Finite Group Databasebeta

← C53×#10170 #27217682

Subgroups — C53×#10170 #27217682

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3206C211yesmaximal, characteristic
2C53×#3152C211yesmaximal, characteristic
3C2×C53×#487C211yesmaximal, characteristic, complemented
4C53×#1997C211yesmaximal, characteristic, complemented
5C53×#1546C211yesmaximal, characteristic, complemented
6C53×#1492C211yesmaximal, characteristic
7C53×#1278C211yesmaximal, characteristic, complemented
8C2×#137×C53—22no—
9C53×#532C2211yescharacteristic
10C2×C4⋊Q8×C53C2211yescharacteristic
11C2×C4⋊Q8×C53C2211yescharacteristic
12C53×#497—22no—
13C53×#494—21no—
14C53×#492—21no—
15C53×#492—21no—
16C53×#487—21no—
17C53×#487—22no—
18C53×#487—22no—
19C53×#461—22no—
20C53×#448—22no—
21C53×#431C2211yescharacteristic
22C53×#423C2211yescharacteristic
23C2×#103×C53—22no—
24C4×M16×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C53×#343—22no—
27C53×#332—22no—
28C53×D4⋊C8—22no—
29C2×#41×C53—21no—
30C22×Q8×C53—21no—
31C22×Q8×C53D412yes—
32#137×C53—44no—
33#137×C53—44no—
34C2×Q16×C53—24no—
35C2×SD16×C53—24no—
36C2×SD16×C53—24no—
37C2×M16×C53—21no—
38C2×M16×C53D412yes—
39C22×C8×C53—21no—
40C4⋊Q8×C53—21no—
41C4⋊Q8×C53—21no—
42C4⋊Q8×C53—21no—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53—21no—
45C4⋊Q8×C53—21no—
46C4⋊Q8×C53—21no—
47C4⋊Q8×C53D411yescharacteristic
48C4⋊Q8×C53D412yes—
49#127×C53D411yescharacteristic
50#124×C53—21no—
51#121×C53—21no—
52#121×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53D412yes—
63#103×C53—28no—
64#103×C53—24no—
65C22⋊C8×C53—21no—
66#97×C53—21no—
67C4×C8×C53—21no—
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70C4×C8×C53D412yes—
71#41×C53—42no—
72#41×C53—42no—
73C2×Q8×C53—22no—
74C2×Q8×C53—22no—
75C2×Q8×C53—21no—
76C2×Q8×C53—22no—
77C2×Q8×C53—44no—
78C2×Q8×C53—41no—
79C2×Q8×C53—42no—
80C2×Q8×C53—42no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C22×C4×C53—22no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91Q16×C53—48no—
92SD16×C53—48no—
93SD16×C53—84no—
94M16×C53—22no—
95M16×C53—42no—
96M16×C53SD1614yescomplemented
97C2×C8×C53—22no—
98C2×C8×C53—22no—
99C2×C8×C53—22no—
100C2×C8×C53—42no—
101C2×C8×C53—41no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C23×C53—41no—
119C23×C53#12111yescharacteristic
120Q8×C53—44no—
121Q8×C53—44no—
122Q8×C53—42no—
123Q8×C53—44no—
124Q8×C53—81no—
125Q8×C53—81no—
126D4×C53—42no—
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—82no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—42no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—82no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53—82no—
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12711yescharacteristic
145C2×C4×C53C2×SD1612yes—
146C2×C4×C53AGL(1,ℤ/8)12yes—
147C8×C53—24no—
148C8×C53—42no—
149C8×C53—42no—
150C8×C53—81no—
151#10170C5311yesmaximal, characteristic, direct product factor, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53#39211yescharacteristic
158C22×C53#46012yes—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—82no—
162C4×C53—81no—
163C4×C53—82no—
164C4×C53#45012yes—
165C4×C53#45512yes—
166C4×C53#47312yes—
167C4×C53#49211yescharacteristic
168C4×C53#49611yescharacteristic
169#3206C2×C5311yescharacteristic
170#3152C2×C5311yescharacteristic
171C2×#487C2×C5311yescharacteristic, complemented
172#1997C2×C5311yescharacteristic, complemented
173#1546C2×C5311yescharacteristic, complemented
174#1492C2×C5311yescharacteristic
