Finite Group Databasebeta

← C53×#7618 #27215130

Subgroups — C53×#7618 #27215130

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#124×C53C211yesmaximal, characteristic
2C53×#1831C211yesmaximal, characteristic, complemented
3C53×#1745C211yesmaximal, characteristic, complemented
4C53×#1272C212yesmaximal, complemented
5C53×#1270C212yesmaximal
6C2×#124×C53C411yescharacteristic
7C2×C4×Q8×C53C411yescharacteristic
8C2×C4×D4×C53C411yescharacteristic
9C2×C4⋊C8×C53—21no—
10C2×#97×C53—21no—
11C53×#391C2211yescharacteristic
12C53×#390C2211yescharacteristic
13C53×#388—21no—
14C53×#388—21no—
15C53×#382C411yescharacteristic
16C4×C22⋊C4×C53—21no—
17C4×C22⋊C4×C53—21no—
18C43×C53C2211yescharacteristic
19C53×#336—21no—
20C53×#336—21no—
21C53×#336C2212yes—
22C53×#336C2212yes—
23C22×Q8×C53C2×C411yescharacteristic
24C22×D4×C53C2×C411yescharacteristic
25C23×C4×C53—21no—
26C22×C8×C53—22no—
27C22×C8×C53—22no—
28#124×C53—22no—
29#124×C53—22no—
30C4×Q8×C53—42no—
31C4×D4×C53—42no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C4⋊C4×C53C2×C411yescharacteristic
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—21no—
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—21no—
40C2×C42×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—42no—
49#97×C53—24no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54C24×C53—21no—
55C2×Q8×C53—21no—
56C2×Q8×C53—41no—
57C2×Q8×C53—41no—
58C2×Q8×C53M1612yes—
59C2×D4×C53—21no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53M1612yes—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C2×C8×C53—24no—
77C2×C8×C53—42no—
78C2×C8×C53—42no—
79C2×C8×C53—42no—
80C2×C8×C53—42no—
81C2×C8×C53—42no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—42no—
85C22⋊C4×C53—24no—
86C22⋊C4×C53—24no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C42×C53—22no—
92C42×C53—22no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—22no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—41no—
99C42×C53—41no—
100C42×C53—42no—
101C23×C53—22no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53C2×C22⋊C411yescharacteristic
109Q8×C53—42no—
110Q8×C53—42no—
111D4×C53—42no—
112D4×C53—42no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—22no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—84no—
139C2×C4×C53#10012yes—
140C2×C4×C53#10412yes—
141C2×C4×C53C2×M1611yescharacteristic
142C2×C4×C53C2×M1611yescharacteristic
143C2×C4×C53#13512yes—
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53#13711yescharacteristic
146C8×C53—44no—
147C8×C53—82no—
148#7618C5311yesmaximal, characteristic, direct product factor, Sylow
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—81no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53#40711yescharacteristic
157C22×C53#41011yescharacteristic
158C22×C53C2×#10011yescharacteristic
159C22×C53#41611yescharacteristic
160C22×C53#41711yescharacteristic
161C22×C53C2×#10411yescharacteristic
162C22×C53#42111yescharacteristic
163C4×C53—22no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—22no—
170C4×C53—42no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—81no—
174C4×C53—81no—
175C4×#124C2×C5311yescharacteristic
176#1831C2×C5311yescharacteristic, complemented
177#1745C2×C5311yescharacteristic, complemented
178#1272C2×C5312yescomplemented
179#1270C2×C5312yes—
180C2×C53—82no—
181C2×C53#146611yescharacteristic
182C2×C53#148811yescharacteristic
183C2×C53#148911yescharacteristic
184C2×C53#150011yescharacteristic
185C2×C53#151211yescharacteristic
186C2×C53#152711yescharacteristic
187C2×C53#155111yescharacteristic
188C2×#124C4×C5311yescharacteristic
189C2×C4×Q8C4×C5311yescharacteristic
190C2×C4×D4C4×C5311yescharacteristic
191C2×C4⋊C8—21no—
192C2×#97—21no—
193#391C22×C5311yescharacteristic
194#390C22×C5311yescharacteristic
195#388—21no—
196#388—21no—
197#382C4×C5311yescharacteristic
198C4×C22⋊C4—21no—
199C4×C22⋊C4—21no—
200C43C22×C5311yescharacteristic
201#336—21no—
202#336—21no—
203#336C22×C5312yes—
204#336C22×C5312yes—
205C53#761811yescharacteristic, direct product factor, Sylow
206C22×Q8C2×C4×C5311yescharacteristic
207C22×D4C2×C4×C5311yescharacteristic
208C23×C4—21no—
209C22×C8—22no—
210C22×C8—22no—
211#124—22no—
212#124—22no—
213C4×Q8—42no—
214C4×D4—42no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2×C4×C5311yescharacteristic
217C2×C4⋊C4C2×C4×C5311yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42—21no—
224C2×C42C2×C4×C5311yescharacteristic
225C2×C42C2×C4×C5311yescharacteristic
226C2×C42D4×C5311yescharacteristic
227C2×C42D4×C5311yescharacteristic
228C2×C42D4×C5311yescharacteristic
229C2×C42D4×C5311yescharacteristic
230C2×C42C23×C5311yescharacteristic
231C4⋊C8—42no—
232#97—24no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237C24—21no—
238C2×Q8—21no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8M16×C5312yes—
242C2×D4—21no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4M16×C5312yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4C22⋊C4×C5311yescharacteristic
253C22×C4C22⋊C4×C5311yescharacteristic
254C22×C4C22⋊C4×C5311yescharacteristic
255C22×C4C22⋊C4×C5311yescharacteristic
256C22×C4C22×C4×C5311yescharacteristic
257C22×C4C2×D4×C5311yescharacteristic
258C22×C4C2×D4×C5311yescharacteristic
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42—22no—
275C42—22no—
276C42—21no—
277C42—21no—
278C42—22no—
279C42—21no—
280C42—21no—
281C42—41no—
282C42—41no—
283C42—42no—
284C23—22no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊C4×C5311yescharacteristic
292Q8—42no—
293Q8—42no—
294D4—42no—
295D4—42no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—84no—
322C2×C4#100×C5312yes—
323C2×C4#104×C5312yes—
324C2×C4C2×M16×C5311yescharacteristic
325C2×C4C2×M16×C5311yescharacteristic
326C2×C4#135×C5312yes—
327C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
328C2×C4#137×C5311yescharacteristic
329C8—44no—
330C8—82no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22C53×#40711yescharacteristic
339C22C53×#41011yescharacteristic
340C22C2×#100×C5311yescharacteristic
341C22C53×#41611yescharacteristic
342C22C53×#41711yescharacteristic
343C22C2×#104×C5311yescharacteristic
344C22C53×#42111yescharacteristic
345C4—21no—
346C4—21no—
347C4—22no—
348C4—21no—
349C4—21no—
350C4—22no—
351C4—22no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C2—82no—
358C2C53×#146611yescharacteristic
359C2C53×#148811yescharacteristic
360C2C53×#148911yescharacteristic
361C2C53×#150011yescharacteristic
362C2C53×#151211yescharacteristic
363C2C53×#152711yescharacteristic
364C2C53×#155111yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C43×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1831C2semidirect
3. C53×#1745C2semidirect
4. C53×#1272C2semidirect
148. #7618C53direct (N × H)
176. #1831C2×C53semidirect
177. #1745C2×C53semidirect
178. #1272C2×C53semidirect
205. C53#7618direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#124×C532
C53×#18312
C53×#17452
C53×#12722
C53×#12702
#761853