Finite Group Databasebeta

← #27530972

Subgroups — #27530972

Order 13568.

628 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C53⋊#2104C212yesmaximal, complemented
2#20275892C212yesmaximal, complemented
3#20275882C211yesmaximal, characteristic, complemented
4#20275258C212yesmaximal
5#20275174C212yesmaximal, complemented
6#20275129C212yesmaximal, complemented
7#20274821C212yesmaximal, complemented
8#20274790C211yesmaximal, characteristic, complemented
9C53×#2022C211yesmaximal, characteristic
10#16811249C2211yescharacteristic
11#16811098C2212yes—
12C2×#13133176C2211yescharacteristic
13C2×#13133132C2212yes—
14C2×#13133132C2212yes—
15C2×#13133132C2212yes—
16#16810486C2211yescharacteristic
17#16810486C2212yes—
18#16810390C2212yes—
19#16810390C2212yes—
20#16810390C2211yescharacteristic
21#16810389C2212yes—
22#16810389C2212yes—
23#16810389C2212yes—
24#16810389C2212yes—
25#16810389C2212yes—
26C53×#534C2211yescharacteristic
27C2×#123×C53C2212yes—
28C2×#99×C53C2212yes—
29C53×#381C2211yescharacteristic
30C53×#342C2211yescharacteristic
31#13133196—41no—
32#13133194—42no—
33D4×Dic53—42no—
34D4×Dic53—42no—
35#13133176—22no—
36#13133176—41no—
37#13133176—41no—
38C2×C4⋊Dic53—21no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53C2312yes—
41C2×C22⋊Dic53C2312yes—
42C2×C22⋊Dic53C2312yes—
43C2×C22⋊Dic53C2312yes—
44C2×C4×Dic53—21no—
45#13133132—42no—
46#13133132—42no—
47#13133132—42no—
48#13133124—41no—
49#13133124—42no—
50C22×D4×C53C2312yes—
51C23×C4×C53C2311yescharacteristic
52#123×C53—22no—
53#123×C53—22no—
54#123×C53—22no—
55#123×C53—42no—
56#121×C53—41no—
57#120×C53—22no—
58C2×C4⋊C4×C53—22no—
59C2×C22⋊C4×C53—22no—
60C2×C22⋊C4×C53C2312yes—
61C2×C22⋊C4×C53C2311yescharacteristic
62C2×C22⋊C4×C53C2311yescharacteristic
63#99×C53—42no—
64#95×C53—41no—
65C24×C53#4112yes—
66C24×C53C2411yescharacteristic
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—22no—
73C2×D4×C53—41no—
74C22×C4×C53—22no—
75C22×C4×C53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—21no—
78C22×C4×C53—22no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53#4112yes—
82C22×C4×C53#4111yescharacteristic
83C22×C4×C53#4111yescharacteristic
84C22×Dic53—22no—
85C22×Dic53—22no—
86C22×Dic53—22no—
87C22×Dic53—22no—
88C22×Dic53—42no—
89C2×Dic106—41no—
90C4⋊C4×C53—42no—
91C4⋊C4×C53—42no—
92C4⋊C4×C53—41no—
93C22⋊C4×C53—22no—
94C22⋊C4×C53—22no—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—42no—
99C22⋊C4×C53—42no—
100C22⋊Dic53—22no—
101C22⋊Dic53—22no—
102C22⋊Dic53—22no—
103C22⋊Dic53—22no—
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106C22⋊Dic53—22no—
107C22⋊Dic53—22no—
108C22⋊Dic53—42no—
109C4⋊Dic53—42no—
110C4⋊Dic53—42no—
111C4⋊Dic53—41no—
112#11719919—42no—
113C4×Dic53—42no—
114C4×Dic53—42no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—22no—
118C23×C53—81no—
119C23×C53—82no—
120C23×C53C2×#4111yescharacteristic
121C23×C53C2×#4112yes—
122C23×C53H(5,𝔽2)12yes—
123C23×C53H(5,𝔽2)11yescharacteristic
124C23×C53H(5,𝔽2)11yescharacteristic
125D4×C53—42no—
126D4×C53—42no—
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—161no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—81no—
144C2×C4×C53—82no—
145C2×C4×C53—81no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—82no—
151C2×Dic53—82no—
152C2×Dic53—82no—
153C2×Dic53—82no—
154C2×Dic53—162no—
155Dic106—161no—
156#32701—531nomaximal, Sylow
157C22×C53—41no—
158C22×C53—42no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—82no—
164C22×C53—82no—
165C22×C53—82no—
166C22×C53#53811yescharacteristic
167C22×C53#53812yes—
168C22×C53#55112yes—
169C22×C53#55111yescharacteristic
170C22×C53#56011yescharacteristic
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—42no—
174C4×C53—81no—
175C4×C53—82no—
176C4×C53—81no—
177Dic53—82no—
178Dic53—82no—
179Dic53—82no—
180Dic53—82no—
181#2104—532no—
182#2032—532no—
