Finite Group Databasebeta

← C53×#9356 #27216868

Subgroups — C53×#9356 #27216868

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3079C211yesmaximal, characteristic
2C53×#2991C211yesmaximal, characteristic
3C53×#1997C211yesmaximal, characteristic, complemented
4C53×#1546C211yesmaximal, characteristic, complemented
5C53×#1489C211yesmaximal, characteristic
6C53×#1309C211yesmaximal, characteristic, complemented
7C53×#1299C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C53C2211yescharacteristic
9C4×#41×C53C2211yescharacteristic
10C53×#477—21no—
11C53×#476—21no—
12C53×#475—21no—
13C53×#474—21no—
14C8×Q8×C53—21no—
15C53×#436—22no—
16C53×#436—21no—
17C53×#431—21no—
18C53×#431C2211yescharacteristic
19C53×#428C412yes—
20C53×#423—21no—
21C53×#423C2211yescharacteristic
22C53×#423C2211yescharacteristic
23C53×#423C2211yescharacteristic
24C53×#417C412yes—
25C4×M16×C53C2211yescharacteristic
26C53×#343—22no—
27C53×#332—21no—
28C53×#332—21no—
29C53×Q8⋊C8—21no—
30C53×D4⋊C8—21no—
31C2×#41×C53C2×C411yescharacteristic
32C22×Q8×C53—21no—
33C2×M16×C53—22no—
34C2×M16×C53—21no—
35C2×M16×C53—21no—
36C2×M16×C53C2×C412yes—
37C2×M16×C53D412yes—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—21no—
40C4⋊Q8×C53—21no—
41C4⋊Q8×C53—21no—
42C4×Q8×C53D411yescharacteristic
43C4×Q8×C53D411yescharacteristic
44C4×D4×C53—21no—
45C4×D4×C53—21no—
46C4×D4×C53D411yescharacteristic
47C4×D4×C53D411yescharacteristic
48#117×C53—21no—
49#117×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53C2×C412yes—
52C2×C42×C53—21no—
53C2×C42×C53C2311yescharacteristic
54#107×C53—22no—
55#106×C53—22no—
56C4⋊C8×C53—22no—
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53D411yescharacteristic
64C4⋊C8×C53D411yescharacteristic
65#103×C53—22no—
66#102×C53—22no—
67C22⋊C8×C53—22no—
68C22⋊C8×C53—21no—
69#97×C53—21no—
70C4×C8×C53—22no—
71C4×C8×C53—21no—
72C4×C8×C53—21no—
73#41×C53—22no—
74#41×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—81no—
78C2×Q8×C53—81no—
79C2×Q8×C53C22⋊C411yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53C22⋊C411yescharacteristic
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89M16×C53—22no—
90M16×C53—41no—
91M16×C53—42no—
92M16×C53—42no—
93M16×C53—82no—
94M16×C53—81no—
95C2×C8×C53—22no—
96C2×C8×C53—21no—
97C2×C8×C53—22no—
98C2×C8×C53—21no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53C22⋊C411yescharacteristic
108C4⋊C4×C53C22⋊C411yescharacteristic
109C22⋊C4×C53—21no—
110C22⋊C4×C53—21no—
111C22⋊C4×C53—21no—
112C42×C53—21no—
113C42×C53—21no—
114C42×C53—21no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119C23×C53—21no—
120C23×C53C4×D411yescharacteristic
121Q8×C53—21no—
122Q8×C53—82no—
123Q8×C53—81no—
124Q8×C53—81no—
125Q8×C53#10112yes—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—42no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
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—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—82no—
145C2×C4×C53C2×C22⋊C411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12311yescharacteristic
151C8×C53—42no—
152C8×C53—42no—
153C8×C53—42no—
154C8×C53—41no—
155C8×C53—41no—
156C8×C53—81no—
157#9356C5311yesmaximal, characteristic, direct product factor, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53#38611yescharacteristic
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—22no—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—82no—
172C4×C53C2×#10111yescharacteristic
173C4×C53#41311yescharacteristic
174C4×C53#45511yescharacteristic
175C4×C53#45611yescharacteristic
176C4×C53#47111yescharacteristic
177C4×C53#47211yescharacteristic
178#3079C2×C5311yescharacteristic
179#2991C2×C5311yescharacteristic
180#1997C2×C5311yescharacteristic, complemented
181#1546C2×C5311yescharacteristic, complemented
