Finite Group Databasebeta

← C53×#8999 #27216511

Subgroups — C53×#8999 #27216511

Order 13568.

530 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
1C2×C53×#489C211yesmaximal, characteristic
2C53×#2379C211yesmaximal, characteristic
3C53×#1993C211yesmaximal, characteristic, complemented
4C53×#1960C211yesmaximal, characteristic, complemented
5C2×C53×#329C211yesmaximal, characteristic
6C53×#1295C211yesmaximal, characteristic, complemented
7C53×#1295C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C53C2211yescharacteristic
9C2×#124×C53C2211yescharacteristic
10C53×#489—22no—
11C53×#489C414yescomplemented
12C2×#103×C53—21no—
13C2×#103×C53—21no—
14C2×#103×C53—21no—
15C2×#102×C53—21no—
16C2×#97×C53C2211yescharacteristic
17C2×#97×C53C2211yescharacteristic
18C2×#97×C53C2211yescharacteristic
19C53×#391—21no—
20C53×#388—21no—
21C53×#388—21no—
22C53×#386—21no—
23C53×#385—21no—
24C53×#385—21no—
25C53×#382C2211yescharacteristic
26C53×#382C2211yescharacteristic
27C53×#340—21no—
28C53×#340—21no—
29C53×#329—22no—
30C22×Q8×C53—21no—
31C22×Q8×C53D411yescharacteristic
32C22×D4×C53D411yescharacteristic
33C23×C4×C53—21no—
34C22×C8×C53—21no—
35C22×C8×C53—21no—
36C22×C8×C53—21no—
37C4⋊Q8×C53—21no—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—21no—
40C4⋊Q8×C53C2×C412yescomplemented
41#124×C53—21no—
42#124×C53—21no—
43#124×C53C2×C412yes—
44#124×C53D412yes—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C42×C53—21no—
58C2×C42×C53C2311yescharacteristic
59#103×C53—24no—
60#103×C53—42no—
61#103×C53—42no—
62#102×C53—24no—
63#97×C53—21no—
64#97×C53—21no—
65#97×C53—21no—
66#97×C53—21no—
67#97×C53—21no—
68#97×C53C2×C412yescomplemented
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72#95×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75C24×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—22no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×Q8×C53—41no—
84C2×Q8×C53—41no—
85C2×Q8×C53C22⋊C412yes—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53C22⋊C412yes—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—41no—
96C22×C4×C53—81no—
97C22×C4×C53—81no—
98C22×C4×C53—81no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C2×C8×C53—41no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—81no—
120C22⋊C4×C53—22no—
121C22⋊C4×C53—22no—
122C22⋊C4×C53—22no—
123C22⋊C4×C53—22no—
124C22⋊C4×C53—81no—
125C22⋊C4×C53—81no—
126C22⋊C4×C53—81no—
127C42×C53—81no—
128C42×C53C22×C411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53#4111yescharacteristic
132C23×C53—22no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53#12011yescharacteristic
140Q8×C53—42no—
141Q8×C53—42no—
142Q8×C53—42no—
143Q8×C53—81no—
144Q8×C53—81no—
145Q8×C53—81no—
146D4×C53—42no—
147D4×C53—81no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—22no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—84no—
171C2×C4×C53—81no—
172C2×C4×C53—81no—
173C2×C4×C53C2×C22⋊C411yescharacteristic
174C2×C4×C53C4×D411yescharacteristic
175C2×C4×C53C4×D411yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12311yescharacteristic
179C8×C53—42no—
180C8×C53—42no—
181C8×C53—42no—
182#8999C5311yesmaximal, characteristic, direct product factor, Sylow
183C22×C53—42no—
184C22×C53—42no—
185C22×C53—42no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53#38611yescharacteristic
191C22×C53#45311yescharacteristic
192C22×C53#45411yescharacteristic
193C22×C53#45511yescharacteristic
194C22×C53#45511yescharacteristic
195C22×C53#45511yescharacteristic
196C22×C53#45611yescharacteristic
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—42no—
204C4×C53—82no—
205C4×C53—81no—
206C4×C53—81no—
207C4×C53—81no—
208C4×C53—81no—
209C2×#489C2×C5311yescharacteristic
210#2379C2×C5311yescharacteristic
211#1993C2×C5311yescharacteristic, complemented
212#1960C2×C5311yescharacteristic, complemented
213C2×#329C2×C5311yescharacteristic
214#1295C2×C5311yescharacteristic, complemented
