Finite Group Databasebeta

← #27293265

Subgroups — #27293265

Order 13568.

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

normal subgroups only

This group has 462 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#20277194C211yesmaximal, characteristic
2#20273026C211yesmaximal, characteristic
3#20273026C211yesmaximal, characteristic
4#20272664C211yesmaximal, characteristic
5#20272663C211yesmaximal, characteristic
6#20272662C211yesmaximal, characteristic
7C53×#1326C211yesmaximal, characteristic
8C2×#13133173C411yescharacteristic
9C2×#13133173C2211yescharacteristic
10C22×C4×Dic53C411yescharacteristic
11#16810673—21no—
12#16810673—21no—
13#16810520—21no—
14#16810518—21no—
15#16810498C411yescharacteristic
16#16810496C411yescharacteristic
17#16810495C2211yescharacteristic
18C4⋊C4×Dic53—21no—
19#16810474—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×#11719919D411yescharacteristic
38C2×#11719919D411yescharacteristic
39C2×C22⋊Dic53C2×C411yescharacteristic
40C2×C22⋊Dic53C2×C411yescharacteristic
41C2×C4×Dic53—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53—41no—
45C2×C4×Dic53—41no—
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—42no—
68C22×C4×C53—41no—
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—42no—
77C22×Dic53—41no—
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—42no—
87C22⋊C4×C53—21no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53C2×C812yes—
90C22⋊Dic53—21no—
91C22⋊Dic53—41no—
92C22⋊Dic53C2×C812yes—
93C4⋊Dic53—42no—
94#11719919—24no—
95#11719919—41no—
96#11719919—41no—
97#11719919—42no—
98#11719919—42no—
99C4×Dic53—24no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—42no—
105C4×Dic53—41no—
106C4×Dic53—41no—
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—22no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—42no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—42no—
146C2×Dic53C22⋊C812yes—
147C2×Dic53C22⋊C812yes—
148C2×Dic53#9911yescharacteristic
149C2×Dic53#9911yescharacteristic
150C2×Dic53#10112yes—
151C8×C53—81no—
152C8×C53—81no—
153#88322—81no—
154#88322—81no—
155#11452—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—42no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53C2×C22⋊C811yescharacteristic
166C22×C53C2×#9911yescharacteristic
167C22×C53C2×#10111yescharacteristic
168C4×C53—42no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—42no—
172Dic53—22no—
173Dic53—42no—
174Dic53—41no—
175Dic53—41no—
176Dic53—42no—
177Dic53#32412yes—
178Dic53#32412yes—
179#2296—531no—
180#1458—531no—
181#1458—531no—
182#1326—531no—
183#1326—531no—
184#1326—531no—
185#1326D5311yescharacteristic
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—41no—
189C2×C53—41no—
190C2×C53C2×#32411yescharacteristic
191C2×C53#212011yescharacteristic
192C2×C53#212011yescharacteristic
193C2×#117—531no—
194C2×#117—531no—
195C22×C42—531no—
196#431—1061no—
197#431—1061no—
198#389—1061no—
199#389—1061no—
200#385—531no—
201#385—531no—
202#385—531no—
203#385C2×D5311yescharacteristic
204C4×C4⋊C4—1061no—
205C4×C4⋊C4—1061no—
206#324—1061no—
207#324—1061no—
208#323—531no—
209#323—531no—
210#323C2×D5311yescharacteristic
211#323C2×D5311yescharacteristic
212C53#1145211yescharacteristic, complemented, Sylow
213C23×C4—531no—
214C23×C4—531no—
215C23×C4C4×D5311yescharacteristic
216#117—534no—
217#117—1062no—
218#117—2121no—
219#117—2121no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C22⋊C4—531no—
228C2×C22⋊C4—531no—
229C2×C22⋊C4C4×D5311yescharacteristic
230C2×C22⋊C4C22×D5311yescharacteristic
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—1061no—
234C2×C42—1061no—
235C2×C42—1061no—
236C2×C42—2121no—
237C2×C42—2121no—
238C4⋊C8—2121no—
239C4⋊C8—2121no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—1061no—
243C22⋊C8—1061no—
244C4×C8—2121no—
245C4×C8—2121no—
246#95—21no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250C24C2×C4×D5311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—532no—
258C22×C4—532no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1062no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—2122no—
267C22×C4—2121no—
268C22×C4D4×D5311yescharacteristic
269C22×C4D4×D5311yescharacteristic
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—2121no—
273C2×C8—2121no—
274C4⋊C4—42no—
275C4⋊C4—1064no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C4⋊C4—2122no—
279C4⋊C4—2122no—
280C4⋊C4—2122no—
281C22⋊C4—21no—
282C22⋊C4—41no—
283C22⋊C4—532no—
284C22⋊C4—1061no—
285C22⋊C4—2121no—
286C22⋊C4C8×D5312yes—
287C42—1061no—
288C42—1061no—
289C42—1061no—
290C42—1061no—
291C42—1064no—
292C42—2121no—
293C42—2121no—
294C42—2122no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23C22⋊C4×D5311yescharacteristic
300C23C2×C8×D5311yescharacteristic
301C23M16×D5311yescharacteristic
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—532no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—532no—
318C2×C4—532no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2122no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2122no—
337C8—81no—
338C8—81no—
339C8—4241no—
340C8—4241no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22C22⋊C8×D5311yescharacteristic
351C22#99×D5311yescharacteristic
352C22#101×D5311yescharacteristic
353C4—42no—
354C4—41no—
355C4—42no—
356C4—41no—
357C4—532no—
358C4—532no—
359C4—1062no—
360C4—2122no—
361C4—2121no—
362C4—2122no—
363C4—2121no—
364C2—21no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2#324×D5311yescharacteristic
369C2#2027635311yescharacteristic
370C2#2027635511yescharacteristic
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#11452
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
212. C53#11452semidirect

Maximal subgroups

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

subgroupindex
#202771942
#202730262
#202726642
#202726632
#202726622
C53×#13262
#1145253