Finite Group Databasebeta

← #27280387

Subgroups — #27280387

Order 13568.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277180C211yesmaximal, characteristic, complemented
2#20275544C211yesmaximal, characteristic, complemented
3#20273210C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic, complemented
5#20272617C211yesmaximal, characteristic, complemented
6#20272547C211yesmaximal, characteristic, complemented
7C53×#3100C211yesmaximal, characteristic
8C2×#13133176—21no—
9#16810676C2211yescharacteristic
10#16810647C2211yescharacteristic
11C2×#13133144—21no—
12C2×#13133144—21no—
13#16810526—21no—
14#16810522—21no—
15#16810516—22no—
16C4×C4⋊Dic53C2211yescharacteristic
17C4×C22⋊Dic53—21no—
18C42×Dic53C2211yescharacteristic
19#16810357—22no—
20D4⋊#88322—22no—
21C53×#532C2211yescharacteristic
22C2×C4×D4×C53C2211yescharacteristic
23C53×#480—21no—
24C53×#474C412yescomplemented
25C53×#459C412yescomplemented
26C53×#447—21no—
27C4×D8×C53—22no—
28C53×#431C2211yescharacteristic
29C2×#11719917—21no—
30#13133176—42no—
31#13133176—42no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53—21no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—22no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44#13133144—42no—
45#13133144—42no—
46C22⋊#88322—21no—
47C4×#88322—21no—
48#13133124—22no—
49#13133124D412yes—
50C2×#41×C53—21no—
51C22×D4×C53D411yescharacteristic
52C23×C4×C53—21no—
53C2×D8×C53—22no—
54C4⋊Q8×C53C2×C411yescharacteristic
55#127×C53C2×C411yescharacteristic
56#124×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53C2×C412yes—
64C4×D4×C53D412yes—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C22⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68#107×C53—22no—
69C4⋊C8×C53C2×C412yes—
70#102×C53—22no—
71#102×C53—22no—
72C22⋊C8×C53—21no—
73C4×C8×C53—21no—
74C24×C53—21no—
75#41×C53—41no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79C2×Q8×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—42no—
85C2×D4×C53—41no—
86C2×D4×C53C22⋊C412yes—
87C22×C4×C53—21no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×Dic53—21no—
99C22×Dic53—21no—
100C22×Dic53—22no—
101D8×C53—44no—
102C2×C8×C53—22no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53C22⋊C412yes—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C42×C53C22×C411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114C42×C53#4111yescharacteristic
115C22⋊Dic53—44no—
116C22⋊Dic53—44no—
117C4⋊Dic53—44no—
118C4⋊Dic53—42no—
119C4×Dic53—21no—
120C4×Dic53—21no—
121C4×Dic53—21no—
122C4×Dic53—42no—
123C4×Dic53—42no—
124C4×Dic53—41no—
125C4×Dic53—41no—
126C4×Dic53D812yescomplemented
127#11719917—41no—
128#11719917—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C23×C53—22no—
132C23×C53—42no—
133C23×C53—42no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53#12011yescharacteristic
138Q8×C53—41no—
139Q8×C53—41no—
140D4×C53—22no—
141D4×C53—22no—
142D4×C53—42no—
143D4×C53—42no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53—81no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—41no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53C2×C22⋊C411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53C4×D411yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12311yescharacteristic
167C2×C4×C53C2×D811yescharacteristic
168C2×C4×C53#13511yescharacteristic
169C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×Dic53—22no—
172C2×Dic53—22no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C8×C53—42no—
181#88322—81no—
182#88322—81no—
183#9237—531nomaximal, Sylow
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—82no—
190C22×C53—82no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53#38611yescharacteristic
195C22×C53#44711yescharacteristic
196C22×C53#44911yescharacteristic
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—42no—
201C4×C53—81no—
202C4×C53#42112yes—
203C4×C53C4×D811yescharacteristic
204C4×C53#44211yescharacteristic
205C4×C53#45911yescharacteristic
206C4×C53#46311yescharacteristic
207Dic53—42no—
208Dic53—42no—
209Dic53—82no—
210#3100D5311yescharacteristic
211#2293—531no—
212#1962—531no—
213#1502—531no—
214#1486—531no—
215#1303—531no—
216#1276—531no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—82no—
