Finite Group Databasebeta

← #27277945

Subgroups — #27277945

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810679C211yesmaximal, characteristic
2#20274692C211yesmaximal, characteristic
3C4×#13133129C211yesmaximal, characteristic
4C2×#16810373C211yesmaximal, characteristic, complemented
5#20272550C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#335C211yesmaximal, characteristic, complemented
8C2×C42×D53C2211yescharacteristic
9#16810679—22no—
10#16810679—22no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133129—21no—
14C2×#13133129—21no—
15C2×#13133129C411yescharacteristic
16C2×#13133129C411yescharacteristic
17C2×#13133127—21no—
18C2×C4×#88322C2211yescharacteristic
19#16810377—21no—
20#16810377C411yescharacteristic
21#16810377C411yescharacteristic
22#16810373—22no—
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C53×#335—22no—
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53Q811yescharacteristic
30C22×#88322—21no—
31C22×#88322—21no—
32C22×#88322C2×C411yescharacteristic
33C22×#88322C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53Q811yescharacteristic
37C42×D53—21no—
38C42×D53—21no—
39C42×D53—21no—
40C42×D53—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322D412yes—
45#13133129—24no—
46#13133129—22no—
47#13133129—22no—
48#13133129—42no—
49#13133127—24no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C22×C8×C53C2×C411yescharacteristic
55C22×C8×C53C2×C411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53D412yes—
61C4×C8×C53—21no—
62C4×C8×C53—21no—
63C23×D53C4⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×Q811yescharacteristic
67C22×Dic53C4⋊C411yescharacteristic
68C22×Dic53C4⋊C411yescharacteristic
69C22×Dic53C4⋊C411yescharacteristic
70C2×C4×D53—22no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—41no—
86C2×C8×C53—21no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—22no—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53M1612yes—
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53#4111yescharacteristic
101C42×C53#4111yescharacteristic
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322—22no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322M1612yes—
120C23×C53C2×C4⋊C411yescharacteristic
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53#10812yes—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53C22⋊Q811yescharacteristic
134C2×C4×C53C22⋊Q811yescharacteristic
135C2×C4×C53#12311yescharacteristic
136C2×C4×C53C2×M1611yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C2×C4×C53#13111yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×C4×C53#13511yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C53#13711yescharacteristic
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C4×D53—42no—
154C4×D53—42no—
155C4×D53—41no—
156C4×D53—41no—
157C4×D53—41no—
158C4×D53—41no—
159C4×D53—41no—
160C4×D53—41no—
161C4×D53—41no—
162C4×D53—41no—
163C8×C53—22no—
164C8×C53—41no—
165C8×C53—42no—
166C8×C53—81no—
167#88322—22no—
168#88322—41no—
169#88322—42no—
170#88322—81no—
171#8834—531nomaximal, Sylow
172C22×C53#38511yescharacteristic
173C22×C53#42311yescharacteristic
174C22×C53#42411yescharacteristic
175C22×C53#42711yescharacteristic
176C22×C53#42811yescharacteristic
177C22×C53C2×#10811yescharacteristic
178C22×C53#43011yescharacteristic
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53#43511yescharacteristic
189C4×C53#43511yescharacteristic
190C4×C53#43611yescharacteristic
191C4×C53#43611yescharacteristic
192C4×C53#44911yescharacteristic
193C4×C53#45211yescharacteristic
194C4×C53#48211yescharacteristic
195C4×C53#48511yescharacteristic
196Dic53—41no—
197Dic53—41no—
198Dic53—41no—
199C2×#432—531no—
200#1797—531no—
201C4×C22⋊C8—531no—
202C2×#335D5311yescharacteristic, complemented
203C2×#334—531no—
204#1277—531no—
205#1277—531no—
206C2×C53#156811yescharacteristic
207C2×C53#156911yescharacteristic
208C2×C53#157211yescharacteristic
209C2×C53#157311yescharacteristic
210C2×C53#185111yescharacteristic
211C2×C53#185411yescharacteristic
212C2×C53#185511yescharacteristic
213D53—42no—
214C22×C42—531no—
215#432—1062no—
216#432—1062no—
