Finite Group Databasebeta

← C53×#33638 #27241150

Subgroups — C53×#33638 #27241150

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3163C211yesmaximal, characteristic, complemented
2C53×#2957C211yesmaximal, characteristic
3C53×#2231C214yesmaximal, complemented
4C53×#2230C214yesmaximal, complemented
5C53×#2155C211yesmaximal, characteristic, complemented
6C2×C53×#365C211yesmaximal, characteristic, complemented
7C53×#2153C211yesmaximal, characteristic, complemented
8C53×#2145C211yesmaximal, characteristic
9C2×C53×#360C211yesmaximal, characteristic
10C2×#135×C53C2211yescharacteristic
11C53×#568—21no—
12C53×#544—21no—
13C2×M32×C53C2211yescharacteristic
14C2×M32×C53C2211yescharacteristic
15C53×#501—22no—
16C53×#499C2212yescomplemented
17C53×#498C2212yescomplemented
18C53×#444—22no—
19C53×#443C2214yes—
20C2×#108×C53C2211yescharacteristic
21C53×#428—21no—
22C53×#427C2211yescharacteristic
23C2×#106×C53C2211yescharacteristic
24C53×#421—21no—
25C53×#405C2211yescharacteristic
26C53×#365C2212yescomplemented
27C53×#365C2214yescomplemented
28C53×#365C2212yescomplemented
29C53×#364C2214yescomplemented
30C53×#360C2212yes—
31C53×#360C2214yes—
32C53×#360C2212yes—
33C2×#41×C53—21no—
34#135×C53—21no—
35#135×C53—21no—
36#135×C53Q812yescomplemented
37#135×C53C2312yes—
38C2×Q16×C53Q811yescharacteristic, complemented
39C2×SD16×C53—21no—
40C2×D8×C53Q811yescharacteristic, complemented
41#131×C53—24no—
42#131×C53—42no—
43C2×M16×C53—21no—
44C2×M16×C53—42no—
45C2×M16×C53D411yescharacteristic
46C22×C8×C53C2311yescharacteristic
47C4⋊Q8×C53—22no—
48C4⋊Q8×C53—41no—
49#125×C53—22no—
50#125×C53—41no—
51C22⋊Q8×C53—42no—
52#117×C53—21no—
53#117×C53D411yescharacteristic
54C2×C4⋊C4×C53—21no—
55M32×C53—21no—
56M32×C53—21no—
57M32×C53C2312yes—
58M32×C53C2312yes—
59C2×C16×C53—21no—
60C2×C16×C53—21no—
61#108×C53—21no—
62#108×C53C2312yes—
63#107×C53—22no—
64#107×C53—21no—
65#106×C53—22no—
66#106×C53—21no—
67#106×C53C2311yescharacteristic
68#106×C53C2312yes—
69#106×C53C2311yescharacteristic
70#104×C53—24no—
71#97×C53D412yes—
72C4×C8×C53C2312yes—
73#41×C53—22no—
74#41×C53—22no—
75#41×C53—41no—
76#41×C53—41no—
77C2×Q8×C53—21no—
78C2×Q8×C53—42no—
79C2×D4×C53—21no—
80C2×D4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84Q16×C53—21no—
85Q16×C53C2×Q812yes—
86SD16×C53—22no—
87SD16×C53—22no—
88D8×C53—21no—
89D8×C53C2×Q812yes—
90M16×C53—22no—
91M16×C53—41no—
92M16×C53—44no—
93M16×C53SD1612yes—
94M16×C53SD1612yes—
95C2×C8×C53—41no—
96C2×C8×C53—44no—
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D412yes—
99C2×C8×C53C2×Q811yescharacteristic
100C2×C8×C53C2×Q811yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C2×C8×C53#4111yescharacteristic
103C2×C8×C53C2411yescharacteristic
104C4⋊C4×C53—21no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—42no—
109C4⋊C4×C53—42no—
110C4⋊C4×C53—42no—
111C4⋊C4×C53—41no—
112C22⋊C4×C53—21no—
113C22⋊C4×C53—41no—
114C42×C53—41no—
115C42×C53C2×D412yes—
116C16×C53—22no—
117C16×C53—22no—
118C23×C53—41no—
119C23×C53AGL(1,ℤ/8)11yescharacteristic
