Finite Group Databasebeta

← C53×#19436 #27226948

Subgroups — C53×#19436 #27226948

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3210C211yesmaximal, characteristic
2C53×#3113C211yesmaximal, characteristic, complemented
3C53×#3103C211yesmaximal, characteristic
4C53×#2983C211yesmaximal, characteristic, complemented
5C53×#1702C212yesmaximal, complemented
6C53×#1700C212yesmaximal, complemented
7C53×#1690C212yesmaximal, complemented
8C53×#1686C212yesmaximal, complemented
9C53×#1572C211yesmaximal, characteristic, complemented
10C53×#1553C211yesmaximal, characteristic
11C53×#1544C211yesmaximal, characteristic, complemented
12C53×#533C2211yescharacteristic
13C53×#532C2211yescharacteristic
14C2×C4×Q8×C53C2211yescharacteristic
15C53×#494C2211yescharacteristic
16C53×#492C2211yescharacteristic
17C53×#487C2211yescharacteristic
18C53×#486C2211yescharacteristic
19C53×#483—21no—
20C53×#481—21no—
21C53×#475C2212yes—
22C53×#475C2212yes—
23C53×#464—22no—
24C53×#462C2212yes—
25C53×#460C2212yes—
26C53×#452—22no—
27C53×#451—21no—
28C53×#448—21no—
29C53×#446—22no—
30C53×#446C2212yes—
31C53×#446C2212yes—
32C53×#441—22no—
33C4×SD16×C53—22no—
34C53×#432—22no—
35C53×#432C2211yescharacteristic
36C53×#431C2211yescharacteristic
37C53×#431C2211yescharacteristic
38C53×#431C2211yescharacteristic
39C53×#335C2212yes—
40C53×#334C2212yes—
41C53×#332C2212yes—
42C53×#332C2212yes—
43C53×#332C2212yes—
44C53×#331C2212yes—
45C2×#41×C53—21no—
46C22×Q8×C53D411yescharacteristic
47C2×Q16×C53—22no—
48C2×Q16×C53—22no—
49C2×SD16×C53—22no—
50C2×SD16×C53—22no—
51C4⋊Q8×C53C2311yescharacteristic
52C4⋊Q8×C53C2311yescharacteristic
53C4⋊Q8×C53C2311yescharacteristic
54#127×C53C2311yescharacteristic
55#125×C53—21no—
56#124×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59#121×C53—21no—
60#121×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×Q8×C53D412yes—
66C4×Q8×C53C2312yes—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69#117×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C42×C53—21no—
73C2×C42×C53C2311yescharacteristic
74#106×C53—22no—
75#106×C53—22no—
76C4⋊C8×C53—22no—
77C4⋊C8×C53—22no—
78C4⋊C8×C53D412yes—
79C4⋊C8×C53C2312yes—
80C4⋊C8×C53C2312yes—
81C4⋊C8×C53C2312yes—
82#103×C53—22no—
83#103×C53—22no—
84#103×C53—22no—
85#103×C53—22no—
86#102×C53—22no—
87#102×C53—22no—
88C22⋊C8×C53—21no—
89C22⋊C8×C53—21no—
90C22⋊C8×C53—21no—
91C22⋊C8×C53D412yes—
92#97×C53—22no—
93#97×C53—22no—
94#97×C53—21no—
95C4×C8×C53—21no—
96C4×C8×C53—21no—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99#41×C53—82no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×Q8×C53—21no—
103C2×Q8×C53—21no—
104C2×Q8×C53—42no—
105C2×Q8×C53C2×D412yes—
106C2×D4×C53—21no—
107C2×D4×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53—41no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—41no—
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53#4111yescharacteristic
116Q16×C53—44no—
117Q16×C53—82no—
118SD16×C53—44no—
119SD16×C53—82no—
120C2×C8×C53—22no—
121C2×C8×C53—22no—
122C2×C8×C53—22no—
123C2×C8×C53C2×D412yes—
124C2×C8×C53C2×D412yes—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53C2×D412yes—
137C22⋊C4×C53—41no—
138C22⋊C4×C53—41no—
139C22⋊C4×C53—41no—
140C42×C53—22no—
141C42×C53—22no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147C23×C53—41no—
148C23×C53#12111yescharacteristic
149Q8×C53—22no—
150Q8×C53—22no—
151Q8×C53—42no—
152Q8×C53—42no—
153Q8×C53—42no—
154Q8×C53—42no—
155D4×C53—42no—
156D4×C53—42no—
157D4×C53—81no—
158D4×C53—82no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—22no—
162C2×C4×C53—22no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—82no—
169C2×C4×C53—81no—
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C2×#4111yescharacteristic
176C8×C53—22no—
177C8×C53—22no—
178C8×C53—42no—
179C8×C53—42no—
180C8×C53—42no—
181#19436C5311yesmaximal, characteristic, direct product factor, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53C2×#12111yescharacteristic
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—42no—
194C4×C53—42no—
195C4×C53—81no—
196C4×C53—81no—
197C4×C53—81no—
198C4×C53#57512yes—
199C4×C53#57712yes—
200#3210C2×C5311yescharacteristic
201#3113C2×C5311yescharacteristic, complemented
202#3103C2×C5311yescharacteristic
203#2983C2×C5311yescharacteristic, complemented
204#1702C2×C5312yescomplemented
205#1700C2×C5312yescomplemented
206#1690C2×C5312yescomplemented
207#1686C2×C5312yescomplemented
208#1572C2×C5311yescharacteristic, complemented
