Finite Group Databasebeta

← #27277626

Subgroups — #27277626

Order 13568.

438 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 99 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#20277333C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic
3#20272622C212yesmaximal
4#20272620C212yesmaximal
5C53×#1463C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7C2×#13133176C411yescharacteristic
8#16810587C2212yes—
9#16810522—21no—
10#16810522—21no—
11#16810522—21no—
12#16810522—21no—
13#16810483C411yescharacteristic
14#16810483C411yescharacteristic
15#16810483C2211yescharacteristic
16#16810483C2211yescharacteristic
17#16810349—21no—
18#16810349—21no—
19C22×C4⋊C4×C53C2211yescharacteristic
20C53×#407C2212yes—
21C53×#324—21no—
22C53×#324—21no—
23C2×#11719917—22no—
24#13133176—42no—
25#13133176—42no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C22⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53C2×C411yescharacteristic
34C2×C22⋊Dic53C2×C411yescharacteristic
35C2×C22⋊Dic53C2×C411yescharacteristic
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C22⋊#88322—22no—
39C22⋊#88322—22no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53C2311yescharacteristic
47C2×M16×C53—22no—
48C2×C4⋊C4×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C22⋊C8×C53—22no—
55C22⋊C8×C53—22no—
56C24×C53C22×C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—81no—
64C22×C4×C53—81no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53#4112yes—
72C22×C4×C53#4112yes—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77M16×C53—82no—
78C2×C8×C53—42no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C22⋊Dic53—41no—
84C22⋊Dic53—41no—
85C22⋊Dic53—41no—
86C22⋊Dic53—41no—
87C4⋊Dic53—42no—
88C4⋊Dic53—42no—
89C4⋊Dic53—81no—
90C4⋊Dic53—81no—
91C4⋊Dic53—81no—
92C4⋊Dic53—81no—
93C4×Dic53—42no—
94C4×Dic53—42no—
95#11719917—82no—
96C2×#88322—42no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53C2×C22⋊C411yescharacteristic
102C23×C53#11711yescharacteristic
103C23×C53#11711yescharacteristic
104C2×C4×C53—24no—
105C2×C4×C53—24no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—81no—
119C2×C4×C53—81no—
120C2×C4×C53#10012yes—
121C2×C4×C53#10112yes—
122C2×C4×C53#12312yes—
123C2×C4×C53#12312yes—
124C2×Dic53—41no—
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—
135C2×Dic53—41no—
136C8×C53—82no—
137#88322—82no—
138#8757—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53#38111yescharacteristic
150C22×C53C2×#10011yescharacteristic
151C22×C53C2×#10111yescharacteristic
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—22no—
156C4×C53—41no—
157C4×C53—41no—
158Dic53—81no—
159Dic53—81no—
160Dic53—81no—
161Dic53—81no—
162#2316—531no—
163#1463—531no—
164#1463D5311yescharacteristic
165#1305—532no—
166#1305—532no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53—41no—
170C2×C53—41no—
171C2×C53#182111yescharacteristic
172C2×C53#182811yescharacteristic
173C2×C53#183011yescharacteristic
174C2×#117—531no—
175C2×#117—531no—
176C22×C4⋊C4C2×D5311yescharacteristic
177#407—532no—
178#407C2×D5312yes—
179#389—1061no—
180#389—1061no—
181#389—1061no—
182#389—1061no—
183#380—531no—
184#380—531no—
185#380—531no—
186#380—531no—
187#324—21no—
188#324—21no—
189#324—1061no—
190#324—1061no—
191C53#875711yescharacteristic, complemented, Sylow
192C23×C4C4×D5311yescharacteristic
193C23×C4C4×D5311yescharacteristic
194C23×C4C22×D5311yescharacteristic
195C2×M16—22no—
196C2×M16—1062no—
197#117—2122no—
198#117—2122no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4D21211yescharacteristic
208C2×C4⋊C4D21211yescharacteristic
209C2×C4⋊C4C22⋊D5311yescharacteristic
210C2×C4⋊C4C22⋊D5311yescharacteristic
211C2×C22⋊C4—531no—
212C2×C22⋊C4—531no—
213C2×C22⋊C4—531no—
214C2×C22⋊C4—531no—
215C2×C42—1061no—
216C2×C42—1061no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—1062no—
220C22⋊C8—1062no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225C24C2×C4×D5311yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4#1171992111yescharacteristic
239C22×C4#1171992111yescharacteristic
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4C2×D21211yescharacteristic
243C22×C4#1171994712yes—
244C22×C4#1171994712yes—
245C22×C4C2×C22⋊D5311yescharacteristic
246M16—82no—
247M16—4242no—
248C2×C8—42no—
249C2×C8—2122no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—4241no—
255C4⋊C4—4241no—
256C4⋊C4—4241no—
257C4⋊C4—4241no—
258C4⋊C4—2122no—
259C4⋊C4—2122no—
260C22⋊C4—2121no—
261C22⋊C4—2121no—
262C22⋊C4—2121no—
263C22⋊C4—2121no—
264C42—2122no—
265C42—2122no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23C2×#1171992111yescharacteristic
271C23#1313317311yescharacteristic
272C23#1313317311yescharacteristic
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
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×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4#1313313312yes—
302C2×C4#1313313512yes—
303C2×C4#1313319812yes—
304C2×C4#1313320112yes—
305C8—82no—
306C8—4242no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22#1681048411yescharacteristic
318C22C2×#1313313311yescharacteristic
319C22C2×#1313313511yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—41no—
326C4—4241no—
327C4—4241no—
328C4—4241no—
329C4—4241no—
330C2—21no—
331C2—21no—
332C2—41no—
333C2—41no—
334C2#2027479311yescharacteristic
335C2#2027482111yescharacteristic
336C2#2027482711yescharacteristic
subgroupG/H
centerC22#16810484
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1463C2
socleC22×C53#381
cosocleC23×C4×C53C23
hypercenter#1463D53
hypocenterC53#8757
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
191. C53#8757semidirect

Maximal subgroups

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

subgroupindex
#202773332
#202730452
#202726222
#202726202
C53×#14632
#875753