Finite Group Databasebeta

← #28299544

Subgroups — #28299544

Order 13568.

654 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 110 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#20290572C212yesmaximal, complemented
2#20289535C212yesmaximal, complemented
3#20289532C212yesmaximal, complemented
4#20277204C211yesmaximal, characteristic
5C2×#13133284—22no—
6C2×#13133274—22no—
7#16811489C2214yes—
8C22×#11719949C411yescharacteristic
9C2×#13133175C411yescharacteristic
10C2×C42×D53C2211yescharacteristic
11#16810530C2211yescharacteristic
12#16810525C411yescharacteristic
13#16810524C2211yescharacteristic
14#16810501—22no—
15#16810470—22no—
16C53×#391C411yescharacteristic
17C22×#88324—24no—
18#13133284—44no—
19#13133274—44no—
20C2×#11719949—22no—
21C2×#11719949—42no—
22C2×#11719949—42no—
23C2×#11719949C814yescomplemented
24C22×D212—21no—
25C22×D212C2×C411yescharacteristic
26C22×C4×D53—21no—
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D412yes—
29#13133175—42no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719921—21no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53D412yes—
36C2×C4×Dic53C2311yescharacteristic
37C42×D53—42no—
38#13133124—22no—
39C22×Q8×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42#95×C53—22no—
43C23×D53—21no—
44C23×D53C22⋊C411yescharacteristic
45C2×Q8×C53—21no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53C2×C812yes—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C2×D412yes—
56#11719949—48no—
57#11719949—48no—
58C2×D212—24no—
59C2×D212—21no—
60C2×D212—41no—
61C2×D212—41no—
62C2×D212—42no—
63C2×D212—42no—
64C2×D212C2×C812yes—
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53—24no—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—42no—
71C2×C4×D53—42no—
72C2×C4×D53—42no—
73C2×C4×D53—42no—
74C2×C4×D53C2×C812yes—
75C2×C4×D53M1612yes—
76C2×#88324—44no—
77C2×#88324—44no—
78C2×#88324—44no—
79C4⋊C4×C53—41no—
80C42×C53—41no—
81#11719921—44no—
82#11719921—42no—
83#11719921—42no—
84#11719921—42no—
85C4⋊Dic53—41no—
86C4×Dic53—24no—
87C4×Dic53—41no—
88C4×Dic53—42no—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—41no—
96C22×D53—41no—
97C22×D53—42no—
98C22×D53—42no—
99C22×D53C22⋊C812yes—
100Q8×C53—42no—
101Q8×C53—42no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53C22×C811yescharacteristic
113C2×C4×C53C2×M1611yescharacteristic
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—41no—
117C2×Dic53—42no—
118C2×Dic53C22⋊C812yes—
119C2×Dic53#9912yes—
120C2×Dic53#10012yes—
121C2×Dic53AGL(1,ℤ/8)12yes—
122C2×Dic53#13712yes—
123D212—42no—
124D212—42no—
125D212—48no—
126C4×D53—24no—
127C4×D53—24no—
128C4×D53—44no—
129C4×D53—44no—
130C4×D53—48no—
131#88324—84no—
132#7597—531nomaximal, Sylow
133C22×C53C2×C22⋊C811yescharacteristic
134C22×C53C2×#9911yescharacteristic
135C22×C53C2×#10011yescharacteristic
136C22×C53#41712yes—
137C22×C53#42112yes—
138C2×D53—22no—
139C2×D53—24no—
140C2×D53—41no—
141C2×D53—41no—
142C2×D53—42no—
143C2×D53—42no—
144C2×D53—44no—
145C2×D53—81no—
146C2×D53—81no—
147C2×D53—82no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—42no—
151C4×C53—42no—
152C4×C53—81no—
153Dic53—22no—
154Dic53—22no—
155Dic53—81no—
156Dic53#32314yes—
157#2297—531no—
158#1797—532no—
159#1272—532no—
160#1270—532no—
161C2×C53C2×#32311yescharacteristic
