Finite Group Databasebeta

← #27293759

Subgroups — #27293759

Order 13568.

374 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 73 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#20279921C211yesmaximal, characteristic
2#20273057C211yesmaximal, characteristic
3#20273037C211yesmaximal, characteristic
4#20272661C211yesmaximal, characteristic
5#20272660C211yesmaximal, characteristic
6#20272660C211yesmaximal, characteristic
7C53×#1325C211yesmaximal, characteristic
8C2×#13133216C411yescharacteristic
9C2×#13133182C2211yescharacteristic
10#16810677—21no—
11#16810673—21no—
12#16810543—21no—
13#16810543—21no—
14#16810529C411yescharacteristic
15#16810528C411yescharacteristic
16#16810518—21no—
17#16810518—21no—
18#16810492C2211yescharacteristic
19C4×#11719919C411yescharacteristic
20#16810349C2211yescharacteristic
21#16810349C2211yescharacteristic
22#16810348—21no—
23#16810348—21no—
24C53×#384C2211yescharacteristic
25C53×#324C2211yescharacteristic
26C53×#324C2211yescharacteristic
27C22×Dic106—21no—
28C22×Dic106C2×C411yescharacteristic
29#13133216—41no—
30#13133216—41no—
31#13133182—41no—
32#13133182—41no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919D411yescharacteristic
39C2×#11719919D411yescharacteristic
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44#13133146—41no—
45#13133146—41no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133127—41no—
49C8×Dic53—41no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2311yescharacteristic
59C2×C42×C53C2×C411yescharacteristic
60C22⋊C8×C53—21no—
61C22⋊C8×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C411yescharacteristic
73C2×Dic106—21no—
74C2×Dic106—21no—
75C2×Dic106—22no—
76C2×Dic106—41no—
77C2×Dic106—42no—
78C2×Dic106—81no—
79C2×Dic106—81no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—81no—
85C4⋊C4×C53—81no—
86C42×C53—21no—
87C42×C53—21no—
88C4⋊Dic53—41no—
89#11719919—24no—
90#11719919—41no—
91#11719919—41no—
92#11719919—41no—
93#11719919—41no—
94#11719919—81no—
95#11719919—81no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C2×#88322—41no—
105C2×#88322—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53#13111yescharacteristic
120C2×C4×C53#13111yescharacteristic
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53#9911yescharacteristic
136C2×Dic53#9911yescharacteristic
137C2×Dic53#10112yes—
138Dic106—42no—
139Dic106—42no—
140Dic106—84no—
141C8×C53—81no—
142C8×C53—81no—
143#88322—81no—
144#88322—81no—
145#11574—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53#40811yescharacteristic
149C22×C53C2×#9911yescharacteristic
150C22×C53C2×#10111yescharacteristic
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—42no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—81no—
157Dic53—21no—
158Dic53—22no—
159Dic53—42no—
160Dic53—41no—
161Dic53—41no—
162Dic53—81no—
163Dic53#35612yes—
164#2567—531no—
165#1467—531no—
166#1461—531no—
167#1325—531no—
168#1325—531no—
169#1325—531no—
170#1325D5311yescharacteristic
171C2×C53—21no—
172C2×C53—21no—
173C2×C53#147211yescharacteristic
174C2×C53#212811yescharacteristic
175C2×C53C2×#35611yescharacteristic
176C2×C4⋊Q8—531no—
177C2×C4×Q8—531no—
178#432—1061no—
179#431—1061no—
180#395—1061no—
181#395—1061no—
182#391—531no—
183#391—531no—
184#389—1061no—
185#389—1061no—
186#384—531no—
187#384C2×D5311yescharacteristic
188C4×C4⋊C4—531no—
189#324—531no—
190#324—531no—
191#324—1061no—
192#324—1061no—
193#324C2×D5311yescharacteristic
194#324C2×D5311yescharacteristic
195C53#1157411yescharacteristic, complemented, Sylow
196C22×Q8—531no—
197C22×Q8—1061no—
198C4⋊Q8—2121no—
199C4⋊Q8—2121no—
200C4×Q8—2121no—
201C4×Q8—2121no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4C4×D5311yescharacteristic
212C2×C4⋊C4C22×D5311yescharacteristic
213C2×C42—531no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42C4×D5311yescharacteristic
218C4⋊C8—2121no—
219C4⋊C8—2121no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222C22⋊C8—1061no—
223C22⋊C8—1061no—
224#97—2121no—
225C4×C8—2121no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231C2×Q8—1062no—
232C2×Q8—1061no—
233C2×Q8—1061no—
234C2×Q8—4241no—
235C2×Q8—4241no—
236C2×Q8—2121no—
237C2×Q8—2122no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4C2×C4×D5311yescharacteristic
247C22×C4D4×D5311yescharacteristic
248C22×C4D4×D5311yescharacteristic
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—2121no—
252C2×C8—2121no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—1064no—
258C4⋊C4—4241no—
259C4⋊C4—4241no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C42—21no—
266C42—21no—
267C42—1061no—
268C42—1061no—
269C42—1061no—
270C42—1061no—
271C42—1061no—
272C42—1061no—
273C42—1061no—
274C42—1061no—
275C23C22⋊C4×D5311yescharacteristic
276Q8—4244no—
277Q8—2122no—
278Q8—2122no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—531no—
292C2×C4—531no—
293C2×C4—532no—
294C2×C4—1062no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1062no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4#1313322311yescharacteristic
309C2×C4#1313322511yescharacteristic
310C8—81no—
311C8—81no—
312C8—4241no—
313C8—4241no—
314C22—21no—
315C22—21no—
316C22#1681058811yescharacteristic
317C22#99×D5311yescharacteristic
318C22#101×D5311yescharacteristic
319C4—21no—
320C4—21no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—532no—
326C4—1061no—
327C4—1062no—
328C4—4241no—
329C4—2122no—
330C4—2121no—
331C4—2121no—
332C2—21no—
333C2—21no—
334C2#2027307711yescharacteristic
335C2#2027638511yescharacteristic
336C2#356×D5311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1325C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1325D53
hypocenterC53#11574
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
195. C53#11574semidirect

Maximal subgroups

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

subgroupindex
#202799212
#202730572
#202730372
#202726612
#202726602
C53×#13252
#1157453