Finite Group Databasebeta

← C53×#13411 #27220923

Subgroups — C53×#13411 #27220923

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3155C211yesmaximal, characteristic
2C2×C53×#361C211yesmaximal, characteristic, complemented
3C2×C53×#359C211yesmaximal, characteristic, complemented
4C53×#2139C211yesmaximal, characteristic, complemented
5C2×C53×#357C211yesmaximal, characteristic, complemented
6C53×#1952C211yesmaximal, characteristic, complemented
7C53×#1379C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C53—21no—
9C2×#135×C53C2211yescharacteristic
10C22×D8×C53C2211yescharacteristic
11C53×#535—21no—
12C2×M32×C53C2211yescharacteristic
13C22×C16×C53C2211yescharacteristic
14C53×#496—22no—
15C53×#495—21no—
16C53×#493—21no—
17C2×#108×C53C2211yescharacteristic
18C53×#428—21no—
19C2×#107×C53C2211yescharacteristic
20C53×#405C2211yescharacteristic
21C53×#361—22no—
22C53×#359—22no—
23C53×#358—21no—
24C53×#357—22no—
25C53×#357—21no—
26C53×#343—21no—
27C2×#41×C53—21no—
28C22×D4×C53—21no—
29AGL(1,ℤ/8)×C53—42no—
30AGL(1,ℤ/8)×C53—42no—
31#135×C53—21no—
32#135×C53—21no—
33#135×C53—21no—
34#135×C53—21no—
35C2×Q16×C53D411yescharacteristic
36C2×SD16×C53—22no—
37C2×SD16×C53—21no—
38C2×D8×C53—22no—
39C2×D8×C53—21no—
40C2×D8×C53—21no—
41C2×D8×C53—21no—
42C2×D8×C53—21no—
43C2×D8×C53—21no—
44C2×D8×C53D411yescharacteristic
45C2×D8×C53D412yes—
46C2×M16×C53—21no—
47C2×M16×C53D411yescharacteristic
48C22×C8×C53C2311yescharacteristic
49#127×C53—22no—
50#124×C53—22no—
51#121×C53—41no—
52#121×C53—41no—
53#120×C53—42no—
54#117×C53D411yescharacteristic
55C2×C4⋊C4×C53—21no—
56M32×C53—21no—
57M32×C53—21no—
58C2×C16×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62#108×C53—21no—
63#108×C53D412yescomplemented
64#107×C53—21no—
65#107×C53—21no—
66#107×C53—41no—
67#106×C53—41no—
68#97×C53—21no—
69C4×C8×C53—21no—
70C24×C53—41no—
71#41×C53—41no—
72#41×C53—41no—
73#41×C53—41no—
74#41×C53—41no—
75C2×Q8×C53—21no—
76C2×D4×C53—22no—
77C2×D4×C53—21no—
78C2×D4×C53—42no—
79C2×D4×C53—42no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89Q16×C53—21no—
90Q16×C53—21no—
91SD16×C53—44no—
92SD16×C53—41no—
93SD16×C53—41no—
94D8×C53—22no—
95D8×C53—21no—
96D8×C53—22no—
97D8×C53—21no—
98D8×C53—44no—
99D8×C53—41no—
100D8×C53—41no—
101M16×C53—21no—
102M16×C53—22no—
103M16×C53—41no—
104M16×C53SD1612yes—
105C2×C8×C53—21no—
106C2×C8×C53—41no—
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C2×C8×C53#4111yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53D811yescharacteristic
116C4⋊C4×C53D811yescharacteristic
117C22⋊C4×C53—21no—
118C22⋊C4×C53—42no—
119C22⋊C4×C53—42no—
120C42×C53—21no—
121C16×C53—41no—
122C16×C53—41no—
123C23×C53—41no—
124C23×C53—42no—
125C23×C53—81no—
126C23×C53—81no—
127C23×C53—161no—
128C23×C53—161no—
129C23×C53AGL(1,ℤ/8)11yescharacteristic
130Q8×C53—41no—
131Q8×C53—41no—
132D4×C53—42no—
133D4×C53—41no—
134D4×C53—42no—
135D4×C53—41no—
136D4×C53—81no—
137D4×C53—81no—
138D4×C53—81no—
139D4×C53—81no—
140D4×C53—162no—
141D4×C53—162no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—41no—
145C2×C4×C53—81no—
146C2×C4×C53—81no—
147C2×C4×C53—81no—
148C2×C4×C53—161no—
149C2×C4×C53#12011yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12411yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C2×SD1611yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C8×C53—22no—
156C8×C53—42no—
157C8×C53#12111yescharacteristic
158C8×C53#12111yescharacteristic
159C8×C53#12311yescharacteristic
160C8×C53#12311yescharacteristic
161#13411C5311yesmaximal, characteristic, direct product factor, Sylow
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—81no—
165C22×C53—82no—
166C22×C53—81no—
167C22×C53—82no—
168C22×C53—81no—
169C22×C53—82no—
170C22×C53—162no—
171C22×C53—162no—
172C22×C53#44711yescharacteristic
173C22×C53#46011yescharacteristic
174C22×C53#49011yescharacteristic
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—81no—
178C4×C53—161no—
179C4×C53#39411yescharacteristic
180C4×C53#45011yescharacteristic
181C4×C53#45911yescharacteristic
182C4×C53#48611yescharacteristic
183#3155C2×C5311yescharacteristic
184C2×#361C2×C5311yescharacteristic, complemented
185C2×#359C2×C5311yescharacteristic, complemented
186#2139C2×C5311yescharacteristic, complemented
187C2×#357C2×C5311yescharacteristic, complemented
188#1952C2×C5311yescharacteristic, complemented
