Finite Group Databasebeta

← C53×#10392 #27217904

Subgroups — C53×#10392 #27217904

Order 13568.

746 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2842C211yesmaximal, characteristic, complemented
2C53×#1781C211yesmaximal, characteristic
3C53×#1781C212yesmaximal
4C53×#1304C211yesmaximal, characteristic, complemented
5C53×#1304C212yesmaximal, complemented
6C23×D4×C53C2211yescharacteristic
7C2×#99×C53—24no—
8C2×#99×C53—22no—
9C53×#399—22no—
10C53×#396—24no—
11C53×#379C2212yes—
12C53×#379C2211yescharacteristic
13C53×#379C2211yescharacteristic
14C53×#379C2212yes—
15C25×C53Q811yescharacteristic
16C25×C53C2311yescharacteristic
17C22×D4×C53—22no—
18C22×D4×C53—21no—
19C22×D4×C53—22no—
20C22×D4×C53—21no—
21C22×D4×C53—44no—
22C23×C4×C53Q811yescharacteristic
23C2×C4⋊C4×C53—24no—
24C2×C4⋊C4×C53—22no—
25C2×C22⋊C4×C53—22no—
26C2×C22⋊C4×C53—22no—
27C2×C22⋊C4×C53—22no—
28C2×C22⋊C4×C53—22no—
29C2×C22⋊C4×C53—21no—
30C2×C22⋊C4×C53—22no—
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53D414yes—
34C2×C22⋊C4×C53D412yes—
35#99×C53—48no—
36#99×C53—44no—
37#95×C53—22no—
38C24×C53—22no—
39C24×C53—21no—
40C24×C53—42no—
41C24×C53—42no—
42C24×C53—82no—
43C24×C53—82no—
44C24×C53C2×D412yes—
45C24×C53C2×D411yescharacteristic
46C24×C53C2×Q811yescharacteristic
47C24×C53#4112yes—
48C24×C53#4111yescharacteristic
49C2×D4×C53—28no—
50C2×D4×C53—24no—
51C2×D4×C53—44no—
52C2×D4×C53—42no—
53C2×D4×C53—44no—
54C2×D4×C53—42no—
55C2×D4×C53—44no—
56C2×D4×C53—42no—
57C2×D4×C53—41no—
58C2×D4×C53—42no—
59C2×D4×C53—41no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—24no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53—41no—
67C4⋊C4×C53—84no—
68C4⋊C4×C53—82no—
69C22⋊C4×C53—44no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—44no—
72C22⋊C4×C53—42no—
73C22⋊C4×C53—82no—
74C22⋊C4×C53—82no—
75C22⋊C4×C53—82no—
76C22⋊C4×C53—82no—
77C22⋊C4×C53—82no—
78C22⋊C4×C53—82no—
79C23×C53—24no—
80C23×C53—22no—
81C23×C53—24no—
82C23×C53—22no—
83C23×C53—41no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53—41no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—82no—
97C23×C53—82no—
98C23×C53—82no—
99C23×C53—88no—
100C23×C53—88no—
101C23×C53#12011yescharacteristic
102C23×C53C22⋊Q812yes—
103C23×C53C22⋊Q811yescharacteristic
104C23×C53#12412yes—
105C23×C53#12411yescharacteristic
106D4×C53—48no—
107D4×C53—44no—
108D4×C53—44no—
109C2×C4×C53—24no—
110C2×C4×C53—22no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—44no—
115C2×C4×C53—44no—
116C2×C4×C53—44no—
117C2×C4×C53—42no—
118C2×C4×C53—44no—
119C2×C4×C53—82no—
120C2×C4×C53—81no—
121C2×C4×C53—82no—
122C2×C4×C53—82no—
123C2×C4×C53—82no—
124#10392C5311yesmaximal, characteristic, direct product factor, Sylow
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—21no—
128C22×C53—22no—
129C22×C53—21no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—44no—
137C22×C53—42no—
138C22×C53—84no—
139C22×C53—82no—
140C22×C53—82no—
141C22×C53—84no—
142C22×C53—84no—
143C22×C53—82no—
144C22×C53—84no—
145C22×C53—82no—
146C22×C53—82no—
147C22×C53#39311yescharacteristic
148C22×C53B(4,𝔽2)12yes—
149C4×C53—42no—
150C4×C53—82no—
151C4×C53—84no—
152C4×C53—162no—
153C4×C53—161no—
154#2842C2×C5311yescharacteristic, complemented
155#1781C2×C5311yescharacteristic
156#1781C2×C5312yes—
157#1304C2×C5311yescharacteristic, complemented
158#1304C2×C5312yescomplemented
159C2×C53—22no—
160C2×C53—42no—
161C2×C53—41no—
162C2×C53—42no—
163C2×C53—41no—
164C2×C53—82no—
165C2×C53#202911yescharacteristic
166C2×C53#203212yes—
167C23×D4C22×C5311yescharacteristic
168C2×#99—24no—
169C2×#99—22no—
170#399—22no—
171#396—24no—
172#379C22×C5312yes—
173#379C22×C5311yescharacteristic
174#379C22×C5311yescharacteristic
175#379C22×C5312yes—
176C53#1039211yescharacteristic, direct product factor, Sylow
177C25Q8×C5311yescharacteristic
178C25C23×C5311yescharacteristic
179C22×D4—22no—
180C22×D4—21no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—44no—
184C23×C4Q8×C5311yescharacteristic
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4D4×C5314yes—
196C2×C22⋊C4D4×C5312yes—
197#99—48no—
198#99—44no—
199#95—22no—
200C24—22no—
201C24—21no—
202C24—42no—
203C24—42no—
204C24—82no—
205C24—82no—
206C24C2×D4×C5312yes—
207C24C2×D4×C5311yescharacteristic
208C24C2×Q8×C5311yescharacteristic
209C24#41×C5312yes—
210C24#41×C5311yescharacteristic
211C2×D4—28no—
212C2×D4—24no—
213C2×D4—44no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C4⋊C4—84no—
230C4⋊C4—82no—
231C22⋊C4—44no—
232C22⋊C4—42no—
233C22⋊C4—44no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C22⋊C4—82no—
241C23—24no—
242C23—22no—
243C23—24no—
244C23—22no—
245C23—41no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—41no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23—82no—
259C23—82no—
260C23—82no—
261C23—88no—
262C23—88no—
263C23#120×C5311yescharacteristic
264C23C22⋊Q8×C5312yes—
265C23C22⋊Q8×C5311yescharacteristic
266C23#124×C5312yes—
267C23#124×C5311yescharacteristic
268D4—48no—
269D4—44no—
270D4—44no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C22—22no—
287C22—22no—
288C22—21no—
289C22—22no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—44no—
298C22—42no—
299C22—84no—
300C22—82no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22—82no—
305C22—84no—
306C22—82no—
307C22—82no—
308C22C53×#39311yescharacteristic
309C22C53×B(4,𝔽2)12yes—
310C4—42no—
311C4—82no—
312C4—84no—
313C4—162no—
314C4—161no—
315C2—22no—
316C2—42no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2—82no—
321C2C53×#202911yescharacteristic
322C2C53×#203212yes—
subgroupG/H
center, socleC22×C53#393
commutator, Frattini, cosocleC25C23×C53
prime radicalC23×D4×C53C22
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C25×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2842C2semidirect
4. C53×#1304C2semidirect
5. C53×#1304C2semidirect
124. #10392C53direct (N × H)
154. #2842C2×C53semidirect
157. #1304C2×C53semidirect
158. #1304C2×C53semidirect
176. C53#10392direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28422
C53×#17812
C53×#13042
#1039253