Finite Group Databasebeta

← #28299647

Subgroups — #28299647

Order 13568.

594 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 116 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#20290441C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3#20289533C214yesmaximal, complemented
4C2×#16810522C211yesmaximal, characteristic
5C2×C22⋊#88324—21no—
6C2×C22⋊#88324—21no—
7C2×#13133272—21no—
8C2×#13133272—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C22×C4⋊Dic53—21no—
12C22×C4⋊Dic53C411yescharacteristic
13C22×C4×Dic53C2211yescharacteristic
14#16810522—21no—
15#16810522—21no—
16#16810522C412yes—
17#16810522C2212yes—
18C2×#13133124—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22C22⋊#88324—24no—
23C22⋊#88324—24no—
24#13133272—42no—
25#13133272—42no—
26C23×Dic53—21no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—44no—
33C2×C4⋊Dic53—42no—
34C2×C4⋊Dic53C2×C412yes—
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—24no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C412yes—
47C24×C53C22⋊C411yescharacteristic
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—24no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—41no—
63C22×Dic53—41no—
64C22×Dic53—42no—
65C22×Dic53—42no—
66C22×Dic53—42no—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C22×Dic53C2×D411yescharacteristic
71C2×#88324—24no—
72C2×#88324—24no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53M1614yes—
80C4⋊Dic53—44no—
81C4⋊Dic53—42no—
82C4⋊Dic53—44no—
83C4⋊Dic53—42no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53#10112yes—
93C23×C53#10412yes—
94C23×C53#10412yes—
95C23×C53C2×C22⋊C411yescharacteristic
96C2×C4×C53—28no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—24no—
102C2×C4×C53—48no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53#9912yes—
106C2×C4×C53C2×M1612yes—
107C2×Dic53—21no—
108C2×Dic53—21no—
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—44no—
118C2×Dic53—44no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—84no—
123C2×Dic53AGL(1,ℤ/8)12yes—
124C2×Dic53#13712yes—
125#88324—44no—
126#88324—44no—
127#7627—531nomaximal, Sylow
128C22×C53—22no—
129C22×C53—22no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53#33014yes—
135C22×C53#40711yescharacteristic
136C22×C53C2×#9911yescharacteristic
137C22×C53C2×#10111yescharacteristic
138C22×C53#41811yescharacteristic
139C22×C53#41911yescharacteristic
140C22×C53C2×#10411yescharacteristic
141C22×C53C2×#10411yescharacteristic
142C4×C53—22no—
143C4×C53—22no—
144C4×C53—44no—
145Dic53—22no—
146Dic53—22no—
147Dic53—41no—
148Dic53—41no—
149Dic53—82no—
150C2×#389—531no—
151#1749—531no—
152#1749—531no—
153#1271—534no—
154C2×C53—24no—
155C2×C53#146311yescharacteristic
156C2×C53#149011yescharacteristic
157C2×C53#149111yescharacteristic
158C2×C53#150711yescharacteristic
159C2×C53#150811yescharacteristic
160C2×C53C2×#33011yescharacteristic
161C2×C53#155611yescharacteristic
162C22×C4⋊C4—531no—
163C22×C4⋊C4—1061no—
164C22×C4⋊C4#615711yescharacteristic
165C22×C42—531no—
166C2×C22⋊C8—1061no—
167C2×C22⋊C8—1061no—
168C2×#97—1061no—
169C2×#97—1061no—
170#389—532no—
171#389—532no—
172#389—1061no—
173#389—1061no—
174C2×#95—1061no—
175#336—532no—
176#336—532no—
177C53#762711yescharacteristic, complemented, Sylow
178C23×C4—21no—
179C23×C4—531no—
180C23×C4—531no—
181C23×C4—1061no—
182C23×C4C2×#615711yescharacteristic
183C22×C8—1062no—
184C22×C8—1062no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—532no—
188C2×C4⋊C4—1062no—
189C2×C4⋊C4—1062no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—2124no—
192C2×C4⋊C4—2122no—
193C2×C4⋊C4C2×#615712yes—
194C2×C42—531no—
195C2×C42—531no—
196C2×C42—531no—
197C2×C42—531no—
198C2×C42—2122no—
199C22⋊C8—1064no—
200C22⋊C8—1064no—
201#97—2122no—
202#97—2122no—
203#95—1062no—
204#95—1062no—
205C24C22⋊#615711yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—1062no—
218C22×C4—1062no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4—1062no—
222C22×C4—2121no—
223C22×C4—2121no—
224C22×C4—2122no—
225C22×C4—2122no—
226C22×C4—2122no—
227C22×C4C22⋊#615711yescharacteristic
228C22×C4C22×#615711yescharacteristic
229C2×C8—1064no—
230C2×C8—1064no—
231C2×C8—2122no—
232C2×C8—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—2122no—
238C4⋊C4—2124no—
239C4⋊C4—2124no—
240C4⋊C4—2122no—
241C4⋊C4#1171994014yes—
242C42—1062no—
243C42—1062no—
244C42—2121no—
245C42—2121no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1313327812yes—
251C23#1313328112yes—
252C23#1313328212yes—
253C23C2×C22⋊#615711yescharacteristic
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—532no—
264C2×C4—532no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—4244no—
276C2×C4—2124no—
277C2×C4—2124no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4#1313327612yes—
282C2×C4C2×#1171994012yes—
283C8—2124no—
284C8—2124no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22#1681148014yes—
292C22#1681159311yescharacteristic
293C22C2×#1313327611yescharacteristic
294C22C2×#1313327811yescharacteristic
295C22#1681161511yescharacteristic
296C22#1681161611yescharacteristic
297C22C2×#1313328111yescharacteristic
298C22C2×#1313328211yescharacteristic
299C4—22no—
300C4—22no—
301C4—44no—
302C4—1062no—
303C4—1062no—
304C4—4242no—
305C4—2121no—
306C4—2121no—
307C2—24no—
308C2#2028996311yescharacteristic
309C2#2029005411yescharacteristic
310C2#2029005711yescharacteristic
311C2#2029011211yescharacteristic
312C2#2029011411yescharacteristic
313C2C2×#1681148011yescharacteristic
314C2#2029024411yescharacteristic
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#7627
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. #20289533C2semidirect
177. C53#7627semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202895332
C2×#168105222
#762753