217C2×C4⋊C8—531no—
218C2×C4⋊C8—531no—
219C2×C4⋊C8C2×D5311yescharacteristic
220C2×C4⋊C8C2×D5311yescharacteristic
221C2×C22⋊C8—531no—
222C2×C22⋊C8—531no—
223C2×C22⋊C8—1061no—
224C2×C22⋊C8—1061no—
225C2×#97—1061no—
226C2×C4×C8—531no—
227C2×C4×C8C2×D5311yescharacteristic
228#336—531no—
229#336—531no—
230#336—1061no—
231#335—22no—
232#334—1062no—
233C53#883411yescharacteristic, complemented, Sylow
234C23×C4—531no—
235C23×C4—531no—
236C23×C4—531no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—531no—
240C22×C8—531no—
241C22×C8—1061no—
242C22×C8—1061no—
243C22×C8C4×D5311yescharacteristic
244C22×C8C4×D5311yescharacteristic
245C2×C42—531no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—1061no—
249C2×C42—1061no—
250C2×C42—1061no—
251C2×C42—1061no—
252C2×C42C22×D5311yescharacteristic
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—532no—
257C4⋊C8—1061no—
258C4⋊C8—1061no—
259C4⋊C8—1061no—
260C4⋊C8C22⋊D5312yes—
261C22⋊C8—1062no—
262C22⋊C8—1064no—
263C22⋊C8—1062no—
264C22⋊C8—2122no—
265#97—1064no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—1061no—
269C4×C8—1061no—
270C24—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—1062no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—2121no—
290C22×C4C2×C4×D5311yescharacteristic
291C22×C4D4×D5311yescharacteristic
292C22×C4Q8×D5311yescharacteristic
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—532no—
305C2×C8—1061no—
306C2×C8—1061no—
307C2×C8—1061no—
308C2×C8—1062no—
309C2×C8—1061no—
310C2×C8—1061no—
311C2×C8—2121no—
312C2×C8—2121no—
313C2×C8—2121no—
314C2×C8—2121no—
315C2×C8—2121no—
316C2×C8#1171991212yes—
317C42—1061no—
318C42—1061no—
319C42—1061no—
320C42—1061no—
321C42—1061no—
322C42—1061no—
323C42#1171994511yescharacteristic
324C42D4×D5311yescharacteristic
325C42#1171994911yescharacteristic
326C42C2×C22⋊D5311yescharacteristic
327C23—532no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23C4⋊C4×D5311yescharacteristic
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4C4×C22⋊D5311yescharacteristic
357C2×C4#1313317911yescharacteristic
358C2×C4#1313318511yescharacteristic
359C2×C4#1313319311yescharacteristic
360C2×C4#1313319511yescharacteristic
361C2×C4#1313320211yescharacteristic
362C2×C4M16×D5311yescharacteristic
363C2×C4C2×#1171991211yescharacteristic
364C2×C4#1313322311yescharacteristic
365C2×C4#1313322511yescharacteristic
366C2×C4#1313323611yescharacteristic
367C2×C4#1313324011yescharacteristic
368C2×C4#1313324111yescharacteristic
369C2×C4#1313324911yescharacteristic
370C8—22no—
371C8—42no—
372C8—41no—
373C8—81no—
374C8—1062no—
375C8—4241no—
376C8—2122no—
377C8—2121no—
378C22—1062no—
379C22—1062no—
380C22—1062no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22—2121no—
385C22#1681049411yescharacteristic
386C22#1681064511yescharacteristic
387C22#1681064811yescharacteristic
388C22#1681066011yescharacteristic
389C22#1681066411yescharacteristic
390C22#108×D5311yescharacteristic
391C22#1681067011yescharacteristic
392C4—21no—
393C4—21no—
394C4—2121no—
395C4—2121no—
396C4—2121no—
397C4#1681069311yescharacteristic
398C4#1681069411yescharacteristic
399C4#1681069911yescharacteristic
400C4#1681070011yescharacteristic
401C4#1681076511yescharacteristic
402C4#1681078411yescharacteristic
403C4#1681095011yescharacteristic
404C4#1681096611yescharacteristic
405C2—2122no—
406C2#2027347711yescharacteristic
407C2#2027348111yescharacteristic
408C2#2027349411yescharacteristic
409C2#2027349911yescharacteristic
410C2#2027489611yescharacteristic
411C2#2027491111yescharacteristic
412C2#2027491811yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#335C2
socleC23×C53C2×C4⋊C4
prime radicalC23×D53C4⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#335D53
hypocenterC53#8834
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
4. C2×#16810373C2semidirect
5. #20272550C2semidirect
6. #20272549C2semidirect
7. C2×C53×#335C2semidirect
202. C2×#335D53semidirect
233. C53#8834semidirect

Maximal subgroups

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

subgroupindex
C2×#168106792
#202746922
C4×#131331292
C2×#168103732
#202725502
#202725492
C2×C53×#3352
#883453