Finite Group Databasebeta

← C53×#10193 #27217705

Subgroups — C53×#10193 #27217705

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3246C211yesmaximal, characteristic, complemented
2C53×#3205C211yesmaximal, characteristic
3C53×#3138C211yesmaximal, characteristic, complemented
4C53×#1982C211yesmaximal, characteristic, complemented
5C53×#1980C211yesmaximal, characteristic, complemented
6C53×#1503C211yesmaximal, characteristic, complemented
7C53×#1283C211yesmaximal, characteristic, complemented
8C53×#533C2211yescharacteristic
9C53×#532C2211yescharacteristic
10C2×#127×C53C2211yescharacteristic
11C53×#493—21no—
12C53×#490—22no—
13C53×#486—21no—
14C53×#486—21no—
15C53×#486—21no—
16C53×#476—21no—
17C53×#475—21no—
18C53×#465—22no—
19C53×#461—21no—
20C53×#460—21no—
21C53×#459—22no—
22C53×#447—22no—
23C53×#431C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×#104×C53—22no—
26C53×#418—22no—
27C2×#102×C53—22no—
28C4×M16×C53C2211yescharacteristic
29C4×C4⋊C4×C53C2211yescharacteristic
30C53×#328—22no—
31C2×#41×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53—21no—
34C22×D4×C53—21no—
35C2×SD16×C53—22no—
36C2×D8×C53—24no—
37C2×M16×C53D412yes—
38C22×C8×C53D412yes—
39C4⋊Q8×C53D411yescharacteristic
40C4⋊Q8×C53D411yescharacteristic
41C4⋊Q8×C53D411yescharacteristic
42#127×C53—21no—
43#127×C53—21no—
44#127×C53—21no—
45#127×C53D412yes—
46#127×C53D411yescharacteristic
47#125×C53—21no—
48#124×C53—21no—
49C22⋊Q8×C53—21no—
50C22⋊Q8×C53—21no—
51#121×C53—21no—
52#121×C53—21no—
53C4×Q8×C53—21no—
54C4×Q8×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53—21no—
57#117×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C42×C53—21no—
60C2×C42×C53C2311yescharacteristic
61#106×C53—22no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65#104×C53—44no—
66#103×C53—22no—
67#102×C53—24no—
68#102×C53—24no—
69#102×C53—24no—
70#102×C53—22no—
71#102×C53—44no—
72C22⋊C8×C53D412yes—
73#97×C53—21no—
74C4×C8×C53—21no—
75C4×C8×C53—21no—
76#95×C53—21no—
77C24×C53—41no—
78#41×C53—42no—
79#41×C53—42no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×D4×C53—22no—
84C2×D4×C53—21no—
85C2×D4×C53—22no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—22no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—42no—
96C2×D4×C53—42no—
97C2×D4×C53—82no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103SD16×C53—82no—
104D8×C53—84no—
105M16×C53—22no—
106M16×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53—22no—
109C2×C8×C53—22no—
110C2×C8×C53—42no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C42×C53—41no—
125C42×C53—42no—
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53#4111yescharacteristic
130C23×C53—41no—
131C23×C53—42no—
132C23×C53—82no—
133C23×C53—82no—
134C23×C53—81no—
135C23×C53—81no—
136C23×C53#12711yescharacteristic
137Q8×C53—41no—
138Q8×C53—41no—
139Q8×C53—42no—
140Q8×C53—81no—
141D4×C53—44no—
142D4×C53—44no—
143D4×C53—42no—
144D4×C53—44no—
145D4×C53—41no—
146D4×C53—41no—
147D4×C53—42no—
148D4×C53—82no—
149D4×C53—81no—
150D4×C53—81no—
151D4×C53—81no—
152D4×C53—81no—
153D4×C53—82no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53—82no—
164C2×C4×C53#12011yescharacteristic
165C2×C4×C53#12011yescharacteristic
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#13512yes—
171C2×C4×C53AGL(1,ℤ/8)12yes—
172C8×C53—42no—
173C8×C53—42no—
174C8×C53—42no—
175#10193C5311yesmaximal, characteristic, direct product factor, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—82no—
179C22×C53—81no—
180C22×C53—82no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53—82no—
184C22×C53—162no—
185C22×C53—162no—
186C22×C53#39211yescharacteristic
187C22×C53#49511yescharacteristic
188C22×C53#49611yescharacteristic
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—82no—
193C4×C53—81no—
194C4×C53—81no—
195C4×C53—81no—
196C4×C53#44912yes—
197C4×C53#45311yescharacteristic
198C4×C53#45411yescharacteristic
199C4×C53#46312yes—
200#3246C2×C5311yescharacteristic, complemented
201#3205C2×C5311yescharacteristic
202#3138C2×C5311yescharacteristic, complemented
203#1982C2×C5311yescharacteristic, complemented