220C2×C53—161no—
221C2×C53#187911yescharacteristic
222C2×C53#189711yescharacteristic
223C2×C53#190611yescharacteristic
224#532C2×D5311yescharacteristic
225C2×C4×D4C2×D5311yescharacteristic
226C2×#117—1061no—
227#480—21no—
228#474Dic5312yescomplemented
229#459Dic5312yescomplemented
230#447—21no—
231C4×D8—22no—
232#431—531no—
233#431C2×D5311yescharacteristic
234#423—531no—
235C2×#104—1061no—
236C2×#104—1061no—
237#390—1061no—
238#389—1061no—
239#388—1062no—
240C4×C4⋊C4—531no—
241C4×C22⋊C4—1061no—
242C43—531no—
243#327—1062no—
244D4⋊C8—1062no—
245C53#923711yescharacteristic, complemented, Sylow
246C2×#41—21no—
247C22×D4C22⋊D5311yescharacteristic
248C23×C4—21no—
249C2×D8—22no—
250C2×M16—1061no—
251C4⋊Q8C2×Dic5311yescharacteristic
252#127C2×Dic5311yescharacteristic
253#124—21no—
254#121—21no—
255#121—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C2×Dic5312yes—
261C4×D4C22⋊D5312yes—
262#117—2122no—
263#117—2122no—
264C2×C4⋊C4—1062no—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4C22⋊D5311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1062no—
269C2×C22⋊C4—1062no—
270C2×C42—531no—
271C2×C42—531no—
272C2×C42—1061no—
273C2×C42—1061no—
274C2×C42—1061no—
275C2×C42C22×D5311yescharacteristic
276#107—22no—
277C4⋊C8—1061no—
278C4⋊C8—1061no—
279C4⋊C8—1061no—
280C4⋊C8C2×Dic5312yes—
281#104—2122no—
282#104—2122no—
283#102—22no—
284#102—22no—
285C22⋊C8—21no—
286C22⋊C8—1061no—
287C4×C8—21no—
288C4×C8—1061no—
289#95—532no—
290#95—1062no—
291C24—21no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4C22⋊Dic5312yes—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—1061no—
313C22×C4—1061no—
314C22×C4—1062no—
315C22×C4D4×D5311yescharacteristic
316C22×C4D4×D5311yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318D8—44no—
319M16—2121no—
320M16—2121no—
321C2×C8—22no—
322C2×C8—2121no—
323C2×C8—2121no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—2122no—
328C4⋊C4—2124no—
329C4⋊C4C22⋊Dic5312yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—2124no—
335C22⋊C4—2124no—
336C42—532no—
337C42—1061no—
338C42—1061no—
339C42—1061no—
340C42—2122no—
341C42—2122no—
342C42—2121no—
343C42—2121no—
344C42#1171994711yescharacteristic
345C42#1171994711yescharacteristic
346C42C22×Dic5311yescharacteristic
347C42C2×C22⋊D5311yescharacteristic
348C23—22no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23#1313318511yescharacteristic
355Q8—41no—
356Q8—41no—
357D4—22no—
358D4—22no—
359D4—42no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—1062no—
379C2×C4—1062no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2121no—
386C2×C4—2121no—
387C2×C4C2×C22⋊Dic5311yescharacteristic
388C2×C4D4×Dic5311yescharacteristic
389C2×C4D4×Dic5311yescharacteristic
390C2×C4#1313318911yescharacteristic
391C2×C4#1313318911yescharacteristic
392C2×C4#1313319811yescharacteristic
393C2×C4D8×D5311yescharacteristic
394C2×C4#1313323611yescharacteristic
395C2×C4#1313324111yescharacteristic
396C2×C4#1313324111yescharacteristic
397C8—42no—
398C8—4241no—
399C8—4241no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#1681050311yescharacteristic
411C22#1681075511yescharacteristic
412C22#1681076511yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—81no—
418C4—4242no—
419C4—2122no—
420C4—2122no—
421C4#1681064012yes—
422C4D8×Dic5311yescharacteristic
423C4#1681073511yescharacteristic
424C4#1681081611yescharacteristic
425C4#1681083611yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2#2027506511yescharacteristic
431C2#2027516411yescharacteristic
432C2#2027521811yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3100C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3100D53
hypocenterC53#9237
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #20277180C2semidirect
2. #20275544C2semidirect
3. #20273210C2semidirect
4. #20273140C2semidirect
5. #20272617C2semidirect
6. #20272547C2semidirect
24. C53×#474C4semidirect
25. C53×#459C4semidirect
126. C4×Dic53D8semidirect
228. #474Dic53semidirect
229. #459Dic53semidirect
245. C53#9237semidirect

Maximal subgroups

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

subgroupindex
#202771802
#202755442
#202732102
#202731402
#202726172
#202725472
C53×#31002
#923753