182#1489C2×C5311yescharacteristic
183#1309C2×C5311yescharacteristic, complemented
184#1299C2×C5311yescharacteristic, complemented
185C2×C53—41no—
186C2×C53—81no—
187C2×C53#188211yescharacteristic
188C2×C53#190111yescharacteristic
189C2×C53#190711yescharacteristic
190C2×C4⋊Q8C22×C5311yescharacteristic
191C4×#41C22×C5311yescharacteristic
192#477—21no—
193#476—21no—
194#475—21no—
195#474—21no—
196C8×Q8—21no—
197#436—22no—
198#436—21no—
199#431—21no—
200#431C22×C5311yescharacteristic
201#428C4×C5312yes—
202#423—21no—
203#423C22×C5311yescharacteristic
204#423C22×C5311yescharacteristic
205#423C22×C5311yescharacteristic
206#417C4×C5312yes—
207C4×M16C22×C5311yescharacteristic
208#343—22no—
209#332—21no—
210#332—21no—
211Q8⋊C8—21no—
212D4⋊C8—21no—
213C53#935611yescharacteristic, direct product factor, Sylow
214C2×#41C2×C4×C5311yescharacteristic
215C22×Q8—21no—
216C2×M16—22no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16C2×C4×C5312yes—
220C2×M16D4×C5312yes—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4×Q8D4×C5311yescharacteristic
226C4×Q8D4×C5311yescharacteristic
227C4×D4—21no—
228C4×D4—21no—
229C4×D4D4×C5311yescharacteristic
230C4×D4D4×C5311yescharacteristic
231#117—21no—
232#117C2×C4×C5311yescharacteristic
233C2×C4⋊C4—21no—
234C2×C4⋊C4C2×C4×C5312yes—
235C2×C42—21no—
236C2×C42C23×C5311yescharacteristic
237#107—22no—
238#106—22no—
239C4⋊C8—22no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8D4×C5311yescharacteristic
247C4⋊C8D4×C5311yescharacteristic
248#103—22no—
249#102—22no—
250C22⋊C8—22no—
251C22⋊C8—21no—
252#97—21no—
253C4×C8—22no—
254C4×C8—21no—
255C4×C8—21no—
256#41—22no—
257#41—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—81no—
261C2×Q8—81no—
262C2×Q8C22⋊C4×C5311yescharacteristic
263C2×D4—21no—
264C2×D4C22⋊C4×C5311yescharacteristic
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4C22×C4×C5311yescharacteristic
270C22×C4C2×D4×C5311yescharacteristic
271C22×C4#41×C5311yescharacteristic
272M16—22no—
273M16—41no—
274M16—42no—
275M16—42no—
276M16—82no—
277M16—81no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4C22⋊C4×C5311yescharacteristic
291C4⋊C4C22⋊C4×C5311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42C2×D4×C5311yescharacteristic
299C42C2×D4×C5311yescharacteristic
300C42C2×D4×C5311yescharacteristic
301C42#41×C5311yescharacteristic
302C23—21no—
303C23C4×D4×C5311yescharacteristic
304Q8—21no—
305Q8—82no—
306Q8—81no—
307Q8—81no—
308Q8#101×C5312yes—
309D4—21no—
310D4—21no—
311D4—42no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4C2×C22⋊C4×C5311yescharacteristic
329C2×C4C4×D4×C5311yescharacteristic
330C2×C4#120×C5311yescharacteristic
331C2×C4#121×C5311yescharacteristic
332C2×C4#121×C5311yescharacteristic
333C2×C4#123×C5311yescharacteristic
334C8—42no—
335C8—42no—
336C8—42no—
337C8—41no—
338C8—41no—
339C8—81no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C53×#38611yescharacteristic
347C4—21no—
348C4—21no—
349C4—22no—
350C4—21no—
351C4—41no—
352C4—41no—
353C4—82no—
354C4C2×#101×C5311yescharacteristic
355C4C53×#41311yescharacteristic
356C4C53×#45511yescharacteristic
357C4C53×#45611yescharacteristic
358C4C53×#47111yescharacteristic
359C4C53×#47211yescharacteristic
360C2—41no—
361C2—81no—
362C2C53×#188211yescharacteristic
363C2C53×#190111yescharacteristic
364C2C53×#190711yescharacteristic
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×#1546C2semidirect
6. C53×#1309C2semidirect
7. C53×#1299C2semidirect
157. #9356C53direct (N × H)
180. #1997C2×C53semidirect
181. #1546C2×C53semidirect
183. #1309C2×C53semidirect
184. #1299C2×C53semidirect
213. C53#9356direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#30792
C53×#29912
C53×#19972
C53×#15462
C53×#14892
C53×#13092
C53×#12992
#935653