Finite Group Databasebeta

← C53×#12721 #27220233

Subgroups — C53×#12721 #27220233

Order 13568.

694 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3510C211yesmaximal, characteristic
2C53×#2135C211yesmaximal, characteristic, complemented
3C53×#2129C211yesmaximal, characteristic, complemented
4C53×#2122C212yesmaximal
5C53×#1827C212yesmaximal, complemented
6C53×#559—22no—
7C53×#556—22no—
8C53×#551—24no—
9C53×#542—22no—
10C53×#536C2211yescharacteristic
11C53×#535C2211yescharacteristic
12C53×#533C411yescharacteristic, complemented
13C53×#529—22no—
14C2×#124×C53C411yescharacteristic, complemented
15C53×#519C411yescharacteristic, complemented
16C53×#518C411yescharacteristic, complemented
17C4×#41×C53C2211yescharacteristic
18C53×#413C2212yes—
19C53×#410C2212yes—
20C53×#356—22no—
21C53×#354—22no—
22C53×AΣL(1,𝔽16)—24no—
23C53×#339—22no—
24C53×H(3,ℤ/4)—22no—
25C2×#41×C53C2311yescharacteristic
26C22×Q8×C53C2×C411yescharacteristic
27C22×D4×C53C2×C411yescharacteristic
28C2×M16×C53—22no—
29C4⋊Q8×C53—21no—
30C4⋊Q8×C53D412yes—
31#127×C53—21no—
32#126×C53—24no—
33#126×C53—24no—
34#125×C53—21no—
35#124×C53—24no—
36#124×C53—21no—
37#124×C53—21no—
38#124×C53—22no—
39#124×C53—21no—
40#124×C53D412yes—
41#123×C53—24no—
42#123×C53—24no—
43#123×C53—22no—
44C22⋊Q8×C53—24no—
45C22⋊Q8×C53—22no—
46C22⋊Q8×C53—22no—
47#121×C53—22no—
48#120×C53D414yes—
49C4×Q8×C53—22no—
50C4×Q8×C53—22no—
51C4×Q8×C53—22no—
52C4×Q8×C53—21no—
53C4×D4×C53—24no—
54C4×D4×C53—22no—
55C4×D4×C53—22no—
56C4×D4×C53—22no—
57C4×D4×C53—21no—
58#117×C53—24no—
59#117×C53—22no—
60#117×C53—22no—
61#117×C53C2×C411yescharacteristic
62#117×C53C2×C411yescharacteristic
63#117×C53C2×C411yescharacteristic
64#117×C53D412yes—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—22no—
67C2×C42×C53C2×C411yescharacteristic
68C2×C42×C53D412yes—
69#101×C53—22no—
70#100×C53—22no—
71#99×C53—22no—
72#99×C53—22no—
73C24×C53C22⋊C412yes—
74#41×C53—42no—
75C2×Q8×C53—22no—
76C2×Q8×C53—21no—
77C2×Q8×C53—42no—
78C2×Q8×C53C22⋊C412yes—
79C2×Q8×C53C2×D411yescharacteristic
80C2×D4×C53—24no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53C2×D411yescharacteristic
85C2×D4×C53C2×D412yes—
86C22×C4×C53—21no—
87C22×C4×C53—24no—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D412yes—
92M16×C53—42no—
93M16×C53—42no—
94C2×C8×C53—42no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—24no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—42no—
105C22⋊C4×C53—24no—
106C22⋊C4×C53—24no—
107C22⋊C4×C53—24no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—21no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—24no—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—42no—
114C22⋊C4×C53C22⋊C412yes—
115C42×C53—24no—
116C42×C53—21no—
117C42×C53—21no—
118C42×C53—22no—
119C42×C53—42no—
120C42×C53C22⋊C412yes—
121C42×C53C22⋊C411yescharacteristic
122C42×C53C22⋊C411yescharacteristic
123C42×C53C22⋊C412yes—
124C23×C53—22no—
125C23×C53—42no—
126C23×C53—82no—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#12012yes—
129Q8×C53—41no—
130Q8×C53—42no—
131Q8×C53—42no—
132Q8×C53—82no—
133D4×C53—42no—
134D4×C53—41no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—84no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—42no—
148C2×C4×C53—82no—
149C2×C4×C53—84no—
150C2×C4×C53—81no—
151C2×C4×C53—82no—
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53#12012yes—
155C8×C53—82no—
156#12721C5311yesmaximal, characteristic, direct product factor, Sylow
157C22×C53—21no—
158C22×C53—22no—
159C22×C53—42no—
160C22×C53—41no—
161C22×C53—44no—
162C22×C53—42no—
163C22×C53—162no—
164C22×C53#37911yescharacteristic
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—42no—
168C4×C53—41no—
169C4×C53—42no—
170C4×C53—41no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53—162no—
174C4×C53#41012yes—
175#3510C2×C5311yescharacteristic
