Finite Group Databasebeta

← C53×#9371 #27216883

Subgroups — C53×#9371 #27216883

Order 13568.

442 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3214C211yesmaximal, characteristic
2C53×#2981C211yesmaximal, characteristic, complemented
3C53×#1551C211yesmaximal, characteristic, complemented
4C53×#1536C211yesmaximal, characteristic, complemented
5C53×#1480C211yesmaximal, characteristic, complemented
6C53×#1290C211yesmaximal, characteristic, complemented
7C53×#1287C211yesmaximal, characteristic, complemented
8C53×#532C2211yescharacteristic
9C53×#532C2211yescharacteristic
10C53×#496C412yes—
11C53×#489C412yes—
12C53×#461—21no—
13C53×#459—21no—
14C53×#449—22no—
15C53×#436—21no—
16C53×#435—22no—
17C8×D4×C53—21no—
18C53×#432C2211yescharacteristic
19C53×#432C2211yescharacteristic
20C53×#432C2211yescharacteristic
21C53×#431C2211yescharacteristic
22C2×C4⋊C8×C53C2211yescharacteristic
23C53×#408—22no—
24C53×#333—21no—
25C53×#331—21no—
26C53×#329—22no—
27C53×D4⋊C8—21no—
28C53×D4⋊C8—21no—
29C2×#41×C53—21no—
30C2×#41×C53D412yes—
31C2×SD16×C53—22no—
32C2×D8×C53—22no—
33C2×M16×C53—22no—
34C22×C8×C53—22no—
35C22×C8×C53—21no—
36C4⋊Q8×C53C2×C411yescharacteristic
37C4⋊Q8×C53D411yescharacteristic
38#127×C53C2×C411yescharacteristic
39#127×C53D411yescharacteristic
40#124×C53—21no—
41#124×C53C2×C412yes—
42#121×C53—21no—
43#121×C53—21no—
44#121×C53—21no—
45#121×C53—21no—
46C4×D4×C53—21no—
47C4×D4×C53—21no—
48C4×D4×C53—21no—
49C4×D4×C53—21no—
50C2×C42×C53C2311yescharacteristic
51C4⋊C8×C53—21no—
52C4⋊C8×C53—21no—
53C4⋊C8×C53—21no—
54C4⋊C8×C53—21no—
55C4⋊C8×C53—21no—
56C4⋊C8×C53—21no—
57#103×C53—22no—
58#102×C53—22no—
59C22⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—21no—
62C22⋊C8×C53—21no—
63C22⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65#97×C53—21no—
66#97×C53C2×C412yes—
67#97×C53D412yes—
68C4×C8×C53D411yescharacteristic
69C4×C8×C53D411yescharacteristic
70#41×C53—81no—
71#41×C53—81no—
72#41×C53—82no—
73C2×Q8×C53—21no—
74C2×Q8×C53C22⋊C412yes—
75C2×D4×C53—22no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—41no—
82C2×D4×C53C22⋊C412yes—
83C22×C4×C53—22no—
84C22×C4×C53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×C4×C53#4111yescharacteristic
88SD16×C53—82no—
89D8×C53—82no—
90M16×C53—82no—
91C2×C8×C53—22no—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—22no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—82no—
98C2×C8×C53—81no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C22⋊C4×C53—22no—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53C22×C411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C23×C53—22no—
112C23×C53—41no—
113C23×C53#12311yescharacteristic
114Q8×C53—42no—
115Q8×C53—81no—
116D4×C53—42no—
117D4×C53—42no—
118D4×C53—41no—
119D4×C53—41no—
120D4×C53—41no—
121D4×C53—41no—
122D4×C53—81no—
123D4×C53—81no—
124D4×C53—82no—
125D4×C53—82no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—22no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—41no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C8×C53—41no—
143C8×C53—42no—
144C8×C53—41no—
145C8×C53—42no—
146C8×C53—81no—
147C8×C53—81no—
148#9371C5311yesmaximal, characteristic, direct product factor, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—42no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53#38611yescharacteristic
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—42no—
165C4×C53—81no—
166C4×C53#47111yescharacteristic
167C4×C53#47211yescharacteristic
168#3214C2×C5311yescharacteristic
169#2981C2×C5311yescharacteristic, complemented
170#1551C2×C5311yescharacteristic, complemented
171#1536C2×C5311yescharacteristic, complemented
