Finite Group Databasebeta

← C53×#17632 #27225144

Subgroups — C53×#17632 #27225144

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3502C211yesmaximal, characteristic
2C53×#2983C211yesmaximal, characteristic
3C53×#2356C211yesmaximal, characteristic
4C53×#1518C212yesmaximal
5C53×#1515C214yesmaximal
6C53×#1506C214yesmaximal
7C53×#1503C212yesmaximal
8Q82×C53C414yescomplemented
9C53×#557—24no—
10C53×#557C414yescomplemented
11C53×#544—21no—
12C53×#544—22no—
13C53×#533C2212yes—
14C53×#533C2214yes—
15C53×#519—21no—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#446—24no—
18C53×#432C2214yes—
19C53×#431C2212yes—
20C53×#391—21no—
21C53×#387—22no—
22C53×#383C2212yes—
23C53×#382C2211yescharacteristic
24C4×C4⋊C4×C53—21no—
25C4×C4⋊C4×C53C2211yescharacteristic
26C53×#376C2212yes—
27C53×#328C2214yes—
28C53×#328C2214yes—
29C53×#328C2218yes—
30C22×Q8×C53D411yescharacteristic
31C4⋊Q8×C53—28no—
32C4⋊Q8×C53—24no—
33C4⋊Q8×C53C2×C412yes—
34C4⋊Q8×C53C2×C414yes—
35C4⋊Q8×C53C2×C412yes—
36C4⋊Q8×C53C2×C414yes—
37#125×C53—28no—
38#125×C53—24no—
39#125×C53—22no—
40#125×C53—24no—
41C22⋊Q8×C53—24no—
42C22⋊Q8×C53—22no—
43C22⋊Q8×C53C2314yes—
44C4×Q8×C53—22no—
45C4×Q8×C53—24no—
46C4×Q8×C53—24no—
47C4×Q8×C53—24no—
48C4×Q8×C53C2×C414yes—
49C4×Q8×C53C2×C414yes—
50C4×Q8×C53C2×C418yes—
51C4×Q8×C53D414yes—
52#117×C53—24no—
53#117×C53—22no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C4⋊C4×C53D412yes—
58C2×C42×C53—22no—
59C2×C42×C53—21no—
60C2×C42×C53C2312yes—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—22no—
63C4⋊C8×C53—24no—
64C22⋊C8×C53C2314yes—
65#97×C53—24no—
66C4×C8×C53—22no—
67#95×C53—22no—
68#95×C53—22no—
69#95×C53—21no—
70#95×C53—22no—
71#95×C53C2311yescharacteristic
72#95×C53C2312yes—
73#95×C53C2311yescharacteristic
74C2×Q8×C53—24no—
75C2×Q8×C53—22no—
76C2×Q8×C53—22no—
77C2×Q8×C53—22no—
78C2×Q8×C53C22⋊C414yes—
79C2×Q8×C53C22×C414yes—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—21no—
83C22×C4×C53C2×D412yes—
84C22×C4×C53C2×D412yes—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2411yescharacteristic
88C2×C8×C53—24no—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—24no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—28no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—42no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53C22⋊C418yes—
101C4⋊C4×C53C22⋊C414yes—
102C4⋊C4×C53C22×C414yes—
103C22⋊C4×C53—24no—
104C42×C53—28no—
105C42×C53—24no—
106C42×C53—44no—
107C42×C53—42no—
108C42×C53C22×C414yes—
109C42×C53C22×C412yes—
110C42×C53C2×D412yes—
111C42×C53C2×D414yes—
112C23×C53C22×D411yescharacteristic
113Q8×C53—28no—
114Q8×C53—44no—
115Q8×C53—84no—
116Q8×C53—82no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—24no—
121C2×C4×C53—24no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53C2×C22⋊C414yes—
134C2×C4×C53C2×C22⋊C412yes—
135C2×C4×C53C2×C22⋊C414yes—
136C2×C4×C53C2×C22⋊C412yes—
137C2×C4×C53#13712yes—
138C2×C4×C53#13712yes—
139C2×C4×C53C23×C411yescharacteristic
140C2×C4×C53C22×D411yescharacteristic
141C8×C53—44no—
142#17632C5311yesmaximal, characteristic, direct product factor, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53C22×C22⋊C411yescharacteristic
146C22×C53C2×#13711yescharacteristic
147C22×C53C2×#13711yescharacteristic
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—24no—
151C4×C53—24no—
152C4×C53—22no—
153C4×C53—42no—
154C4×C53—44no—
155C4×C53—44no—
156C4×C53—81no—
157C4×C53—82no—
158C4×C53#41914yes—
159#3502C2×C5311yescharacteristic
160#2983C2×C5311yescharacteristic
161#2356C2×C5311yescharacteristic
162#1518C2×C5312yes—
163#1515C2×C5314yes—
