Finite Group Databasebeta

← C53×#19300 #27226812

Subgroups — C53×#19300 #27226812

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#487C211yesmaximal, characteristic, complemented
2C53×#3079C211yesmaximal, characteristic
3C53×#3039C211yesmaximal, characteristic
4C53×#2965C211yesmaximal, characteristic
5C53×#1700C214yesmaximal, complemented
6C53×#1699C214yesmaximal, complemented
7C53×#1567C211yesmaximal, characteristic, complemented
8C53×#1549C211yesmaximal, characteristic, complemented
9C53×#1546C211yesmaximal, characteristic, complemented
10C2×#137×C53—21no—
11C2×C4⋊Q8×C53C2211yescharacteristic
12C2×C4⋊Q8×C53C2211yescharacteristic
13C4×#41×C53C2211yescharacteristic
14C53×#487—21no—
15C53×#487—21no—
16C53×#487C2214yes—
17C53×#477—21no—
18C53×#476—21no—
19C53×#475C2212yes—
20C53×#474C2212yes—
21C53×#462—21no—
22C53×#462C2212yes—
23C53×#461—21no—
24C53×#461C2212yes—
25C53×#446C2212yes—
26C53×#436C2212yes—
27C53×#435—22no—
28C53×#431—21no—
29C53×#431C2211yescharacteristic
30C53×#428—21no—
31C53×#423C2211yescharacteristic
32C53×#423C2211yescharacteristic
33C53×#419—21no—
34C53×#417—21no—
35C2×#103×C53—22no—
36C4×M16×C53—21no—
37C2×C4×C8×C53C2211yescharacteristic
38C53×#335C2214yes—
39C53×#332C2214yes—
40C53×#332C2214yes—
41C2×#41×C53D411yescharacteristic
42C22×Q8×C53—21no—
43C22×Q8×C53—21no—
44#137×C53—44no—
45C2×Q16×C53—22no—
46C2×SD16×C53—22no—
47C2×M16×C53—22no—
48C2×M16×C53—21no—
49C2×M16×C53—21no—
50C22×C8×C53D412yes—
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53C2312yes—
58C4⋊Q8×C53C2312yes—
59C4×Q8×C53D411yescharacteristic
60C4×Q8×C53C2311yescharacteristic
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53D411yescharacteristic
64C4×D4×C53C2311yescharacteristic
65#117×C53—21no—
66#117×C53D411yescharacteristic
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73#107×C53—22no—
74#106×C53—22no—
75C4⋊C8×C53—22no—
76C4⋊C8×C53—21no—
77C4⋊C8×C53—21no—
78C4⋊C8×C53C2312yes—
79C4⋊C8×C53C2312yes—
80C4⋊C8×C53C2312yes—
81#103×C53—24no—
82#103×C53—24no—
83#103×C53—22no—
84#103×C53—22no—
85#103×C53—22no—
86#102×C53—22no—
87C22⋊C8×C53—22no—
88C22⋊C8×C53—21no—
89#97×C53—22no—
90C4×C8×C53—21no—
91C4×C8×C53D411yescharacteristic
92C4×C8×C53D411yescharacteristic
93C4×C8×C53C2312yes—
94#41×C53—22no—
95#41×C53—22no—
96C2×Q8×C53—22no—
97C2×Q8×C53—22no—
98C2×Q8×C53—44no—
99C2×Q8×C53—41no—
100C2×Q8×C53—41no—
101C2×Q8×C53—42no—
102C2×Q8×C53—42no—
103C2×Q8×C53C2×D411yescharacteristic
104C2×D4×C53—21no—
105C2×D4×C53C2×D411yescharacteristic
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—41no—
110C22×C4×C53—41no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114Q16×C53—44no—
115SD16×C53—44no—
116M16×C53—44no—
117M16×C53—42no—
118M16×C53—42no—
119C2×C8×C53—22no—
120C2×C8×C53—22no—
121C2×C8×C53—22no—
122C2×C8×C53—42no—
123C2×C8×C53C2×D412yes—
124C2×C8×C53C2×D412yes—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53C2×D411yescharacteristic
136C4⋊C4×C53C2×D411yescharacteristic
137C22⋊C4×C53—21no—
138C22⋊C4×C53—21no—
139C22⋊C4×C53—21no—
140C42×C53—21no—
141C42×C53—21no—
142C42×C53—21no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147C23×C53—21no—
148C23×C53#12111yescharacteristic
149Q8×C53—21no—
150Q8×C53—21no—
151Q8×C53—42no—
152Q8×C53—42no—
153Q8×C53—44no—
154Q8×C53—82no—
155Q8×C53—82no—
156D4×C53—21no—
157D4×C53—21no—
158D4×C53—42no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—82no—
170C2×C4×C53—82no—
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53AGL(1,ℤ/8)12yes—
175C2×C4×C53#13712yes—
176C2×C4×C53C22×D411yescharacteristic
177C2×C4×C53C22×D411yescharacteristic
178C2×C4×C53C2×#4111yescharacteristic
179C8×C53—22no—
180C8×C53—22no—
181C8×C53—42no—
182C8×C53—42no—
183C8×C53—42no—
184#19300C5311yesmaximal, characteristic, direct product factor, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53#46812yes—
191C22×C53C2×#12111yescharacteristic
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—41no—
195C4×C53—41no—
196C4×C53—41no—
197C4×C53—82no—
198C4×C53—82no—
199C4×C53#46812yes—
200C4×C53#47312yes—
201C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
202C4×C53C2×#13711yescharacteristic
203C4×C53#57512yes—
204C2×#487C2×C5311yescharacteristic, complemented
205#3079C2×C5311yescharacteristic
206#3039C2×C5311yescharacteristic
207#2965C2×C5311yescharacteristic
208#1700C2×C5314yescomplemented
209#1699C2×C5314yescomplemented
210#1567C2×C5311yescharacteristic, complemented
