Finite Group Databasebeta

← #27293271

Subgroups — #27293271

Order 13568.

434 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 93 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#20277334C211yesmaximal, characteristic
2#20273026C211yesmaximal, characteristic
3#20273026C211yesmaximal, characteristic
4#20272664C211yesmaximal, characteristic
5#20272663C211yesmaximal, characteristic
6#20272662C211yesmaximal, characteristic
7C53×#1326C211yesmaximal, characteristic
8C2×#13133173C411yescharacteristic
9C2×#13133173C2211yescharacteristic
10C22×#11719919C411yescharacteristic
11#16810673—21no—
12#16810673—21no—
13#16810520—21no—
14#16810518—21no—
15#16810498C411yescharacteristic
16#16810496C411yescharacteristic
17#16810495C2211yescharacteristic
18#16810492—21no—
19#16810487—21no—
20#16810348—21no—
21#16810348—21no—
22#16810347C2211yescharacteristic
23#16810347C2211yescharacteristic
24C53×#385C2211yescharacteristic
25C53×#323C2211yescharacteristic
26C53×#323C2211yescharacteristic
27C23×Dic53C2×C411yescharacteristic
28C23×Dic53C2×C411yescharacteristic
29#13133173—22no—
30#13133173—41no—
31#13133173—41no—
32#13133173C814yescomplemented
33C2×C4⋊Dic53—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—41no—
40C2×#11719919—41no—
41C2×#11719919D411yescharacteristic
42C2×#11719919D411yescharacteristic
43C2×C22⋊Dic53C2×C411yescharacteristic
44C2×C22⋊Dic53C2×C411yescharacteristic
45C2×C4×Dic53—21no—
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48#13133146—41no—
49#13133146—41no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C8×Dic53—41no—
53C8×Dic53—41no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C23×C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53—21no—
59C2×C22⋊C4×C53C2×C411yescharacteristic
60C2×C22⋊C4×C53C2311yescharacteristic
61C22⋊C8×C53—21no—
62C22⋊C8×C53—21no—
63#95×C53—21no—
64C24×C53C22×C411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—41no—
68C22×C4×C53—81no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—22no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—41no—
77C22×Dic53—81no—
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C22⋊C411yescharacteristic
80C22×Dic53C22⋊C411yescharacteristic
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C2×C812yes—
83C22×Dic53M1612yes—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C4⋊C4×C53—81no—
87C22⋊C4×C53—21no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53C2×C812yes—
90C22⋊Dic53—21no—
91C22⋊Dic53—41no—
92C22⋊Dic53C2×C812yes—
93C4⋊Dic53—81no—
94#11719919—24no—
95#11719919—41no—
96#11719919—41no—
97#11719919—42no—
98#11719919—41no—
99#11719919—41no—
100#11719919—81no—
101#11719919—81no—
102C4×Dic53—24no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C2×#88322—41no—
108C2×#88322—41no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C22×C811yescharacteristic
115C23×C53C2×M1611yescharacteristic
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—82no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—82no—
144C2×Dic53C22⋊C812yes—
145C2×Dic53C22⋊C812yes—
146C2×Dic53#9911yescharacteristic
147C2×Dic53#9911yescharacteristic
148C2×Dic53#10112yes—
149C8×C53—81no—
150C8×C53—81no—
151#88322—81no—
152#88322—81no—
153#11454—531nomaximal, Sylow
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—42no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53C2×C22⋊C811yescharacteristic
164C22×C53C2×#9911yescharacteristic
165C22×C53C2×#10111yescharacteristic
166C4×C53—42no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—81no—
170Dic53—22no—
171Dic53—42no—
172Dic53—41no—
173Dic53—41no—
174Dic53—81no—
175Dic53#32412yes—
176Dic53#32412yes—
177#2317—531no—
178#1458—531no—
179#1458—531no—
180#1326—531no—
181#1326—531no—
182#1326—531no—
183#1326D5311yescharacteristic
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—41no—
187C2×C53—41no—
188C2×C53C2×#32411yescharacteristic
189C2×C53#212211yescharacteristic
190C2×C53#212311yescharacteristic
191C2×#117—531no—
192C2×#117—531no—
193C22×C4⋊C4—531no—
194#431—1061no—
195#431—1061no—
196#389—1061no—
197#389—1061no—
198#385—531no—
199#385—531no—
200#385—531no—
201#385C2×D5311yescharacteristic
202#384—1061no—
203#382—1061no—
204#324—1061no—
205#324—1061no—
206#323—531no—
207#323—531no—
208#323C2×D5311yescharacteristic
209#323C2×D5311yescharacteristic
210C53#1145411yescharacteristic, complemented, Sylow
211C23×C4—531no—
212C23×C4—531no—
213C23×C4C4×D5311yescharacteristic
214#117—534no—
215#117—1062no—
216#117—2121no—
217#117—2121no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—2121no—
228C2×C4⋊C4—2121no—
229C2×C22⋊C4—531no—
230C2×C22⋊C4—531no—
231C2×C22⋊C4C4×D5311yescharacteristic
232C2×C22⋊C4C22×D5311yescharacteristic
233C2×C42—531no—
234C2×C42—531no—
235C2×C42—1061no—
236C4⋊C8—2121no—
237C4⋊C8—2121no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—1061no—
241C22⋊C8—1061no—
242C4×C8—2121no—
243C4×C8—2121no—
244#95—21no—
245#95—1061no—
246#95—1061no—
247#95—1061no—
248C24C2×C4×D5311yescharacteristic
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—81no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—532no—
256C22×C4—532no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1062no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—4241no—
265C22×C4—2121no—
266C22×C4D4×D5311yescharacteristic
267C22×C4D4×D5311yescharacteristic
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—2121no—
271C2×C8—2121no—
272C4⋊C4—81no—
273C4⋊C4—1064no—
274C4⋊C4—4241no—
275C4⋊C4—4241no—
276C4⋊C4—4241no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C4⋊C4—2122no—
282C22⋊C4—21no—
283C22⋊C4—41no—
284C22⋊C4—532no—
285C22⋊C4—1061no—
286C22⋊C4—2121no—
287C22⋊C4C8×D5312yes—
288C42—1061no—
289C42—1061no—
290C42—1061no—
291C42—1061no—
292C42—1064no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23C22⋊C4×D5311yescharacteristic
298C23C2×C8×D5311yescharacteristic
299C23M16×D5311yescharacteristic
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—82no—
311C2×C4—532no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—532no—
315C2×C4—532no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1062no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—4242no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C8—81no—
334C8—81no—
335C8—4241no—
336C8—4241no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22C22⋊C8×D5311yescharacteristic
347C22#99×D5311yescharacteristic
348C22#101×D5311yescharacteristic
349C4—42no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—532no—
354C4—532no—
355C4—1062no—
356C4—4241no—
357C4—2122no—
358C4—2121no—
359C4—2121no—
360C2—21no—
361C2—21no—
362C2—41no—
363C2—41no—
364C2#324×D5311yescharacteristic
365C2#2027636311yescharacteristic
366C2#2027636711yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC23×C53C22×C8
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1326C2
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1326D53
hypocenterC53#11454
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1326

Complements

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

NH ≅ G/Ntype
32. #13133173C8semidirect
210. C53#11454semidirect

Maximal subgroups

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

subgroupindex
#202773342
#202730262
#202726642
#202726632
#202726622
C53×#13262
#1145453