Finite Group Databasebeta

← C53×#9314 #27216826

Subgroups — C53×#9314 #27216826

Order 13568.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3116C211yesmaximal, characteristic
2C53×#2923C211yesmaximal, characteristic
3C53×#1531C211yesmaximal, characteristic, complemented
4C53×#1488C211yesmaximal, characteristic
5C53×#1485C211yesmaximal, characteristic, complemented
6C53×#1302C211yesmaximal, characteristic, complemented
7C53×#1291C211yesmaximal, characteristic, complemented
8C53×#533C2211yescharacteristic
9C4×#41×C53C2211yescharacteristic
10C53×#485—21no—
11C53×#477C411yescharacteristic
12C53×#476C411yescharacteristic
13C53×#462C411yescharacteristic
14C53×#461C411yescharacteristic
15C53×#452—21no—
16C53×#441—21no—
17C53×#440—21no—
18C53×#433—22no—
19C53×#432—22no—
20C53×#432C2211yescharacteristic
21C53×#432C2211yescharacteristic
22C53×#431C2211yescharacteristic
23C53×#424—21no—
24C53×#408—21no—
25C4×M16×C53C2211yescharacteristic
26C2×#97×C53C2211yescharacteristic
27C53×#330—22no—
28C53×Q8⋊C8—21no—
29C53×Q8⋊C8—22no—
30C53×D4⋊C8—21no—
31C2×#41×C53D411yescharacteristic
32C2×Q16×C53—21no—
33C2×SD16×C53—21no—
34C2×M16×C53—21no—
35C22×C8×C53—21no—
36C4⋊Q8×C53C2×C411yescharacteristic
37C4⋊Q8×C53C2×C411yescharacteristic
38#125×C53—21no—
39C22⋊Q8×C53—21no—
40C22⋊Q8×C53—21no—
41C4×Q8×C53—21no—
42C4×Q8×C53—21no—
43C4×Q8×C53C2×C411yescharacteristic
44C4×Q8×C53D411yescharacteristic
45C4×D4×C53—21no—
46C4×D4×C53—21no—
47C4×D4×C53C2×C411yescharacteristic
48C4×D4×C53D411yescharacteristic
49#117×C53—21no—
50#117×C53—21no—
51#117×C53D411yescharacteristic
52C2×C42×C53—21no—
53C2×C42×C53C2311yescharacteristic
54#107×C53—21no—
55#106×C53—21no—
56C4⋊C8×C53—24no—
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53C2×C411yescharacteristic
60C4⋊C8×C53C2×C411yescharacteristic
61#103×C53—21no—
62#103×C53—21no—
63#103×C53—21no—
64#102×C53—21no—
65C22⋊C8×C53—24no—
66C22⋊C8×C53—21no—
67C22⋊C8×C53—21no—
68C22⋊C8×C53—21no—
69#97×C53—21no—
70#97×C53—21no—
71#97×C53—21no—
72#97×C53—21no—
73#97×C53D412yes—
74C4×C8×C53—21no—
75C4×C8×C53D412yes—
76#41×C53—41no—
77#41×C53—41no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53C22⋊C411yescharacteristic
81C2×D4×C53—21no—
82C2×D4×C53C22⋊C411yescharacteristic
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×C4×C53#4111yescharacteristic
89Q16×C53—42no—
90SD16×C53—42no—
91M16×C53—41no—
92M16×C53—41no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53C22⋊C411yescharacteristic
110C4⋊C4×C53C22⋊C411yescharacteristic
111C22⋊C4×C53—21no—
112C22⋊C4×C53—21no—
113C22⋊C4×C53—21no—
114C22⋊C4×C53—41no—
115C42×C53—21no—
116C42×C53—21no—
117C42×C53—21no—
118C42×C53—41no—
119C42×C53C22×C411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C23×C53—21no—
124C23×C53#12311yescharacteristic
125Q8×C53—21no—
126Q8×C53—21no—
127Q8×C53—42no—
128Q8×C53—41no—
129Q8×C53—41no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—42no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53C2×C22⋊C411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53#12011yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C8×C53—42no—
152C8×C53—42no—
153C8×C53—41no—
154C8×C53—41no—
155C8×C53—81no—
156C8×C53—81no—
157#9314C5311yesmaximal, characteristic, direct product factor, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53#38611yescharacteristic
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—81no—
174C4×C53—81no—
175C4×C53#42111yescharacteristic
176C4×C53#42111yescharacteristic
177#3116C2×C5311yescharacteristic
178#2923C2×C5311yescharacteristic
179#1531C2×C5311yescharacteristic, complemented
180#1488C2×C5311yescharacteristic
181#1485C2×C5311yescharacteristic, complemented
182#1302C2×C5311yescharacteristic, complemented
