Finite Group Databasebeta

← #27287863

Subgroups — #27287863

Order 13568.

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

normal subgroups only

This group has 398 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#20284514C211yesmaximal, characteristic
2#20274726C211yesmaximal, characteristic
3#20273373C211yesmaximal, characteristic, complemented
4#20273280C211yesmaximal, characteristic, complemented
5#20273035C211yesmaximal, characteristic
6#20272575C211yesmaximal, characteristic, complemented
7C53×#1571C211yesmaximal, characteristic, complemented
8#16811180C2211yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810924—22no—
11#16810914—21no—
12#16810904—21no—
13#16810676C2211yescharacteristic
14#16810665—21no—
15#16810629—21no—
16C2×#13133139—21no—
17#16810489C2211yescharacteristic
18#16810386—21no—
19#16810371—21no—
20#16810368—21no—
21#16810361—22no—
22#16810346—22no—
23C53×#431C2211yescharacteristic
24C53×#423C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×#335—22no—
27C22×D212D411yescharacteristic
28C22×C4×D53—21no—
29#13133217D411yescharacteristic
30#13133217D411yescharacteristic
31#13133209—21no—
32#13133196—21no—
33#13133196—21no—
34C4×Dic106—21no—
35C4×Dic106—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40#13133176—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53D411yescharacteristic
43C2×#11719921—21no—
44#13133151—22no—
45#13133149—22no—
46C4⋊#88322—21no—
47#13133139—22no—
48#13133139—22no—
49#13133139—42no—
50C22⋊#88322—21no—
51C4×#88322—21no—
52#13133124—21no—
53#13133124—21no—
54C2×M16×C53—21no—
55C22×C8×C53—21no—
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53D412yes—
62C22⋊C8×C53D412yes—
63C4×C8×C53—21no—
64C23×D53—21no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4111yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—41no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212Q1612yes—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×C4×D53—82no—
79C2×Dic106—21no—
80C2×Dic106—21no—
81M16×C53—41no—
82M16×C53—41no—
83C2×C8×C53—22no—
84C2×C8×C53—22no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C42×C53C2×D411yescharacteristic
89C42×C53C2×D411yescharacteristic
90C42×C53#4111yescharacteristic
91C42×C53#4111yescharacteristic
92C22⋊Dic53—41no—
93#11719921—41no—
94#11719921—41no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103#11719919—41no—
104#11719919—41no—
105C4×Dic53—41no—
106C2×#88322—41no—
107C23×C53#12111yescharacteristic
108C22×D53—22no—
109C22×D53—41no—
110C22×D53—41no—
111C22×D53—81no—
112C22×D53—81no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53#12011yescharacteristic
116C2×C4×C53#12111yescharacteristic
117C2×C4×C53#12111yescharacteristic
118C2×C4×C53#12311yescharacteristic
119C2×C4×C53#12311yescharacteristic
120C2×C4×C53#12411yescharacteristic
121C2×C4×C53C2×Q1611yescharacteristic
122C2×C4×C53#13511yescharacteristic
123C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C53#13711yescharacteristic
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—81no—
129C2×Dic53—81no—
130C2×Dic53—81no—
131C2×Dic53—81no—
132D212—22no—
133D212—22no—
134D212—41no—
135D212—41no—
136C4×D53—82no—
137C4×D53—82no—
138Dic106—81no—
139Dic106—81no—
140C8×C53—42no—
141C8×C53—42no—
142C8×C53—81no—
143#88322—81no—
144#10469—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—21no—
147C22×C53#39411yescharacteristic
148C22×C53#46711yescharacteristic
149C22×C53#46811yescharacteristic
150C2×D53—42no—
151C2×D53—82no—
152C2×D53—81no—
153C2×D53—81no—
154C2×D53—82no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53#44911yescharacteristic
160C4×C53#45111yescharacteristic
161C4×C53#48411yescharacteristic
162C4×C53#48511yescharacteristic
163Dic53—81no—
164Dic53—81no—
165Dic53—81no—
166Dic53—161no—
167#3077—531no—
168#1804—531no—
169#1571D5311yescharacteristic, complemented
170#1542—531no—
