Finite Group Databasebeta

← C53×#9274 #27216786

Subgroups — C53×#9274 #27216786

Order 13568.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3208C211yesmaximal, characteristic
2C53×#2962C211yesmaximal, characteristic, complemented
3C53×#1965C211yesmaximal, characteristic
4C53×#1550C211yesmaximal, characteristic, complemented
5C2×C53×#331C211yesmaximal, characteristic
6C53×#1301C211yesmaximal, characteristic, complemented
7C53×#1283C211yesmaximal, characteristic, complemented
8C53×#533C2211yescharacteristic
9C2×#127×C53C2211yescharacteristic
10C53×#492—21no—
11C53×#486—21no—
12C53×#460—22no—
13C53×#450C414yes—
14C53×#446—22no—
15C53×#446—21no—
16C8×Q8×C53—21no—
17C53×#431C2211yescharacteristic
18C53×#424—21no—
19C53×#423C2211yescharacteristic
20C2×C4⋊C8×C53C2211yescharacteristic
21C2×C4⋊C8×C53C2211yescharacteristic
22C2×#100×C53—21no—
23C2×#100×C53—21no—
24C53×#405—21no—
25C4×M16×C53C2211yescharacteristic
26C53×#333—21no—
27C53×#332—21no—
28C53×#331—21no—
29C53×#331—21no—
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53C2×C412yes—
33C2×SD16×C53—24no—
34C2×M16×C53—21no—
35C2×M16×C53D411yescharacteristic
36C2×M16×C53D411yescharacteristic
37C22×C8×C53—21no—
38C22×C8×C53—21no—
39C4⋊Q8×C53D411yescharacteristic
40C4⋊Q8×C53D411yescharacteristic
41#127×C53—21no—
42#127×C53—21no—
43#127×C53—21no—
44#127×C53—21no—
45#125×C53—21no—
46C22⋊Q8×C53—21no—
47C22⋊Q8×C53C2×C412yes—
48C4×Q8×C53—21no—
49C4×Q8×C53D412yes—
50#117×C53—21no—
51C2×C42×C53C2311yescharacteristic
52C4⋊C8×C53—22no—
53C4⋊C8×C53—22no—
54C4⋊C8×C53—21no—
55C4⋊C8×C53—21no—
56C4⋊C8×C53—21no—
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53D411yescharacteristic
61C4⋊C8×C53D411yescharacteristic
62#102×C53—24no—
63#100×C53—42no—
64#100×C53—42no—
65C22⋊C8×C53C2×C412yes—
66#97×C53—22no—
67#97×C53—21no—
68C4×C8×C53—22no—
69C4×C8×C53—21no—
70C4×C8×C53—21no—
71C24×C53—22no—
72C2×Q8×C53—21no—
73C2×Q8×C53C22⋊C412yes—
74C2×D4×C53—22no—
75C2×D4×C53—22no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—81no—
85C2×D4×C53—81no—
86C2×D4×C53—82no—
87C22×C4×C53C22×C411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90SD16×C53—84no—
91M16×C53—21no—
92M16×C53—21no—
93M16×C53—21no—
94M16×C53—41no—
95M16×C53—41no—
96M16×C53SD1612yescomplemented
97C2×C8×C53—21no—
98C2×C8×C53—22no—
99C2×C8×C53—21no—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—81no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53C22⋊C412yes—
113C22⋊C4×C53—22no—
114C42×C53—22no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—82no—
123C23×C53—82no—
124C23×C53C4×D411yescharacteristic
125Q8×C53—42no—
126Q8×C53—41no—
127Q8×C53—41no—
128D4×C53—44no—
129D4×C53—82no—
130D4×C53—82no—
131D4×C53—81no—
132D4×C53—81no—
133D4×C53—81no—
134D4×C53—81no—
135D4×C53—81no—
136D4×C53—81no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—41no—
142C2×C4×C53C2×C22⋊C411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12311yescharacteristic
148C2×C4×C53C2×SD1611yescharacteristic
149C2×C4×C53#13511yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C8×C53—22no—
153C8×C53—42no—
154C8×C53—41no—
155C8×C53—41no—
156C8×C53—41no—
157C8×C53—81no—
158C8×C53—81no—
159#9274C5311yesmaximal, characteristic, direct product factor, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—82no—
163C22×C53—82no—
164C22×C53—82no—
165C22×C53—162no—
166C22×C53—162no—
167C22×C53#38611yescharacteristic
168C22×C53C4×SD1611yescharacteristic
169C22×C53#44011yescharacteristic
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—42no—
174C4×C53—42no—
175C4×C53#44811yescharacteristic
176C4×C53#44911yescharacteristic
177C4×C53#46011yescharacteristic
178C4×C53#46411yescharacteristic
179C4×C53#47111yescharacteristic
180C4×C53#47211yescharacteristic
181#3208C2×C5311yescharacteristic
182#2962C2×C5311yescharacteristic, complemented
183#1965C2×C5311yescharacteristic
184#1550C2×C5311yescharacteristic, complemented
185C2×#331C2×C5311yescharacteristic
186#1301C2×C5311yescharacteristic, complemented
