Finite Group Databasebeta

← C53×#9353 #27216865

Subgroups — C53×#9353 #27216865

Order 13568.

472 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3039C211yesmaximal, characteristic
2C53×#2990C211yesmaximal, characteristic
3C53×#1997C211yesmaximal, characteristic, complemented
4C53×#1548C211yesmaximal, characteristic, complemented
5C53×#1489C211yesmaximal, characteristic
6C53×#1309C211yesmaximal, characteristic, complemented
7C53×#1299C211yesmaximal, characteristic, complemented
8C2×#137×C53C412yes—
9C2×C4⋊Q8×C53C2211yescharacteristic
10C4×#41×C53C2211yescharacteristic
11C53×#462—21no—
12C53×#462—21no—
13C53×#461—21no—
14C53×#461—21no—
15C53×#446—21no—
16C53×#435—22no—
17C8×D4×C53—21no—
18C53×#432—21no—
19C53×#431C2211yescharacteristic
20C53×#423C2211yescharacteristic
21C53×#423C2211yescharacteristic
22C53×#423C2211yescharacteristic
23C2×C4⋊C8×C53—21no—
24C53×#419C412yes—
25C4×M16×C53C2211yescharacteristic
26C53×#343—22no—
27C53×#333—21no—
28C53×#333—21no—
29C53×Q8⋊C8—21no—
30C53×D4⋊C8—21no—
31C2×#41×C53C2×C411yescharacteristic
32C22×Q8×C53C2×C412yes—
33#137×C53—44no—
34C2×Q16×C53—22no—
35C2×SD16×C53—22no—
36C2×M16×C53—21no—
37C2×M16×C53—21no—
38C2×M16×C53C2×C412yes—
39C2×M16×C53D412yes—
40C22×C8×C53—22no—
41C4⋊Q8×C53—21no—
42C4⋊Q8×C53—21no—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53—21no—
45C4×Q8×C53D411yescharacteristic
46C4×Q8×C53D411yescharacteristic
47C4×D4×C53—21no—
48C4×D4×C53—21no—
49C4×D4×C53D411yescharacteristic
50C4×D4×C53D411yescharacteristic
51#117×C53—21no—
52#117×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C42×C53—21no—
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—22no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53D411yescharacteristic
65C4⋊C8×C53D411yescharacteristic
66#103×C53—22no—
67#103×C53—22no—
68C22⋊C8×C53—22no—
69C22⋊C8×C53—21no—
70#97×C53—22no—
71#97×C53—21no—
72C4×C8×C53—21no—
73C4×C8×C53—21no—
74#41×C53—22no—
75#41×C53—41no—
76C2×Q8×C53—22no—
77C2×Q8×C53—22no—
78C2×Q8×C53—42no—
79C2×Q8×C53—42no—
80C2×Q8×C53C22⋊C411yescharacteristic
81C2×D4×C53—21no—
82C2×D4×C53C22⋊C411yescharacteristic
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53C22×C411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90Q16×C53—44no—
91SD16×C53—44no—
92M16×C53—22no—
93M16×C53—41no—
94M16×C53—42no—
95M16×C53—42no—
96M16×C53—81no—
97C2×C8×C53—22no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—22no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53—82no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53C22⋊C411yescharacteristic
111C4⋊C4×C53C22⋊C411yescharacteristic
112C22⋊C4×C53—21no—
113C22⋊C4×C53—21no—
114C22⋊C4×C53—21no—
115C42×C53—21no—
116C42×C53—21no—
117C42×C53—21no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122C23×C53—21no—
123C23×C53C4×D411yescharacteristic
124Q8×C53—21no—
125Q8×C53—21no—
126Q8×C53—42no—
127Q8×C53—42no—
128Q8×C53—82no—
129D4×C53—21no—
130D4×C53—42no—
131D4×C53#10012yes—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—82no—
148C2×C4×C53C2×C22⋊C411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12311yescharacteristic
154C8×C53—42no—
155C8×C53—42no—
156C8×C53—41no—
157C8×C53—42no—
158C8×C53—41no—
159C8×C53—81no—
160#9353C5311yesmaximal, characteristic, direct product factor, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—22no—
164C22×C53—21no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53#38611yescharacteristic
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—82no—
175C4×C53C2×#10011yescharacteristic
176C4×C53#41311yescharacteristic
177C4×C53#45511yescharacteristic
178C4×C53#45611yescharacteristic
179C4×C53#47111yescharacteristic
180C4×C53#47311yescharacteristic
181#3039C2×C5311yescharacteristic
182#2990C2×C5311yescharacteristic
183#1997C2×C5311yescharacteristic, complemented
184#1548C2×C5311yescharacteristic, complemented
