Finite Group Databasebeta

← #27293465

Subgroups — #27293465

Order 13568.

510 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 87 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#20277299C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272644C214yesmaximal
4C53×#1464C211yesmaximal, characteristic
5C23×Dic106C411yescharacteristic
6C2×#13133176C411yescharacteristic
7#16810680—21no—
8#16810529—21no—
9#16810529—21no—
10#16810528—21no—
11#16810528—21no—
12#16810484C412yes—
13#16810484C2212yes—
14#16810349—21no—
15#16810347C2212yes—
16C2×#117×C53C2211yescharacteristic
17C53×#432—21no—
18C53×#324—21no—
19C53×#323C2212yes—
20C22×Dic106—21no—
21C22×Dic106—21no—
22C22×Dic106—24no—
23C22×Dic106—41no—
24C22×Dic106—41no—
25C22×Dic106—42no—
26C23×Dic53C2×C412yes—
27#13133176—42no—
28C2×C4⋊Dic53—21no—
29C2×C22⋊Dic53C2×C412yes—
30C2×C4×Dic53—21no—
31C4⋊#88322—41no—
32C22⋊#88322—22no—
33#13133128—41no—
34#13133124—22no—
35#13133124—22no—
36#13133124—22no—
37#13133124—22no—
38C23×C4×C53C2×C411yescharacteristic
39#117×C53—41no—
40#117×C53—41no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53C2311yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C4⋊C8×C53—41no—
48C22⋊C8×C53—22no—
49#97×C53—41no—
50C24×C53C22×C411yescharacteristic
51C22×C4×C53—22no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—42no—
65C22×Dic53—42no—
66C22×Dic53M1614yes—
67C2×Dic106—24no—
68C2×Dic106—24no—
69C2×Dic106—48no—
70C2×Dic106—48no—
71C2×Dic106—42no—
72C2×Dic106—42no—
73C2×Dic106—41no—
74C2×Dic106—41no—
75C2×Dic106—42no—
76C2×C8×C53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C22⋊Dic53—42no—
86C4⋊Dic53—42no—
87C4×Dic53—42no—
88C2×#88322—42no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53C2×M1612yes—
95C2×C4×C53—24no—
96C2×C4×C53—24no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
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—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#10012yes—
112C2×Dic53—24no—
113C2×Dic53—44no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—44no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120Dic106—44no—
121Dic106—44no—
122Dic106—48no—
123C8×C53—82no—
124#88322—82no—
125#11504—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—22no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—41no—
132C22×C53—41no—
133C22×C53#40711yescharacteristic
134C22×C53C2×#9911yescharacteristic
135C22×C53C2×#10011yescharacteristic
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—22no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—41no—
143Dic53—44no—
144Dic53—82no—
145#2310—531no—
146#1464—531no—
147#1464D5311yescharacteristic
148#1317—534no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—42no—
152C2×C53#146211yescharacteristic
153C2×C53#212311yescharacteristic
154C2×C53#212311yescharacteristic
155C23×Q8—531no—
156C2×#117—531no—
157C2×#117C2×D5311yescharacteristic
158#432—21no—
159#432—1061no—
160#391—1061no—
161#391—1061no—
162#391—1061no—
163#391—1061no—
164#381—532no—
165#381—532no—
166#324—21no—
167#324—1061no—
168#323—532no—
169#323C2×D5312yes—
170C53#1150411yescharacteristic, complemented, Sylow
171C22×Q8—1064no—
172C22×Q8—1061no—
173C22×Q8—1061no—
174C22×Q8—2121no—
175C22×Q8—2121no—
176C22×Q8—2122no—
177C23×C4—532no—
178C23×C4C4×D5311yescharacteristic
179#117—41no—
180#117—41no—
181#117—2122no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4D21211yescharacteristic
184C2×C4⋊C4C22⋊D5311yescharacteristic
185C2×C22⋊C4—532no—
186C2×C22⋊C4C4×D5311yescharacteristic
187C2×C22⋊C4C22×D5311yescharacteristic
188C2×C42—1061no—
189C2×C42D21211yescharacteristic
190C2×C42C22⋊D5311yescharacteristic
191C4⋊C8—41no—
192C4⋊C8—2121no—
193C22⋊C8—22no—
194C22⋊C8—1062no—
195#97—41no—
196#97—2121no—
197#95—1062no—
198#95—1062no—
199#95—1062no—
200#95—1062no—
201C24C2×C4×D5311yescharacteristic
202C2×Q8—1064no—
203C2×Q8—1064no—
204C2×Q8—2128no—
205C2×Q8—2128no—
206C2×Q8—2122no—
207C2×Q8—2122no—
208C2×Q8—2121no—
209C2×Q8—2121no—
210C2×Q8—2122no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—534no—
216C22×C4—1062no—
217C22×C4—1062no—
218C22×C4—1062no—
219C22×C4—2122no—
220C22×C4—2122no—
221C22×C4#1171992111yescharacteristic
222C22×C4#1171992111yescharacteristic
223C22×C4#1171992111yescharacteristic
224C22×C4#1171992111yescharacteristic
225C22×C4C2×D21211yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227C2×C8—42no—
228C2×C8—2122no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—2122no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—2122no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—2122no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23C2×#1171992111yescharacteristic
245C23M16×D5312yes—
246Q8—2124no—
247Q8—2124no—
248Q8—2128no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—1064no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2124no—
268C2×C4—2122no—
269C2×C4—2122no—
270C2×C4—2124no—
271C2×C4#1313313111yescharacteristic
272C2×C4#1313313111yescharacteristic
273C2×C4#1313313312yes—
274C8—82no—
275C8—4242no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22#1681058611yescharacteristic
284C22C2×#1313313111yescharacteristic
285C22C2×#1313313311yescharacteristic
286C4—21no—
287C4—21no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—4242no—
294C4—2124no—
295C2—21no—
296C2—21no—
297C2—42no—
298C2#2027304111yescharacteristic
299C2#2027636811yescharacteristic
300C2#2027637011yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutator, prime radicalC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1464C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#1464D53
hypocenterC53#11504
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
170. C53#11504semidirect

Maximal subgroups

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

subgroupindex
#202772992
#202730492
#202726442
C53×#14642
#1150453