Finite Group Databasebeta

← C53×#10074 #27217586

Subgroups — C53×#10074 #27217586

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#498C211yesmaximal, characteristic, complemented
2C2×C53×#474C211yesmaximal, characteristic
3C53×#2547C211yesmaximal, characteristic
4C53×#1981C211yesmaximal, characteristic, complemented
5C2×C53×#332C211yesmaximal, characteristic, complemented
6C2×C53×D4⋊C8C211yesmaximal, characteristic
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C53C2211yescharacteristic
9C2×C4⋊Q8×C53C2211yescharacteristic
10C2×#124×C53—21no—
11C2×C4×D4×C53C2211yescharacteristic
12C53×#498—22no—
13C53×#498—22no—
14C53×#474—22no—
15C53×#474—22no—
16C2×#107×C53—21no—
17C2×#106×C53—22no—
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×C4⋊C8×C53C2211yescharacteristic
20C2×#103×C53—22no—
21C2×#102×C53—21no—
22C2×C4×C8×C53C2211yescharacteristic
23C53×#397—22no—
24C53×#391—21no—
25C53×#380—22no—
26C4×C4⋊C4×C53C2211yescharacteristic
27C53×#332—22no—
28C53×D4⋊C8—22no—
29C22×Q8×C53—21no—
30C22×Q8×C53—21no—
31C22×D4×C53D411yescharacteristic
32C23×C4×C53—21no—
33C22×C8×C53—21no—
34C22×C8×C53—21no—
35C22×C8×C53—21no—
36C4⋊Q8×C53—21no—
37C4⋊Q8×C53—21no—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—21no—
40C4⋊Q8×C53—21no—
41C4⋊Q8×C53—21no—
42C4⋊Q8×C53—21no—
43C4⋊Q8×C53D412yes—
44#124×C53—42no—
45#124×C53—42no—
46C4×D4×C53—21no—
47C4×D4×C53—21no—
48C4×D4×C53—21no—
49C4×D4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D412yes—
55C2×C4⋊C4×C53D412yes—
56C2×C22⋊C4×C53—24no—
57C2×C22⋊C4×C53—21no—
58C2×C42×C53—21no—
59C2×C42×C53C2311yescharacteristic
60#107×C53—24no—
61#106×C53—28no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53D412yes—
66#103×C53—44no—
67#102×C53—24no—
68C4×C8×C53—21no—
69C4×C8×C53D412yes—
70#95×C53—24no—
71#95×C53—21no—
72C24×C53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53—42no—
82C2×Q8×C53—42no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53SD1612yes—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—42no—
96C22×C4×C53—42no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C2×C8×C53—24no—
101C2×C8×C53—22no—
102C2×C8×C53—22no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—42no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C22⋊C4×C53—48no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128C23×C53—22no—
129C23×C53—41no—
130C23×C53—42no—
131C23×C53—41no—
132C23×C53—42no—
133C23×C53#12011yescharacteristic
134Q8×C53—44no—
135Q8×C53—81no—
136Q8×C53—81no—
137Q8×C53—81no—
138Q8×C53—81no—
139Q8×C53—81no—
140Q8×C53—81no—
141D4×C53—22no—
142D4×C53—22no—
143D4×C53—41no—
144D4×C53—41no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—82no—
156C2×C4×C53—82no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12711yescharacteristic
165C2×C4×C53C2×SD1611yescharacteristic
166C2×C4×C53#13511yescharacteristic
167C2×C4×C53#13512yes—
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C53AGL(1,ℤ/8)12yes—
170C2×C4×C53#13711yescharacteristic
171C8×C53—24no—
172C8×C53—42no—
173C8×C53—81no—
174#10074C5311yesmaximal, characteristic, direct product factor, Sylow
175C22×C53—44no—
176C22×C53—42no—
177C22×C53—82no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53#39211yescharacteristic
181C22×C53#44912yes—
182C22×C53#45011yescharacteristic
183C22×C53#45211yescharacteristic
184C22×C53#45612yes—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—82no—
188C4×C53—82no—
189C4×C53—81no—
190C4×C53—82no—
191C4×C53#44811yescharacteristic
192C4×C53#44911yescharacteristic
193C4×C53#46312yes—
194C4×C53#46511yescharacteristic
195C4×C53#46811yescharacteristic
196C4×C53#49511yescharacteristic
197C4×C53#49611yescharacteristic
198C2×#498C2×C5311yescharacteristic, complemented
199C2×#474C2×C5311yescharacteristic
200#2547C2×C5311yescharacteristic
