Finite Group Databasebeta

← C53×#13215 #27220727

Subgroups — C53×#13215 #27220727

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3155C211yesmaximal, characteristic
2C2×C53×#361C211yesmaximal, characteristic, complemented
3C53×#2145C211yesmaximal, characteristic, complemented
4C53×#2143C211yesmaximal, characteristic, complemented
5C53×#2140C211yesmaximal, characteristic, complemented
6C53×#2110C211yesmaximal, characteristic, complemented
7C53×#1833C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C53—21no—
9C2×#135×C53C2211yescharacteristic
10C22×D8×C53C2211yescharacteristic
11C53×#535—21no—
12C2×M32×C53C2211yescharacteristic
13C2×M32×C53C2211yescharacteristic
14C53×#496—22no—
15C53×#495C412yescomplemented
16C53×#493C412yescomplemented
17C2×#108×C53C2211yescharacteristic
18C53×#427C2211yescharacteristic
19C53×#405C2211yescharacteristic
20C53×#361—22no—
21C53×#360—22no—
22C53×#359—22no—
23C53×#357—22no—
24C53×#347—22no—
25C53×#344—21no—
26C53×H(3,ℤ/4)—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×C53C2×C411yescharacteristic
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×C53C2×C411yescharacteristic
45C2×D8×C53C2×C412yes—
46C2×M16×C53—21no—
47C2×M16×C53D411yescharacteristic
48C22×C8×C53C2311yescharacteristic
49#127×C53—22no—
50#124×C53—22no—
51#121×C53—42no—
52#120×C53—42no—
53#117×C53—21no—
54#117×C53D411yescharacteristic
55M32×C53—21no—
56M32×C53—21no—
57M32×C53—21no—
58M32×C53D412yescomplemented
59C2×C16×C53—21no—
60C2×C16×C53D412yescomplemented
61#108×C53—21no—
62#108×C53—21no—
63#107×C53—21no—
64#106×C53—21no—
65#97×C53D412yes—
66C4×C8×C53C2×C412yes—
67C24×C53—41no—
68#41×C53—41no—
69#41×C53—41no—
70#41×C53—41no—
71#41×C53—41no—
72C2×Q8×C53—21no—
73C2×D4×C53—22no—
74C2×D4×C53—21no—
75C2×D4×C53—44no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84Q16×C53—21no—
85Q16×C53—21no—
86SD16×C53—44no—
87SD16×C53—41no—
88SD16×C53—41no—
89D8×C53—22no—
90D8×C53—22no—
91D8×C53—21no—
92D8×C53—21no—
93D8×C53—44no—
94D8×C53—41no—
95D8×C53—41no—
96M16×C53—21no—
97M16×C53—21no—
98M16×C53—41no—
99M16×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53C22⋊C412yes—
102C2×C8×C53C22×C411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53#4111yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C4⋊C4×C53—21no—
109C4⋊C4×C53—41no—
110C22⋊C4×C53—21no—
111C22⋊C4×C53—44no—
112C22⋊C4×C53—41no—
113C42×C53—41no—
114C42×C53C22⋊C412yes—
115C16×C53—22no—
116C16×C53—22no—
117C23×C53—41no—
118C23×C53—42no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53—162no—
122C23×C53AGL(1,ℤ/8)11yescharacteristic
123Q8×C53—41no—
124Q8×C53—41no—
125D4×C53—42no—
126D4×C53—41no—
127D4×C53—42no—
128D4×C53—41no—
129D4×C53—81no—
130D4×C53—81no—
131D4×C53—81no—
132D4×C53—81no—
133D4×C53—164no—
134C2×C4×C53—22no—
135C2×C4×C53—41no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53C2×C22⋊C411yescharacteristic
140C2×C4×C53#12411yescharacteristic
141C2×C4×C53#12711yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C8×C53—22no—
146C8×C53—81no—
147C8×C53C4×D411yescharacteristic
148C8×C53C4×D411yescharacteristic
149C8×C53#12111yescharacteristic
150C8×C53#12111yescharacteristic
151#13215C5311yesmaximal, characteristic, direct product factor, Sylow
152C22×C53—41no—
153C22×C53—82no—
154C22×C53—84no—
155C22×C53—81no—
156C22×C53—82no—
157C22×C53—162no—
158C22×C53—162no—
159C22×C53#41811yescharacteristic
160C22×C53#49011yescharacteristic
161C22×C53#49611yescharacteristic
162C4×C53—42no—
163C4×C53—81no—
164C4×C53—161no—
165C4×C53#39011yescharacteristic
166C4×C53#41811yescharacteristic
167C4×C53#49011yescharacteristic
168C4×C53#49611yescharacteristic
169#3155C2×C5311yescharacteristic
170C2×#361C2×C5311yescharacteristic, complemented
171#2145C2×C5311yescharacteristic, complemented
172#2143C2×C5311yescharacteristic, complemented
173#2140C2×C5311yescharacteristic, complemented
174#2110C2×C5311yescharacteristic, complemented
175#1833C2×C5311yescharacteristic, complemented
176C2×C53—21no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—161no—
180C2×C53—162no—
181C2×C53#196111yescharacteristic
182C2×AGL(1,ℤ/8)—21no—