120Q8×C53—22no—
121Q8×C53—41no—
122Q8×C53—162no—
123D4×C53—22no—
124D4×C53—42no—
125D4×C53—42no—
126D4×C53—41no—
127C2×C4×C53—22no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—161no—
135C2×C4×C53C22⋊Q811yescharacteristic
136C2×C4×C53C22⋊Q811yescharacteristic
137C2×C4×C53C2×SD1611yescharacteristic
138C2×C4×C53C2×SD1611yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53C22×D411yescharacteristic
141C8×C53—42no—
142C8×C53C22⋊Q811yescharacteristic
143C8×C53C22⋊Q811yescharacteristic
144C8×C53C2×SD1614yes—
145C8×C53C22×Q811yescharacteristic
146C8×C53C2×#4111yescharacteristic
147#33638C5311yesmaximal, characteristic, direct product factor, Sylow
148C22×C53—41no—
149C22×C53—42no—
150C22×C53—161no—
151C22×C53#47611yescharacteristic
152C22×C53#47611yescharacteristic
153C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
154C4×C53—42no—
155C4×C53—42no—
156C4×C53—81no—
157C4×C53—82no—
158C4×C53—81no—
159C4×C53#47611yescharacteristic
160C4×C53#47611yescharacteristic
161C4×C53C2×C22⋊Q811yescharacteristic
162C4×C53C22×SD1611yescharacteristic
163#3163C2×C5311yescharacteristic, complemented
164#2957C2×C5311yescharacteristic
165#2231C2×C5314yescomplemented
166#2230C2×C5314yescomplemented
167#2155C2×C5311yescharacteristic, complemented
168C2×#365C2×C5311yescharacteristic, complemented
169#2153C2×C5311yescharacteristic, complemented
170#2145C2×C5311yescharacteristic
171C2×#360C2×C5311yescharacteristic
172C2×C53—21no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—82no—
176C2×C53C2×#47611yescharacteristic
177C2×#135C22×C5311yescharacteristic
178#568—21no—
179#544—21no—
180C2×M32C22×C5311yescharacteristic
181C2×M32C22×C5311yescharacteristic
182#501—22no—
183#499C22×C5312yescomplemented
184#498C22×C5312yescomplemented
185#444—22no—
186#443C22×C5314yes—
187C2×#108C22×C5311yescharacteristic
188#428—21no—
189#427C22×C5311yescharacteristic
190C2×#106C22×C5311yescharacteristic
191#421—21no—
192#405C22×C5311yescharacteristic
193#365C22×C5314yescomplemented
194#365C22×C5312yescomplemented
195#365C22×C5312yescomplemented
196#364C22×C5314yescomplemented
197#360C22×C5314yes—
198#360C22×C5312yes—
199#360C22×C5312yes—
200C53#3363811yescharacteristic, direct product factor, Sylow
201C2×#41—21no—
202#135—21no—
203#135—21no—
204#135Q8×C5312yescomplemented
205#135C23×C5312yes—
206C2×Q16Q8×C5311yescharacteristic, complemented
207C2×SD16—21no—
208C2×D8Q8×C5311yescharacteristic, complemented
209#131—24no—
210#131—42no—
211C2×M16—21no—
212C2×M16—42no—
213C2×M16D4×C5311yescharacteristic
214C22×C8C23×C5311yescharacteristic
215C4⋊Q8—22no—
216C4⋊Q8—41no—
217#125—22no—
218#125—41no—
219C22⋊Q8—42no—
220#117—21no—
221#117D4×C5311yescharacteristic
222C2×C4⋊C4—21no—
223M32—21no—
224M32—21no—
225M32C23×C5312yes—
226M32C23×C5312yes—
227C2×C16—21no—
228C2×C16—21no—
229#108—21no—
230#108C23×C5312yes—
231#107—21no—
232#107—22no—
233#106—21no—
234#106—22no—
235#106C23×C5312yes—
236#106C23×C5311yescharacteristic
237#106C23×C5311yescharacteristic
238#104—24no—
