Finite Group Databasebeta

← #28299577

Subgroups — #28299577

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—44no—
35C2×C4×Dic53—42no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53C2311yescharacteristic
40#13133124—22no—
41#13133124—22no—
42C23×C4×C53—21no—
43C23×C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—42no—
46C2×C4⋊C4×C53C2×C412yes—
47C24×C53C22⋊C411yescharacteristic
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—42no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53—22no—
59C22×Dic53—24no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—41no—
64C22×Dic53—42no—
65C22×Dic53—42no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53SD1612yes—
69C22×Dic53Q1612yes—
70C22×Dic53C2×D411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72#11719940—48no—
73C2×#88324—24no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C4⋊Dic53—24no—
82C4⋊Dic53—22no—
83C4⋊Dic53M1614yescomplemented
84#11719919—44no—
85#11719919—44no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53#9912yes—
95C23×C53#10312yes—
96C23×C53#10412yes—
97C23×C53C2×C22⋊C411yescharacteristic
98C2×C4×C53—28no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—84no—
107C2×C4×C53#10112yes—
108C2×C4×C53C2×M1612yes—
109C2×Dic53—24no—
110C2×Dic53—24no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—24no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—44no—
122C2×Dic53—48no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53#10312yes—
126C2×Dic53C2×SD1611yescharacteristic
127C2×Dic53C2×Q1611yescharacteristic
128C2×Dic53AGL(1,ℤ/8)12yes—
129#88324—44no—
130#88324—82no—
131#7607—531nomaximal, Sylow
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—22no—
138C22×C53#32814yes—
139C22×C53#40711yescharacteristic
140C22×C53C2×#9911yescharacteristic
141C22×C53C2×#10111yescharacteristic
142C22×C53C2×#10311yescharacteristic
143C22×C53#41811yescharacteristic
144C22×C53C2×#10411yescharacteristic
145C22×C53#42111yescharacteristic
146C4×C53—22no—
147C4×C53—22no—
148C4×C53—82no—
149Dic53—21no—
150Dic53—22no—
151Dic53—21no—
152Dic53—22no—
153Dic53—44no—
154Dic53—41no—
155C2×#389—531no—
156#1817—531no—
157#1728—531no—
158#1271—534no—
159C2×C53—24no—
160C2×C53#146311yescharacteristic
161C2×C53#148711yescharacteristic
162C2×C53#149011yescharacteristic
163C2×C53C2×#32811yescharacteristic
164C2×C53#151011yescharacteristic
165C2×C53#152811yescharacteristic
166C2×C53#154911yescharacteristic
167C22×M16—1061no—
168C22×C4⋊C4—531no—
169C22×C4⋊C4—1061no—
170C22×C4⋊C4#615711yescharacteristic
171C22×C42—531no—
172C2×C22⋊C8—1061no—
173#389—532no—
174#389—532no—
175#389—1061no—
176#389—1061no—
177C2×#95—1061no—
178#336—532no—
179#336—532no—
180#336—1061no—
181#336—1061no—
182C53#760711yescharacteristic, complemented, Sylow
183C23×C4—21no—
184C23×C4—531no—
185C23×C4—531no—
186C23×C4—1061no—
187C23×C4C2×#615711yescharacteristic
188C2×M16—1064no—
189C2×M16—2122no—
190C2×M16—2122no—
191C22×C8—1062no—
192C22×C8—1062no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—532no—
196C2×C4⋊C4—1064no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1062no—
199C2×C4⋊C4—1062no—
200C2×C4⋊C4—2124no—
201C2×C4⋊C4C2×#615712yes—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—2122no—
207C22⋊C8—2122no—
208#95—1062no—
209#95—1062no—
210C24C22⋊#615711yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—532no—
219C22×C4—532no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—1062no—
225C22×C4—1062no—
226C22×C4—1062no—
227C22×C4—1064no—
228C22×C4—1062no—
229C22×C4—1062no—
230C22×C4—2121no—
231C22×C4—2122no—
232C22×C4—2122no—
233C22×C4C22⋊#615711yescharacteristic
234C22×C4C22×#615711yescharacteristic
235M16—2128no—
236C2×C8—1064no—
237C2×C8—2122no—
238C2×C8—2122no—
239C2×C8—2122no—
240C2×C8—2122no—
241C2×C8—2122no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—534no—
245C4⋊C4—1064no—
246C4⋊C4—1062no—
247C4⋊C4—2124no—
248C4⋊C4—2124no—
249C42—2121no—
250C42—2121no—
251C42—2121no—
252C42—2121no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23#1313327512yes—
258C23#1313328012yes—
259C23#1313328212yes—
260C23C2×C22⋊#615711yescharacteristic
261C2×C4—28no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—84no—
270C2×C4—532no—
271C2×C4—531no—
272C2×C4—531no—
273C2×C4—532no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1064no—
281C2×C4—1064no—
282C2×C4—1064no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—2124no—
287C2×C4—2128no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4#1313327812yes—
291C2×C4D53⋊M1612yes—
292C8—4242no—
293C8—2124no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22#1681147714yes—
301C22#1681159511yescharacteristic
302C22C2×#1313327511yescharacteristic
303C22C2×#1313327811yescharacteristic
304C22C2×#1313328011yescharacteristic
305C22#1681161511yescharacteristic
306C22C2×#1313328211yescharacteristic
307C22#1681162111yescharacteristic
308C4—22no—
309C4—22no—
310C4—82no—
311C4—1061no—
312C4—1062no—
313C4—1061no—
314C4—1062no—
315C4—2124no—
316C4—2121no—
317C2—24no—
318C2#2028996511yescharacteristic
319C2#2029004511yescharacteristic
320C2#2029005611yescharacteristic
321C2C2×#1681147711yescharacteristic
322C2#2029012011yescharacteristic
323C2#2029017911yescharacteristic
324C2#2029023011yescharacteristic
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#7607
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
83. C4⋊Dic53M16semidirect
182. C53#7607semidirect

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
#760753