Finite Group Databasebeta

← #27285188

Subgroups — #27285188

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810969C211yesmaximal, characteristic, complemented
2#20279668C211yesmaximal, characteristic, complemented
3#20275643C211yesmaximal, characteristic, complemented
4C2×#16810368C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×#459C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×#13133196—22no—
10#16810969—22no—
11#16810969—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133138—22no—
14C2×#13133138—22no—
15C2×C4×#88322C2211yescharacteristic
16#16810559—22no—
17#16810526—21no—
18#16810492—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810368—22no—
21D4⋊#88322—22no—
22C22×D8×C53—21no—
23C2×#127×C53C2211yescharacteristic
24C2×C4×D4×C53C2211yescharacteristic
25C53×#459—22no—
26C53×#459—22no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#102×C53—21no—
29C22×Dic106—21no—
30C22×#88322—21no—
31C22×#88322—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133196—44no—
37#13133196—44no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53D412yes—
40C2×C4⋊Dic53D412yes—
41C2×#11719919—24no—
42C2×C22⋊Dic53—24no—
43C2×C4×Dic53—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46#13133138—28no—
47#13133138—44no—
48C4×#88322—21no—
49C4×#88322D412yes—
50#13133124—21no—
51C22×D4×C53—21no—
52C22×D4×C53—21no—
53C22×D4×C53—21no—
54C22×D4×C53D411yescharacteristic
55C23×C4×C53—21no—
56C2×D8×C53—24no—
57C2×D8×C53—42no—
58C2×D8×C53—42no—
59C22×C8×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53—21no—
63#127×C53D412yes—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C22⋊C4×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—21no—
72C4⋊C8×C53D412yes—
73#102×C53—24no—
74C24×C53—21no—
75C24×C53—41no—
76C2×D4×C53—21no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53—21no—
80C2×D4×C53—22no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—44no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—42no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—42no—
93C2×D4×C53—41no—
94C2×D4×C53SD1612yes—
95C22×C4×C53—21no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—42no—
99C22×C4×C53—42no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×Dic53—22no—
104C22×Dic53—22no—
105C2×Dic106—41no—
106C2×Dic106—41no—
107C2×Dic106—42no—
108C2×Dic106—42no—
109D8×C53—48no—
110C2×C8×C53—22no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C22⋊Dic53—48no—
122C4⋊Dic53—22no—
123C4⋊Dic53—22no—
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53D814yescomplemented
128#11719919—48no—
129C4×Dic53—41no—
130C4×Dic53—41no—
131C2×#88322—22no—
132C2×#88322—42no—
133C2×#88322—41no—
134C2×#88322—42no—
135C23×C53—22no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53—41no—
139C23×C53—42no—
140C23×C53—42no—
141C23×C53—82no—
142C23×C53—81no—
143C23×C53—81no—
144C23×C53—82no—
145C23×C53#12011yescharacteristic
146D4×C53—22no—
147D4×C53—22no—
148D4×C53—44no—
149D4×C53—42no—
150D4×C53—41no—
151D4×C53—42no—
152D4×C53—44no—
153D4×C53—41no—
154D4×C53—82no—
155D4×C53—81no—
156D4×C53—81no—
157D4×C53—81no—
158D4×C53—81no—
159C2×C4×C53—22no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—82no—
163C2×C4×C53—81no—
164C2×C4×C53—81no—
165C2×C4×C53#12011yescharacteristic
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12711yescharacteristic
171C2×C4×C53C2×D812yes—
172C2×C4×C53C2×SD1611yescharacteristic
173C2×C4×C53#13511yescharacteristic
174C2×C4×C53AGL(1,ℤ/8)12yes—
175C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C53#13711yescharacteristic
177C2×Dic53—24no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—42no—
181C2×Dic53—44no—
182Dic106—82no—
183Dic106—81no—
184Dic106—81no—
185C8×C53—42no—
186#88322—42no—
187#88322—81no—
188#10052—531nomaximal, Sylow
189C22×C53—44no—
190C22×C53—42no—
191C22×C53—82no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—82no—
195C22×C53—82no—
196C22×C53—82no—
197C22×C53—162no—
198C22×C53—162no—
199C22×C53#39211yescharacteristic
200C22×C53#44712yes—
201C22×C53#44811yescharacteristic
202C22×C53#44911yescharacteristic
203C22×C53#45412yes—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—81no—
207C4×C53#45011yescharacteristic
208C4×C53#45211yescharacteristic
209C4×C53#45912yes—
210C4×C53#46511yescharacteristic
211C4×C53#46811yescharacteristic
212C4×C53#49311yescharacteristic
213C4×C53#49611yescharacteristic
214Dic53—44no—
215Dic53—82no—
216C2×#486—531no—
217C2×#459D5311yescharacteristic
218#2542—531no—
219#1982—531no—
220C2×#332—531no—
221C2×D4⋊C8—531no—
222#1279—531no—
223C2×C53—82no—
224C2×C53—162no—
225C2×C53#162512yes—
226C2×C53#162712yes—
