Finite Group Databasebeta

← C53×#10197 #27217709

Subgroups — C53×#10197 #27217709

Order 13568.

524 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3246C211yesmaximal, characteristic
2C53×#3206C211yesmaximal, characteristic
3C53×#3165C211yesmaximal, characteristic, complemented
4C53×#1987C211yesmaximal, characteristic, complemented
5C53×#1980C211yesmaximal, characteristic
6C53×#1503C211yesmaximal, characteristic, complemented
7C53×#1283C211yesmaximal, characteristic, complemented
8C53×#533C2211yescharacteristic
9C53×#532C2211yescharacteristic
10C2×C4⋊Q8×C53C2211yescharacteristic
11C53×#501—22no—
12C53×#500—21no—
13C53×#498—21no—
14C53×#492—21no—
15C53×#487—21no—
16C53×#476—21no—
17C53×#475—21no—
18C53×#465—22no—
19C53×#461—22no—
20C53×#461—21no—
21C53×#460—21no—
22C53×#448—22no—
23C53×#431C2211yescharacteristic
24C53×#428—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×#104×C53—22no—
27C4×M16×C53C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C53×#340—22no—
30C53×#328—22no—
31C2×#41×C53—21no—
32C22×Q8×C53—21no—
33C2×SD16×C53—24no—
34C2×SD16×C53—22no—
35C2×M16×C53D412yes—
36C22×C8×C53D412yes—
37C4⋊Q8×C53—21no—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—21no—
40C4⋊Q8×C53D411yescharacteristic
41C4⋊Q8×C53D411yescharacteristic
42C4⋊Q8×C53D411yescharacteristic
43C4⋊Q8×C53D412yes—
44#127×C53D411yescharacteristic
45#125×C53—21no—
46#124×C53—21no—
47C22⋊Q8×C53—21no—
48C22⋊Q8×C53—21no—
49#121×C53—21no—
50#121×C53—21no—
51C4×Q8×C53—21no—
52C4×Q8×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55#117×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C42×C53—21no—
60C2×C42×C53C2311yescharacteristic
61#107×C53—24no—
62#106×C53—24no—
63#106×C53—22no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67#104×C53—44no—
68#103×C53—24no—
69#103×C53—22no—
70#102×C53—22no—
71C22⋊C8×C53D412yes—
72#97×C53—21no—
73C4×C8×C53—21no—
74C4×C8×C53—21no—
75#95×C53—21no—
76#41×C53—42no—
77#41×C53—42no—
78C2×Q8×C53—21no—
79C2×Q8×C53—22no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—41no—
83C2×Q8×C53—82no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94SD16×C53—84no—
95SD16×C53—82no—
96M16×C53—22no—
97M16×C53SD1614yescomplemented
98C2×C8×C53—22no—
99C2×C8×C53—22no—
100C2×C8×C53—22no—
101C2×C8×C53—42no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C42×C53—41no—
120C42×C53—42no—
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125C23×C53—41no—
126C23×C53#12711yescharacteristic
127Q8×C53—44no—
128Q8×C53—41no—
129Q8×C53—41no—
130Q8×C53—42no—
131Q8×C53—82no—
132Q8×C53—81no—
133D4×C53—42no—
134D4×C53—41no—
135D4×C53—41no—
136D4×C53—42no—
137D4×C53—81no—
138D4×C53—81no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—81no—
148C2×C4×C53—82no—
149C2×C4×C53—81no—
150C2×C4×C53—82no—
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53C2×SD1612yes—
158C2×C4×C53#13712yes—
159C8×C53—24no—
160C8×C53—42no—
161C8×C53—42no—
162#10197C5311yesmaximal, characteristic, direct product factor, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53#39211yescharacteristic
169C22×C53#49211yescharacteristic
170C22×C53#49711yescharacteristic
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—82no—
175C4×C53—81no—
176C4×C53—82no—
177C4×C53—81no—
178C4×C53—81no—
179C4×C53#44812yes—
180C4×C53#45511yescharacteristic
181C4×C53#45611yescharacteristic
182C4×C53#46112yes—
183#3246C2×C5311yescharacteristic
184#3206C2×C5311yescharacteristic
185#3165C2×C5311yescharacteristic, complemented
186#1987C2×C5311yescharacteristic, complemented
187#1980C2×C5311yescharacteristic
188#1503C2×C5311yescharacteristic, complemented
189#1283C2×C5311yescharacteristic, complemented
