2211yescharacteristic
21C53×#433—24no—
22C53×#423C2212yes—
23C53×#407—21no—
24C4×M16×C53—21no—
25C22×Dic106C2×C411yescharacteristic
26C4×Dic106—22no—
27C4×Dic106—22no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C4⋊#88322—22no—
38C22⋊#88322—22no—
39C4×#88322—22no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×C53—22no—
48#117×C53—22no—
49#117×C53—22no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—22no—
57C4⋊C8×C53D414yes—
58C22⋊C8×C53—24no—
59#97×C53—22no—
60C4×C8×C53—22no—
61C24×C53C22⋊C411yescharacteristic
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C2×Dic106—22no—
77C2×Dic106—42no—
78M16×C53—42no—
79M16×C53—42no—
80C2×C8×C53—24no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53C2×D412yes—
88C42×C53#4112yes—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C4⋊Dic53—22no—
92C4⋊Dic53—42no—
93#11719919—42no—
94#11719919—42no—
95#11719919—42no—
96C4×Dic53—41no—
97C4×Dic53—41no—
98C4×Dic53—42no—
99C2×#88322—42no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53C4×D412yes—
118C2×C4×C53#12011yescharacteristic
119C2×C4×C53#12112yes—
120C2×C4×C53#12311yescharacteristic
121C2×C4×C53#13512yes—
122C2×C4×C53#13712yes—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135Dic106—24no—
136Dic106—42no—
137C8×C53—44no—
138#88322—82no—
139#9034—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—42no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—81no—
147C22×C53#38611yescharacteristic
148C22×C53#41611yescharacteristic
149C22×C53#41911yescharacteristic
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—41no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53#45212yes—
156C4×C53#48512yes—
157Dic53—42no—
158Dic53—42no—
159Dic53—81no—
160Dic53—81no—
161#2921D5311yescharacteristic
162#2396—531no—
163#1487—532no—
164#1458—531no—
165#1303—532no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—81no—
169C2×C53#187311yescharacteristic
170C2×C53#188411yescharacteristic
171C2×C53#189111yescharacteristic
172C2×C4×Q8—531no—
173C2×#117C2×D5311yescharacteristic
174#433—24no—
175#431—532no—
176#423C2×D5312yes—
177#407—21no—
178C4×M16—21no—
179#400—531no—
180#399—531no—
181#398—1061no—
182#396—1061no—
183#395—531no—
184#393—531no—
185C4×C4⋊C4—531no—
186C4×C4⋊C4—1061no—
187C4×C22⋊C4—1061no—
188Q8⋊C8—1064no—
189#324—1061no—
190#323—1061no—
191C53#903411yescharacteristic, complemented, Sylow
192C22×Q8—531no—
193C23×C4C22⋊D5311yescharacteristic
194C2×M16—22no—
195C4×Q8—1062no—
196C4×Q8—1062no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4C4×D5311yescharacteristic
205C2×C4⋊C4D21211yescharacteristic
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4C4×D5311yescharacteristic
209C2×C22⋊C4D21211yescharacteristic
210C2×C42—1061no—
211C2×C42—1061no—
212C2×C42C22⋊D5311yescharacteristic
213C2×C42C22×D5311yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—1062no—
216C4⋊C8C22⋊D5314yes—
217C22⋊C8—24no—
218C22⋊C8—1062no—
219#97—22no—
220C4×C8—22no—
221C4×C8—1062no—
222#95—531no—
223#95—531no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228C24#1171992111yescharacteristic
229C2×Q8—1062no—
230C2×Q8—2122no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4#1171992111yescharacteristic
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4C2×C4×D5311yescharacteristic
243C22×C4C2×D21211yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—24no—
248C2×C8—2122no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—1062no—
252C4⋊C4—2122no—
253C4⋊C4—2122no—
254C4⋊C4—2122no—
255C4⋊C4—2122no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—2122no—
259C22⋊C4—2122no—
260C42—21no—
261C42—21no—
262C42—2121no—
263C42—2121no—
264C42—2122no—
265C42#1171994512yes—
266C42C2×C22⋊D5312yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23C2×#1171992111yescharacteristic
273Q8—1064no—
274Q8—2122no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—1062no—
287C2×C4—1062no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4C4×C22⋊D5312yes—
299C2×C4#1313318611yescharacteristic
300C2×C4#1313319112yes—
301C2×C4#1313320011yescharacteristic
302C2×C4#1313323712yes—
303C2×C4#1313324712yes—
304C8—44no—
305C8—4242no—
306C22—21no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22#1681050211yescharacteristic
314C22#1681061711yescharacteristic
315C22#1681063111yescharacteristic
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—4241no—
322C4—4241no—
323C4—2122no—
324C4—2122no—
325C4#1681078512yes—
326C4#1681096412yes—
327C2—21no—
328C2—21no—
329C2—81no—
330C2#2027502711yescharacteristic
331C2#2027509811yescharacteristic
332C2#2027513311yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2921C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2921D53
hypocenterC53#9034
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #20273143C2semidirect
4. #20272616C2semidirect
191. C53#9034semidirect

Maximal subgroups

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

subgroupindex
#202779832
#202731432
#202730292
#202726162
C53×#29212
#903453