Finite Group Databasebeta

← #27288507

Subgroups — #27288507

Order 13568.

604 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 121 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#20274907C212yesmaximal
2#20272979C211yesmaximal, characteristic
3#20272593C212yesmaximal
4#20272558C211yesmaximal, characteristic
5C53×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7C22×#11719919—21no—
8C2×#13133148—22no—
9C2×C22⋊#88322C2212yes—
10C2×#13133124—21no—
11C2×#13133124C2211yescharacteristic
12#16810385—22no—
13C2×C22⋊Q8×C53—21no—
14C53×#398—22no—
15C53×#395—22no—
16C53×#380—21no—
17C53×#380C2212yes—
18C2×#95×C53C2211yescharacteristic
19C23×Dic53—21no—
20C23×Dic53—21no—
21C22×#88322—22no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—42no—
24C2×#11719919—42no—
25C2×#11719919—42no—
26C2×#11719919—42no—
27#13133148—44no—
28C22⋊#88322—22no—
29C22⋊#88322Q814yescomplemented
30#13133124—21no—
31#13133124—42no—
32#13133124Q812yescomplemented
33C22×Q8×C53—21no—
34C23×C4×C53D411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37C22⋊Q8×C53—42no—
38C22⋊Q8×C53—42no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53D412yes—
44C2×C4⋊C4×C53D412yes—
45C2×C22⋊C4×C53—22no—
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—22no—
49C24×C53C2×D411yescharacteristic
50C2×Q8×C53—42no—
51C2×Q8×C53—81no—
52C2×Q8×C53—81no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53SD1612yes—
66C22×C4×C53SD1612yes—
67C22×C4×C53C2×D412yes—
68C22×C4×C53C2×Q812yes—
69C22×C4×C53C2×Q811yescharacteristic
70C22×C4×C53#4111yescharacteristic
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C4⋊C4×C53—44no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—42no—
89C4⋊C4×C53—81no—
90C4⋊C4×C53—81no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C4⋊Dic53—24no—
94C4⋊Dic53—42no—
95#11719919—48no—
96C2×#88322—24no—
97C2×#88322—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53#12011yescharacteristic
104C23×C53C22⋊Q811yescharacteristic
105C23×C53C4⋊Q811yescharacteristic
106C23×C53C2×SD1611yescharacteristic
107C23×C53C2×SD1611yescharacteristic
108C23×C53AGL(1,ℤ/8)11yescharacteristic
109C23×C53#13711yescharacteristic
110Q8×C53—84no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53C22⋊Q812yes—
133C2×C4×C53C4⋊Q812yes—
134C2×Dic53—24no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148#88322—44no—
149#10570—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53#39511yescharacteristic
163C22×C53#44811yescharacteristic
164C22×C53#45011yescharacteristic
165C22×C53#45311yescharacteristic
166C22×C53#45611yescharacteristic
167C22×C53B(4,𝔽2)11yescharacteristic
168C22×C53#45811yescharacteristic
169C22×C53#47511yescharacteristic
170C22×C53#47611yescharacteristic
171C22×C53#49811yescharacteristic
172C22×C53#50111yescharacteristic
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—42no—
176C4×C53—42no—
177C4×C53—82no—
178C4×C53—82no—
179Dic53—44no—
180Dic53—82no—
181#2659D5311yescharacteristic
182#1853—532no—
183#1444—531no—
184#1294—532no—
185#1280—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53#159611yescharacteristic
191C2×C53#160411yescharacteristic
192C2×C53#161111yescharacteristic
193C2×C53#161311yescharacteristic
194C2×C53#205811yescharacteristic
195C2×C53#206011yescharacteristic
196C2×C53#206211yescharacteristic
197C2×C22⋊Q8—21no—
198C22×C4⋊C4—531no—
199C22×C4⋊C4—1061no—
200C2×#106—1062no—
201C2×C22⋊C8—532no—
202#398—22no—
203#395—22no—
204#380—21no—
205#380C2×D5312yes—
206C2×#95—531no—
207C2×#95—1061no—
208C2×#95C2×D5311yescharacteristic
209#340—1062no—
210C53#1057011yescharacteristic, complemented, Sylow
211C22×Q8—21no—
212C23×C4—1061no—
213C23×C4—1061no—
214C23×C4C22⋊D5311yescharacteristic
215C23×C4C22⋊D5311yescharacteristic
216C23×C4C22×D5311yescharacteristic
217C22×C8—1062no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—1062no—
225C2×C4⋊C4—2122no—
226C2×C4⋊C4—2122no—
227C2×C4⋊C4—2122no—
228C2×C4⋊C4—2122no—
229C2×C4⋊C4C22⋊D5312yes—
230C2×C4⋊C4C22⋊D5312yes—
231C2×C22⋊C4—22no—
232#106—2124no—
233C22⋊C8—534no—
234C22⋊C8—1062no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—532no—
239#95—1061no—
240#95—2122no—
241C24C2×C22⋊D5311yescharacteristic
242C2×Q8—42no—
243C2×Q8—81no—
244C2×Q8—81no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—1062no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4—2121no—
266C22×C4—2121no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4#1171992312yes—
272C22×C4#1171992412yes—
273C22×C4Q8×D5312yes—
274C22×C4Q8×D5311yescharacteristic
275C22×C4#1171994911yescharacteristic
276C22×C4C2×C22⋊D5312yes—
277C2×C8—1064no—
278C2×C8—2122no—
279C2×C8—2122no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—1064no—
287C4⋊C4—2122no—
288C4⋊C4—2128no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1313318611yescharacteristic
296C23#1313319311yescharacteristic
297C23#1313321511yescharacteristic
298C23C2×#1171992411yescharacteristic
299C23C2×#1171992311yescharacteristic
300C23#1313324311yescharacteristic
301C23#1313324811yescharacteristic
302Q8—84no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—1064no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—4241no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4#1313319312yes—
339C2×C4#1313321512yes—
340C8—2124no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1681054211yescharacteristic
354C22#1681076211yescharacteristic
355C22#1681077411yescharacteristic
356C22#1681079111yescharacteristic
357C22#1681080611yescharacteristic
358C22#1681081011yescharacteristic
359C22#1681081311yescharacteristic
360C22#1681090511yescharacteristic
361C22#1681091011yescharacteristic
362C22#1681101911yescharacteristic
363C22#1681102911yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—82no—
370C4—4242no—
371C4—2124no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2027363811yescharacteristic
377C2#2027368411yescharacteristic
378C2#2027372311yescharacteristic
379C2#2027373511yescharacteristic
380C2#2027604511yescharacteristic
381C2#2027605711yescharacteristic
382C2#2027606711yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2659C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#2659D53
hypocenterC53#10570
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
29. C22⋊#88322Q8semidirect
32. #13133124Q8semidirect
210. C53#10570semidirect

Maximal subgroups

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

subgroupindex
#202749072
#202729792
#202725932
#202725582
C53×#26592
#1057053