Finite Group Databasebeta

← C53×#12921 #27220433

Subgroups — C53×#12921 #27220433

Order 13568.

442 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#427C211yesmaximal, characteristic
2C2×C53×#367C211yesmaximal, characteristic, complemented
3C2×C53×#366C211yesmaximal, characteristic, complemented
4C2×C53×C22⋊C16C211yesmaximal, characteristic
5C53×#1808C211yesmaximal, characteristic
6C53×#1380C211yesmaximal, characteristic, complemented
7C53×#1380C211yesmaximal, characteristic, complemented
8C23×C8×C53C2211yescharacteristic
9C2×#117×C53—21no—
10C22×C16×C53C2211yescharacteristic
11C22×C16×C53C2211yescharacteristic
12C53×#427—22no—
13C53×#427—22no—
14C2×#107×C53C2211yescharacteristic
15C2×#107×C53C2211yescharacteristic
16C2×#107×C53C2211yescharacteristic
17C2×#107×C53C2211yescharacteristic
18C2×#106×C53—21no—
19C53×#380—21no—
20C53×#367—22no—
21C53×#366—22no—
22C53×C22⋊C16C414yescomplemented
23C53×#340—21no—
24C23×C4×C53D411yescharacteristic
25C22×C8×C53—21no—
26C22×C8×C53—21no—
27C22×C8×C53—21no—
28C22×C8×C53—21no—
29C22×C8×C53C2×C412yes—
30C22×C8×C53D411yescharacteristic
31C22×C8×C53Q812yes—
32C22×C8×C53C2311yescharacteristic
33#117×C53—24no—
34#117×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C22⋊C4×C53—41no—
40C2×C22⋊C4×C53—41no—
41C2×C42×C53—41no—
42C2×C16×C53—21no—
43C2×C16×C53—21no—
44C2×C16×C53—21no—
45C2×C16×C53—21no—
46C2×C16×C53C2×C412yes—
47C2×C16×C53C2×C412yes—
48#107×C53—21no—
49#107×C53—21no—
50#107×C53—21no—
51#107×C53—21no—
52#107×C53—21no—
53#107×C53—21no—
54#107×C53D412yes—
55#107×C53D412yes—
56#106×C53—24no—
57#95×C53—41no—
58C24×C53D811yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53C4⋊C412yes—
68C22×C4×C53D811yescharacteristic
69C22×C4×C53Q1612yes—
70C22×C4×C53C2×D411yescharacteristic
71C2×C8×C53—21no—
72C2×C8×C53—21no—
73C2×C8×C53—22no—
74C2×C8×C53—22no—
75C2×C8×C53—22no—
76C2×C8×C53—22no—
77C2×C8×C53C4⋊C412yes—
78C2×C8×C53C22×C411yescharacteristic
79C2×C8×C53C2×D411yescharacteristic
80C2×C8×C53C2×D411yescharacteristic
81C2×C8×C53C2×Q811yescharacteristic
82C2×C8×C53#4111yescharacteristic
83C2×C8×C53#4111yescharacteristic
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C22⋊C4×C53—44no—
93C22⋊C4×C53—82no—
94C42×C53—44no—
95C16×C53—22no—
96C16×C53—22no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53#10712yes—
104C23×C53C2×D811yescharacteristic
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—81no—
114C2×C4×C53—81no—
115C2×C4×C53—81no—
116C2×C4×C53—81no—
117C2×C4×C53—81no—
118C2×C4×C53—81no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53—81no—
122C2×C4×C53—81no—
123C2×C4×C53#10712yes—
124C2×C4×C53C2×C4⋊C411yescharacteristic
125C2×C4×C53#12011yescharacteristic
126C2×C4×C53#12311yescharacteristic
127C2×C4×C53C2×Q1611yescharacteristic
128C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C53#13711yescharacteristic
130C8×C53—22no—
131C8×C53C4×D411yescharacteristic
132C8×C53C4×D411yescharacteristic
133C8×C53C22⋊Q811yescharacteristic
134C8×C53C22⋊Q811yescharacteristic
135#12921C5311yesmaximal, characteristic, direct product factor, Sylow
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53C2×#10711yescharacteristic
144C22×C53#44711yescharacteristic
145C22×C53#48011yescharacteristic
146C22×C53#50811yescharacteristic
147C22×C53#50811yescharacteristic
148C22×C53#50911yescharacteristic
149C22×C53#51011yescharacteristic
150C4×C53—22no—
151C4×C53—82no—
152C4×C53—82no—
153C4×C53—161no—
154C4×C53—161no—
155C4×C53#38511yescharacteristic
156C4×C53#42811yescharacteristic
157C4×C53#45111yescharacteristic
158C4×C53#48411yescharacteristic
159C2×#427C2×C5311yescharacteristic
160C2×#367C2×C5311yescharacteristic, complemented
161C2×#366C2×C5311yescharacteristic, complemented
162C2×C22⋊C16C2×C5311yescharacteristic
163#1808C2×C5311yescharacteristic
164#1380C2×C5311yescharacteristic, complemented
