Finite Group Databasebeta

← C53×#11809 #27219321

Subgroups — C53×#11809 #27219321

Order 13568.

616 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×C53×#399C211yesmaximal, characteristic
2C53×#1819C211yesmaximal, characteristic
3C53×#1462C211yesmaximal, characteristic
4C53×#1352C214yesmaximal, complemented
5C22×C4⋊C4×C53—21no—
6C22×C4⋊C4×C53C411yescharacteristic, complemented
7C22×C22⋊C4×C53—21no—
8C22×C22⋊C4×C53C2211yescharacteristic
9C53×#407—21no—
10C53×#399—21no—
11C53×#399—21no—
12C53×#399C412yescomplemented
13C53×#399C2212yes—
14C53×#379—21no—
15C2×#95×C53C411yescharacteristic, complemented
16C53×#342—21no—
17C53×#342C2212yes—
18C53×#323—21no—
19C53×#323C2212yes—
20C25×C53D411yescharacteristic
21C23×C4×C53—21no—
22C23×C4×C53—21no—
23C23×C4×C53C2×C411yescharacteristic
24C23×C4×C53D411yescharacteristic
25C2×M16×C53—41no—
26C2×C4⋊C4×C53—22no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—44no—
30C2×C4⋊C4×C53—41no—
31C2×C4⋊C4×C53—41no—
32C2×C4⋊C4×C53C2×C412yes—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—41no—
37C2×C22⋊C4×C53—41no—
38C2×C22⋊C4×C53—44no—
39C2×C22⋊C4×C53—41no—
40C2×C22⋊C4×C53—41no—
41C2×C22⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53C2311yescharacteristic
45C22⋊C8×C53—22no—
46C22⋊C8×C53—41no—
47#95×C53—21no—
48#95×C53C2×C412yes—
49C24×C53—22no—
50C24×C53—21no—
51C24×C53—21no—
52C24×C53—41no—
53C24×C53—42no—
54C24×C53—42no—
55C24×C53C22⋊C411yescharacteristic
56C24×C53C22⋊C411yescharacteristic
57C24×C53C2×D411yescharacteristic
58C22×C4×C53—24no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—42no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—42no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78M16×C53—82no—
79C2×C8×C53—42no—
80C4⋊C4×C53—44no—
81C4⋊C4×C53—44no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—42no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—44no—
89C22⋊C4×C53—44no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—82no—
93C22⋊C4×C53—82no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—48no—
110C23×C53—42no—
111C23×C53—42no—
112C23×C53—81no—
113C23×C53—81no—
114C23×C53—84no—
115C23×C53#9911yescharacteristic
116C23×C53#9911yescharacteristic
117C23×C53C2×C22⋊C411yescharacteristic
118C23×C53#13712yes—
119C2×C4×C53—24no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—48no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—81no—
138C2×C4×C53—82no—
139C2×C4×C53—82no—
140C2×C4×C53—82no—
141C2×C4×C53—82no—
142C2×C4×C53—82no—
143C8×C53—82no—
144#11809C5311yesmaximal, characteristic, direct product factor, Sylow
145C22×C53—24no—
146C22×C53—24no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—42no—
155C22×C53—42no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—81no—
162C22×C53—84no—
163C22×C53—84no—
164C22×C53—81no—
165C22×C53#35212yes—
166C22×C53#35312yes—
167C22×C53C2×#9911yescharacteristic
168C22×C53#41911yescharacteristic
169C22×C53#42111yescharacteristic
170C4×C53—44no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—82no—
176C4×C53—162no—
177C2×#399C2×C5311yescharacteristic
178#1819C2×C5311yescharacteristic
179#1462C2×C5311yescharacteristic
180#1352C2×C5314yescomplemented
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—22no—
184C2×C53—22no—
185C2×C53—42no—
186C2×C53—81no—
187C2×C53#151111yescharacteristic
188C2×C53C2×#35211yescharacteristic
189C2×C53C2×#35311yescharacteristic
190C22×C4⋊C4—21no—
191C22×C4⋊C4C4×C5311yescharacteristic, complemented
192C22×C22⋊C4—21no—
193C22×C22⋊C4C22×C5311yescharacteristic
194#407—21no—
195#399—21no—
196#399—21no—
197#399C4×C5312yescomplemented
198#399C22×C5312yes—
199#379—21no—
200C2×#95C4×C5311yescharacteristic, complemented
201#342—21no—
202#342C22×C5312yes—
203#323—21no—
204#323C22×C5312yes—
205C53#1180911yescharacteristic, direct product factor, Sylow
206C25D4×C5311yescharacteristic
207C23×C4—21no—
208C23×C4—21no—
209C23×C4C2×C4×C5311yescharacteristic
210C23×C4D4×C5311yescharacteristic
211C2×M16—41no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C2×C4×C5312yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4C2×C4×C5311yescharacteristic
228C2×C22⋊C4D4×C5311yescharacteristic
229C2×C22⋊C4D4×C5311yescharacteristic
230C2×C22⋊C4C23×C5311yescharacteristic
231C22⋊C8—22no—
232C22⋊C8—41no—
233#95—21no—
234#95C2×C4×C5312yes—
235C24—22no—
236C24—21no—
237C24—21no—
238C24—41no—
239C24—42no—
240C24—42no—
241C24C22⋊C4×C5311yescharacteristic
242C24C22⋊C4×C5311yescharacteristic
243C24C2×D4×C5311yescharacteristic
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4C22⋊C4×C5311yescharacteristic
261C22×C4C22⋊C4×C5311yescharacteristic
262C22×C4C22×C4×C5311yescharacteristic
263C22×C4C2×D4×C5311yescharacteristic
264M16—82no—
265C2×C8—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—82no—
279C22⋊C4—82no—
280C23—22no—
281C23—22no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—42no—
290C23—42no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—48no—
296C23—42no—
297C23—42no—
298C23—81no—
299C23—81no—
300C23—84no—
301C23#99×C5311yescharacteristic
302C23#99×C5311yescharacteristic
303C23C2×C22⋊C4×C5311yescharacteristic
304C23#137×C5312yes—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—48no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C8—82no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—84no—
348C22—84no—
349C22—81no—
350C22C53×#35212yes—
351C22C53×#35312yes—
352C22C2×#99×C5311yescharacteristic
353C22C53×#41911yescharacteristic
354C22C53×#42111yescharacteristic
355C4—44no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—82no—
361C4—162no—
362C2—21no—
363C2—21no—
364C2—22no—
365C2—22no—
366C2—42no—
367C2—81no—
368C2C53×#151111yescharacteristic
369C2C2×C53×#35211yescharacteristic
370C2C2×C53×#35311yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC25×C53D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C22×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1352C2semidirect
6. C22×C4⋊C4×C53C4semidirect
12. C53×#399C4semidirect
15. C2×#95×C53C4semidirect
144. #11809C53direct (N × H)
180. #1352C2×C53semidirect
191. C22×C4⋊C4C4×C53semidirect
197. #399C4×C53semidirect
200. C2×#95C4×C53semidirect
205. C53#11809direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#3992
C53×#18192
C53×#14622
C53×#13522
#1180953