Finite Group Databasebeta

← C53×#10550 #27218062

Subgroups — C53×#10550 #27218062

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3152C211yesmaximal, characteristic
2C53×#1950C211yesmaximal, characteristic, complemented
3C53×#1547C211yesmaximal, characteristic
4C53×#1546C211yesmaximal, characteristic, complemented
5C2×C53×#332C211yesmaximal, characteristic, complemented
6C53×#1480C211yesmaximal, characteristic
7C53×#1283C211yesmaximal, characteristic, complemented
8C2×#137×C53—21no—
9C2×#137×C53—21no—
10C53×#532C2211yescharacteristic
11C2×C4⋊Q8×C53C2211yescharacteristic
12C53×#497—22no—
13C53×#494—21no—
14C53×#492—21no—
15C53×#431C2211yescharacteristic
16C2×#108×C53—21no—
17C53×#423C2211yescharacteristic
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×C4⋊C8×C53C2211yescharacteristic
20C4×M16×C53C2211yescharacteristic
21C53×#341—21no—
22C53×#332—22no—
23C53×#332—21no—
24C53×#332—22no—
25C53×#331—21no—
26C53×D4⋊C8—22no—
27C2×#41×C53—21no—
28C22×Q8×C53—21no—
29#137×C53—42no—
30#137×C53—42no—
31#137×C53—42no—
32#137×C53—42no—
33C2×Q16×C53—22no—
34C2×Q16×C53—22no—
35C2×SD16×C53—22no—
36C2×SD16×C53—22no—
37C2×M16×C53—21no—
38C2×M16×C53D411yescharacteristic
39C2×M16×C53D411yescharacteristic
40C22×C8×C53—21no—
41C22×C8×C53—21no—
42C4⋊Q8×C53—21no—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53—21no—
45C4⋊Q8×C53D411yescharacteristic
46C4⋊Q8×C53D412yes—
47#127×C53D411yescharacteristic
48#124×C53—21no—
49#121×C53—21no—
50#121×C53—21no—
51C4×D4×C53—21no—
52C4×D4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C42×C53C2311yescharacteristic
56#108×C53—42no—
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×C53D412yes—
64C22⋊C8×C53—21no—
65#97×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68#41×C53—41no—
69#41×C53—41no—
70#41×C53—41no—
71#41×C53—41no—
72C2×Q8×C53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×C4×C53#4111yescharacteristic
88Q16×C53—44no—
89Q16×C53—44no—
90SD16×C53—44no—
91SD16×C53—44no—
92M16×C53—21no—
93M16×C53—21no—
94M16×C53—41no—
95M16×C53—41no—
96M16×C53D812yes—
97M16×C53SD1612yes—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—22no—
101C2×C8×C53—41no—
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—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119C23×C53—41no—
120C23×C53#12411yescharacteristic
121Q8×C53—42no—
122Q8×C53—41no—
123Q8×C53—42no—
124Q8×C53—41no—
125Q8×C53—81no—
126Q8×C53—81no—
127D4×C53—41no—
128D4×C53—41no—
129D4×C53—41no—
130D4×C53—41no—
131D4×C53—81no—
132D4×C53—81no—
133D4×C53—81no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53—81no—
142C2×C4×C53—81no—
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12311yescharacteristic
148C2×C4×C53#12311yescharacteristic
149C2×C4×C53C2×D811yescharacteristic
150C2×C4×C53C2×SD1611yescharacteristic
151C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
153C8×C53—22no—
154C8×C53—22no—
155C8×C53—42no—
156C8×C53—81no—
157C8×C53—81no—
158C8×C53—81no—
159#10550C5311yesmaximal, characteristic, direct product factor, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53#39411yescharacteristic
166C22×C53#48611yescharacteristic
167C22×C53#49011yescharacteristic
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—81no—
172C4×C53—82no—
173C4×C53#44711yescharacteristic
174C4×C53#45011yescharacteristic
175C4×C53#45911yescharacteristic
176C4×C53#46011yescharacteristic
177C4×C53#47111yescharacteristic
178C4×C53#47311yescharacteristic
179#3152C2×C5311yescharacteristic
180#1950C2×C5311yescharacteristic, complemented
181#1547C2×C5311yescharacteristic
182#1546C2×C5311yescharacteristic, complemented
183C2×#332C2×C5311yescharacteristic, complemented
184#1480C2×C5311yescharacteristic
185#1283C2×C5311yescharacteristic, complemented
