Finite Group Databasebeta

← #27273147

Subgroups — #27273147

Order 13568.

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

normal subgroups only

This group has 462 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#20281087C211yesmaximal, characteristic
2#20276272C211yesmaximal, characteristic, complemented
3#20276192C211yesmaximal, characteristic, complemented
4#20272537C211yesmaximal, characteristic, complemented
5#20272537C211yesmaximal, characteristic, complemented
6#20272536C211yesmaximal, characteristic
7C53×#2071C211yesmaximal, characteristic
8C2×#13133194—21no—
9C2×#13133194—21no—
10C2×#13133148—21no—
11#16810546—21no—
12#16810546—21no—
13#16810543—21no—
14#16810521C2211yescharacteristic
15#16810521C2211yescharacteristic
16#16810503—21no—
17#16810385—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C2×#102×C53—21no—
23C53×#390C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C53×#340—21no—
26C53×#336C2211yescharacteristic
27C22×Dic106—21no—
28C23×Dic53—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133194—42no—
32#13133194—42no—
33#13133194—42no—
34#13133194—42no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53—21no—
46#13133148—42no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×D4×C53D411yescharacteristic
51C22×C8×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C42×C53C2311yescharacteristic
59#102×C53—42no—
60C24×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53SD1612yes—
65C22×C4×C53—21no—
66C22×C4×C53—41no—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76C22×Dic53—81no—
77C2×Dic106—42no—
78C2×Dic106—42no—
79C2×Dic106—81no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C2×C8×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C42×C53—41no—
94C22⋊Dic53—42no—
95C22⋊Dic53—42no—
96C22⋊Dic53—81no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53D812yes—
102C4⋊Dic53SD1612yes—
103#11719919—42no—
104#11719919—42no—
105#11719919—41no—
106#11719919—41no—
107#11719919—41no—
108#11719919—41no—
109C2×#88322—22no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322—41no—
114C2×#88322—41no—
115C23×C53—22no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53#12011yescharacteristic
123D4×C53—42no—
124D4×C53—81no—
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—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53C2×D811yescharacteristic
138C2×C4×C53C2×SD1611yescharacteristic
139C2×C4×C53C2×SD1611yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53#13711yescharacteristic
143C2×Dic53—22no—
144C2×Dic53—22no—
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—
154C2×Dic53—41no—
155C2×Dic53—84no—
156Dic106—84no—
157C8×C53—81no—
158#88322—42no—
159#88322—81no—
160#7791—531nomaximal, Sylow
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53#44711yescharacteristic
169C22×C53#44811yescharacteristic
170C22×C53#45011yescharacteristic
171C22×C53#45411yescharacteristic
172C22×C53#45511yescharacteristic
173C22×C53#45611yescharacteristic
174C22×C53#45811yescharacteristic
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—81no—
182C4×C53—161no—
183Dic53—42no—
184Dic53—42no—
185Dic53—81no—
186Dic53—81no—
187#2683—531no—
188#2099—531no—
189#2084—531no—
190#2071D5311yescharacteristic
191#1271—531no—
192#1271—531no—
193#1270—531no—
194C2×C53—82no—
195C2×C53#160811yescharacteristic
196C2×C53#161011yescharacteristic
197C2×C53#161611yescharacteristic
198C2×C53#162711yescharacteristic
199C2×C53#163911yescharacteristic
200C2×C53#165111yescharacteristic
201C2×C53#166411yescharacteristic
202C2×C22⋊Q8—1061no—
203C2×C22⋊Q8—1061no—
204C2×#106—1061no—
205C2×#102—21no—
206#396—1061no—
207#396—1061no—
208#395—1061no—
209#390C2×D5311yescharacteristic
210#389—531no—
211#389—531no—
212#386—1061no—
213#384C2×D5311yescharacteristic
214#340—21no—
215#340—1061no—
216#340—1061no—
217#340—1061no—
218#336—531no—
219#336—531no—
220#336C2×D5311yescharacteristic
221C53#779111yescharacteristic, complemented, Sylow
222C22×Q8—1061no—
223C22×D4C22⋊D5311yescharacteristic
224C23×C4—1061no—
225C22×C8—21no—
226C22×C8—1061no—
227C22×C8—1061no—
228C22⋊Q8—2122no—
229C22⋊Q8—2122no—
230C22⋊Q8—2122no—
231C22⋊Q8—2122no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—1061no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4C22⋊D5311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—1061no—
247C2×C22⋊C4—1061no—
248C2×C22⋊C4—1061no—
249C2×C42C22×D5311yescharacteristic
250#106—2122no—
251#102—42no—
252#95—1061no—
253#95—1061no—
254#95—1061no—
255C24—21no—
256C2×Q8—4241no—
257C2×Q8—2122no—
258C2×Q8—2122no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4#1171992312yes—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—4241no—
270C22×C4—2122no—
271C22×C4—2122no—
272C22×C4D4×D5311yescharacteristic
273C22×C4D4×D5311yescharacteristic
274C22×C4C2×C22⋊D5311yescharacteristic
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—1062no—
279C2×C8—2121no—
280C2×C8—2121no—
281C2×C8—2121no—
282C2×C8—2121no—
283C2×C8—2121no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—532no—
291C4⋊C4—532no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—2121no—
295C4⋊C4—2121no—
296C4⋊C4—2122no—
297C4⋊C4—2122no—
298C4⋊C4—2121no—
299C4⋊C4—2121no—
300C4⋊C4—2121no—
301C4⋊C4—2121no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—4241no—
307C22⋊C4—2122no—
308C22⋊C4—2122no—
309C42—41no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#1313318511yescharacteristic
318Q8—4244no—
319D4—42no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—1062no—
334C2×C4—1062no—
335C2×C4—4244no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4D8×D5311yescharacteristic
347C2×C4SD16×D5311yescharacteristic
348C2×C4C2×#1171992311yescharacteristic
349C2×C4#1313324111yescharacteristic
350C2×C4#1313324411yescharacteristic
351C2×C4#1313324711yescharacteristic
352C8—81no—
353C8—4241no—
354C8—2122no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22#1681075511yescharacteristic
363C22#1681076111yescharacteristic
364C22#1681077211yescharacteristic
365C22#1681079411yescharacteristic
366C22#1681080011yescharacteristic
367C22#1681080411yescharacteristic
368C22#1681081211yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—81no—
376C4—161no—
377C4—4241no—
378C4—4241no—
379C4—2122no—
380C4—2122no—
381C2—82no—
382C2#2027370511yescharacteristic
383C2#2027371611yescharacteristic
384C2#2027375011yescharacteristic
385C2#2027381711yescharacteristic
386C2#2027389111yescharacteristic
387C2#2027397611yescharacteristic
388C2#2027404811yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2071C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2071D53
hypocenterC53#7791
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. #20276272C2semidirect
3. #20276192C2semidirect
4. #20272537C2semidirect
5. #20272537C2semidirect
221. C53#7791semidirect

Maximal subgroups

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

subgroupindex
#202810872
#202762722
#202761922
#202725372
#202725362
C53×#20712
#779153