239#97D4×C5312yes—
240C4×C8C23×C5312yes—
241#41—22no—
242#41—22no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—42no—
247C2×D4—21no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C2×D4×C5311yescharacteristic
252Q16—21no—
253Q16C2×Q8×C5312yes—
254SD16—22no—
255SD16—22no—
256D8—21no—
257D8C2×Q8×C5312yes—
258M16—22no—
259M16—41no—
260M16—44no—
261M16SD16×C5312yes—
262M16SD16×C5312yes—
263C2×C8—41no—
264C2×C8—44no—
265C2×C8C2×D4×C5311yescharacteristic
266C2×C8C2×D4×C5312yes—
267C2×C8C2×Q8×C5311yescharacteristic
268C2×C8C2×Q8×C5311yescharacteristic
269C2×C8#41×C5311yescharacteristic
270C2×C8#41×C5311yescharacteristic
271C2×C8C24×C5311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C22⋊C4—21no—
281C22⋊C4—41no—
282C42—41no—
283C42C2×D4×C5312yes—
284C16—22no—
285C16—22no—
286C23—41no—
287C23AGL(1,ℤ/8)×C5311yescharacteristic
288Q8—22no—
289Q8—41no—
290Q8—162no—
291D4—22no—
292D4—42no—
293D4—42no—
294D4—41no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—161no—
303C2×C4C22⋊Q8×C5311yescharacteristic
304C2×C4C22⋊Q8×C5311yescharacteristic
305C2×C4C2×SD16×C5311yescharacteristic
306C2×C4C2×SD16×C5311yescharacteristic
307C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
308C2×C4C22×D4×C5311yescharacteristic
309C8—42no—
310C8C22⋊Q8×C5311yescharacteristic
311C8C22⋊Q8×C5311yescharacteristic
312C8C2×SD16×C5314yes—
313C8C22×Q8×C5311yescharacteristic
314C8C2×#41×C5311yescharacteristic
315C22—41no—
316C22—42no—
317C22—161no—
318C22C53×#47611yescharacteristic
319C22C53×#47611yescharacteristic
320C22C2×AGL(1,ℤ/8)×C5311yescharacteristic
321C4—42no—
322C4—42no—
323C4—82no—
324C4—81no—
325C4—81no—
326C4C53×#47611yescharacteristic
327C4C53×#47611yescharacteristic
328C4C2×C22⋊Q8×C5311yescharacteristic
329C4C22×SD16×C5311yescharacteristic
330C2—21no—
331C2—21no—
332C2—21no—
333C2—82no—
334C2C2×C53×#47611yescharacteristic
subgroupG/H
center, socleC2×C53C2×#476
commutator, Frattini, cosocleC2×C8C24×C53
prime radicalC2×#135×C53C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3163C2semidirect
3. C53×#2231C2semidirect
4. C53×#2230C2semidirect
5. C53×#2155C2semidirect
6. C2×C53×#365C2semidirect
7. C53×#2153C2semidirect
16. C53×#499C22semidirect
17. C53×#498C22semidirect
26. C53×#365C22semidirect
27. C53×#365C22semidirect
28. C53×#365C22semidirect
29. C53×#364C22semidirect
36. #135×C53Q8semidirect
38. C2×Q16×C53Q8semidirect
40. C2×D8×C53Q8semidirect
147. #33638C53direct (N × H)
163. #3163C2×C53semidirect
165. #2231C2×C53semidirect
166. #2230C2×C53semidirect
167. #2155C2×C53semidirect
168. C2×#365C2×C53semidirect
169. #2153C2×C53semidirect
183. #499C22×C53semidirect
184. #498C22×C53semidirect
193. #365C22×C53semidirect
194. #365C22×C53semidirect
195. #365C22×C53semidirect
196. #364C22×C53semidirect
200. C53#33638direct (N × H)
204. #135Q8×C53semidirect
206. C2×Q16Q8×C53semidirect
208. C2×D8Q8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#31632
C53×#29572
C53×#22312
C53×#22302
C53×#21552
C2×C53×#3652
C53×#21532
C53×#21452
C2×C53×#3602
#3363853