172#1480C2×C5311yescharacteristic, complemented
173#1290C2×C5311yescharacteristic, complemented
174#1287C2×C5311yescharacteristic, complemented
175C2×C53—41no—
176C2×C53—82no—
177C2×C53—161no—
178C2×C53#188111yescharacteristic
179C2×C53#190711yescharacteristic
180C2×C53#191111yescharacteristic
181#532C22×C5311yescharacteristic
182#532C22×C5311yescharacteristic
183#496C4×C5312yes—
184#489C4×C5312yes—
185#461—21no—
186#459—21no—
187#449—22no—
188#436—21no—
189#435—22no—
190C8×D4—21no—
191#432C22×C5311yescharacteristic
192#432C22×C5311yescharacteristic
193#432C22×C5311yescharacteristic
194#431C22×C5311yescharacteristic
195C2×C4⋊C8C22×C5311yescharacteristic
196#408—22no—
197#333—21no—
198#331—21no—
199#329—22no—
200D4⋊C8—21no—
201D4⋊C8—21no—
202C53#937111yescharacteristic, direct product factor, Sylow
203C2×#41—21no—
204C2×#41D4×C5312yes—
205C2×SD16—22no—
206C2×D8—22no—
207C2×M16—22no—
208C22×C8—22no—
209C22×C8—21no—
210C4⋊Q8C2×C4×C5311yescharacteristic
211C4⋊Q8D4×C5311yescharacteristic
212#127C2×C4×C5311yescharacteristic
213#127D4×C5311yescharacteristic
214#124—21no—
215#124C2×C4×C5312yes—
216#121—21no—
217#121—21no—
218#121—21no—
219#121—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C2×C42C23×C5311yescharacteristic
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231#103—22no—
232#102—22no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239#97—21no—
240#97C2×C4×C5312yes—
241#97D4×C5312yes—
242C4×C8D4×C5311yescharacteristic
243C4×C8D4×C5311yescharacteristic
244#41—81no—
245#41—81no—
246#41—82no—
247C2×Q8—21no—
248C2×Q8C22⋊C4×C5312yes—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4C22⋊C4×C5312yes—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4C2×D4×C5311yescharacteristic
260C22×C4#41×C5311yescharacteristic
261C22×C4#41×C5311yescharacteristic
262SD16—82no—
263D8—82no—
264M16—82no—
265C2×C8—22no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—82no—
272C2×C8—81no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C42C22×C4×C5311yescharacteristic
282C42C2×D4×C5311yescharacteristic
283C42C2×D4×C5311yescharacteristic
284C42C2×D4×C5311yescharacteristic
285C23—22no—
286C23—41no—
287C23#123×C5311yescharacteristic
288Q8—42no—
289Q8—81no—
290D4—42no—
291D4—41no—
292D4—41no—
293D4—42no—
294D4—41no—
295D4—41no—
296D4—81no—
297D4—81no—
298D4—82no—
299D4—82no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4C2×C22⋊C4×C5311yescharacteristic
311C2×C4C4×D4×C5311yescharacteristic
312C2×C4C4×D4×C5311yescharacteristic
313C2×C4#120×C5311yescharacteristic
314C2×C4#121×C5311yescharacteristic
315C2×C4#121×C5311yescharacteristic
316C8—41no—
317C8—42no—
318C8—41no—
319C8—42no—
320C8—81no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22C53×#38611yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—42no—
338C4—81no—
339C4C53×#47111yescharacteristic
340C4C53×#47211yescharacteristic
341C2—41no—
342C2—82no—
343C2—161no—
344C2C53×#188111yescharacteristic
345C2C53×#190711yescharacteristic
346C2C53×#191111yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3214C2
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
2. C53×#2981C2semidirect
3. C53×#1551C2semidirect
4. C53×#1536C2semidirect
5. C53×#1480C2semidirect
6. C53×#1290C2semidirect
7. C53×#1287C2semidirect
148. #9371C53direct (N × H)
169. #2981C2×C53semidirect
170. #1551C2×C53semidirect
171. #1536C2×C53semidirect
172. #1480C2×C53semidirect
173. #1290C2×C53semidirect
174. #1287C2×C53semidirect
202. C53#9371direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32142
C53×#29812
C53×#15512
C53×#15362
C53×#14802
C53×#12902
C53×#12872
#937153