Finite Group Databasebeta

← C53×#12966 #27220478

Subgroups — C53×#12966 #27220478

Order 13568.

580 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2380C211yesmaximal, characteristic
2C2×C53×#353C211yesmaximal, characteristic
3C53×#1887C211yesmaximal, characteristic
4C53×#1867C212yesmaximal, complemented
5C53×#1390C212yesmaximal, complemented
6C2×#127×C53C2211yescharacteristic
7C53×#518C2211yescharacteristic
8C53×#418—21no—
9C2×#100×C53C2211yescharacteristic
10C2×#99×C53C2212yes—
11C53×#386—21no—
12C53×#386—22no—
13C53×#386—22no—
14C53×#386C2212yes—
15C53×#385—22no—
16C53×#383—21no—
17C53×#353C414yescomplemented
18C53×#342—22no—
19C53×#339—21no—
20C22×D4×C53—22no—
21C22×D4×C53D411yescharacteristic
22C22×D4×C53C2311yescharacteristic
23C23×C4×C53—21no—
24C23×C4×C53D412yes—
25C2×M16×C53—21no—
26#127×C53—22no—
27#127×C53C2×C412yes—
28#127×C53Q812yes—
29C4×D4×C53—42no—
30#117×C53—21no—
31C2×C4⋊C4×C53—42no—
32C2×C22⋊C4×C53—21no—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53D412yes—
38C2×C42×C53D411yescharacteristic
39#102×C53—42no—
40#100×C53—22no—
41#99×C53—22no—
42#99×C53C2×C414yescomplemented
43#95×C53—21no—
44#95×C53—22no—
45#95×C53—21no—
46#95×C53—22no—
47#95×C53—42no—
48C24×C53—21no—
49C24×C53C2×D412yes—
50C2×D4×C53—24no—
51C2×D4×C53—24no—
52C2×D4×C53—21no—
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53—82no—
58C2×D4×C53—82no—
59C2×D4×C53C4⋊C412yescomplemented
60C2×D4×C53C22×C411yescharacteristic
61C2×D4×C53C2×Q811yescharacteristic
62C2×D4×C53#4112yes—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—82no—
71C22×C4×C53—82no—
72C22×C4×C53—81no—
73C22×C4×C53C2×D411yescharacteristic
74M16×C53—41no—
75M16×C53—41no—
76C2×C8×C53—41no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—84no—
79C22⋊C4×C53—24no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—82no—
85C22⋊C4×C53—82no—
86C22⋊C4×C53—81no—
87C42×C53—21no—
88C42×C53—41no—
89C42×C53C4⋊C412yescomplemented
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53C4×D412yes—
100C23×C53#12011yescharacteristic
101C23×C53C22⋊Q812yes—
102D4×C53—41no—
103D4×C53—41no—
104D4×C53—42no—
105D4×C53—42no—
106D4×C53—81no—
107D4×C53—84no—
108D4×C53—84no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—44no—
118C2×C4×C53—44no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—82no—
123C2×C4×C53—82no—
124C2×C4×C53—84no—
125C2×C4×C53—82no—
126C2×C4×C53—82no—
127C2×C4×C53—82no—
128C2×C4×C53C2×C4⋊C411yescharacteristic
129C2×C4×C53#12311yescharacteristic
130C8×C53—81no—
131#12966C5311yesmaximal, characteristic, direct product factor, Sylow
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—44no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—82no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53—82no—
145C22×C53#38511yescharacteristic
146C22×C53B(4,𝔽2)11yescharacteristic
147C22×C53#45811yescharacteristic
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—42no—
151C4×C53—82no—
152C4×C53—84no—
153C4×C53—82no—
154C4×C53—82no—
155#2380C2×C5311yescharacteristic
156C2×#353C2×C5311yescharacteristic
157#1887C2×C5311yescharacteristic
158#1867C2×C5312yescomplemented
159#1390C2×C5312yescomplemented
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—42no—
163C2×C53—42no—
164C2×C53—82no—
165C2×C53#186711yescharacteristic
166C2×C53#220012yes—
167C2×#127C22×C5311yescharacteristic
168#518C22×C5311yescharacteristic
169#418—21no—
170C2×#100C22×C5311yescharacteristic
171C2×#99C22×C5312yes—
172#386—21no—
173#386—22no—
174#386—22no—
175#386C22×C5312yes—
176#385—22no—
177#383—21no—
178#353C4×C5314yescomplemented
179#342—22no—
180#339—21no—
181C53#1296611yescharacteristic, direct product factor, Sylow
182C22×D4—22no—
183C22×D4D4×C5311yescharacteristic
184C22×D4C23×C5311yescharacteristic
185C23×C4—21no—
186C23×C4D4×C5312yes—
187C2×M16—21no—
188#127—22no—
189#127C2×C4×C5312yes—
190#127Q8×C5312yes—
191C4×D4—42no—
192#117—21no—
193C2×C4⋊C4—42no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4D4×C5312yes—
200C2×C42D4×C5311yescharacteristic
201#102—42no—
202#100—22no—
203#99—22no—
204#99C2×C4×C5314yescomplemented
205#95—21no—
206#95—22no—
207#95—21no—
208#95—22no—
209#95—42no—
210C24—21no—
211C24C2×D4×C5312yes—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—21no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4C4⋊C4×C5312yescomplemented
222C2×D4C22×C4×C5311yescharacteristic
223C2×D4C2×Q8×C5311yescharacteristic
224C2×D4#41×C5312yes—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—81no—
235C22×C4C2×D4×C5311yescharacteristic
236M16—41no—
237M16—41no—
238C2×C8—41no—
239C4⋊C4—22no—
240C4⋊C4—84no—
241C22⋊C4—24no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—81no—
249C42—21no—
250C42—41no—
251C42C4⋊C4×C5312yescomplemented
252C23—22no—
253C23—22no—
254C23—22no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—41no—
261C23C4×D4×C5312yes—
262C23#120×C5311yescharacteristic
263C23C22⋊Q8×C5312yes—
264D4—41no—
265D4—41no—
266D4—42no—
267D4—42no—
268D4—81no—
269D4—84no—
270D4—84no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4C2×C4⋊C4×C5311yescharacteristic
291C2×C4#123×C5311yescharacteristic
292C8—81no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—82no—
303C22—81no—
304C22—81no—
305C22—82no—
306C22C53×#38511yescharacteristic
307C22C53×B(4,𝔽2)11yescharacteristic
308C22C53×#45811yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—82no—
313C4—84no—
314C4—82no—
315C4—82no—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—42no—
320C2—82no—
321C2C53×#186711yescharacteristic
322C2C53×#220012yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutatorC2×D4C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC2×#127×C53C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C22×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1867C2semidirect
5. C53×#1390C2semidirect
17. C53×#353C4semidirect
42. #99×C53C2×C4semidirect
59. C2×D4×C53C4⋊C4semidirect
89. C42×C53C4⋊C4semidirect
131. #12966C53direct (N × H)
158. #1867C2×C53semidirect
159. #1390C2×C53semidirect
178. #353C4×C53semidirect
181. C53#12966direct (N × H)
204. #99C2×C4×C53semidirect
221. C2×D4C4⋊C4×C53semidirect
251. C42C4⋊C4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#23802
C2×C53×#3532
C53×#18872
C53×#18672
C53×#13902
#1296653