211#1549C2×C5311yescharacteristic, complemented
212#1546C2×C5311yescharacteristic, complemented
213C2×C53—22no—
214C2×C53—81no—
215C2×C53#304011yescharacteristic
216C2×C53C2×#46811yescharacteristic
217C2×C53C2×#47311yescharacteristic
218C2×#137—21no—
219C2×C4⋊Q8C22×C5311yescharacteristic
220C2×C4⋊Q8C22×C5311yescharacteristic
221C4×#41C22×C5311yescharacteristic
222#487—21no—
223#487—21no—
224#487C22×C5314yes—
225#477—21no—
226#476—21no—
227#475C22×C5312yes—
228#474C22×C5312yes—
229#462—21no—
230#462C22×C5312yes—
231#461—21no—
232#461C22×C5312yes—
233#446C22×C5312yes—
234#436C22×C5312yes—
235#435—22no—
236#431—21no—
237#431C22×C5311yescharacteristic
238#428—21no—
239#423C22×C5311yescharacteristic
240#423C22×C5311yescharacteristic
241#419—21no—
242#417—21no—
243C2×#103—22no—
244C4×M16—21no—
245C2×C4×C8C22×C5311yescharacteristic
246#335C22×C5314yes—
247#332C22×C5314yes—
248#332C22×C5314yes—
249C53#1930011yescharacteristic, direct product factor, Sylow
250C2×#41D4×C5311yescharacteristic
251C22×Q8—21no—
252C22×Q8—21no—
253#137—44no—
254C2×Q16—22no—
255C2×SD16—22no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—22no—
259C22×C8D4×C5312yes—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8C23×C5312yes—
267C4⋊Q8C23×C5312yes—
268C4×Q8D4×C5311yescharacteristic
269C4×Q8C23×C5311yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4D4×C5311yescharacteristic
273C4×D4C23×C5311yescharacteristic
274#117—21no—
275#117D4×C5311yescharacteristic
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C42—21no—
281C2×C42C23×C5311yescharacteristic
282#107—22no—
283#106—22no—
284C4⋊C8—22no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8C23×C5312yes—
288C4⋊C8C23×C5312yes—
289C4⋊C8C23×C5312yes—
290#103—22no—
291#103—24no—
292#103—24no—
293#103—22no—
294#103—22no—
295#102—22no—
296C22⋊C8—21no—
297C22⋊C8—22no—
298#97—22no—
299C4×C8—21no—
300C4×C8D4×C5311yescharacteristic
301C4×C8D4×C5311yescharacteristic
302C4×C8C23×C5312yes—
303#41—22no—
304#41—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—44no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8C2×D4×C5311yescharacteristic
313C2×D4—21no—
314C2×D4C2×D4×C5311yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4C2×D4×C5311yescharacteristic
321C22×C4C2×D4×C5311yescharacteristic
322C22×C4#41×C5311yescharacteristic
323Q16—44no—
324SD16—44no—
325M16—44no—
326M16—42no—
327M16—42no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—42no—
332C2×C8C2×D4×C5312yes—
333C2×C8C2×D4×C5312yes—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4C2×D4×C5311yescharacteristic
345C4⋊C4C2×D4×C5311yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42C2×D4×C5311yescharacteristic
353C42C2×D4×C5311yescharacteristic
354C42#41×C5311yescharacteristic
355C42C24×C5311yescharacteristic
356C23—21no—
357C23#121×C5311yescharacteristic
358Q8—21no—
359Q8—21no—
360Q8—42no—
361Q8—42no—
362Q8—44no—
363Q8—82no—
364Q8—82no—
365D4—21no—
366D4—21no—
367D4—42no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4#121×C5311yescharacteristic
381C2×C4#121×C5311yescharacteristic
382C2×C4#121×C5311yescharacteristic
383C2×C4AGL(1,ℤ/8)×C5312yes—
384C2×C4#137×C5312yes—
385C2×C4C22×D4×C5311yescharacteristic
386C2×C4C22×D4×C5311yescharacteristic
387C2×C4C2×#41×C5311yescharacteristic
388C8—22no—
389C8—22no—
390C8—42no—
391C8—42no—
392C8—42no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22C53×#46812yes—
399C22C2×#121×C5311yescharacteristic
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—82no—
406C4—82no—
407C4C53×#46812yes—
408C4C53×#47312yes—
409C4C2×AGL(1,ℤ/8)×C5311yescharacteristic
410C4C2×#137×C5311yescharacteristic
411C4C53×#57512yes—
412C2—22no—
413C2—81no—
414C2C53×#304011yescharacteristic
415C2C2×C53×#46811yescharacteristic
416C2C2×C53×#47311yescharacteristic
subgroupG/H
center, socleC22×C53C2×#121
commutator, Frattini, cosocleC42C24×C53
prime radicalC2×#41×C53D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#487C2semidirect
5. C53×#1700C2semidirect
6. C53×#1699C2semidirect
7. C53×#1567C2semidirect
8. C53×#1549C2semidirect
9. C53×#1546C2semidirect
184. #19300C53direct (N × H)
204. C2×#487C2×C53semidirect
208. #1700C2×C53semidirect
209. #1699C2×C53semidirect
210. #1567C2×C53semidirect
211. #1549C2×C53semidirect
212. #1546C2×C53semidirect
249. C53#19300direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#4872
C53×#30792
C53×#30392
C53×#29652
C53×#17002
C53×#16992
C53×#15672
C53×#15492
C53×#15462
#1930053