164#1506C2×C5314yes—
165#1503C2×C5312yes—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53#288411yescharacteristic
169C2×C53C2×#41911yescharacteristic
170C2×C53#290111yescharacteristic
171Q82C4×C5314yescomplemented
172#557—24no—
173#557C4×C5314yescomplemented
174#544—21no—
175#544—22no—
176#533C22×C5312yes—
177#533C22×C5314yes—
178#519—21no—
179C2×C4×Q8C22×C5311yescharacteristic
180#446—24no—
181#432C22×C5314yes—
182#431C22×C5312yes—
183#391—21no—
184#387—22no—
185#383C22×C5312yes—
186#382C22×C5311yescharacteristic
187C4×C4⋊C4—21no—
188C4×C4⋊C4C22×C5311yescharacteristic
189#376C22×C5312yes—
190#328C22×C5314yes—
191#328C22×C5314yes—
192#328C22×C5318yes—
193C53#1763211yescharacteristic, direct product factor, Sylow
194C22×Q8D4×C5311yescharacteristic
195C4⋊Q8—28no—
196C4⋊Q8—24no—
197C4⋊Q8C2×C4×C5312yes—
198C4⋊Q8C2×C4×C5314yes—
199C4⋊Q8C2×C4×C5312yes—
200C4⋊Q8C2×C4×C5314yes—
201#125—28no—
202#125—24no—
203#125—22no—
204#125—24no—
205C22⋊Q8—24no—
206C22⋊Q8—22no—
207C22⋊Q8C23×C5314yes—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—22no—
211C4×Q8—24no—
212C4×Q8C2×C4×C5314yes—
213C4×Q8C2×C4×C5318yes—
214C4×Q8C2×C4×C5314yes—
215C4×Q8D4×C5314yes—
216#117—24no—
217#117—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4D4×C5311yescharacteristic
221C2×C4⋊C4D4×C5312yes—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42C23×C5312yes—
225C2×C42C23×C5311yescharacteristic
226C4⋊C8—22no—
227C4⋊C8—24no—
228C22⋊C8C23×C5314yes—
229#97—24no—
230C4×C8—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95C23×C5311yescharacteristic
236#95C23×C5312yes—
237#95C23×C5311yescharacteristic
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8C22⋊C4×C5314yes—
243C2×Q8C22×C4×C5314yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4C2×D4×C5312yes—
248C22×C4C2×D4×C5312yes—
249C22×C4C2×D4×C5311yescharacteristic
250C22×C4C2×D4×C5311yescharacteristic
251C22×C4C24×C5311yescharacteristic
252C2×C8—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—28no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4C22⋊C4×C5318yes—
265C4⋊C4C22⋊C4×C5314yes—
266C4⋊C4C22×C4×C5314yes—
267C22⋊C4—24no—
268C42—28no—
269C42—24no—
270C42—44no—
271C42—42no—
272C42C22×C4×C5314yes—
273C42C22×C4×C5312yes—
274C42C2×D4×C5312yes—
275C42C2×D4×C5314yes—
276C23C22×D4×C5311yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—84no—
280Q8—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4C2×C22⋊C4×C5314yes—
298C2×C4C2×C22⋊C4×C5312yes—
299C2×C4C2×C22⋊C4×C5314yes—
300C2×C4C2×C22⋊C4×C5312yes—
301C2×C4#137×C5312yes—
302C2×C4#137×C5312yes—
303C2×C4C23×C4×C5311yescharacteristic
304C2×C4C22×D4×C5311yescharacteristic
305C8—44no—
306C22—21no—
307C22—21no—
308C22C22×C22⋊C4×C5311yescharacteristic
309C22C2×#137×C5311yescharacteristic
310C22C2×#137×C5311yescharacteristic
311C4—21no—
312C4—21no—
313C4—24no—
314C4—24no—
315C4—22no—
316C4—42no—
317C4—44no—
318C4—44no—
319C4—81no—
320C4—82no—
321C4C53×#41914yes—
322C2—21no—
323C2—21no—
324C2C53×#288411yescharacteristic
325C2C2×C53×#41911yescharacteristic
326C2C53×#290111yescharacteristic
subgroupG/H
center, socleC22×C53C22×C22⋊C4
commutatorC2×C4C23×C4×C53
Frattini, cosocleC22×C4C24×C53
prime radicalC23×C53C22×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C4×Q8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
8. Q82×C53C4semidirect
10. C53×#557C4semidirect
142. #17632C53direct (N × H)
171. Q82C4×C53semidirect
173. #557C4×C53semidirect
193. C53#17632direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35022
C53×#29832
C53×#23562
C53×#15182
C53×#15152
C53×#15062
C53×#15032
#1763253