175#1278C2×C5311yescharacteristic, complemented
176C2×C53—22no—
177C2×C53—161no—
178C2×C53#200411yescharacteristic
179C2×C53#201411yescharacteristic
180C2×C53#202011yescharacteristic
181C2×#137—22no—
182#532C22×C5311yescharacteristic
183C2×C4⋊Q8C22×C5311yescharacteristic
184C2×C4⋊Q8C22×C5311yescharacteristic
185#497—22no—
186#494—21no—
187#492—21no—
188#492—21no—
189#487—22no—
190#487—21no—
191#487—22no—
192#461—22no—
193#448—22no—
194#431C22×C5311yescharacteristic
195#423C22×C5311yescharacteristic
196C2×#103—22no—
197C4×M16C22×C5311yescharacteristic
198C2×C4×C8C22×C5311yescharacteristic
199#343—22no—
200#332—22no—
201D4⋊C8—22no—
202C53#1017011yescharacteristic, direct product factor, Sylow
203C2×#41—21no—
204C22×Q8—21no—
205C22×Q8D4×C5312yes—
206#137—44no—
207#137—44no—
208C2×Q16—24no—
209C2×SD16—24no—
210C2×SD16—24no—
211C2×M16—21no—
212C2×M16D4×C5312yes—
213C22×C8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8D4×C5312yes—
222C4⋊Q8D4×C5311yescharacteristic
223#127D4×C5311yescharacteristic
224#124—21no—
225#121—21no—
226#121—21no—
227C4×D4—21no—
228C4×D4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C42C23×C5311yescharacteristic
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8D4×C5312yes—
237#103—28no—
238#103—24no—
239C22⋊C8—21no—
240#97—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8D4×C5312yes—
245#41—42no—
246#41—42no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8—22no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C2×D4×C5311yescharacteristic
263C22×C4C2×D4×C5311yescharacteristic
264C22×C4#41×C5311yescharacteristic
265Q16—48no—
266SD16—48no—
267SD16—84no—
268M16—22no—
269M16—42no—
270M16SD16×C5314yescomplemented
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—42no—
275C2×C8—41no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42C2×D4×C5311yescharacteristic
289C42C2×D4×C5311yescharacteristic
290C42C2×D4×C5311yescharacteristic
291C42C2×D4×C5311yescharacteristic
292C23—41no—
293C23#121×C5311yescharacteristic
294Q8—44no—
295Q8—44no—
296Q8—42no—
297Q8—44no—
298Q8—81no—
299Q8—81no—
300D4—42no—
301D4—42no—
302D4—42no—
303D4—82no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4#120×C5311yescharacteristic
314C2×C4#120×C5311yescharacteristic
315C2×C4#120×C5311yescharacteristic
316C2×C4#121×C5311yescharacteristic
317C2×C4#121×C5311yescharacteristic
318C2×C4#127×C5311yescharacteristic
319C2×C4C2×SD16×C5312yes—
320C2×C4AGL(1,ℤ/8)×C5312yes—
321C8—24no—
322C8—42no—
323C8—42no—
324C8—81no—
325C22—21no—
326C22—21no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C53×#39211yescharacteristic
331C22C53×#46012yes—
332C4—21no—
333C4—21no—
334C4—82no—
335C4—81no—
336C4—82no—
337C4C53×#45012yes—
338C4C53×#45512yes—
339C4C53×#47312yes—
340C4C53×#49211yescharacteristic
341C4C53×#49611yescharacteristic
342C2—22no—
343C2—161no—
344C2C53×#200411yescharacteristic
345C2C53×#201411yescharacteristic
346C2C53×#202011yescharacteristic
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. C2×C53×#487C2semidirect
4. C53×#1997C2semidirect
5. C53×#1546C2semidirect
7. C53×#1278C2semidirect
96. M16×C53SD16semidirect
151. #10170C53direct (N × H)
171. C2×#487C2×C53semidirect
172. #1997C2×C53semidirect
173. #1546C2×C53semidirect
175. #1278C2×C53semidirect
202. C53#10170direct (N × H)
270. M16SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#32062
C53×#31522
C2×C53×#4872
C53×#19972
C53×#15462
C53×#14922
C53×#12782
#1017053