Finite Group Databasebeta

← C53×#13387 #27220899

Subgroups — C53×#13387 #27220899

Order 13568.

428 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3058C211yesmaximal, characteristic
2C53×#2159C211yesmaximal, characteristic, complemented
3C2×C53×#361C211yesmaximal, characteristic, complemented
4C2×C53×#357C211yesmaximal, characteristic, complemented
5C2×C53×#350C211yesmaximal, characteristic, complemented
6C53×#1816C211yesmaximal, characteristic, complemented
7C53×#1382C211yesmaximal, characteristic, complemented
8C22×D8×C53C2211yescharacteristic
9C2×#131×C53C2211yescharacteristic
10C53×#535—21no—
11C2×M32×C53C2211yescharacteristic
12C22×C16×C53C2211yescharacteristic
13C53×#469—22no—
14C53×#466—21no—
15C53×#466—21no—
16C2×#108×C53C2211yescharacteristic
17C53×#427C2211yescharacteristic
18C2×#107×C53C2211yescharacteristic
19C53×#418—21no—
20C53×#417—21no—
21C53×#367—21no—
22C53×#366—21no—
23C53×#361—22no—
24C53×#357—22no—
25C53×#350—22no—
26C53×#343—21no—
27C2×#41×C53D411yescharacteristic
28C22×D4×C53—21no—
29C2×D8×C53—21no—
30C2×D8×C53—21no—
31C2×D8×C53—21no—
32C2×D8×C53—21no—
33C2×D8×C53—21no—
34C2×D8×C53D412yes—
35#131×C53—21no—
36#131×C53—21no—
37#131×C53—21no—
38#131×C53—21no—
39C2×M16×C53—21no—
40C2×M16×C53—21no—
41C2×M16×C53D411yescharacteristic
42C22×C8×C53—21no—
43C22×C8×C53C2311yescharacteristic
44#127×C53—41no—
45#124×C53—41no—
46#121×C53—22no—
47#121×C53—22no—
48#120×C53—42no—
49#117×C53—21no—
50C2×C4⋊C4×C53—21no—
51M32×C53—21no—
52M32×C53D412yescomplemented
53C2×C16×C53—21no—
54C2×C16×C53—21no—
55C2×C16×C53—21no—
56C2×C16×C53—21no—
57#108×C53—21no—
58#108×C53—21no—
59#107×C53—21no—
60#107×C53—21no—
61#107×C53D411yescharacteristic
62#107×C53D411yescharacteristic
63#106×C53—21no—
64#103×C53—41no—
65#102×C53—22no—
66#102×C53—22no—
67#102×C53—41no—
68C24×C53—41no—
69#41×C53—21no—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53—21no—
73C2×Q8×C53D811yescharacteristic
74C2×D4×C53—21no—
75C2×D4×C53—22no—
76C2×D4×C53—42no—
77C2×D4×C53—42no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53D811yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87D8×C53—22no—
88D8×C53—22no—
89D8×C53—41no—
90D8×C53—41no—
91M16×C53—21no—
92M16×C53—21no—
93M16×C53—41no—
94M16×C53—41no—
95M16×C53—41no—
96M16×C53—41no—
97C2×C8×C53—21no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C2×C8×C53#4111yescharacteristic
106C2×C8×C53#4111yescharacteristic
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—42no—
113C22⋊C4×C53—41no—
114C42×C53—41no—
115C16×C53—22no—
116C16×C53—41no—
117C23×C53—21no—
118C23×C53—42no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53—161no—
122C23×C53—161no—
123C23×C53AGL(1,ℤ/8)11yescharacteristic
124Q8×C53—21no—
125Q8×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—41no—
129D4×C53—41no—
130D4×C53—42no—
131D4×C53—42no—
132D4×C53—81no—
133D4×C53—81no—
134D4×C53—162no—
135D4×C53—162no—
136C2×C4×C53—21no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—81no—
142C2×C4×C53—161no—
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C8×C53—41no—
150C8×C53—81no—
151C8×C53#12111yescharacteristic
152C8×C53#12111yescharacteristic
153C8×C53#12311yescharacteristic
154C8×C53#12411yescharacteristic
155#13387C5311yesmaximal, characteristic, direct product factor, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—82no—
162C22×C53—82no—
163C22×C53—82no—
164C22×C53—162no—
165C22×C53—162no—
166C22×C53#44711yescharacteristic
167C22×C53#46811yescharacteristic
168C22×C53#48211yescharacteristic
169C4×C53—41no—
170C4×C53—81no—
171C4×C53—81no—
172C4×C53—161no—
173C4×C53#39411yescharacteristic
174C4×C53#45211yescharacteristic
175C4×C53#46611yescharacteristic
176C4×C53#48011yescharacteristic
177#3058C2×C5311yescharacteristic
178#2159C2×C5311yescharacteristic, complemented
179C2×#361C2×C5311yescharacteristic, complemented
180C2×#357C2×C5311yescharacteristic, complemented
181C2×#350C2×C5311yescharacteristic, complemented
182#1816C2×C5311yescharacteristic, complemented