204#1980C2×C5311yescharacteristic, complemented
205#1503C2×C5311yescharacteristic, complemented
206#1283C2×C5311yescharacteristic, complemented
207C2×C53—21no—
208C2×C53—21no—
209C2×C53—161no—
210C2×C53—162no—
211C2×C53#200611yescharacteristic
212C2×C53#200811yescharacteristic
213C2×C53#201211yescharacteristic
214#533C22×C5311yescharacteristic
215#532C22×C5311yescharacteristic
216C2×#127C22×C5311yescharacteristic
217#493—21no—
218#490—22no—
219#486—21no—
220#486—21no—
221#486—21no—
222#476—21no—
223#475—21no—
224#465—22no—
225#461—21no—
226#460—21no—
227#459—22no—
228#447—22no—
229#431C22×C5311yescharacteristic
230C2×C4⋊C8C22×C5311yescharacteristic
231C2×#104—22no—
232#418—22no—
233C2×#102—22no—
234C4×M16C22×C5311yescharacteristic
235C4×C4⋊C4C22×C5311yescharacteristic
236#328—22no—
237C53#1019311yescharacteristic, direct product factor, Sylow
238C2×#41—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C2×SD16—22no—
243C2×D8—24no—
244C2×M16D4×C5312yes—
245C22×C8D4×C5312yes—
246C4⋊Q8D4×C5311yescharacteristic
247C4⋊Q8D4×C5311yescharacteristic
248C4⋊Q8D4×C5311yescharacteristic
249#127—21no—
250#127—21no—
251#127—21no—
252#127D4×C5312yes—
253#127D4×C5311yescharacteristic
254#125—21no—
255#124—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258#121—21no—
259#121—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—21no—
264#117—21no—
265C2×C4⋊C4—21no—
266C2×C42—21no—
267C2×C42C23×C5311yescharacteristic
268#106—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272#104—44no—
273#103—22no—
274#102—24no—
275#102—24no—
276#102—24no—
277#102—22no—
278#102—44no—
279C22⋊C8D4×C5312yes—
280#97—21no—
281C4×C8—21no—
282C4×C8—21no—
283#95—21no—
284C24—41no—
285#41—42no—
286#41—42no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—22no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—82no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4C2×D4×C5311yescharacteristic
308C22×C4C2×D4×C5311yescharacteristic
309C22×C4C2×D4×C5311yescharacteristic
310SD16—82no—
311D8—84no—
312M16—22no—
313M16—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—42no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C42—41no—
332C42—42no—
333C42C2×D4×C5311yescharacteristic
334C42C2×D4×C5311yescharacteristic
335C42C2×D4×C5311yescharacteristic
336C42#41×C5311yescharacteristic
337C23—41no—
338C23—42no—
339C23—82no—
340C23—82no—
341C23—81no—
342C23—81no—
343C23#127×C5311yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—42no—
347Q8—81no—
348D4—42no—
349D4—44no—
350D4—44no—
351D4—44no—
352D4—41no—
353D4—41no—
354D4—42no—
355D4—82no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—82no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4#120×C5311yescharacteristic
372C2×C4#120×C5311yescharacteristic
373C2×C4#120×C5311yescharacteristic
374C2×C4#121×C5311yescharacteristic
375C2×C4#121×C5311yescharacteristic
376C2×C4#121×C5311yescharacteristic
377C2×C4#135×C5312yes—
378C2×C4AGL(1,ℤ/8)×C5312yes—
379C8—42no—
380C8—42no—
381C8—42no—
382C22—21no—
383C22—21no—
384C22—82no—
385C22—81no—
386C22—82no—
387C22—81no—
388C22—81no—
389C22—82no—
390C22—162no—
391C22—162no—
392C22C53×#39211yescharacteristic
393C22C53×#49511yescharacteristic
394C22C53×#49611yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4C53×#44912yes—
403C4C53×#45311yescharacteristic
404C4C53×#45411yescharacteristic
405C4C53×#46312yes—
406C2—21no—
407C2—21no—
408C2—162no—
409C2—161no—
410C2C53×#200611yescharacteristic
411C2C53×#200811yescharacteristic
412C2C53×#201211yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3205C2
LCSG ▷ C2×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. C53×#3246C2semidirect
3. C53×#3138C2semidirect
4. C53×#1982C2semidirect
5. C53×#1980C2semidirect
6. C53×#1503C2semidirect
7. C53×#1283C2semidirect
175. #10193C53direct (N × H)
200. #3246C2×C53semidirect
202. #3138C2×C53semidirect
203. #1982C2×C53semidirect
204. #1980C2×C53semidirect
205. #1503C2×C53semidirect
206. #1283C2×C53semidirect
237. C53#10193direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32462
C53×#32052
C53×#31382
C53×#19822
C53×#19802
C53×#15032
C53×#12832
#1019353