Finite Group Databasebeta

← #28308183

Subgroups — #28308183

Order 13568.

574 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 76 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#20289963C212yesmaximal, complemented
2#20289646C212yesmaximal, complemented
3#20289644C212yesmaximal, complemented
4#20277303C211yesmaximal, characteristic
5#16811593—22no—
6#16811468—22no—
7#16811468C2214yes—
8C22×#11719945C411yescharacteristic
9C22×C4⋊Dic53C2211yescharacteristic
10#16810529C411yescharacteristic
11#16810529C2211yescharacteristic
12#16810525C411yescharacteristic
13#16810525C2211yescharacteristic
14#16810500—22no—
15#16810480—22no—
16C2×#117×C53C411yescharacteristic
17C2×#11719940—42no—
18C22⋊#88324—24no—
19C22⋊#88324—42no—
20C2×#11719945—22no—
21C2×#11719945—22no—
22C2×#11719945—42no—
23C2×#11719945—42no—
24C22×Dic106C2×C411yescharacteristic
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C23×Dic53D412yes—
28C22×C4×D53—21no—
29C2×C4⋊Dic53—42no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53D412yes—
32C2×C4⋊Dic53C2311yescharacteristic
33C2×#11719921—22no—
34C2×#11719921—22no—
35#13133124—22no—
36#13133124—22no—
37C23×C4×C53C2×C411yescharacteristic
38#117×C53—42no—
39C2×C4⋊C4×C53—21no—
40C2×C22⋊C4×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C24×C53C22⋊C411yescharacteristic
44C23×D53—21no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C2×C22⋊D53—24no—
54C2×C22⋊D53—42no—
55C2×C22⋊D53—42no—
56C22×Dic53—22no—
57C22×Dic53—21no—
58C22×Dic53—22no—
59C22×Dic53—82no—
60C22×Dic53C22⋊C412yes—
61C22×Dic53C2×D412yes—
62#11719945—48no—
63#11719945—48no—
64C2×D212—21no—
65C2×D212—41no—
66C2×D212—41no—
67C2×D212M1612yescomplemented
68C2×C4×D53—24no—
69C2×C4×D53—42no—
70C2×C4×D53—42no—
71C2×Dic106—21no—
72C2×Dic106—41no—
73C2×Dic106—41no—
74C2×Dic106M1612yescomplemented
75#11719940—84no—
76C2×#88324—44no—
77C4⋊C4×C53—42no—
78C22⋊C4×C53—42no—
79C42×C53—41no—
80C42×C53—41no—
81#11719921—82no—
82#11719921—82no—
83C4⋊Dic53—41no—
84C4⋊Dic53—42no—
85C4⋊Dic53—41no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—41no—
89C23×C53—41no—
90C23×C53#9912yes—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—22no—
93C22×D53—42no—
94C22×D53—42no—
95C22×D53—41no—
96C22×D53—41no—
97C2×C4×C53—24no—
98C2×C4×C53—24no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53#10012yes—
110C2×C4×C53C2×M1611yescharacteristic
111C2×C4×C53C2×M1611yescharacteristic
112C22⋊D53—48no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—44no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—82no—
122D212—42no—
123D212—42no—
124C4×D53—48no—
125Dic106—42no—
126Dic106—42no—
127#88324—84no—
128#11508—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53#40711yescharacteristic
137C22×C53C2×#9911yescharacteristic
138C22×C53C2×#10011yescharacteristic
139C2×D53—44no—
140C2×D53—42no—
141C2×D53—82no—
142C2×D53—81no—
143C2×D53—81no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—22no—
147C4×C53—81no—
148C4×C53—81no—
149Dic53—42no—
150Dic53—42no—
151Dic53—42no—
152#2311—531no—
153#1463—532no—
154#1319—532no—
155#1318—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—42no—
159C2×C53#146211yescharacteristic
160C2×C53#212711yescharacteristic
161C2×C53#213311yescharacteristic
162D53—82no—
163C22×#41—531no—
164C2×#117#615711yescharacteristic
165C22×C4⋊C4—531no—
166#407—1062no—
167#391—531no—
168#391—531no—
169#390—531no—
170#390—531no—
171#386—1062no—
172#380—1062no—
173#324—534no—
174#324—1062no—
175C53#1150811yescharacteristic, complemented, Sylow
176C2×#41—1062no—
177C2×#41—1062no—
178C2×#41—2122no—
179C2×#41—2122no—
180C22×Q8—531no—
181C22×D4—531no—
182C22×D4—1061no—
183C23×C4—532no—
184C23×C4—1061no—
185C23×C4C2×#615711yescharacteristic
186C2×M16—2122no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—532no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—2122no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1062no—
196C2×C42C2×#615711yescharacteristic
197C2×C42C2×#615711yescharacteristic
198C22⋊C8—1064no—
199C22⋊C8—2122no—
200#95—1062no—
201#95—1062no—
202C24—1061no—
203C24C22⋊#615711yescharacteristic
204#41—2128no—
205#41—2128no—
206C2×Q8—532no—
207C2×Q8—1061no—
208C2×Q8—2121no—
209C2×Q8—2121no—
210C2×D4—532no—
211C2×D4—1064no—
212C2×D4—1061no—
213C2×D4—2121no—
214C2×D4—2121no—
215C2×D4—2122no—
216C2×D4—2122no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—532no—
224C22×C4—532no—
225C22×C4—1064no—
226C22×C4—1062no—
227C22×C4—1062no—
228C22×C4—1061no—
229C22×C4—4242no—
230C22×C4—2122no—
231C22×C4—2122no—
232C22×C4C22⋊#615711yescharacteristic
233C22×C4C22×#615711yescharacteristic
234M16—4244no—
235C2×C8—2124no—
236C4⋊C4—42no—
237C4⋊C4—2122no—
238C4⋊C4—2121no—
239C4⋊C4—2121no—
240C22⋊C4—42no—
241C22⋊C4—4242no—
242C22⋊C4—4242no—
243C42—41no—
244C42—41no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—1062no—
250C23—2122no—
251C23—2122no—
252C23—2121no—
253C23—2121no—
254C23#1313327612yes—
255C23C2×C22⋊#615711yescharacteristic
256Q8—2122no—
257Q8—2122no—
258D4—2122no—
259D4—2122no—
260D4—2128no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—4242no—
277C2×C4—2128no—
278C2×C4—2124no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4#1313327712yes—
284C2×C4D53⋊M1611yescharacteristic
285C2×C4D53⋊M1611yescharacteristic
286C8—4244no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—4242no—
295C22—4241no—
296C22—4241no—
297C22—2124no—
298C22—2122no—
299C22#1681159511yescharacteristic
300C22C2×#1313327611yescharacteristic
301C22C2×#1313327711yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—81no—
306C4—81no—
307C4—2122no—
308C4—2122no—
309C4—2122no—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—4242no—
314C2#2028996211yescharacteristic
315C2#2029099211yescharacteristic
316C2#2029101211yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC22×#11719945C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11508
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #20289963C2semidirect
2. #20289646C2semidirect
3. #20289644C2semidirect
67. C2×D212M16semidirect
74. C2×Dic106M16semidirect
175. C53#11508semidirect

Maximal subgroups

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

subgroupindex
#202899632
#202896462
#202896442
#202773032
#1150853