165#1380C2×C5311yescharacteristic, complemented
166C2×C53—42no—
167C2×C53#185211yescharacteristic
168C2×C53#215811yescharacteristic
169C2×C53#215911yescharacteristic
170C2×C53#218611yescharacteristic
171C2×C53#218911yescharacteristic
172C2×C53#223411yescharacteristic
173C2×C53#223711yescharacteristic
174C23×C8C22×C5311yescharacteristic
175C2×#117—21no—
176C22×C16C22×C5311yescharacteristic
177C22×C16C22×C5311yescharacteristic
178#427—22no—
179#427—22no—
180C2×#107C22×C5311yescharacteristic
181C2×#107C22×C5311yescharacteristic
182C2×#107C22×C5311yescharacteristic
183C2×#107C22×C5311yescharacteristic
184C2×#106—21no—
185#380—21no—
186#367—22no—
187#366—22no—
188C22⋊C16C4×C5314yescomplemented
189#340—21no—
190C53#1292111yescharacteristic, direct product factor, Sylow
191C23×C4D4×C5311yescharacteristic
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C2×C4×C5312yes—
197C22×C8D4×C5311yescharacteristic
198C22×C8Q8×C5312yes—
199C22×C8C23×C5311yescharacteristic
200#117—24no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—41no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16C2×C4×C5312yes—
214C2×C16C2×C4×C5312yes—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107D4×C5312yes—
222#107D4×C5312yes—
223#106—24no—
224#95—41no—
225C24D8×C5311yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4C4⋊C4×C5312yes—
235C22×C4D8×C5311yescharacteristic
236C22×C4Q16×C5312yes—
237C22×C4C2×D4×C5311yescharacteristic
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8C4⋊C4×C5312yes—
245C2×C8C22×C4×C5311yescharacteristic
246C2×C8C2×D4×C5311yescharacteristic
247C2×C8C2×D4×C5311yescharacteristic
248C2×C8C2×Q8×C5311yescharacteristic
249C2×C8#41×C5311yescharacteristic
250C2×C8#41×C5311yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C22⋊C4—44no—
260C22⋊C4—82no—
261C42—44no—
262C16—22no—
263C16—22no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23#107×C5312yes—
271C23C2×D8×C5311yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4#107×C5312yes—
291C2×C4C2×C4⋊C4×C5311yescharacteristic
292C2×C4#120×C5311yescharacteristic
293C2×C4#123×C5311yescharacteristic
294C2×C4C2×Q16×C5311yescharacteristic
295C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
296C2×C4#137×C5311yescharacteristic
297C8—22no—
298C8C4×D4×C5311yescharacteristic
299C8C4×D4×C5311yescharacteristic
300C8C22⋊Q8×C5311yescharacteristic
301C8C22⋊Q8×C5311yescharacteristic
302C22—22no—
303C22—22no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×#107×C5311yescharacteristic
310C22C53×#44711yescharacteristic
311C22C53×#48011yescharacteristic
312C22C53×#50811yescharacteristic
313C22C53×#50811yescharacteristic
314C22C53×#50911yescharacteristic
315C22C53×#51011yescharacteristic
316C4—22no—
317C4—82no—
318C4—82no—
319C4—161no—
320C4—161no—
321C4C53×#38511yescharacteristic
322C4C53×#42811yescharacteristic
323C4C53×#45111yescharacteristic
324C4C53×#48411yescharacteristic
325C2—42no—
326C2C53×#185211yescharacteristic
327C2C53×#215811yescharacteristic
328C2C53×#215911yescharacteristic
329C2C53×#218611yescharacteristic
330C2C53×#218911yescharacteristic
331C2C53×#223411yescharacteristic
332C2C53×#223711yescharacteristic
subgroupG/H
center, socleC23×C53C2×D8
commutatorC2×C8C22×C4×C53
Frattini, cosocleC22×C8C23×C53
prime radicalC24×C53D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C53 ◁ C23×C4×C53 ◁ C23×C8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C53×#367C2semidirect
3. C2×C53×#366C2semidirect
6. C53×#1380C2semidirect
7. C53×#1380C2semidirect
22. C53×C22⋊C16C4semidirect
135. #12921C53direct (N × H)
160. C2×#367C2×C53semidirect
161. C2×#366C2×C53semidirect
164. #1380C2×C53semidirect
165. #1380C2×C53semidirect
188. C22⋊C16C4×C53semidirect
190. C53#12921direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#4272
C2×C53×#3672
C2×C53×#3662
C2×C53×C22⋊C162
C53×#18082
C53×#13802
#1292153