201#1981C2×C5311yescharacteristic, complemented
202C2×#332C2×C5311yescharacteristic, complemented
203C2×D4⋊C8C2×C5311yescharacteristic
204#1279C2×C5311yescharacteristic, complemented
205C2×C53—82no—
206C2×C53#165212yes—
207C2×C53#165412yes—
208C2×C53#200011yescharacteristic
209C2×C53#200511yescharacteristic
210C2×C53#201211yescharacteristic
211C2×C4⋊Q8C22×C5311yescharacteristic
212C2×C4⋊Q8C22×C5311yescharacteristic
213C2×#124—21no—
214C2×C4×D4C22×C5311yescharacteristic
215#498—22no—
216#498—22no—
217#474—22no—
218#474—22no—
219C2×#107—21no—
220C2×#106—22no—
221C2×C4⋊C8C22×C5311yescharacteristic
222C2×C4⋊C8C22×C5311yescharacteristic
223C2×#103—22no—
224C2×#102—21no—
225C2×C4×C8C22×C5311yescharacteristic
226#397—22no—
227#391—21no—
228#380—22no—
229C4×C4⋊C4C22×C5311yescharacteristic
230#332—22no—
231D4⋊C8—22no—
232C53#1007411yescharacteristic, direct product factor, Sylow
233C22×Q8—21no—
234C22×Q8—21no—
235C22×D4D4×C5311yescharacteristic
236C23×C4—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8D4×C5312yes—
248#124—42no—
249#124—42no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D4×C5311yescharacteristic
258C2×C4⋊C4D4×C5312yes—
259C2×C4⋊C4D4×C5312yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—21no—
262C2×C42—21no—
263C2×C42C23×C5311yescharacteristic
264#107—24no—
265#106—28no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8D4×C5312yes—
270#103—44no—
271#102—24no—
272C4×C8—21no—
273C4×C8D4×C5312yes—
274#95—24no—
275#95—21no—
276C24—21no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4SD16×C5312yes—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C2×D4×C5311yescharacteristic
302C22×C4C2×D4×C5311yescharacteristic
303C22×C4C2×D4×C5311yescharacteristic
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—48no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C42—41no—
327C42—41no—
328C42C2×D4×C5311yescharacteristic
329C42C2×D4×C5311yescharacteristic
330C42C2×D4×C5311yescharacteristic
331C42#41×C5311yescharacteristic
332C23—22no—
333C23—41no—
334C23—42no—
335C23—41no—
336C23—42no—
337C23#120×C5311yescharacteristic
338Q8—44no—
339Q8—81no—
340Q8—81no—
341Q8—81no—
342Q8—81no—
343Q8—81no—
344Q8—81no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4#120×C5311yescharacteristic
364C2×C4#120×C5311yescharacteristic
365C2×C4#121×C5311yescharacteristic
366C2×C4#121×C5311yescharacteristic
367C2×C4#121×C5311yescharacteristic
368C2×C4#127×C5311yescharacteristic
369C2×C4C2×SD16×C5311yescharacteristic
370C2×C4#135×C5312yes—
371C2×C4#135×C5311yescharacteristic
372C2×C4AGL(1,ℤ/8)×C5312yes—
373C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
374C2×C4#137×C5311yescharacteristic
375C8—24no—
376C8—42no—
377C8—81no—
378C22—42no—
379C22—44no—
380C22—81no—
381C22—81no—
382C22—82no—
383C22C53×#39211yescharacteristic
384C22C53×#44912yes—
385C22C53×#45011yescharacteristic
386C22C53×#45211yescharacteristic
387C22C53×#45612yes—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—81no—
392C4—82no—
393C4—82no—
394C4C53×#44811yescharacteristic
395C4C53×#44911yescharacteristic
396C4C53×#46312yes—
397C4C53×#46511yescharacteristic
398C4C53×#46811yescharacteristic
399C4C53×#49511yescharacteristic
400C4C53×#49611yescharacteristic
401C2—82no—
402C2C53×#165212yes—
403C2C53×#165412yes—
404C2C53×#200011yescharacteristic
405C2C53×#200511yescharacteristic
406C2C53×#201211yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#498C2semidirect
4. C53×#1981C2semidirect
5. C2×C53×#332C2semidirect
7. C53×#1279C2semidirect
174. #10074C53direct (N × H)
198. C2×#498C2×C53semidirect
201. #1981C2×C53semidirect
202. C2×#332C2×C53semidirect
204. #1279C2×C53semidirect
232. C53#10074direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#4982
C2×C53×#4742
C53×#25472
C53×#19812
C2×C53×#3322
C2×C53×D4⋊C82
C53×#12792
#1007453