Finite Group Databasebeta

← #28299566

Subgroups — #28299566

Order 13568.

440 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 106 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#20290643C211yesmaximal, characteristic
2#20290643C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5C4×#13133216C211yesmaximal, characteristic
6#16811489—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133216C411yescharacteristic
13C2×C4×Dic106—21no—
14#16810527—21no—
15#16810522C2211yescharacteristic
16#16810520—21no—
17#16810519C2211yescharacteristic
18#16810492C411yescharacteristic
19C4×#11719919—21no—
20C4×C4⋊Dic53C411yescharacteristic
21C42×Dic53C2211yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22×Dic106—21no—
26#13133216—22no—
27#13133216C814yescomplemented
28C4×Dic106—44no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C42×C53—21no—
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C2×Dic106—24no—
63C2×Dic106—42no—
64C2×Dic106—42no—
65C2×#88324—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C4⋊C4×C53—21no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53C2×C812yes—
74C42×C53—22no—
75C42×C53—42no—
76C4⋊Dic53—21no—
77C4⋊Dic53—42no—
78C4⋊Dic53—41no—
79C4⋊Dic53C2×C812yes—
80#11719919—24no—
81#11719919—42no—
82#11719919—42no—
83#11719919—42no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C4×Dic53—41no—
89C4×Dic53—42no—
90C4×Dic53C2×C812yes—
91C4×Dic53M1612yes—
92C23×C53C2×C22⋊C411yescharacteristic
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53#10112yes—
107C2×C4×C53C22×C811yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53C22⋊C812yes—
124C2×Dic53C22⋊C812yes—
125C2×Dic53AGL(1,ℤ/8)12yes—
126C2×Dic53#13712yes—
127Dic106—48no—
128#88324—82no—
129#88324—82no—
130#7604—531nomaximal, Sylow
131C22×C53C2×C22⋊C811yescharacteristic
132C22×C53#41011yescharacteristic
133C22×C53C2×#10111yescharacteristic
134C22×C53#41811yescharacteristic
135C22×C53#41911yescharacteristic
136C22×C53#42111yescharacteristic
137C22×C53#42111yescharacteristic
138C4×C53—22no—
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—41no—
142C4×C53—42no—
143C4×C53—81no—
144Dic53—22no—
145Dic53—22no—
146Dic53—22no—
147Dic53—22no—
148Dic53—42no—
149Dic53—81no—
150C4×C4⋊Q8—531no—
151#1839—531no—
152#1839—531no—
153#1271—532no—
154#1271—532no—
155C2×C53#146111yescharacteristic
156C2×C53#147811yescharacteristic
157C2×C53#147911yescharacteristic
158C2×C53#151411yescharacteristic
159C2×C53#151511yescharacteristic
160C2×C53#153711yescharacteristic
161C2×C53#155611yescharacteristic
162C2×C4⋊Q8—531no—
163C2×C4×Q8—1061no—
164#391—1061no—
165#389—531no—
166#389—531no—
167#389—1061no—
168#384—531no—
169C4×C4⋊C4—531no—
170C4×C4⋊C4—1061no—
171C4×C4⋊C4#615711yescharacteristic
172C43—531no—
173#336—532no—
174#336—532no—
175#336—1061no—
176#336—1061no—
177#336—1061no—
178#336—1061no—
179C53#760411yescharacteristic, complemented, Sylow
180C22×Q8—1061no—
181C22×C8—1062no—
182C22×C8—1062no—
183C4⋊Q8—534no—
184C4⋊Q8—1062no—
185C4×Q8—2124no—
186C2×C4⋊C4—531no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4C2×#615711yescharacteristic
194C2×C4⋊C4C2×#615711yescharacteristic
195C2×C42—21no—
196C2×C42—531no—
197C2×C42—531no—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—1061no—
203C2×C42C2×#615711yescharacteristic
204#95—21no—
205#95—1061no—
206#95—1061no—
207#95—1061no—
208C2×Q8—1064no—
209C2×Q8—2122no—
210C2×Q8—2122no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4C22⋊#615711yescharacteristic
220C22×C4C22⋊#615711yescharacteristic
221C22×C4C22×#615711yescharacteristic
222C2×C8—2122no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C4⋊C4—21no—
229C4⋊C4—41no—
230C4⋊C4—532no—
231C4⋊C4—1064no—
232C4⋊C4—1061no—
233C4⋊C4—2121no—
234C4⋊C4—2122no—
235C4⋊C4—2122no—
236C4⋊C4—2122no—
237C4⋊C4—2122no—
238C4⋊C4D53⋊C812yes—
239C42—22no—
240C42—42no—
241C42—532no—
242C42—532no—
243C42—1062no—
244C42—1062no—
245C42—2121no—
246C42—2121no—
247C42—2121no—
248C42—2122no—
249C23C2×C22⋊#615711yescharacteristic
250Q8—2128no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—532no—
265C2×C4—532no—
266C2×C4—532no—
267C2×C4—532no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1062no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4#1313327812yes—
283C2×C4C2×D53⋊C811yescharacteristic
284C2×C4D53⋊M1611yescharacteristic
285C8—4242no—
286C8—4242no—
287C22D53⋊(C22⋊C8)11yescharacteristic
288C22#1681160211yescharacteristic
289C22C2×#1313327811yescharacteristic
290C22#1681161511yescharacteristic
291C22#1681161611yescharacteristic
292C22#1681162011yescharacteristic
293C22#1681162011yescharacteristic
294C4—21no—
295C4—21no—
296C4—22no—
297C4—41no—
298C4—42no—
299C4—81no—
300C4—1062no—
301C4—1062no—
302C4—1062no—
303C4—1062no—
304C4—4241no—
305C4—2122no—
306C2#2028995711yescharacteristic
307C2#2029001511yescharacteristic
308C2#2029001811yescharacteristic
309C2#2029013411yescharacteristic
310C2#2029013711yescharacteristic
311C2#2029020611yescharacteristic
312C2#2029024411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#7604
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
27. #13133216C8semidirect
179. C53#7604semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202895342
#202895332
C4×#131332162
#760453