171#1518—531no—
172#1460—531no—
173#1286—531no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53#203611yescharacteristic
177C2×C53#204811yescharacteristic
178C2×C53#205311yescharacteristic
179D53—82no—
180#533—531no—
181C2×C4×D4—531no—
182#478—1062no—
183#476—1061no—
184#474—1061no—
185#431—531no—
186#431C2×D5311yescharacteristic
187#428—1061no—
188#423C2×D5311yescharacteristic
189C2×C4⋊C8C2×D5311yescharacteristic
190#418—1061no—
191C2×#102—1061no—
192#382—531no—
193#340—1061no—
194#335—22no—
195#333—1061no—
196#332—1061no—
197#328—1062no—
198#323—1062no—
199C53#1046911yescharacteristic, complemented, Sylow
200C22×D4—531no—
201C23×C4—1061no—
202C2×M16—21no—
203C22×C8—21no—
204C4⋊Q8—531no—
205C4⋊Q8—531no—
206#125—1061no—
207C22⋊Q8—1061no—
208C22⋊Q8—1061no—
209C4×Q8—1061no—
210C4×Q8—1061no—
211C4×D4—1061no—
212C4×D4—1061no—
213C4×D4—1061no—
214C4×D4—1061no—
215#117—1061no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C42C22×D5311yescharacteristic
220#107—1062no—
221#106—1062no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—1061no—
227C4⋊C8C22⋊D5312yes—
228#102—1062no—
229#102—1062no—
230#102—2122no—
231C22⋊C8—1061no—
232C22⋊C8D21212yes—
233C4×C8—21no—
234C4×C8—1061no—
235#95—1061no—
236#95—1061no—
237C24—1061no—
238C2×Q8—1061no—
239C2×Q8—1061no—
240C2×D4—532no—
241C2×D4—1061no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4—1061no—
246C22×C4—1061no—
247C22×C4—4242no—
248C22×C4—2121no—
249C22×C4—2121no—
250C22×C4—2121no—
251C22×C4C2×D21211yescharacteristic
252C22×C4D4×D5311yescharacteristic
253C22×C4#1171994911yescharacteristic
254M16—41no—
255M16—41no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—2121no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—2121no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C22⋊C4—2121no—
273C22⋊C4—2121no—
274C22⋊C4—2121no—
275C42—2121no—
276C42#1171994511yescharacteristic
277C42D4×D5311yescharacteristic
278C42#1171994911yescharacteristic
279C42C2×C22⋊D5311yescharacteristic
280C23—1062no—
281C23—4241no—
282C23—4241no—
283C23—2121no—
284C23—2121no—
285C23#1313319211yescharacteristic
286Q8—4241no—
287Q8—4241no—
288D4—1062no—
289D4—1062no—
290D4—2121no—
291D4—2121no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—4242no—
295C2×C4—4242no—
296C2×C4—4241no—
297C2×C4—4241no—
298C2×C4—4241no—
299C2×C4—4241no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4#1313318511yescharacteristic
304C2×C4#1313319111yescharacteristic
305C2×C4#1313319211yescharacteristic
306C2×C4#1313320211yescharacteristic
307C2×C4#1313320211yescharacteristic
308C2×C4#1313320611yescharacteristic
309C2×C4Q16×D5311yescharacteristic
310C2×C4#1313323611yescharacteristic
311C2×C4#1313324111yescharacteristic
312C2×C4#1313324911yescharacteristic
313C8—42no—
314C8—42no—
315C8—81no—
316C8—4241no—
317C22—21no—
318C22—21no—
319C22—4242no—
320C22—4242no—
321C22—4241no—
322C22—4241no—
323C22—2122no—
324C22#1681054011yescharacteristic
325C22#1681086511yescharacteristic
326C22#1681087011yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—8481no—
332C4—4241no—
333C4—4241no—
334C4—4241no—
335C4#1681076511yescharacteristic
336C4#1681077711yescharacteristic
337C4#1681096011yescharacteristic
338C4#1681096611yescharacteristic
339C2—21no—
340C2—21no—
341C2—4242no—
342C2#2027592011yescharacteristic
343C2#2027599011yescharacteristic
344C2#2027602111yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1571C2
socleC22×C53#394
prime radicalC22×D212D4
hypercenter#1571D53
hypocenterC53#10469
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
3. #20273373C2semidirect
4. #20273280C2semidirect
6. #20272575C2semidirect
7. C53×#1571C2semidirect
169. #1571D53semidirect
199. C53#10469semidirect

Maximal subgroups

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

subgroupindex
#202845142
#202747262
#202733732
#202732802
#202730352
#202725752
C53×#15712
#1046953