215#1295C2×C5311yescharacteristic, complemented
216C2×C53—82no—
217C2×C53#165911yescharacteristic
218C2×C53#166111yescharacteristic
219C2×C53#166211yescharacteristic
220C2×C53#166411yescharacteristic
221C2×C53#188211yescharacteristic
222C2×C53#188311yescharacteristic
223C2×C53#188511yescharacteristic
224C2×C4⋊Q8C22×C5311yescharacteristic
225C2×#124C22×C5311yescharacteristic
226#489—22no—
227#489C4×C5314yescomplemented
228C2×#103—21no—
229C2×#103—21no—
230C2×#103—21no—
231C2×#102—21no—
232C2×#97C22×C5311yescharacteristic
233C2×#97C22×C5311yescharacteristic
234C2×#97C22×C5311yescharacteristic
235#391—21no—
236#388—21no—
237#388—21no—
238#386—21no—
239#385—21no—
240#385—21no—
241#382C22×C5311yescharacteristic
242#382C22×C5311yescharacteristic
243#340—21no—
244#340—21no—
245#329—22no—
246C53#899911yescharacteristic, direct product factor, Sylow
247C22×Q8—21no—
248C22×Q8D4×C5311yescharacteristic
249C22×D4D4×C5311yescharacteristic
250C23×C4—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8C2×C4×C5312yescomplemented
258#124—21no—
259#124—21no—
260#124C2×C4×C5312yes—
261#124D4×C5312yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4D4×C5311yescharacteristic
266C2×C4⋊C4D4×C5311yescharacteristic
267C2×C4⋊C4D4×C5311yescharacteristic
268C2×C4⋊C4D4×C5311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42C23×C5311yescharacteristic
276#103—24no—
277#103—42no—
278#103—42no—
279#102—24no—
280#97—21no—
281#97—21no—
282#97—21no—
283#97—21no—
284#97—21no—
285#97C2×C4×C5312yescomplemented
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292C24—21no—
293C2×Q8—21no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8C22⋊C4×C5312yes—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4C22⋊C4×C5312yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4C2×D4×C5311yescharacteristic
317C22×C4C2×D4×C5311yescharacteristic
318C22×C4C2×D4×C5311yescharacteristic
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—81no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—81no—
342C22⋊C4—81no—
343C22⋊C4—81no—
344C42—81no—
345C42C22×C4×C5311yescharacteristic
346C42C2×D4×C5311yescharacteristic
347C42#41×C5311yescharacteristic
348C42#41×C5311yescharacteristic
349C23—22no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23#120×C5311yescharacteristic
357Q8—42no—
358Q8—42no—
359Q8—42no—
360Q8—81no—
361Q8—81no—
362Q8—81no—
363D4—42no—
364D4—81no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—84no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4C2×C22⋊C4×C5311yescharacteristic
391C2×C4C4×D4×C5311yescharacteristic
392C2×C4C4×D4×C5311yescharacteristic
393C2×C4#121×C5311yescharacteristic
394C2×C4#121×C5311yescharacteristic
395C2×C4#123×C5311yescharacteristic
396C8—42no—
397C8—42no—
398C8—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22C53×#38611yescharacteristic
407C22C53×#45311yescharacteristic
408C22C53×#45411yescharacteristic
409C22C53×#45511yescharacteristic
410C22C53×#45511yescharacteristic
411C22C53×#45511yescharacteristic
412C22C53×#45611yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—82no—
425C2—82no—
426C2C53×#165911yescharacteristic
427C2C53×#166111yescharacteristic
428C2C53×#166211yescharacteristic
429C2C53×#166411yescharacteristic
430C2C53×#188211yescharacteristic
431C2C53×#188311yescharacteristic
432C2C53×#188511yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1993C2semidirect
4. C53×#1960C2semidirect
6. C53×#1295C2semidirect
7. C53×#1295C2semidirect
11. C53×#489C4semidirect
40. C4⋊Q8×C53C2×C4semidirect
68. #97×C53C2×C4semidirect
182. #8999C53direct (N × H)
211. #1993C2×C53semidirect
212. #1960C2×C53semidirect
214. #1295C2×C53semidirect
215. #1295C2×C53semidirect
227. #489C4×C53semidirect
246. C53#8999direct (N × H)
257. C4⋊Q8C2×C4×C53semidirect
285. #97C2×C4×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#4892
C53×#23792
C53×#19932
C53×#19602
C2×C53×#3292
C53×#12952
#899953