183#2030—531no—
184#2022D5311yescharacteristic
185#1913—532no—
186#1899—532no—
187#1890—532no—
188#1828—532no—
189#1820—531no—
190C2×C53—21no—
191C2×C53—22no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—22no—
195C2×C53—82no—
196C2×C53#276011yescharacteristic
197#543—531no—
198#534C2×D5311yescharacteristic
199C2×#123C2×D5312yes—
200#518—532no—
201C2×#117—531no—
202C2×#99—532no—
203C2×#99—532no—
204C2×#99—532no—
205C2×#99C2×D5312yes—
206#381—531no—
207#381—532no—
208#381C2×D5311yescharacteristic
209#342—532no—
210#342—532no—
211#342—531no—
212#342—532no—
213#342—532no—
214#342—532no—
215#342—532no—
216#342—532no—
217#342C2×D5311yescharacteristic
218C53#3270111yescharacteristic, complemented, Sylow
219C22×D4C22×D5312yes—
220C23×C4C22×D5311yescharacteristic
221#123—22no—
222#123—22no—
223#123—22no—
224#123—42no—
225C22⋊Q8—2122no—
226C22⋊Q8—2121no—
227#121—41no—
228#120—22no—
229C4×D4—2122no—
230C4×D4—2122no—
231#117—1062no—
232#117—2121no—
233#117—2121no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—1061no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—532no—
238C2×C22⋊C4—532no—
239C2×C22⋊C4—532no—
240C2×C22⋊C4—532no—
241C2×C22⋊C4—1062no—
242C2×C22⋊C4C22×D5312yes—
243C2×C22⋊C4C22×D5311yescharacteristic
244C2×C22⋊C4C22×D5311yescharacteristic
245C2×C42—1061no—
246#99—42no—
247#99—2122no—
248#99—2122no—
249#99—2122no—
250#95—41no—
251#95—2122no—
252#95—2121no—
253C24#1171994712yes—
254C24C23×D5311yescharacteristic
255C2×Q8—2121no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—41no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—1062no—
271C22×C4—1062no—
272C22×C4—1062no—
273C22×C4—1062no—
274C22×C4—2122no—
275C22×C4#1171994711yescharacteristic
276C22×C4#1171994712yes—
277C22×C4#1171994711yescharacteristic
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—2122no—
282C4⋊C4—2122no—
283C4⋊C4—2122no—
284C4⋊C4—2121no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—1062no—
293C22⋊C4—1062no—
294C22⋊C4—1062no—
295C22⋊C4—1062no—
296C22⋊C4—1062no—
297C22⋊C4—1062no—
298C22⋊C4—1062no—
299C22⋊C4—1062no—
300C22⋊C4—2122no—
301C42—2122no—
302C42—2122no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—81no—
307C23—82no—
308C23C2×#1171994711yescharacteristic
309C23C2×#1171994712yes—
310C23#1313326212yes—
311C23#1313326211yescharacteristic
312C23#1313326311yescharacteristic
313Q8—8481no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—161no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—8482no—
336C2×C4—4242no—
337C2×C4—4242no—
338C2×C4—4242no—
339C2×C4—4242no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C22—41no—
345C22—42no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—42no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22#1681121011yescharacteristic
354C22#1681121012yes—
355C22#1681130412yes—
356C22#1681130411yescharacteristic
357C22#1681136011yescharacteristic
358C4—41no—
359C4—41no—
360C4—42no—
361C4—81no—
362C4—82no—
363C4—81no—
364C4—4242no—
365C4—4242no—
366C4—4242no—
367C4—4242no—
368C2—21no—
369C2—21no—
370C2—22no—
371C2—22no—
372C2—21no—
373C2—82no—
374C2#2028209411yescharacteristic
subgroupG/H
centerC2#20282094
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#2022C2
socleC2×C53#2760
prime radicalC53×#534C22
hypercenter#2022D53
hypocenterC53#32701
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ #2022

Complements

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

NH ≅ G/Ntype
1. C53⋊#2104C2semidirect
2. #20275892C2semidirect
3. #20275882C2semidirect
5. #20275174C2semidirect
6. #20275129C2semidirect
7. #20274821C2semidirect
8. #20274790C2semidirect
218. C53#32701semidirect

Maximal subgroups

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

subgroupindex
C53⋊#21042
#202758922
#202758822
#202752582
#202751742
#202751292
#202748212
#202747902
C53×#20222
#3270153