Finite Group Databasebeta

← #28299587

Subgroups — #28299587

Order 13568.

612 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290606C211yesmaximal, characteristic
2#20290378C211yesmaximal, characteristic
3#20289534C214yesmaximal, complemented
4C2×#16810519C211yesmaximal, characteristic
5C22×#11719940—21no—
6C2×C22⋊#88324—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C22×C4⋊Dic53C411yescharacteristic
12C22×#11719919—21no—
13C22×C4×Dic53C2211yescharacteristic
14#16810519—21no—
15#16810519—21no—
16#16810519C412yes—
17#16810519C2212yes—
18C2×#13133124—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C2×#11719940—24no—
21C2×#11719940—42no—
22C2×#11719940—42no—
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—42no—
26C23×Dic53—21no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C2×C4⋊Dic53—24no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C412yes—
32C2×#11719919—22no—
33C2×#11719919—22no—
34C2×#11719919—42no—
35C2×#11719919—42no—
36C2×C4×Dic53—42no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53C2311yescharacteristic
41#13133124—22no—
42#13133124—22no—
43C23×C4×C53—21no—
44C23×C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—42no—
47C2×C4⋊C4×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53—42no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53—24no—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—41no—
65C22×Dic53—42no—
66C22×Dic53—42no—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53D812yes—
70C22×Dic53SD1612yes—
71C22×Dic53C2×D411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73#11719940—48no—
74C2×#88324—24no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C4⋊Dic53—24no—
83C4⋊Dic53—22no—
84C4⋊Dic53M1614yescomplemented
85#11719919—44no—
86#11719919—42no—
87#11719919—42no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53#9912yes—
97C23×C53#10212yes—
98C23×C53#10412yes—
99C23×C53C2×C22⋊C411yescharacteristic
100C2×C4×C53—28no—
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—84no—
109C2×C4×C53#10112yes—
110C2×C4×C53C2×M1612yes—
111C2×Dic53—24no—
112C2×Dic53—24no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—24no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—44no—
124C2×Dic53—48no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53#10212yes—
128C2×Dic53C2×D811yescharacteristic
129C2×Dic53C2×SD1611yescharacteristic
130C2×Dic53#13712yes—
131#88324—44no—
132#88324—82no—
133#7610—531nomaximal, Sylow
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53#32714yes—
141C22×C53#40711yescharacteristic
142C22×C53C2×#9911yescharacteristic
143C22×C53C2×#10111yescharacteristic
144C22×C53C2×#10211yescharacteristic
145C22×C53#41911yescharacteristic
146C22×C53C2×#10411yescharacteristic
147C22×C53#42111yescharacteristic
148C4×C53—22no—
149C4×C53—22no—
150C4×C53—82no—
151Dic53—21no—
152Dic53—22no—
153Dic53—21no—
154Dic53—22no—
155Dic53—44no—
156Dic53—41no—
157C2×#389—531no—
158#1817—531no—
159#1728—531no—
160#1271—534no—
161C2×C53—24no—
162C2×C53#146311yescharacteristic
163C2×C53#148611yescharacteristic
164C2×C53#149111yescharacteristic
165C2×C53C2×#32711yescharacteristic
166C2×C53#151111yescharacteristic
167C2×C53#152811yescharacteristic
168C2×C53#154611yescharacteristic
169C22×M16—1061no—
170C22×C4⋊C4—531no—
171C22×C4⋊C4—1061no—
172C22×C4⋊C4#615711yescharacteristic
173C22×C42—531no—
174C2×C22⋊C8—1061no—
175#389—532no—
176#389—532no—
177#389—1061no—
178#389—1061no—
179C2×#95—1061no—
180#336—532no—
181#336—532no—
182#336—1061no—
183#336—1061no—
184C53#761011yescharacteristic, complemented, Sylow
185C23×C4—21no—
186C23×C4—531no—
187C23×C4—531no—
188C23×C4—1061no—
189C23×C4C2×#615711yescharacteristic
190C2×M16—1064no—
191C2×M16—2122no—
192C2×M16—2122no—
193C22×C8—1062no—
194C22×C8—1062no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—532no—
198C2×C4⋊C4—1064no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—1062no—
201C2×C4⋊C4—1062no—
202C2×C4⋊C4—2122no—
203C2×C4⋊C4—2122no—
204C2×C4⋊C4C2×#615712yes—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—2122no—
210C22⋊C8—2122no—
211#95—1062no—
212#95—1062no—
213C24C22⋊#615711yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—532no—
222C22×C4—532no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—1062no—
228C22×C4—1062no—
229C22×C4—1062no—
230C22×C4—1064no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4—2121no—
234C22×C4—2122no—
235C22×C4—2122no—
236C22×C4C22⋊#615711yescharacteristic
237C22×C4C22×#615711yescharacteristic
238M16—2128no—
239C2×C8—1064no—
240C2×C8—2122no—
241C2×C8—2122no—
242C2×C8—2122no—
243C2×C8—2122no—
244C2×C8—2122no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—534no—
248C4⋊C4—1064no—
249C4⋊C4—1062no—
250C4⋊C4—2124no—
251C4⋊C4—2122no—
252C4⋊C4—2122no—
253C42—2121no—
254C42—2121no—
255C42—2121no—
256C42—2121no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#1313327512yes—
262C23#1313327912yes—
263C23#1313328112yes—
264C23C2×C22⋊#615711yescharacteristic
265C2×C4—28no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—84no—
274C2×C4—532no—
275C2×C4—531no—
276C2×C4—531no—
277C2×C4—532no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1064no—
285C2×C4—1064no—
286C2×C4—1064no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—2124no—
291C2×C4—2128no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4#1313327812yes—
295C2×C4D53⋊M1612yes—
296C8—4242no—
297C8—2124no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22#1681147414yes—
305C22#1681159511yescharacteristic
306C22C2×#1313327511yescharacteristic
307C22C2×#1313327811yescharacteristic
308C22C2×#1313327911yescharacteristic
309C22#1681161611yescharacteristic
310C22C2×#1313328111yescharacteristic
311C22#1681162211yescharacteristic
312C4—22no—
313C4—22no—
314C4—82no—
315C4—1061no—
316C4—1062no—
317C4—1061no—
318C4—1062no—
319C4—2121no—
320C4—2124no—
321C2—24no—
322C2#2028996511yescharacteristic
323C2#2029004211yescharacteristic
324C2#2029005911yescharacteristic
325C2C2×#1681147411yescharacteristic
326C2#2029012511yescharacteristic
327C2#2029017711yescharacteristic
328C2#2029022411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#7610
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
84. C4⋊Dic53M16semidirect
184. C53#7610semidirect

Maximal subgroups

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

subgroupindex
#202906062
#202903782
#202895342
C2×#168105192
#761053