227C2×C53#200011yescharacteristic
228C2×C53#200311yescharacteristic
229C2×C53#201111yescharacteristic
230C22×D8—21no—
231C2×C4⋊Q8—531no—
232C2×#127C2×D5311yescharacteristic
233C2×C22⋊Q8—1062no—
234C2×C4×D4C2×D5311yescharacteristic
235#486—1062no—
236#486—1062no—
237#459—22no—
238#459—22no—
239C2×C4⋊C8—531no—
240C2×C4⋊C8C2×D5311yescharacteristic
241C2×#102—21no—
242C2×#102—1062no—
243C2×#102—1062no—
244C2×C4×C8—531no—
245#399—1062no—
246#390—1061no—
247#384—1061no—
248C4×C4⋊C4—531no—
249#332—1062no—
250D4⋊C8—1062no—
251C53#1005211yescharacteristic, complemented, Sylow
252C22×Q8—1061no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4C22⋊D5311yescharacteristic
257C23×C4—21no—
258C2×D8—24no—
259C2×D8—42no—
260C2×D8—42no—
261C22×C8—21no—
262C22×C8—1061no—
263C22×C8—1061no—
264C4⋊Q8—1061no—
265C4⋊Q8—1061no—
266C4⋊Q8—1061no—
267C4⋊Q8—1061no—
268#127—21no—
269#127—21no—
270#127—21no—
271#127C22⋊D5312yes—
272C22⋊Q8—2124no—
273C22⋊Q8—2124no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C2×C4⋊C4—532no—
279C2×C4⋊C4—532no—
280C2×C4⋊C4—1064no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4C22⋊D5311yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—1064no—
285C2×C42—1061no—
286C2×C42C22×D5311yescharacteristic
287C4⋊C8—21no—
288C4⋊C8—1061no—
289C4⋊C8—1061no—
290C4⋊C8C22⋊D5312yes—
291#102—24no—
292#102—1068no—
293#102—2124no—
294C4×C8—532no—
295C4×C8—1061no—
296#95—1061no—
297C24—21no—
298C24—41no—
299C2×Q8—2121no—
300C2×Q8—2121no—
301C2×Q8—2122no—
302C2×Q8—2122no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—21no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—44no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4#1171992312yes—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—1062no—
328C22×C4—1062no—
329C22×C4D4×D5311yescharacteristic
330C22×C4D4×D5311yescharacteristic
331C22×C4C2×C22⋊D5311yescharacteristic
332D8—48no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—1062no—
337C2×C8—2121no—
338C2×C8—2122no—
339C2×C8—2122no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—534no—
343C4⋊C4—1062no—
344C4⋊C4—1062no—
345C4⋊C4—1062no—
346C4⋊C4—2128no—
347C4⋊C4—2121no—
348C4⋊C4—2121no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—2128no—
352C42—2121no—
353C42—2121no—
354C42D4×D5311yescharacteristic
355C42#1171994711yescharacteristic
356C42C2×C22⋊D5311yescharacteristic
357C42C2×C22⋊D5311yescharacteristic
358C23—22no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—82no—
365C23—81no—
366C23—81no—
367C23—82no—
368C23#1313318511yescharacteristic
369Q8—4241no—
370Q8—4242no—
371Q8—4241no—
372D4—22no—
373D4—22no—
374D4—44no—
375D4—42no—
376D4—41no—
377D4—42no—
378D4—44no—
379D4—41no—
380D4—82no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385C2×C4—22no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—1064no—
392C2×C4—2124no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4—2122no—
396C2×C4#1313318511yescharacteristic
397C2×C4#1313318611yescharacteristic
398C2×C4#1313318811yescharacteristic
399C2×C4#1313318911yescharacteristic
400C2×C4#1313318911yescharacteristic
401C2×C4#1313321311yescharacteristic
402C2×C4D8×D5312yes—
403C2×C4C2×#1171992311yescharacteristic
404C2×C4#1313323711yescharacteristic
405C2×C4#1313324112yes—
406C2×C4#1313324311yescharacteristic
407C2×C4#1313324711yescharacteristic
408C8—42no—
409C8—4241no—
410C8—2122no—
411C22—42no—
412C22—44no—
413C22—81no—
414C22—81no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—162no—
420C22—162no—
421C22#1681053111yescharacteristic
422C22#1681075512yes—
423C22#1681076111yescharacteristic
424C22#1681076611yescharacteristic
425C22#1681079412yes—
426C4—21no—
427C4—21no—
428C4—81no—
429C4—4242no—
430C4—2124no—
431C4#1681077411yescharacteristic
432C4#1681078511yescharacteristic
433C4#1681081612yes—
434C4#1681085011yescharacteristic
435C4#1681086711yescharacteristic
436C4#1681099811yescharacteristic
437C4#1681101111yescharacteristic
438C2—82no—
439C2—162no—
440C2#2027380312yes—
441C2#2027381712yes—
442C2#2027571811yescharacteristic
443C2#2027573411yescharacteristic
444C2#2027578411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#459C2
socleC23×C53#120
hypercenterC2×#459D53
hypocenterC53#10052
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16810969C2semidirect
2. #20279668C2semidirect
3. #20275643C2semidirect
4. C2×#16810368C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272554C2semidirect
127. C4⋊Dic53D8semidirect
251. C53#10052semidirect

Maximal subgroups

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

subgroupindex
C2×#168109692
#202796682
#202756432
C2×#168103682
C2×D4⋊#883222
#202725542
C2×C53×#4592
#1005253