162C2×C53#147712yes—
163C2×C53#151312yes—
164C2×C53#153611yescharacteristic
165C2×C53#156011yescharacteristic
166D53—42no—
167D53—82no—
168C22×#41—531no—
169C2×#117—531no—
170C22×C42—531no—
171C2×C4⋊C8—1062no—
172C2×C22⋊C8—1062no—
173#391—531no—
174#391#615711yescharacteristic
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×Q8C2×#615711yescharacteristic
186C22×D4—531no—
187C22×D4—1061no—
188C23×C4—531no—
189C23×C4—532no—
190C23×C4—1061no—
191C22×C8—1064no—
192#117—2122no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4C2×#615711yescharacteristic
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1061no—
197C2×C22⋊C4—1062no—
198C2×C42—532no—
199C2×C42—531no—
200C2×C42—1061no—
201C2×C42—2122no—
202C2×C42C2×#615711yescharacteristic
203C4⋊C8—2124no—
204C22⋊C8—2124no—
205#95—22no—
206#95—1062no—
207C24—531no—
208C24—1061no—
209#41—2128no—
210#41—2128no—
211C2×Q8—21no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8D53⋊C812yes—
215C2×D4—532no—
216C2×D4—1064no—
217C2×D4—1061no—
218C2×D4—2121no—
219C2×D4—2121no—
220C2×D4—2122no—
221C2×D4—2122no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—531no—
225C22×C4—532no—
226C22×C4—532no—
227C22×C4—532no—
228C22×C4—1062no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4—1062no—
232C22×C4—1064no—
233C22×C4—1061no—
234C22×C4—2122no—
235C22×C4—2122no—
236C22×C4—2122no—
237C22×C4—2122no—
238C22×C4C22⋊#615712yes—
239C22×C4C22×#615711yescharacteristic
240C2×C8—2124no—
241C2×C8—2124no—
242C2×C8—2124no—
243C4⋊C4—41no—
244C4⋊C4—2121no—
245C22⋊C4—2124no—
246C22⋊C4—2122no—
247C22⋊C4—2122no—
248C22⋊C4—2122no—
249C42—41no—
250C42—1064no—
251C42—2121no—
252C42—2122no—
253C23—532no—
254C23—1061no—
255C23—1061no—
256C23—1062no—
257C23—1062no—
258C23—1062no—
259C23—2121no—
260C23—2121no—
261C23—2122no—
262C23—2122no—
263C23C2×C22⋊#615711yescharacteristic
264Q8—42no—
265Q8—42no—
266D4—2122no—
267D4—2122no—
268D4—2128no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—532no—
280C2×C4—532no—
281C2×C4—532no—
282C2×C4—532no—
283C2×C4—532no—
284C2×C4—1064no—
285C2×C4—1064no—
286C2×C4—1062no—
287C2×C4—1062no—
288C2×C4—2128no—
289C2×C4—2124no—
290C2×C4—2124no—
291C2×C4—2121no—
292C2×C4—2122no—
293C2×C4C2×D53⋊C811yescharacteristic
294C2×C4D53⋊M1611yescharacteristic
295C8—4244no—
296C22—1062no—
297C22—1064no—
298C22—4242no—
299C22—4241no—
300C22—4241no—
301C22—2122no—
302C22—2124no—
303C22—2122no—
304C22—2121no—
305C22—2121no—
306C22D53⋊(C22⋊C8)11yescharacteristic
307C22D53⋊#9911yescharacteristic
308C22D53⋊#10011yescharacteristic
309C22#1681161412yes—
310C22#1681162212yes—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—534no—
317C4—1062no—
318C4—1062no—
319C4—4241no—
320C2—4242no—
321C2—2122no—
322C2D53⋊#32311yescharacteristic
323C2#2029001312yes—
324C2#2029013312yes—
325C2#2029020511yescharacteristic
326C2#2029025311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC23×C53C2×C22⋊C4
prime radicalC22×#11719949C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#7597
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #20290572C2semidirect
2. #20289535C2semidirect
3. #20289532C2semidirect
23. C2×#11719949C8semidirect
180. C53#7597semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202895352
#202895322
#202772042
#759753