185#1489C2×C5311yescharacteristic
186#1309C2×C5311yescharacteristic, complemented
187#1299C2×C5311yescharacteristic, complemented
188C2×C53—41no—
189C2×C53—42no—
190C2×C53#188211yescharacteristic
191C2×C53#190011yescharacteristic
192C2×C53#190811yescharacteristic
193C2×#137C4×C5312yes—
194C2×C4⋊Q8C22×C5311yescharacteristic
195C4×#41C22×C5311yescharacteristic
196#462—21no—
197#462—21no—
198#461—21no—
199#461—21no—
200#446—21no—
201#435—22no—
202C8×D4—21no—
203#432—21no—
204#431C22×C5311yescharacteristic
205#423C22×C5311yescharacteristic
206#423C22×C5311yescharacteristic
207#423C22×C5311yescharacteristic
208C2×C4⋊C8—21no—
209#419C4×C5312yes—
210C4×M16C22×C5311yescharacteristic
211#343—22no—
212#333—21no—
213#333—21no—
214Q8⋊C8—21no—
215D4⋊C8—21no—
216C53#935311yescharacteristic, direct product factor, Sylow
217C2×#41C2×C4×C5311yescharacteristic
218C22×Q8C2×C4×C5312yes—
219#137—44no—
220C2×Q16—22no—
221C2×SD16—22no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16C2×C4×C5312yes—
225C2×M16D4×C5312yes—
226C22×C8—22no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4×Q8D4×C5311yescharacteristic
232C4×Q8D4×C5311yescharacteristic
233C4×D4—21no—
234C4×D4—21no—
235C4×D4D4×C5311yescharacteristic
236C4×D4D4×C5311yescharacteristic
237#117—21no—
238#117C2×C4×C5311yescharacteristic
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C42—21no—
242C2×C42C23×C5311yescharacteristic
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8D4×C5311yescharacteristic
251C4⋊C8D4×C5311yescharacteristic
252#103—22no—
253#103—22no—
254C22⋊C8—22no—
255C22⋊C8—21no—
256#97—22no—
257#97—21no—
258C4×C8—21no—
259C4×C8—21no—
260#41—22no—
261#41—41no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8C22⋊C4×C5311yescharacteristic
267C2×D4—21no—
268C2×D4C22⋊C4×C5311yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4C22×C4×C5311yescharacteristic
274C22×C4C2×D4×C5311yescharacteristic
275C22×C4#41×C5311yescharacteristic
276Q16—44no—
277SD16—44no—
278M16—22no—
279M16—41no—
280M16—42no—
281M16—42no—
282M16—81no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—82no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4C22⋊C4×C5311yescharacteristic
297C4⋊C4C22⋊C4×C5311yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42C2×D4×C5311yescharacteristic
305C42C2×D4×C5311yescharacteristic
306C42C2×D4×C5311yescharacteristic
307C42#41×C5311yescharacteristic
308C23—21no—
309C23C4×D4×C5311yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—42no—
313Q8—42no—
314Q8—82no—
315D4—21no—
316D4—42no—
317D4#100×C5312yes—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4C2×C22⋊C4×C5311yescharacteristic
335C2×C4C4×D4×C5311yescharacteristic
336C2×C4#120×C5311yescharacteristic
337C2×C4#121×C5311yescharacteristic
338C2×C4#121×C5311yescharacteristic
339C2×C4#123×C5311yescharacteristic
340C8—42no—
341C8—42no—
342C8—41no—
343C8—42no—
344C8—41no—
345C8—81no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22C53×#38611yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—82no—
360C4C2×#100×C5311yescharacteristic
361C4C53×#41311yescharacteristic
362C4C53×#45511yescharacteristic
363C4C53×#45611yescharacteristic
364C4C53×#47111yescharacteristic
365C4C53×#47311yescharacteristic
366C2—41no—
367C2—42no—
368C2C53×#188211yescharacteristic
369C2C53×#190011yescharacteristic
370C2C53×#190811yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC2×#41×C53C2×C4
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
3. C53×#1997C2semidirect
4. C53×#1548C2semidirect
6. C53×#1309C2semidirect
7. C53×#1299C2semidirect
160. #9353C53direct (N × H)
183. #1997C2×C53semidirect
184. #1548C2×C53semidirect
186. #1309C2×C53semidirect
187. #1299C2×C53semidirect
216. C53#9353direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#30392
C53×#29902
C53×#19972
C53×#15482
C53×#14892
C53×#13092
C53×#12992
#935353