Finite Group Databasebeta

← #28299423

Subgroups — #28299423

Order 13568.

690 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C4×C22⋊#88324C212yesmaximal, complemented
2#20289536C212yesmaximal, complemented
3#20289532C212yesmaximal, complemented
4#20277203C211yesmaximal, characteristic
5C2×C22⋊#88324—22no—
6C2×C4×#88324—22no—
7#16811489C2214yes—
8C22×#11719945C411yescharacteristic
9C22×C4×Dic53C2211yescharacteristic
10#16810528C411yescharacteristic
11#16810528C2211yescharacteristic
12#16810524C411yescharacteristic
13#16810524C2211yescharacteristic
14#16810500—22no—
15C4×#11719921—22no—
16C2×#117×C53C411yescharacteristic
17C22×#88324—24no—
18C22⋊#88324—28no—
19C4×#88324—44no—
20C2×#11719945—22no—
21C2×#11719945—42no—
22C2×#11719945—42no—
23C2×#11719945C814yescomplemented
24C22×Dic106C2×C411yescharacteristic
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C23×Dic53D412yes—
28C22×C4×D53—21no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×C4×Dic53—42no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34C2×C4×Dic53C2311yescharacteristic
35#13133124—22no—
36#13133124—22no—
37C23×C4×C53C2×C411yescharacteristic
38#117×C53—42no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C42×C53—21no—
43C24×C53C22⋊C411yescharacteristic
44C23×D53—21no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C2×C812yes—
51C22×C4×C53M1612yes—
52C22×C4×C53C22×C411yescharacteristic
53C2×C22⋊D53—24no—
54C2×C22⋊D53—42no—
55C2×C22⋊D53—42no—
56C22×Dic53—21no—
57C22×Dic53—24no—
58C22×Dic53—24no—
59C22×Dic53—42no—
60C22×Dic53C22⋊C412yes—
61C22×Dic53C2×D412yes—
62#11719945—48no—
63#11719945—48no—
64C2×D212—21no—
65C2×D212—41no—
66C2×D212—41no—
67C2×D212C2×C812yes—
68C2×C4×D53—24no—
69C2×C4×D53—42no—
70C2×C4×D53—42no—
71C2×Dic106—21no—
72C2×Dic106—41no—
73C2×Dic106—41no—
74C2×Dic106C2×C812yes—
75C2×#88324—28no—
76C2×#88324—44no—
77C2×#88324—44no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—42no—
81C42×C53—42no—
82#11719921—44no—
83#11719921—44no—
84C4×Dic53—24no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53C22⋊C812yes—
92C23×C53C2×C22⋊C411yescharacteristic
93C22×D53—22no—
94C22×D53—41no—
95C22×D53—41no—
96C22×D53—42no—
97C22×D53—42no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—24no—
101C2×C4×C53—24no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53C22⋊C812yes—
107C2×C4×C53#9912yes—
108C2×C4×C53#10012yes—
109C2×C4×C53C22×C811yescharacteristic
110C2×C4×C53C2×M1611yescharacteristic
111C22⋊D53—48no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—48no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53#10414yes—
121C2×Dic53#13514yes—
122D212—42no—
123D212—42no—
124C4×D53—48no—
125Dic106—42no—
126Dic106—42no—
127#88324—48no—
128#7563—531nomaximal, Sylow
129C22×C53—22no—
130C22×C53—24no—
131C22×C53—41no—
132C22×C53—42no—
133C22×C53—41no—
134C22×C53C2×C22⋊C811yescharacteristic
135C22×C53C2×#9911yescharacteristic
136C22×C53C2×#10011yescharacteristic
137C22×C53#41612yes—
138C22×C53C2×#10412yes—
139C2×D53—42no—
140C2×D53—44no—
141C2×D53—81no—
142C2×D53—82no—
143C2×D53—81no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—81no—
147C4×C53—81no—
148C4×C53#32314yes—
149Dic53—24no—
150Dic53—24no—
151Dic53—42no—
152#2297—531no—
153C4×C22⋊C8—532no—
154#1272—532no—
155#1270—532no—
156C2×C53—42no—
157C2×C53C2×#32311yescharacteristic
158C2×C53#147612yes—
159C2×C53#150112yes—
160C2×C53#152511yescharacteristic
161C2×C53#154311yescharacteristic
162D53—82no—
163C22×#41—531no—
164C2×#117#615711yescharacteristic
165C22×C42—531no—
166C2×C22⋊C8—1062no—
167C2×C4×C8—1062no—
168#391—531no—
169#391—531no—
170#390—531no—
171#390—531no—
172#386—1062no—
173C4×C22⋊C4—1062no—
174#336—534no—
175C53#756311yescharacteristic, complemented, Sylow
176C2×#41—534no—
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
186C22×C8—1064no—
187#117—42no—
188C2×C4⋊C4C2×#615711yescharacteristic
189C2×C4⋊C4C2×#615711yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—1062no—
192C2×C22⋊C4—1062no—
193C2×C42—21no—
194C2×C42—532no—
195C2×C42—531no—
196C2×C42—531no—
197C2×C42—2122no—
198C22⋊C8—1068no—
199C4×C8—2124no—
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—532no—
222C22×C4—532no—
223C22×C4—1064no—
224C22×C4—1064no—
225C22×C4—1061no—
226C22×C4—1064no—
227C22×C4—2122no—
228C22×C4—2122no—
229C22×C4—2122no—
230C22×C4D53⋊C812yes—
231C22×C4#1171993612yes—
232C22×C4C22⋊#615711yescharacteristic
233C22×C4C22×#615711yescharacteristic
234C2×C8—1068no—
235C2×C8—2124no—
236C2×C8—2124no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—2124no—
241C22⋊C4—2124no—
242C42—42no—
243C42—1064no—
244C42—2121no—
245C42—2121no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—1062no—
251C23—2121no—
252C23—2121no—
253C23—2122no—
254C23—2122no—
255C23#1313327412yes—
256C23C2×C22⋊#615711yescharacteristic
257Q8—2122no—
258Q8—2122no—
259D4—2122no—
260D4—2122no—
261D4—2128no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—534no—
271C2×C4—534no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—1062no—
277C2×C4—2128no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2128no—
281C2×C4#1313327412yes—
282C2×C4#1313327612yes—
283C2×C4#1313327712yes—
284C2×C4C2×D53⋊C811yescharacteristic
285C2×C4C2×#1171993611yescharacteristic
286C8—2128no—
287C22—22no—
288C22—24no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—4242no—
293C22—4241no—
294C22—4241no—
295C22—2122no—
296C22—2124no—
297C22C2×#1313327411yescharacteristic
298C22C2×#1313327611yescharacteristic
299C22C2×#1313327711yescharacteristic
300C22#1681161312yes—
301C22D53⋊#10412yes—
302C4—21no—
303C4—21no—
304C4—81no—
305C4—81no—
306C4—1064no—
307C4—1064no—
308C4—2122no—
309C4#1681146714yes—
310C2—42no—
311C2—4242no—
312C2C2×#1681146711yescharacteristic
313C2#2029000812yes—
314C2#2029009212yes—
315C2#2029016911yescharacteristic
316C2#2029021811yescharacteristic
subgroupG/H
centerC22×C4C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×#11719945C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#7563
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C4×C22⋊#88324C2semidirect
2. #20289536C2semidirect
3. #20289532C2semidirect
23. C2×#11719945C8semidirect
175. C53#7563semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊#883242
#202895362
#202895322
#202772032
#756353