Finite Group Databasebeta

← C53×#10169 #27217681

Subgroups — C53×#10169 #27217681

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3208C211yesmaximal, characteristic
2C53×#3139C211yesmaximal, characteristic, complemented
3C2×C53×#486C211yesmaximal, characteristic
4C53×#1965C211yesmaximal, characteristic
5C53×#1546C211yesmaximal, characteristic, complemented
6C53×#1493C211yesmaximal, characteristic, complemented
7C53×#1278C211yesmaximal, characteristic, complemented
8C53×#533C2211yescharacteristic
9C2×C4⋊Q8×C53C2211yescharacteristic
10C2×#127×C53C2211yescharacteristic
11C53×#492—21no—
12C53×#491—22no—
13C53×#487—21no—
14C53×#487—21no—
15C53×#486—21no—
16C53×#486—22no—
17C53×#486—22no—
18C53×#460—22no—
19C53×#450—22no—
20C53×#431C2211yescharacteristic
21C53×#423C2211yescharacteristic
22C53×#419—22no—
23C2×#102×C53—22no—
24C2×#100×C53—22no—
25C4×M16×C53C2211yescharacteristic
26C2×C4×C8×C53C2211yescharacteristic
27C53×#332—22no—
28C53×Q8⋊C8—22no—
29C22×Q8×C53—21no—
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53D412yes—
33C2×SD16×C53—24no—
34C2×M16×C53—21no—
35C2×M16×C53D412yes—
36C22×C8×C53—21no—
37C4⋊Q8×C53—21no—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—21no—
40C4⋊Q8×C53D411yescharacteristic
41C4⋊Q8×C53D412yes—
42C4⋊Q8×C53D411yescharacteristic
43#127×C53—21no—
44#127×C53—21no—
45#127×C53—21no—
46#127×C53—21no—
47#125×C53—21no—
48C22⋊Q8×C53—21no—
49C22⋊Q8×C53—21no—
50C4×Q8×C53—21no—
51C4×Q8×C53—21no—
52#117×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—21no—
57C4⋊C8×C53—21no—
58C4⋊C8×C53D412yes—
59#103×C53—24no—
60#103×C53—24no—
61#102×C53—28no—
62#102×C53—24no—
63#100×C53—44no—
64C22⋊C8×C53—21no—
65#97×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C4×C8×C53—21no—
69C4×C8×C53D412yes—
70C24×C53—22no—
71C2×Q8×C53—22no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53—41no—
75C2×Q8×C53—42no—
76C2×Q8×C53—42no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53—44no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—42no—
85C2×D4×C53—42no—
86C2×D4×C53—82no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91SD16×C53—84no—
92M16×C53—22no—
93M16×C53—22no—
94M16×C53—42no—
95C2×C8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53—22no—
98C2×C8×C53—42no—
99C2×C8×C53—41no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C42×C53—41no—
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C23×C53—42no—
117C23×C53—44no—
118C23×C53—82no—
119C23×C53—82no—
120C23×C53#12111yescharacteristic
121Q8×C53—44no—
122Q8×C53—42no—
123Q8×C53—42no—
124Q8×C53—81no—
125Q8×C53—81no—
126D4×C53—44no—
127D4×C53—44no—
128D4×C53—82no—
129D4×C53—82no—
130D4×C53—82no—
131D4×C53—81no—
132D4×C53—81no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—82no—
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12711yescharacteristic
145C2×C4×C53#13512yes—
146C2×C4×C53AGL(1,ℤ/8)12yes—
147C8×C53—42no—
148C8×C53—42no—
149C8×C53—42no—
150C8×C53—81no—
151#10169C5311yesmaximal, characteristic, direct product factor, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—82no—
155C22×C53—82no—
156C22×C53—82no—
157C22×C53—164no—
158C22×C53#39211yescharacteristic
159C22×C53#46312yes—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—81no—
163C4×C53—82no—
164C4×C53—81no—
165C4×C53#44912yes—
166C4×C53#45512yes—
167C4×C53#47112yes—
168C4×C53#49511yescharacteristic
169C4×C53#49611yescharacteristic
170#3208C2×C5311yescharacteristic
171#3139C2×C5311yescharacteristic, complemented
172C2×#486C2×C5311yescharacteristic
173#1965C2×C5311yescharacteristic
174#1546C2×C5311yescharacteristic, complemented
