Finite Group Databasebeta

← #28299543

Subgroups — #28299543

Order 13568.

654 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 124 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#20290569C212yesmaximal, complemented
2#20289536C212yesmaximal, complemented
3#20289532C212yesmaximal, complemented
4#20277203C211yesmaximal, characteristic
5C2×C4⋊#88324—22no—
6C2×C22⋊#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—
18C4⋊#88324—44no—
19C22⋊#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—22no—
58C22×Dic53—22no—
59C22×Dic53—42no—
60C22×Dic53—42no—
61C22×Dic53C22⋊C412yes—
62C22×Dic53C2×D412yes—
63#11719945—48no—
64#11719945—48no—
65C2×D212—21no—
66C2×D212—41no—
67C2×D212—41no—
68C2×D212C2×C812yes—
69C2×C4×D53—24no—
70C2×C4×D53—42no—
71C2×C4×D53—42no—
72C2×Dic106—21no—
73C2×Dic106—41no—
74C2×Dic106—41no—
75C2×Dic106C2×C812yes—
76C2×#88324—44no—
77C2×#88324—44no—
78C2×#88324—44no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C42×C53—42no—
83#11719921—44no—
84#11719921—42no—
85#11719921—42no—
86C4×Dic53—24no—
87C4×Dic53—41no—
88C4×Dic53—41no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53C22⋊C812yes—
94C23×C53C2×C22⋊C411yescharacteristic
95C22×D53—22no—
96C22×D53—41no—
97C22×D53—41no—
98C22×D53—42no—
99C22×D53—42no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—24no—
103C2×C4×C53—24no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53C22⋊C812yes—
109C2×C4×C53#9912yes—
110C2×C4×C53#10012yes—
111C2×C4×C53C22×C811yescharacteristic
112C2×C4×C53C2×M1611yescharacteristic
113C22⋊D53—48no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—44no—
121C2×Dic53—44no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53AGL(1,ℤ/8)12yes—
125C2×Dic53#13712yes—
126D212—42no—
127D212—42no—
128C4×D53—48no—
129Dic106—42no—
130Dic106—42no—
131#88324—84no—
132#7597—531nomaximal, Sylow
133C22×C53—22no—
134C22×C53—24no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—42no—
138C22×C53C2×C22⋊C811yescharacteristic
139C22×C53C2×#9911yescharacteristic
140C22×C53C2×#10011yescharacteristic
141C22×C53#41712yes—
142C22×C53#42112yes—
143C2×D53—42no—
144C2×D53—44no—
145C2×D53—81no—
146C2×D53—81no—
147C2×D53—82no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—81no—
151C4×C53—81no—
152C4×C53#32314yes—
153Dic53—22no—
154Dic53—22no—
155Dic53—42no—
156Dic53—42no—
157#2297—531no—
158#1797—532no—
159#1272—532no—
160#1270—532no—
161C2×C53—42no—
162C2×C53C2×#32311yescharacteristic
163C2×C53#147712yes—
164C2×C53#151312yes—
165C2×C53#153611yescharacteristic
166C2×C53#156011yescharacteristic
167D53—82no—
168C22×#41—531no—
169C2×#117#615711yescharacteristic
170C22×C42—531no—
171C2×C4⋊C8—1062no—
172C2×C22⋊C8—1062no—
173#391—531no—
174#391—531no—
175#390—531no—
176#390—531no—
177#386—1062no—
178C4×C22⋊C4—1062no—
179#336—534no—
180C53#759711yescharacteristic, complemented, Sylow
181C2×#41—534no—
182C2×#41—1062no—
183C2×#41—2122no—
184C2×#41—2122no—
185C22×Q8—531no—
186C22×D4—531no—
187C22×D4—1061no—
188C23×C4—532no—
189C23×C4—1061no—
190C23×C4C2×#615711yescharacteristic
191C22×C8—1064no—
192#117—42no—
193C2×C4⋊C4C2×#615711yescharacteristic
194C2×C4⋊C4C2×#615711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4—1062no—
198C2×C42—21no—
199C2×C42—532no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—2122no—
203C4⋊C8—2124no—
204C22⋊C8—2124no—
205#95—1062no—
206#95—1062no—
207C24—1061no—
208C24C22⋊#615711yescharacteristic
209#41—2128no—
210#41—2128no—
211C2×Q8—532no—
212C2×Q8—1061no—
213C2×Q8—2121no—
214C2×Q8—2121no—
215C2×D4—532no—
216C2×D4—1064no—
217C2×D4—1061no—
218C2×D4—2121no—
219C2×D4—2121no—
220C2×D4—2122no—
221C2×D4—2122no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—532no—
227C22×C4—532no—
228C22×C4—1062no—
229C22×C4—1062no—
230C22×C4—1061no—
231C22×C4—1064no—
232C22×C4—2122no—
233C22×C4—2122no—
234C22×C4—2122no—
235C22×C4—2122no—
236C22×C4D53⋊C812yes—
237C22×C4#1171993612yes—
238C22×C4C22⋊#615711yescharacteristic
239C22×C4C22×#615711yescharacteristic
240C2×C8—2124no—
241C2×C8—2124no—
242C2×C8—2124no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—2122no—
247C22⋊C4—2122no—
248C22⋊C4—2124no—
249C42—42no—
250C42—1064no—
251C42—2121no—
252C42—2121no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—1062no—
258C23—2121no—
259C23—2121no—
260C23—2122no—
261C23—2122no—
262C23#1313327412yes—
263C23C2×C22⋊#615711yescharacteristic
264Q8—2122no—
265Q8—2122no—
266D4—2122no—
267D4—2122no—
268D4—2128no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—532no—
278C2×C4—532no—
279C2×C4—1062no—
280C2×C4—1062no—
281C2×C4—1062no—
282C2×C4—1062no—
283C2×C4—1062no—
284C2×C4—1062no—
285C2×C4—2124no—
286C2×C4—2124no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2128no—
290C2×C4#1313327412yes—
291C2×C4#1313327612yes—
292C2×C4#1313327712yes—
293C2×C4C2×D53⋊C811yescharacteristic
294C2×C4C2×#1171993611yescharacteristic
295C8—4244no—
296C22—22no—
297C22—24no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—4242no—
302C22—4241no—
303C22—4241no—
304C22—2122no—
305C22—2124no—
306C22C2×#1313327411yescharacteristic
307C22C2×#1313327611yescharacteristic
308C22C2×#1313327711yescharacteristic
309C22#1681161412yes—
310C22#1681162012yes—
311C4—21no—
312C4—21no—
313C4—81no—
314C4—81no—
315C4—1062no—
316C4—1062no—
317C4—2122no—
318C4—2122no—
319C4#1681146714yes—
320C2—42no—
321C2—4242no—
322C2C2×#1681146711yescharacteristic
323C2#2029001212yes—
324C2#2029013212yes—
325C2#2029020311yescharacteristic
326C2#2029025211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#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#7597
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #20290569C2semidirect
2. #20289536C2semidirect
3. #20289532C2semidirect
23. C2×#11719945C8semidirect
180. C53#7597semidirect

Maximal subgroups

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

subgroupindex
#202905692
#202895362
#202895322
#202772032
#759753