183C2×#135C22×C5311yescharacteristic
184C22×D8C22×C5311yescharacteristic
185#535—21no—
186C2×M32C22×C5311yescharacteristic
187C2×M32C22×C5311yescharacteristic
188#496—22no—
189#495C4×C5312yescomplemented
190#493C4×C5312yescomplemented
191C2×#108C22×C5311yescharacteristic
192#427C22×C5311yescharacteristic
193#405C22×C5311yescharacteristic
194#361—22no—
195#360—22no—
196#359—22no—
197#357—22no—
198#347—22no—
199#344—21no—
200H(3,ℤ/4)—21no—
201C53#1321511yescharacteristic, direct product factor, Sylow
202C2×#41—21no—
203C22×D4—21no—
204AGL(1,ℤ/8)—42no—
205AGL(1,ℤ/8)—42no—
206#135—21no—
207#135—21no—
208#135—21no—
209#135—21no—
210C2×Q16C2×C4×C5311yescharacteristic
211C2×SD16—21no—
212C2×SD16—22no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—22no—
219C2×D8C2×C4×C5312yes—
220C2×D8C2×C4×C5311yescharacteristic
221C2×M16—21no—
222C2×M16D4×C5311yescharacteristic
223C22×C8C23×C5311yescharacteristic
224#127—22no—
225#124—22no—
226#121—42no—
227#120—42no—
228#117—21no—
229#117D4×C5311yescharacteristic
230M32—21no—
231M32—21no—
232M32—21no—
233M32D4×C5312yescomplemented
234C2×C16—21no—
235C2×C16D4×C5312yescomplemented
236#108—21no—
237#108—21no—
238#107—21no—
239#106—21no—
240#97D4×C5312yes—
241C4×C8C2×C4×C5312yes—
242C24—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247C2×Q8—21no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C22×C4—41no—
258C22×C4C2×D4×C5311yescharacteristic
259Q16—21no—
260Q16—21no—
261SD16—44no—
262SD16—41no—
263SD16—41no—
264D8—22no—
265D8—22no—
266D8—21no—
267D8—21no—
268D8—44no—
269D8—41no—
270D8—41no—
271M16—21no—
272M16—21no—
273M16—41no—
274M16—41no—
275C2×C8—41no—
276C2×C8C22⋊C4×C5312yes—
277C2×C8C22×C4×C5311yescharacteristic
278C2×C8C2×D4×C5311yescharacteristic
279C2×C8C2×D4×C5311yescharacteristic
280C2×C8C2×D4×C5311yescharacteristic
281C2×C8#41×C5311yescharacteristic
282C2×C8#41×C5311yescharacteristic
283C4⋊C4—21no—
284C4⋊C4—41no—
285C22⋊C4—21no—
286C22⋊C4—44no—
287C22⋊C4—41no—
288C42—41no—
289C42C22⋊C4×C5312yes—
290C16—22no—
291C16—22no—
292C23—42no—
293C23—41no—
294C23—81no—
295C23—81no—
296C23—162no—
297C23AGL(1,ℤ/8)×C5311yescharacteristic
298Q8—41no—
299Q8—41no—
300D4—42no—
301D4—41no—
302D4—41no—
303D4—42no—
304D4—81no—
305D4—81no—
306D4—81no—
307D4—81no—
308D4—164no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4C2×C22⋊C4×C5311yescharacteristic
315C2×C4#124×C5311yescharacteristic
316C2×C4#127×C5311yescharacteristic
317C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
318C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
319C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
320C8—22no—
321C8—81no—
322C8C4×D4×C5311yescharacteristic
323C8C4×D4×C5311yescharacteristic
324C8#121×C5311yescharacteristic
325C8#121×C5311yescharacteristic
326C22—41no—
327C22—82no—
328C22—84no—
329C22—81no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22C53×#41811yescharacteristic
334C22C53×#49011yescharacteristic
335C22C53×#49611yescharacteristic
336C4—42no—
337C4—81no—
338C4—161no—
339C4C53×#39011yescharacteristic
340C4C53×#41811yescharacteristic
341C4C53×#49011yescharacteristic
342C4C53×#49611yescharacteristic
343C2—21no—
344C2—21no—
345C2—21no—
346C2—162no—
347C2—161no—
348C2C53×#196111yescharacteristic
subgroupG/H
center, socleC2×C53#1961
commutatorC2×C8C22×C4×C53
Frattini, cosocleC22×C8C23×C53
prime radicalC53×#3155C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×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. C53×#2145C2semidirect
4. C53×#2143C2semidirect
5. C53×#2140C2semidirect
6. C53×#2110C2semidirect
7. C53×#1833C2semidirect
15. C53×#495C4semidirect
16. C53×#493C4semidirect
58. M32×C53D4semidirect
60. C2×C16×C53D4semidirect
151. #13215C53direct (N × H)
170. C2×#361C2×C53semidirect
171. #2145C2×C53semidirect
172. #2143C2×C53semidirect
173. #2140C2×C53semidirect
174. #2110C2×C53semidirect
175. #1833C2×C53semidirect
189. #495C4×C53semidirect
190. #493C4×C53semidirect
201. C53#13215direct (N × H)
233. M32D4×C53semidirect
235. C2×C16D4×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
C53×#21452
C53×#21432
C53×#21402
C53×#21102
C53×#18332
#1321553