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—161no—
189C2×C53#204111yescharacteristic
190C2×C53#204911yescharacteristic
191C2×C53#205211yescharacteristic
192C2×#137—21no—
193C2×#137—21no—
194#532C22×C5311yescharacteristic
195C2×C4⋊Q8C22×C5311yescharacteristic
196#497—22no—
197#494—21no—
198#492—21no—
199#431C22×C5311yescharacteristic
200C2×#108—21no—
201#423C22×C5311yescharacteristic
202C2×C4⋊C8C22×C5311yescharacteristic
203C2×C4⋊C8C22×C5311yescharacteristic
204C4×M16C22×C5311yescharacteristic
205#341—21no—
206#332—22no—
207#332—21no—
208#332—22no—
209#331—21no—
210D4⋊C8—22no—
211C53#1055011yescharacteristic, direct product factor, Sylow
212C2×#41—21no—
213C22×Q8—21no—
214#137—42no—
215#137—42no—
216#137—42no—
217#137—42no—
218C2×Q16—22no—
219C2×Q16—22no—
220C2×SD16—22no—
221C2×SD16—22no—
222C2×M16—21no—
223C2×M16D4×C5311yescharacteristic
224C2×M16D4×C5311yescharacteristic
225C22×C8—21no—
226C22×C8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8D4×C5312yes—
231C4⋊Q8D4×C5311yescharacteristic
232#127D4×C5311yescharacteristic
233#124—21no—
234#121—21no—
235#121—21no—
236C4×D4—21no—
237C4×D4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C42C23×C5311yescharacteristic
241#108—42no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8D4×C5312yes—
249C22⋊C8—21no—
250#97—21no—
251C4×C8—21no—
252C4×C8—21no—
253#41—41no—
254#41—41no—
255#41—41no—
256#41—41no—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C2×D4×C5311yescharacteristic
271C22×C4#41×C5311yescharacteristic
272C22×C4#41×C5311yescharacteristic
273Q16—44no—
274Q16—44no—
275SD16—44no—
276SD16—44no—
277M16—21no—
278M16—21no—
279M16—41no—
280M16—41no—
281M16D8×C5312yes—
282M16SD16×C5312yes—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—81no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42C2×D4×C5311yescharacteristic
301C42C2×D4×C5311yescharacteristic
302C42C2×D4×C5311yescharacteristic
303C42#41×C5311yescharacteristic
304C23—41no—
305C23#124×C5311yescharacteristic
306Q8—42no—
307Q8—41no—
308Q8—42no—
309Q8—41no—
310Q8—81no—
311Q8—81no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—81no—
317D4—81no—
318D4—81no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4#120×C5311yescharacteristic
329C2×C4#121×C5311yescharacteristic
330C2×C4#121×C5311yescharacteristic
331C2×C4#121×C5311yescharacteristic
332C2×C4#123×C5311yescharacteristic
333C2×C4#123×C5311yescharacteristic
334C2×C4C2×D8×C5311yescharacteristic
335C2×C4C2×SD16×C5311yescharacteristic
336C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
337C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
338C8—22no—
339C8—22no—
340C8—42no—
341C8—81no—
342C8—81no—
343C8—81no—
344C22—21no—
345C22—21no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22C53×#39411yescharacteristic
350C22C53×#48611yescharacteristic
351C22C53×#49011yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—82no—
357C4C53×#44711yescharacteristic
358C4C53×#45011yescharacteristic
359C4C53×#45911yescharacteristic
360C4C53×#46011yescharacteristic
361C4C53×#47111yescharacteristic
362C4C53×#47311yescharacteristic
363C2—21no—
364C2—21no—
365C2—161no—
366C2C53×#204111yescharacteristic
367C2C53×#204911yescharacteristic
368C2C53×#205211yescharacteristic
subgroupG/H
center, socleC22×C53#394
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
2. C53×#1950C2semidirect
4. C53×#1546C2semidirect
5. C2×C53×#332C2semidirect
7. C53×#1283C2semidirect
159. #10550C53direct (N × H)
180. #1950C2×C53semidirect
182. #1546C2×C53semidirect
183. C2×#332C2×C53semidirect
185. #1283C2×C53semidirect
211. C53#10550direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31522
C53×#19502
C53×#15472
C53×#15462
C2×C53×#3322
C53×#14802
C53×#12832
#1055053