209#1553C2×C5311yescharacteristic
210#1544C2×C5311yescharacteristic, complemented
211C2×C53—41no—
212C2×C53—161no—
213C2×C53#304311yescharacteristic
214C2×C53#305411yescharacteristic
215C2×C53#306911yescharacteristic
216#533C22×C5311yescharacteristic
217#532C22×C5311yescharacteristic
218C2×C4×Q8C22×C5311yescharacteristic
219#494C22×C5311yescharacteristic
220#492C22×C5311yescharacteristic
221#487C22×C5311yescharacteristic
222#486C22×C5311yescharacteristic
223#483—21no—
224#481—21no—
225#475C22×C5312yes—
226#475C22×C5312yes—
227#464—22no—
228#462C22×C5312yes—
229#460C22×C5312yes—
230#452—22no—
231#451—21no—
232#448—21no—
233#446—22no—
234#446C22×C5312yes—
235#446C22×C5312yes—
236#441—22no—
237C4×SD16—22no—
238#432—22no—
239#432C22×C5311yescharacteristic
240#431C22×C5311yescharacteristic
241#431C22×C5311yescharacteristic
242#431C22×C5311yescharacteristic
243#335C22×C5312yes—
244#334C22×C5312yes—
245#332C22×C5312yes—
246#332C22×C5312yes—
247#332C22×C5312yes—
248#331C22×C5312yes—
249C53#1943611yescharacteristic, direct product factor, Sylow
250C2×#41—21no—
251C22×Q8D4×C5311yescharacteristic
252C2×Q16—22no—
253C2×Q16—22no—
254C2×SD16—22no—
255C2×SD16—22no—
256C4⋊Q8C23×C5311yescharacteristic
257C4⋊Q8C23×C5311yescharacteristic
258C4⋊Q8C23×C5311yescharacteristic
259#127C23×C5311yescharacteristic
260#125—21no—
261#124—21no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264#121—21no—
265#121—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8D4×C5312yes—
271C4×Q8C23×C5312yes—
272C4×D4—21no—
273C4×D4—21no—
274#117—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4D4×C5311yescharacteristic
277C2×C42—21no—
278C2×C42C23×C5311yescharacteristic
279#106—22no—
280#106—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8D4×C5312yes—
284C4⋊C8C23×C5312yes—
285C4⋊C8C23×C5312yes—
286C4⋊C8C23×C5312yes—
287#103—22no—
288#103—22no—
289#103—22no—
290#103—22no—
291#102—22no—
292#102—22no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8D4×C5312yes—
297#97—22no—
298#97—22no—
299#97—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8C23×C5311yescharacteristic
303C4×C8C23×C5311yescharacteristic
304#41—82no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—42no—
310C2×Q8C2×D4×C5312yes—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—41no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4C2×D4×C5311yescharacteristic
319C22×C4C2×D4×C5311yescharacteristic
320C22×C4#41×C5311yescharacteristic
321Q16—44no—
322Q16—82no—
323SD16—44no—
324SD16—82no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×D4×C5312yes—
329C2×C8C2×D4×C5312yes—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4C2×D4×C5312yes—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C42—22no—
346C42—22no—
347C42—41no—
348C42C2×D4×C5311yescharacteristic
349C42C2×D4×C5311yescharacteristic
350C42#41×C5311yescharacteristic
351C42C24×C5311yescharacteristic
352C23—41no—
353C23#121×C5311yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—42no—
357Q8—42no—
358Q8—42no—
359Q8—42no—
360D4—42no—
361D4—42no—
362D4—81no—
363D4—82no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4#121×C5311yescharacteristic
376C2×C4#121×C5311yescharacteristic
377C2×C4#121×C5311yescharacteristic
378C2×C4C22×D4×C5311yescharacteristic
379C2×C4C22×D4×C5311yescharacteristic
380C2×C4C2×#41×C5311yescharacteristic
381C8—22no—
382C8—22no—
383C8—42no—
384C8—42no—
385C8—42no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22C2×#121×C5311yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4C53×#57512yes—
403C4C53×#57712yes—
404C2—41no—
405C2—161no—
406C2C53×#304311yescharacteristic
407C2C53×#305411yescharacteristic
408C2C53×#306911yescharacteristic
subgroupG/H
center, socleC22×C53C2×#121
commutator, Frattini, cosocleC42C24×C53
prime radicalC53×#532C22
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
2. C53×#3113C2semidirect
4. C53×#2983C2semidirect
5. C53×#1702C2semidirect
6. C53×#1700C2semidirect
7. C53×#1690C2semidirect
8. C53×#1686C2semidirect
9. C53×#1572C2semidirect
11. C53×#1544C2semidirect
181. #19436C53direct (N × H)
201. #3113C2×C53semidirect
203. #2983C2×C53semidirect
204. #1702C2×C53semidirect
205. #1700C2×C53semidirect
206. #1690C2×C53semidirect
207. #1686C2×C53semidirect
208. #1572C2×C53semidirect
210. #1544C2×C53semidirect
249. C53#19436direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32102
C53×#31132
C53×#31032
C53×#29832
C53×#17022
C53×#17002
C53×#16902
C53×#16862
C53×#15722
C53×#15532
C53×#15442
#1943653