176#2135C2×C5311yescharacteristic, complemented
177#2129C2×C5311yescharacteristic, complemented
178#2122C2×C5312yes—
179#1827C2×C5312yescomplemented
180C2×C53—21no—
181C2×C53—42no—
182C2×C53—41no—
183C2×C53—82no—
184C2×C53#178211yescharacteristic
185#559—22no—
186#556—22no—
187#551—24no—
188#542—22no—
189#536C22×C5311yescharacteristic
190#535C22×C5311yescharacteristic
191#533C4×C5311yescharacteristic, complemented
192#529—22no—
193C2×#124C4×C5311yescharacteristic, complemented
194#519C4×C5311yescharacteristic, complemented
195#518C4×C5311yescharacteristic, complemented
196C4×#41C22×C5311yescharacteristic
197#413C22×C5312yes—
198#410C22×C5312yes—
199#356—22no—
200#354—22no—
201AΣL(1,𝔽16)—24no—
202#339—22no—
203H(3,ℤ/4)—22no—
204C53#1272111yescharacteristic, direct product factor, Sylow
205C2×#41C23×C5311yescharacteristic
206C22×Q8C2×C4×C5311yescharacteristic
207C22×D4C2×C4×C5311yescharacteristic
208C2×M16—22no—
209C4⋊Q8—21no—
210C4⋊Q8D4×C5312yes—
211#127—21no—
212#126—24no—
213#126—24no—
214#125—21no—
215#124—24no—
216#124—21no—
217#124—22no—
218#124—21no—
219#124—21no—
220#124D4×C5312yes—
221#123—24no—
222#123—24no—
223#123—22no—
224C22⋊Q8—24no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227#121—22no—
228#120D4×C5314yes—
229C4×Q8—21no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×D4—24no—
234C4×D4—22no—
235C4×D4—21no—
236C4×D4—22no—
237C4×D4—22no—
238#117—24no—
239#117—22no—
240#117—22no—
241#117C2×C4×C5311yescharacteristic
242#117C2×C4×C5311yescharacteristic
243#117C2×C4×C5311yescharacteristic
244#117D4×C5312yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C42C2×C4×C5311yescharacteristic
248C2×C42D4×C5312yes—
249#101—22no—
250#100—22no—
251#99—22no—
252#99—22no—
253C24C22⋊C4×C5312yes—
254#41—42no—
255C2×Q8—22no—
256C2×Q8—21no—
257C2×Q8—42no—
258C2×Q8C22⋊C4×C5312yes—
259C2×Q8C2×D4×C5311yescharacteristic
260C2×D4—24no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4C2×D4×C5311yescharacteristic
265C2×D4C2×D4×C5312yes—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4C22×C4×C5311yescharacteristic
271C22×C4C2×D4×C5312yes—
272M16—42no—
273M16—42no—
274C2×C8—42no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4C22⋊C4×C5312yes—
295C42—24no—
296C42—21no—
297C42—21no—
298C42—22no—
299C42—42no—
300C42C22⋊C4×C5311yescharacteristic
301C42C22⋊C4×C5311yescharacteristic
302C42C22⋊C4×C5312yes—
303C42C22⋊C4×C5312yes—
304C23—22no—
305C23—42no—
306C23—82no—
307C23C2×C22⋊C4×C5311yescharacteristic
308C23#120×C5312yes—
309Q8—41no—
310Q8—42no—
311Q8—42no—
312Q8—82no—
313D4—42no—
314D4—41no—
315D4—42no—
316D4—42no—
317D4—84no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4C2×C22⋊C4×C5311yescharacteristic
333C2×C4C2×C22⋊C4×C5311yescharacteristic
334C2×C4#120×C5312yes—
335C8—82no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—41no—
340C22—44no—
341C22—42no—
342C22—162no—
343C22C53×#37911yescharacteristic
344C4—22no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—42no—
352C4—162no—
353C4C53×#41012yes—
354C2—21no—
355C2—42no—
356C2—41no—
357C2—82no—
358C2C53×#178211yescharacteristic
subgroupG/H
center, socleC2×C53#1782
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×#41C23×C53
prime radicalC53×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×#41×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2135C2semidirect
3. C53×#2129C2semidirect
5. C53×#1827C2semidirect
12. C53×#533C4semidirect
14. C2×#124×C53C4semidirect
15. C53×#519C4semidirect
16. C53×#518C4semidirect
156. #12721C53direct (N × H)
176. #2135C2×C53semidirect
177. #2129C2×C53semidirect
179. #1827C2×C53semidirect
191. #533C4×C53semidirect
193. C2×#124C4×C53semidirect
194. #519C4×C53semidirect
195. #518C4×C53semidirect
204. C53#12721direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35102
C53×#21352
C53×#21292
C53×#21222
C53×#18272
#1272153