Finite Group Databasebeta

← C53×#11526 #27219038

Subgroups — C53×#11526 #27219038

Order 13568.

470 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#123×C53C211yesmaximal, characteristic
2C53×#1466C211yesmaximal, characteristic
3C53×#1465C211yesmaximal, characteristic
4C53×#1326C212yesmaximal, complemented
5C53×#1317C212yesmaximal, complemented
6C2×#123×C53C411yescharacteristic
7C2×C4×D4×C53C2211yescharacteristic
8C22×C42×C53C411yescharacteristic
9C53×#432—21no—
10C53×#407—21no—
11C53×#388—21no—
12C53×#387—21no—
13C53×#386C411yescharacteristic
14C53×#385C2211yescharacteristic
15C53×#381C2211yescharacteristic
16C4×C4⋊C4×C53—21no—
17C4×C22⋊C4×C53—21no—
18C4×C22⋊C4×C53C411yescharacteristic
19C53×#324—21no—
20C53×#323—21no—
21C53×#323C2212yes—
22C53×#323C2212yes—
23C22×D4×C53C2×C411yescharacteristic
24C23×C4×C53C2×C411yescharacteristic
25C23×C4×C53C2×C411yescharacteristic
26C23×C4×C53C2×C411yescharacteristic
27C23×C4×C53D411yescharacteristic
28C2×M16×C53—41no—
29#123×C53—22no—
30#123×C53—22no—
31C4×D4×C53—41no—
32C4×D4×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53D411yescharacteristic
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—21no—
38C2×C22⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53C2311yescharacteristic
42C2×C42×C53—21no—
43C2×C42×C53—21no—
44C2×C42×C53—21no—
45C2×C42×C53—41no—
46C2×C42×C53—41no—
47C2×C42×C53D411yescharacteristic
48C4⋊C8×C53—41no—
49C22⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C22⋊C8×C53—41no—
52#97×C53—41no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C22×C411yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53M1612yescomplemented
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85M16×C53—82no—
86C2×C8×C53—42no—
87C2×C8×C53—42no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—42no—
91C22⋊C4×C53—24no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53M1612yescomplemented
98C42×C53—21no—
99C42×C53—21no—
100C42×C53—42no—
101C42×C53—42no—
102C42×C53—41no—
103C42×C53—41no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C2×M1611yescharacteristic
115C23×C53C2×M1611yescharacteristic
116D4×C53—42no—
117D4×C53—42no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—81no—
148C2×C4×C53#9911yescharacteristic
149C2×C4×C53#9911yescharacteristic
150C8×C53—82no—
151C8×C53—82no—
152#11526C5311yesmaximal, characteristic, direct product factor, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—42no—
160C22×C53—42no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—81no—
167C22×C53#40711yescharacteristic
168C22×C53C2×#9911yescharacteristic
169C22×C53#41311yescharacteristic
170C4×C53—21no—
171C4×C53—42no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—42no—
176C4×C53—41no—
177C4×C53—42no—
178C4×C53—81no—
179C4×C53#35212yes—
180C4×#123C2×C5311yescharacteristic
181#1466C2×C5311yescharacteristic
182#1465C2×C5311yescharacteristic
183#1326C2×C5312yescomplemented
184#1317C2×C5312yescomplemented
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—41no—
188C2×C53—41no—
189C2×C53—81no—
190C2×C53#146511yescharacteristic
191C2×C53C2×#35211yescharacteristic
192C2×C53#212011yescharacteristic
193C2×#123C4×C5311yescharacteristic
194C2×C4×D4C22×C5311yescharacteristic
195C22×C42C4×C5311yescharacteristic
196#432—21no—
197#407—21no—
198#388—21no—
199#387—21no—
200#386C4×C5311yescharacteristic
201#385C22×C5311yescharacteristic
202#381C22×C5311yescharacteristic
203C4×C4⋊C4—21no—
204C4×C22⋊C4—21no—
205C4×C22⋊C4C4×C5311yescharacteristic
206#324—21no—
207#323—21no—
208#323C22×C5312yes—
209#323C22×C5312yes—
210C53#1152611yescharacteristic, direct product factor, Sylow
211C22×D4C2×C4×C5311yescharacteristic
212C23×C4C2×C4×C5311yescharacteristic
213C23×C4C2×C4×C5311yescharacteristic
214C23×C4C2×C4×C5311yescharacteristic
215C23×C4D4×C5311yescharacteristic
216C2×M16—41no—
217#123—22no—
218#123—22no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4D4×C5311yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4C2×C4×C5311yescharacteristic
227C2×C22⋊C4C2×C4×C5311yescharacteristic
228C2×C22⋊C4D4×C5311yescharacteristic
229C2×C22⋊C4C23×C5311yescharacteristic
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42D4×C5311yescharacteristic
236C4⋊C8—41no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—41no—
240#97—41no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C24C22⋊C4×C5311yescharacteristic
247C24C22×C4×C5311yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C22⋊C4×C5311yescharacteristic
268C22×C4C22⋊C4×C5311yescharacteristic
269C22×C4C22⋊C4×C5311yescharacteristic
270C22×C4M16×C5312yescomplemented
271C22×C4C2×D4×C5311yescharacteristic
272C22×C4C2×D4×C5311yescharacteristic
273M16—82no—
274C2×C8—42no—
275C2×C8—42no—
276C4⋊C4—24no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C22⋊C4—24no—
280C22⋊C4—21no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—42no—
284C22⋊C4—41no—
285C22⋊C4M16×C5312yescomplemented
286C42—21no—
287C42—21no—
288C42—42no—
289C42—42no—
290C42—41no—
291C42—41no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23C2×C22⋊C4×C5311yescharacteristic
302C23C2×M16×C5311yescharacteristic
303C23C2×M16×C5311yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4#99×C5311yescharacteristic
337C2×C4#99×C5311yescharacteristic
338C8—82no—
339C8—82no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22C53×#40711yescharacteristic
355C22C2×#99×C5311yescharacteristic
356C22C53×#41311yescharacteristic
357C4—21no—
358C4—42no—
359C4—41no—
360C4—42no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—42no—
365C4—81no—
366C4C53×#35212yes—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—81no—
372C2C53×#146511yescharacteristic
373C2C2×C53×#35211yescharacteristic
374C2C53×#212011yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC24C22×C4×C53
Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC22×D4×C53C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1326C2semidirect
5. C53×#1317C2semidirect
82. C22×C4×C53M16semidirect
97. C22⋊C4×C53M16semidirect
152. #11526C53direct (N × H)
183. #1326C2×C53semidirect
184. #1317C2×C53semidirect
210. C53#11526direct (N × H)
270. C22×C4M16×C53semidirect
285. C22⋊C4M16×C53semidirect

Maximal subgroups

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

subgroupindex
C4×#123×C532
C53×#14662
C53×#14652
C53×#13262
C53×#13172
#1152653