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—161no—
193C2×C53#200711yescharacteristic
194C2×C53#200811yescharacteristic
195C2×C53#201311yescharacteristic
196#533C22×C5311yescharacteristic
197#532C22×C5311yescharacteristic
198C2×C4⋊Q8C22×C5311yescharacteristic
199#501—22no—
200#500—21no—
201#498—21no—
202#492—21no—
203#487—21no—
204#476—21no—
205#475—21no—
206#465—22no—
207#461—22no—
208#461—21no—
209#460—21no—
210#448—22no—
211#431C22×C5311yescharacteristic
212#428—22no—
213C2×C4⋊C8C22×C5311yescharacteristic
214C2×#104—22no—
215C4×M16C22×C5311yescharacteristic
216C4×C4⋊C4C22×C5311yescharacteristic
217#340—22no—
218#328—22no—
219C53#1019711yescharacteristic, direct product factor, Sylow
220C2×#41—21no—
221C22×Q8—21no—
222C2×SD16—24no—
223C2×SD16—22no—
224C2×M16D4×C5312yes—
225C22×C8D4×C5312yes—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8D4×C5312yes—
230C4⋊Q8D4×C5311yescharacteristic
231C4⋊Q8D4×C5311yescharacteristic
232C4⋊Q8D4×C5311yescharacteristic
233#127D4×C5311yescharacteristic
234#125—21no—
235#124—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238#121—21no—
239#121—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×D4—21no—
243C4×D4—21no—
244#117—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C42—21no—
249C2×C42C23×C5311yescharacteristic
250#107—24no—
251#106—24no—
252#106—22no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256#104—44no—
257#103—24no—
258#103—22no—
259#102—22no—
260C22⋊C8D4×C5312yes—
261#97—21no—
262C4×C8—21no—
263C4×C8—21no—
264#95—21no—
265#41—42no—
266#41—42no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—41no—
272C2×Q8—82no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4C2×D4×C5311yescharacteristic
281C22×C4C2×D4×C5311yescharacteristic
282C22×C4C2×D4×C5311yescharacteristic
283SD16—84no—
284SD16—82no—
285M16—22no—
286M16SD16×C5314yescomplemented
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—42no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C42—41no—
309C42—42no—
310C42C2×D4×C5311yescharacteristic
311C42C2×D4×C5311yescharacteristic
312C42C2×D4×C5311yescharacteristic
313C42#41×C5311yescharacteristic
314C23—41no—
315C23#127×C5311yescharacteristic
316Q8—44no—
317Q8—41no—
318Q8—41no—
319Q8—42no—
320Q8—81no—
321Q8—82no—
322D4—42no—
323D4—41no—
324D4—41no—
325D4—42no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4#120×C5311yescharacteristic
341C2×C4#120×C5311yescharacteristic
342C2×C4#120×C5311yescharacteristic
343C2×C4#121×C5311yescharacteristic
344C2×C4#121×C5311yescharacteristic
345C2×C4#121×C5311yescharacteristic
346C2×C4C2×SD16×C5312yes—
347C2×C4#137×C5312yes—
348C8—24no—
349C8—42no—
350C8—42no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22C53×#39211yescharacteristic
357C22C53×#49211yescharacteristic
358C22C53×#49711yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—82no—
363C4—81no—
364C4—82no—
365C4—81no—
366C4—81no—
367C4C53×#44812yes—
368C4C53×#45511yescharacteristic
369C4C53×#45611yescharacteristic
370C4C53×#46112yes—
371C2—21no—
372C2—21no—
373C2—161no—
374C2C53×#200711yescharacteristic
375C2C53×#200811yescharacteristic
376C2C53×#201311yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC53×#532C22
LCSG ▷ C2×C42 ▷ 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×#3165C2semidirect
4. C53×#1987C2semidirect
6. C53×#1503C2semidirect
7. C53×#1283C2semidirect
97. M16×C53SD16semidirect
162. #10197C53direct (N × H)
185. #3165C2×C53semidirect
186. #1987C2×C53semidirect
188. #1503C2×C53semidirect
189. #1283C2×C53semidirect
219. C53#10197direct (N × H)
286. M16SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#32462
C53×#32062
C53×#31652
C53×#19872
C53×#19802
C53×#15032
C53×#12832
#1019753