189#1379C2×C5311yescharacteristic, complemented
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—161no—
193C2×C53—162no—
194C2×C53#204111yescharacteristic
195C2×C53#219211yescharacteristic
196C2×C53#219411yescharacteristic
197C2×AGL(1,ℤ/8)—21no—
198C2×#135C22×C5311yescharacteristic
199C22×D8C22×C5311yescharacteristic
200#535—21no—
201C2×M32C22×C5311yescharacteristic
202C22×C16C22×C5311yescharacteristic
203#496—22no—
204#495—21no—
205#493—21no—
206C2×#108C22×C5311yescharacteristic
207#428—21no—
208C2×#107C22×C5311yescharacteristic
209#405C22×C5311yescharacteristic
210#361—22no—
211#359—22no—
212#358—21no—
213#357—22no—
214#357—21no—
215#343—21no—
216C53#1341111yescharacteristic, direct product factor, Sylow
217C2×#41—21no—
218C22×D4—21no—
219AGL(1,ℤ/8)—42no—
220AGL(1,ℤ/8)—42no—
221#135—21no—
222#135—21no—
223#135—21no—
224#135—21no—
225C2×Q16D4×C5311yescharacteristic
226C2×SD16—22no—
227C2×SD16—21no—
228C2×D8—22no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8D4×C5312yes—
235C2×D8D4×C5311yescharacteristic
236C2×M16—21no—
237C2×M16D4×C5311yescharacteristic
238C22×C8C23×C5311yescharacteristic
239#127—22no—
240#124—22no—
241#121—41no—
242#121—41no—
243#120—42no—
244#117D4×C5311yescharacteristic
245C2×C4⋊C4—21no—
246M32—21no—
247M32—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252#108—21no—
253#108D4×C5312yescomplemented
254#107—21no—
255#107—21no—
256#107—41no—
257#106—41no—
258#97—21no—
259C4×C8—21no—
260C24—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265C2×Q8—21no—
266C2×D4—22no—
267C2×D4—21no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×D4×C5311yescharacteristic
279Q16—21no—
280Q16—21no—
281SD16—44no—
282SD16—41no—
283SD16—41no—
284D8—22no—
285D8—21no—
286D8—22no—
287D8—21no—
288D8—44no—
289D8—41no—
290D8—41no—
291M16—21no—
292M16—22no—
293M16—41no—
294M16SD16×C5312yes—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8C2×D4×C5311yescharacteristic
298C2×C8C2×D4×C5311yescharacteristic
299C2×C8C2×D4×C5311yescharacteristic
300C2×C8#41×C5311yescharacteristic
301C2×C8#41×C5311yescharacteristic
302C2×C8#41×C5311yescharacteristic
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4D8×C5311yescharacteristic
306C4⋊C4D8×C5311yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C42—21no—
311C16—41no—
312C16—41no—
313C23—42no—
314C23—41no—
315C23—81no—
316C23—81no—
317C23—161no—
318C23—161no—
319C23AGL(1,ℤ/8)×C5311yescharacteristic
320Q8—41no—
321Q8—41no—
322D4—42no—
323D4—41no—
324D4—41no—
325D4—42no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330D4—162no—
331D4—162no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—161no—
339C2×C4#120×C5311yescharacteristic
340C2×C4#121×C5311yescharacteristic
341C2×C4#124×C5311yescharacteristic
342C2×C4C2×D8×C5311yescharacteristic
343C2×C4C2×SD16×C5311yescharacteristic
344C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
345C8—22no—
346C8—42no—
347C8#121×C5311yescharacteristic
348C8#121×C5311yescharacteristic
349C8#123×C5311yescharacteristic
350C8#123×C5311yescharacteristic
351C22—21no—
352C22—21no—
353C22—81no—
354C22—82no—
355C22—81no—
356C22—82no—
357C22—81no—
358C22—82no—
359C22—162no—
360C22—162no—
361C22C53×#44711yescharacteristic
362C22C53×#46011yescharacteristic
363C22C53×#49011yescharacteristic
364C4—41no—
365C4—41no—
366C4—81no—
367C4—161no—
368C4C53×#39411yescharacteristic
369C4C53×#45011yescharacteristic
370C4C53×#45911yescharacteristic
371C4C53×#48611yescharacteristic
372C2—21no—
373C2—21no—
374C2—162no—
375C2—161no—
376C2C53×#204111yescharacteristic
377C2C53×#219211yescharacteristic
378C2C53×#219411yescharacteristic
subgroupG/H
center, socleC22×C53#447
commutator, Frattini, cosocleC22×C8C23×C53
prime radicalC53×#3155C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C53×#361C2semidirect
3. C2×C53×#359C2semidirect
4. C53×#2139C2semidirect
5. C2×C53×#357C2semidirect
6. C53×#1952C2semidirect
7. C53×#1379C2semidirect
63. #108×C53D4semidirect
161. #13411C53direct (N × H)
184. C2×#361C2×C53semidirect
185. C2×#359C2×C53semidirect
186. #2139C2×C53semidirect
187. C2×#357C2×C53semidirect
188. #1952C2×C53semidirect
189. #1379C2×C53semidirect
216. C53#13411direct (N × H)
253. #108D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#31552
C2×C53×#3612
C2×C53×#3592
C53×#21392
C2×C53×#3572
C53×#19522
C53×#13792
#1341153