Finite Group Databasebeta

← C53×#12123 #27219635

Subgroups — C53×#12123 #27219635

Order 13568.

570 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2840C211yesmaximal, characteristic, complemented
2C53×#2021C211yesmaximal, characteristic
3C53×#1601C212yesmaximal, complemented
4C53×#1351C212yesmaximal, complemented
5C53×#1316C211yesmaximal, characteristic, complemented
6C53×#534C2211yescharacteristic
7C2×#120×C53—21no—
8C53×#448—22no—
9C2×#99×C53—21no—
10C53×#393—22no—
11C53×#393—21no—
12C53×#393—22no—
13C53×#393C2212yes—
14C53×#379—21no—
15C53×#379C2211yescharacteristic
16C53×#342C2211yescharacteristic
17C53×#327—22no—
18C53×#323C2212yes—
19C25×C53D411yescharacteristic
20C22×Q8×C53—21no—
21C22×Q8×C53D412yescomplemented
22C22×D4×C53—21no—
23C22×D4×C53—41no—
24C2×SD16×C53—42no—
25#123×C53—41no—
26#121×C53—22no—
27#120×C53—41no—
28#120×C53—42no—
29#120×C53—42no—
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53—21no—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—41no—
37C2×C22⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53C2311yescharacteristic
39#103×C53—42no—
40#99×C53—42no—
41C22⋊C8×C53—22no—
42#95×C53—21no—
43#95×C53—22no—
44#95×C53—22no—
45#95×C53D412yescomplemented
46C24×C53—21no—
47C24×C53—21no—
48C24×C53—21no—
49C24×C53—41no—
50C24×C53—41no—
51C24×C53—82no—
52C24×C53SD1612yescomplemented
53C24×C53C2×D411yescharacteristic
54C2×Q8×C53—42no—
55C2×Q8×C53—82no—
56C2×Q8×C53—82no—
57C2×Q8×C53—81no—
58C2×D4×C53—22no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—44no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C2×D4×C53—84no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—41no—
72C22×C4×C53C2×D412yes—
73SD16×C53—84no—
74C2×C8×C53—42no—
75C4⋊C4×C53—22no—
76C22⋊C4×C53—22no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—44no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—82no—
83C22⋊C4×C53—82no—
84C22⋊C4×C53—82no—
85C22⋊C4×C53—81no—
86C22⋊C4×C53—81no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—22no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—44no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—81no—
101C23×C53—82no—
102C23×C53—82no—
103C23×C53—82no—
104C23×C53—84no—
105C23×C53—82no—
106C23×C53#12011yescharacteristic
107C23×C53C2×SD1611yescharacteristic
108C23×C53#13711yescharacteristic
109Q8×C53—44no—
110Q8×C53—82no—
111Q8×C53—84no—
112D4×C53—82no—
113D4×C53—81no—
114D4×C53—81no—
115D4×C53—88no—
116D4×C53—81no—
117D4×C53—81no—
118C2×C4×C53—22no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—82no—
133C8×C53—82no—
134#12123C5311yesmaximal, characteristic, direct product factor, Sylow
135C22×C53—22no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—44no—
144C22×C53—42no—
145C22×C53—81no—
146C22×C53—82no—
147C22×C53—82no—
148C22×C53—84no—
149C22×C53—82no—
150C22×C53—82no—
151C22×C53—81no—
152C22×C53—82no—
153C22×C53—161no—
154C22×C53#44811yescharacteristic
155C22×C53#45311yescharacteristic
156C22×C53B(4,𝔽2)11yescharacteristic
157C4×C53—42no—
158C4×C53—81no—
159C4×C53—82no—
160C4×C53—82no—
161C4×C53—81no—
162C4×C53—82no—
163#2840C2×C5311yescharacteristic, complemented
164#2021C2×C5311yescharacteristic
165#1601C2×C5312yescomplemented
166#1351C2×C5312yescomplemented
167#1316C2×C5311yescharacteristic, complemented
168C2×C53—21no—
169C2×C53—21no—
170C2×C53—41no—
171C2×C53—41no—
172C2×C53—82no—
173C2×C53—161no—
174C2×C53#159611yescharacteristic
175C2×C53#219912yes—
176#534C22×C5311yescharacteristic
177C2×#120—21no—
178#448—22no—
179C2×#99—21no—
180#393—22no—
181#393—21no—
182#393—22no—
183#393C22×C5312yes—
184#379—21no—
185#379C22×C5311yescharacteristic
186#342C22×C5311yescharacteristic
187#327—22no—
188#323C22×C5312yes—
189C53#1212311yescharacteristic, direct product factor, Sylow
190C25D4×C5311yescharacteristic
191C22×Q8—21no—
192C22×Q8D4×C5312yescomplemented
193C22×D4—21no—
194C22×D4—41no—
195C2×SD16—42no—
196#123—41no—
197#121—22no—
198#120—42no—
199#120—41no—
200#120—42no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4D4×C5311yescharacteristic
209C2×C22⋊C4C23×C5311yescharacteristic
210#103—42no—
211#99—42no—
212C22⋊C8—22no—
213#95—21no—
214#95—22no—
215#95—22no—
216#95D4×C5312yescomplemented
217C24—21no—
218C24—21no—
219C24—21no—
220C24—41no—
221C24—41no—
222C24—82no—
223C24SD16×C5312yescomplemented
224C24C2×D4×C5311yescharacteristic
225C2×Q8—42no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—81no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—84no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4C2×D4×C5312yes—
244SD16—84no—
245C2×C8—42no—
246C4⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—44no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—41no—
267C23—44no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—81no—
272C23—82no—
273C23—82no—
274C23—82no—
275C23—84no—
276C23—82no—
277C23#120×C5311yescharacteristic
278C23C2×SD16×C5311yescharacteristic
279C23#137×C5311yescharacteristic
280Q8—44no—
281Q8—82no—
282Q8—84no—
283D4—82no—
284D4—81no—
285D4—81no—
286D4—88no—
287D4—81no—
288D4—81no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—82no—
304C8—82no—
305C22—22no—
306C22—21no—
307C22—21no—
308C22—41no—
309C22—41no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—44no—
314C22—42no—
315C22—81no—
316C22—82no—
317C22—82no—
318C22—84no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—81no—
323C22—161no—
324C22C53×#44811yescharacteristic
325C22C53×#45311yescharacteristic
326C22C53×B(4,𝔽2)11yescharacteristic
327C4—42no—
328C4—81no—
329C4—82no—
330C4—82no—
331C4—82no—
332C4—81no—
333C2—21no—
334C2—21no—
335C2—41no—
336C2—41no—
337C2—82no—
338C2—161no—
339C2C53×#159611yescharacteristic
340C2C53×#219912yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC53×#2021C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2840C2semidirect
3. C53×#1601C2semidirect
4. C53×#1351C2semidirect
5. C53×#1316C2semidirect
21. C22×Q8×C53D4semidirect
45. #95×C53D4semidirect
52. C24×C53SD16semidirect
134. #12123C53direct (N × H)
163. #2840C2×C53semidirect
165. #1601C2×C53semidirect
166. #1351C2×C53semidirect
167. #1316C2×C53semidirect
189. C53#12123direct (N × H)
192. C22×Q8D4×C53semidirect
216. #95D4×C53semidirect
223. C24SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#28402
C53×#20212
C53×#16012
C53×#13512
C53×#13162
#1212353