183#1291C2×C5311yescharacteristic, complemented
184C2×C53—41no—
185C2×C53—81no—
186C2×C53#188111yescharacteristic
187C2×C53#189711yescharacteristic
188C2×C53#190211yescharacteristic
189#533C22×C5311yescharacteristic
190C4×#41C22×C5311yescharacteristic
191#485—21no—
192#477C4×C5311yescharacteristic
193#476C4×C5311yescharacteristic
194#462C4×C5311yescharacteristic
195#461C4×C5311yescharacteristic
196#452—21no—
197#441—21no—
198#440—21no—
199#433—22no—
200#432—22no—
201#432C22×C5311yescharacteristic
202#432C22×C5311yescharacteristic
203#431C22×C5311yescharacteristic
204#424—21no—
205#408—21no—
206C4×M16C22×C5311yescharacteristic
207C2×#97C22×C5311yescharacteristic
208#330—22no—
209Q8⋊C8—21no—
210Q8⋊C8—22no—
211D4⋊C8—21no—
212C53#931411yescharacteristic, direct product factor, Sylow
213C2×#41D4×C5311yescharacteristic
214C2×Q16—21no—
215C2×SD16—21no—
216C2×M16—21no—
217C22×C8—21no—
218C4⋊Q8C2×C4×C5311yescharacteristic
219C4⋊Q8C2×C4×C5311yescharacteristic
220#125—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8C2×C4×C5311yescharacteristic
226C4×Q8D4×C5311yescharacteristic
227C4×D4—21no—
228C4×D4—21no—
229C4×D4C2×C4×C5311yescharacteristic
230C4×D4D4×C5311yescharacteristic
231#117—21no—
232#117—21no—
233#117D4×C5311yescharacteristic
234C2×C42—21no—
235C2×C42C23×C5311yescharacteristic
236#107—21no—
237#106—21no—
238C4⋊C8—24no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8C2×C4×C5311yescharacteristic
242C4⋊C8C2×C4×C5311yescharacteristic
243#103—21no—
244#103—21no—
245#103—21no—
246#102—21no—
247C22⋊C8—24no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251#97—21no—
252#97—21no—
253#97—21no—
254#97—21no—
255#97D4×C5312yes—
256C4×C8—21no—
257C4×C8D4×C5312yes—
258#41—41no—
259#41—41no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8C22⋊C4×C5311yescharacteristic
263C2×D4—21no—
264C2×D4C22⋊C4×C5311yescharacteristic
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4C2×D4×C5311yescharacteristic
269C22×C4#41×C5311yescharacteristic
270C22×C4#41×C5311yescharacteristic
271Q16—42no—
272SD16—42no—
273M16—41no—
274M16—41no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4C22⋊C4×C5311yescharacteristic
292C4⋊C4C22⋊C4×C5311yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—41no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—41no—
301C42C22×C4×C5311yescharacteristic
302C42C2×D4×C5311yescharacteristic
303C42C2×D4×C5311yescharacteristic
304C42C2×D4×C5311yescharacteristic
305C23—21no—
306C23#123×C5311yescharacteristic
307Q8—21no—
308Q8—21no—
309Q8—42no—
310Q8—41no—
311Q8—41no—
312D4—21no—
313D4—21no—
314D4—42no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4C2×C22⋊C4×C5311yescharacteristic
328C2×C4C4×D4×C5311yescharacteristic
329C2×C4C4×D4×C5311yescharacteristic
330C2×C4#120×C5311yescharacteristic
331C2×C4#121×C5311yescharacteristic
332C2×C4#121×C5311yescharacteristic
333C8—42no—
334C8—42no—
335C8—41no—
336C8—41no—
337C8—81no—
338C8—81no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22C53×#38611yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4C53×#42111yescharacteristic
357C4C53×#42111yescharacteristic
358C2—41no—
359C2—81no—
360C2C53×#188111yescharacteristic
361C2C53×#189711yescharacteristic
362C2C53×#190211yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC2×#41×C53D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1531C2semidirect
5. C53×#1485C2semidirect
6. C53×#1302C2semidirect
7. C53×#1291C2semidirect
157. #9314C53direct (N × H)
179. #1531C2×C53semidirect
181. #1485C2×C53semidirect
182. #1302C2×C53semidirect
183. #1291C2×C53semidirect
212. C53#9314direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31162
C53×#29232
C53×#15312
C53×#14882
C53×#14852
C53×#13022
C53×#12912
#931453