175#1493C2×C5311yescharacteristic, complemented
176#1278C2×C5311yescharacteristic, complemented
177C2×C53—22no—
178C2×C53—162no—
179C2×C53#200411yescharacteristic
180C2×C53#201211yescharacteristic
181C2×C53#201911yescharacteristic
182#533C22×C5311yescharacteristic
183C2×C4⋊Q8C22×C5311yescharacteristic
184C2×#127C22×C5311yescharacteristic
185#492—21no—
186#491—22no—
187#487—21no—
188#487—21no—
189#486—22no—
190#486—21no—
191#486—22no—
192#460—22no—
193#450—22no—
194#431C22×C5311yescharacteristic
195#423C22×C5311yescharacteristic
196#419—22no—
197C2×#102—22no—
198C2×#100—22no—
199C4×M16C22×C5311yescharacteristic
200C2×C4×C8C22×C5311yescharacteristic
201#332—22no—
202Q8⋊C8—22no—
203C53#1016911yescharacteristic, direct product factor, Sylow
204C22×Q8—21no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4D4×C5312yes—
208C2×SD16—24no—
209C2×M16—21no—
210C2×M16D4×C5312yes—
211C22×C8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8D4×C5312yes—
216C4⋊Q8D4×C5311yescharacteristic
217C4⋊Q8D4×C5311yescharacteristic
218#127—21no—
219#127—21no—
220#127—21no—
221#127—21no—
222#125—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227#117—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C42C23×C5311yescharacteristic
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8D4×C5312yes—
234#103—24no—
235#103—24no—
236#102—28no—
237#102—24no—
238#100—44no—
239C22⋊C8—21no—
240#97—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8D4×C5312yes—
245C24—22no—
246C2×Q8—22no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—44no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—82no—
262C22×C4—41no—
263C22×C4C2×D4×C5311yescharacteristic
264C22×C4C2×D4×C5311yescharacteristic
265C22×C4#41×C5311yescharacteristic
266SD16—84no—
267M16—22no—
268M16—22no—
269M16—42no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—42no—
274C2×C8—41no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—41no—
286C42—41no—
287C42C2×D4×C5311yescharacteristic
288C42C2×D4×C5311yescharacteristic
289C42C2×D4×C5311yescharacteristic
290C42C2×D4×C5311yescharacteristic
291C23—42no—
292C23—44no—
293C23—82no—
294C23—82no—
295C23#121×C5311yescharacteristic
296Q8—44no—
297Q8—42no—
298Q8—42no—
299Q8—81no—
300Q8—81no—
301D4—44no—
302D4—44no—
303D4—82no—
304D4—81no—
305D4—81no—
306D4—82no—
307D4—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4#120×C5311yescharacteristic
315C2×C4#120×C5311yescharacteristic
316C2×C4#120×C5311yescharacteristic
317C2×C4#121×C5311yescharacteristic
318C2×C4#121×C5311yescharacteristic
319C2×C4#127×C5311yescharacteristic
320C2×C4#135×C5312yes—
321C2×C4AGL(1,ℤ/8)×C5312yes—
322C8—42no—
323C8—42no—
324C8—42no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—164no—
332C22C53×#39211yescharacteristic
333C22C53×#46312yes—
334C4—21no—
335C4—21no—
336C4—81no—
337C4—82no—
338C4—81no—
339C4C53×#44912yes—
340C4C53×#45512yes—
341C4C53×#47112yes—
342C4C53×#49511yescharacteristic
343C4C53×#49611yescharacteristic
344C2—22no—
345C2—162no—
346C2C53×#200411yescharacteristic
347C2C53×#201211yescharacteristic
348C2C53×#201911yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC2×#127×C53C22
LCSG ▷ C2×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×#3139C2semidirect
5. C53×#1546C2semidirect
6. C53×#1493C2semidirect
7. C53×#1278C2semidirect
151. #10169C53direct (N × H)
171. #3139C2×C53semidirect
174. #1546C2×C53semidirect
175. #1493C2×C53semidirect
176. #1278C2×C53semidirect
203. C53#10169direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32082
C53×#31392
C2×C53×#4862
C53×#19652
C53×#15462
C53×#14932
C53×#12782
#1016953