Finite Group Databasebeta

← #27294614

Subgroups — #27294614

Order 13568.

488 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 87 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#20280195C211yesmaximal, characteristic
2C2×#16810392C211yesmaximal, characteristic, complemented
3#20272743C212yesmaximal, complemented
4#20272719C212yesmaximal, complemented
5C53×#1463C211yesmaximal, characteristic, complemented
6C22×#11719917—21no—
7C2×#13133191C411yescharacteristic
8#16810562C411yescharacteristic
9#16810562C2211yescharacteristic
10#16810540C411yescharacteristic
11#16810540C2211yescharacteristic
12#16810513—21no—
13#16810513—21no—
14#16810502—21no—
15#16810502—21no—
16C4×C22⋊Dic53C411yescharacteristic
17#16810392—21no—
18#16810392C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C2×#11719917—24no—
26C2×#11719917—22no—
27C2×#11719917—41no—
28C2×#11719917—42no—
29C22×#88322—41no—
30#13133191—41no—
31#13133191—41no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C2×M16×C53—41no—
50C2×C4⋊C4×C53—41no—
51C2×C4⋊C4×C53—41no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53C2311yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C23×D53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—81no—
70C22×C4×C53—81no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C2×C22⋊D53—42no—
78C2×C22⋊D53—42no—
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C2×D212—41no—
85C2×D212—81no—
86M16×C53—82no—
87C2×C8×C53—42no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C22⋊Dic53—42no—
93#11719921—42no—
94#11719921—42no—
95#11719921—41no—
96#11719921—41no—
97#11719921—81no—
98#11719921—81no—
99C4⋊Dic53—41no—
100C4×Dic53—81no—
101#11719917—24no—
102#11719917—44no—
103#11719917—42no—
104C2×#88322—44no—
105C2×#88322—42no—
106C23×C53—22no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53#9911yescharacteristic
111C23×C53#9911yescharacteristic
112C23×C53C2×C22⋊C411yescharacteristic
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—42no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53#10412yes—
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53#13711yescharacteristic
143C22⋊D53—82no—
144C22⋊D53—82no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154D212—82no—
155C8×C53—82no—
156#88322—44no—
157#11766—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53#35512yes—
168C22×C53C2×#9911yescharacteristic
169C22×C53#41711yescharacteristic
170C22×C53C2×#10411yescharacteristic
171C2×D53—81no—
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—21no—
179C4×C53—22no—
180C4×C53—21no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—41no—
185Dic53—81no—
186Dic53—81no—
187Dic53—81no—
188#2591—531no—
189C2×#343—531no—
190#1463D5311yescharacteristic, complemented
191#1353—532no—
192#1345—532no—
193C2×C53—21no—
194C2×C53—21no—
195C2×C53—22no—
196C2×C53—41no—
197C2×C53#150411yescharacteristic
198C2×C53#212111yescharacteristic
199C2×C53C2×#35511yescharacteristic
200D53—161no—
201C22×M16—1061no—
202C2×#121—531no—
203C22×C4⋊C4C2×D5311yescharacteristic
204#407—21no—
205#400—531no—
206#400—531no—
207#394—531no—
208#394—531no—
209#388—1061no—
210#388—1061no—
211#386—1061no—
212#386—1061no—
213C4×C22⋊C4—531no—
214#343—532no—
215#343—1061no—
216#324—21no—
217#324C2×D5312yes—
218C53#1176611yescharacteristic, complemented, Sylow
219C22×D4—531no—
220C22×D4—1061no—
221C23×C4C4×D5311yescharacteristic
222C23×C4D21211yescharacteristic
223C23×C4C22⋊D5311yescharacteristic
224C2×M16—41no—
225C2×M16—1064no—
226C2×M16—1062no—
227C2×M16—2121no—
228C2×M16—2122no—
229C22×C8—2121no—
230#121—2121no—
231#121—2121no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4C4×D5311yescharacteristic
236C2×C4⋊C4D21211yescharacteristic
237C2×C4⋊C4C22⋊D5311yescharacteristic
238C2×C4⋊C4C22×D5311yescharacteristic
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C42—1061no—
246C22⋊C8—22no—
247C22⋊C8—41no—
248#95—531no—
249#95—531no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253#95—1061no—
254C24—1061no—
255C24#1171992111yescharacteristic
256C2×D4—4241no—
257C2×D4—2121no—
258C2×D4—2122no—
259C2×D4—2122no—
260C2×D4—2121no—
261C2×D4—2121no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4#1171992111yescharacteristic
277C22×C4#1171992111yescharacteristic
278C22×C4#1171992111yescharacteristic
279C22×C4C2×C4×D5311yescharacteristic
280C22×C4C2×D21211yescharacteristic
281C22×C4C2×C22⋊D5311yescharacteristic
282M16—82no—
283M16—1064no—
284M16—2124no—
285M16—2122no—
286C2×C8—42no—
287C2×C8—2124no—
288C2×C8—2122no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—2121no—
294C22⋊C4—4241no—
295C22⋊C4—4241no—
296C22⋊C4—2122no—
297C22⋊C4—2122no—
298C22⋊C4—2122no—
299C22⋊C4—2121no—
300C22⋊C4—2121no—
301C42—4241no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—41no—
306C23—2121no—
307C23—2121no—
308C23—2121no—
309C23—2121no—
310C23—2121no—
311C23—2121no—
312C23—2121no—
313C23#1313313111yescharacteristic
314C23#1313313111yescharacteristic
315C23C2×#1171992111yescharacteristic
316D4—4242no—
317D4—4242no—
318D4—4242no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4#1313314312yes—
349C2×C4#1313324511yescharacteristic
350C2×C4#1313325011yescharacteristic
351C8—82no—
352C8—2124no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—42no—
361C22—42no—
362C22—4241no—
363C22—4241no—
364C22—4241no—
365C22—4241no—
366C22—4241no—
367C22—4241no—
368C22—4241no—
369C22#1681041912yes—
370C22C2×#1313313111yescharacteristic
371C22#1681062511yescharacteristic
372C22C2×#1313314311yescharacteristic
373C4—21no—
374C4—22no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—4241no—
381C4—4241no—
382C4—4241no—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—41no—
387C2—8481no—
388C2#2027322011yescharacteristic
389C2#2027635911yescharacteristic
390C2C2×#1681041911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1463C2
socleC22×C53C2×#99
prime radicalC2×#13133191C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1463D53
hypocenterC53#11766
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. C2×#16810392C2semidirect
3. #20272743C2semidirect
4. #20272719C2semidirect
5. C53×#1463C2semidirect
190. #1463D53semidirect
218. C53#11766semidirect

Maximal subgroups

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

subgroupindex
#202801952
C2×#168103922
#202727432
#202727192
C53×#14632
#1176653