183#1382C2×C5311yescharacteristic, complemented
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—81no—
187C2×C53—162no—
188C2×C53#203311yescharacteristic
189C2×C53#219211yescharacteristic
190C2×C53#219411yescharacteristic
191C22×D8C22×C5311yescharacteristic
192C2×#131C22×C5311yescharacteristic
193#535—21no—
194C2×M32C22×C5311yescharacteristic
195C22×C16C22×C5311yescharacteristic
196#469—22no—
197#466—21no—
198#466—21no—
199C2×#108C22×C5311yescharacteristic
200#427C22×C5311yescharacteristic
201C2×#107C22×C5311yescharacteristic
202#418—21no—
203#417—21no—
204#367—21no—
205#366—21no—
206#361—22no—
207#357—22no—
208#350—22no—
209#343—21no—
210C53#1338711yescharacteristic, direct product factor, Sylow
211C2×#41D4×C5311yescharacteristic
212C22×D4—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8D4×C5312yes—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16D4×C5311yescharacteristic
226C22×C8—21no—
227C22×C8C23×C5311yescharacteristic
228#127—41no—
229#124—41no—
230#121—22no—
231#121—22no—
232#120—42no—
233#117—21no—
234C2×C4⋊C4—21no—
235M32—21no—
236M32D4×C5312yescomplemented
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241#108—21no—
242#108—21no—
243#107—21no—
244#107—21no—
245#107D4×C5311yescharacteristic
246#107D4×C5311yescharacteristic
247#106—21no—
248#103—41no—
249#102—22no—
250#102—22no—
251#102—41no—
252C24—41no—
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257C2×Q8D8×C5311yescharacteristic
258C2×D4—21no—
259C2×D4—22no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4D8×C5311yescharacteristic
268C22×C4—21no—
269C22×C4—41no—
270C22×C4C2×D4×C5311yescharacteristic
271D8—22no—
272D8—22no—
273D8—41no—
274D8—41no—
275M16—21no—
276M16—21no—
277M16—41no—
278M16—41no—
279M16—41no—
280M16—41no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8C2×D4×C5311yescharacteristic
286C2×C8C2×D4×C5311yescharacteristic
287C2×C8C2×D4×C5311yescharacteristic
288C2×C8#41×C5311yescharacteristic
289C2×C8#41×C5311yescharacteristic
290C2×C8#41×C5311yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C42—41no—
299C16—22no—
300C16—41no—
301C23—21no—
302C23—42no—
303C23—81no—
304C23—81no—
305C23—161no—
306C23—161no—
307C23AGL(1,ℤ/8)×C5311yescharacteristic
308Q8—21no—
309Q8—21no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—81no—
317D4—81no—
318D4—162no—
319D4—162no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—161no—
327C2×C4#120×C5311yescharacteristic
328C2×C4#121×C5311yescharacteristic
329C2×C4#123×C5311yescharacteristic
330C2×C4C2×D8×C5311yescharacteristic
331C2×C4#135×C5311yescharacteristic
332C2×C4#137×C5311yescharacteristic
333C8—41no—
334C8—81no—
335C8#121×C5311yescharacteristic
336C8#121×C5311yescharacteristic
337C8#123×C5311yescharacteristic
338C8#124×C5311yescharacteristic
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—162no—
348C22—162no—
349C22C53×#44711yescharacteristic
350C22C53×#46811yescharacteristic
351C22C53×#48211yescharacteristic
352C4—41no—
353C4—81no—
354C4—81no—
355C4—161no—
356C4C53×#39411yescharacteristic
357C4C53×#45211yescharacteristic
358C4C53×#46611yescharacteristic
359C4C53×#48011yescharacteristic
360C2—21no—
361C2—21no—
362C2—81no—
363C2—162no—
364C2C53×#203311yescharacteristic
365C2C53×#219211yescharacteristic
366C2C53×#219411yescharacteristic
subgroupG/H
center, socleC22×C53#447
commutator, Frattini, cosocleC22×C8C23×C53
prime radicalC53×#3058C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2159C2semidirect
3. C2×C53×#361C2semidirect
4. C2×C53×#357C2semidirect
5. C2×C53×#350C2semidirect
6. C53×#1816C2semidirect
7. C53×#1382C2semidirect
52. M32×C53D4semidirect
155. #13387C53direct (N × H)
178. #2159C2×C53semidirect
179. C2×#361C2×C53semidirect
180. C2×#357C2×C53semidirect
181. C2×#350C2×C53semidirect
182. #1816C2×C53semidirect
183. #1382C2×C53semidirect
210. C53#13387direct (N × H)
236. M32D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#30582
C53×#21592
C2×C53×#3612
C2×C53×#3572
C2×C53×#3502
C53×#18162
C53×#13822
#1338753