187#1283C2×C5311yescharacteristic, complemented
188C2×C53—21no—
189C2×C53—21no—
190C2×C53—162no—
191C2×C53#188011yescharacteristic
192C2×C53#190711yescharacteristic
193C2×C53#191411yescharacteristic
194#533C22×C5311yescharacteristic
195C2×#127C22×C5311yescharacteristic
196#492—21no—
197#486—21no—
198#460—22no—
199#450C4×C5314yes—
200#446—22no—
201#446—21no—
202C8×Q8—21no—
203#431C22×C5311yescharacteristic
204#424—21no—
205#423C22×C5311yescharacteristic
206C2×C4⋊C8C22×C5311yescharacteristic
207C2×C4⋊C8C22×C5311yescharacteristic
208C2×#100—21no—
209C2×#100—21no—
210#405—21no—
211C4×M16C22×C5311yescharacteristic
212#333—21no—
213#332—21no—
214#331—21no—
215#331—21no—
216C53#927411yescharacteristic, direct product factor, Sylow
217C22×D4—21no—
218C22×D4—21no—
219C22×D4C2×C4×C5312yes—
220C2×SD16—24no—
221C2×M16—21no—
222C2×M16D4×C5311yescharacteristic
223C2×M16D4×C5311yescharacteristic
224C22×C8—21no—
225C22×C8—21no—
226C4⋊Q8D4×C5311yescharacteristic
227C4⋊Q8D4×C5311yescharacteristic
228#127—21no—
229#127—21no—
230#127—21no—
231#127—21no—
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8C2×C4×C5312yes—
235C4×Q8—21no—
236C4×Q8D4×C5312yes—
237#117—21no—
238C2×C42C23×C5311yescharacteristic
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8D4×C5311yescharacteristic
248C4⋊C8D4×C5311yescharacteristic
249#102—24no—
250#100—42no—
251#100—42no—
252C22⋊C8C2×C4×C5312yes—
253#97—22no—
254#97—21no—
255C4×C8—22no—
256C4×C8—21no—
257C4×C8—21no—
258C24—22no—
259C2×Q8—21no—
260C2×Q8C22⋊C4×C5312yes—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—81no—
272C2×D4—81no—
273C2×D4—82no—
274C22×C4C22×C4×C5311yescharacteristic
275C22×C4C2×D4×C5311yescharacteristic
276C22×C4#41×C5311yescharacteristic
277SD16—84no—
278M16—21no—
279M16—21no—
280M16—21no—
281M16—41no—
282M16—41no—
283M16SD16×C5312yescomplemented
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—81no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4C22⋊C4×C5312yes—
300C22⋊C4—22no—
301C42—22no—
302C42C2×D4×C5311yescharacteristic
303C42C2×D4×C5311yescharacteristic
304C42C2×D4×C5311yescharacteristic
305C42#41×C5311yescharacteristic
306C23—42no—
307C23—42no—
308C23—42no—
309C23—82no—
310C23—82no—
311C23C4×D4×C5311yescharacteristic
312Q8—42no—
313Q8—41no—
314Q8—41no—
315D4—44no—
316D4—82no—
317D4—82no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4C2×C22⋊C4×C5311yescharacteristic
330C2×C4C4×D4×C5311yescharacteristic
331C2×C4#120×C5311yescharacteristic
332C2×C4#121×C5311yescharacteristic
333C2×C4#121×C5311yescharacteristic
334C2×C4#123×C5311yescharacteristic
335C2×C4C2×SD16×C5311yescharacteristic
336C2×C4#135×C5311yescharacteristic
337C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
338C2×C4#137×C5311yescharacteristic
339C8—22no—
340C8—42no—
341C8—41no—
342C8—41no—
343C8—41no—
344C8—81no—
345C8—81no—
346C22—21no—
347C22—21no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—162no—
352C22—162no—
353C22C53×#38611yescharacteristic
354C22C4×SD16×C5311yescharacteristic
355C22C53×#44011yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4C53×#44811yescharacteristic
362C4C53×#44911yescharacteristic
363C4C53×#46011yescharacteristic
364C4C53×#46411yescharacteristic
365C4C53×#47111yescharacteristic
366C4C53×#47211yescharacteristic
367C2—21no—
368C2—21no—
369C2—162no—
370C2C53×#188011yescharacteristic
371C2C53×#190711yescharacteristic
372C2C53×#191411yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC2×#127×C53C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2962C2semidirect
4. C53×#1550C2semidirect
6. C53×#1301C2semidirect
7. C53×#1283C2semidirect
96. M16×C53SD16semidirect
159. #9274C53direct (N × H)
182. #2962C2×C53semidirect
184. #1550C2×C53semidirect
186. #1301C2×C53semidirect
187. #1283C2×C53semidirect
216. C53#9274direct (N × H)
283. M16SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#32082
C53×#29622
C53×#19652
C53×#15502
C2×C53×#3312
C53×#13012
C53×#12832
#927453