Finite Group Databasebeta

← C53×#3345 #20272282

Subgroups — C53×#3345 #20272282

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#553C211yesmaximal, characteristic, complemented
2C53×#547C211yesmaximal, characteristic, complemented
3C53×#501C211yesmaximal, characteristic, complemented
4C53×#482C211yesmaximal, characteristic, complemented
5C53×#481C211yesmaximal, characteristic, complemented
6C53×#481C211yesmaximal, characteristic, complemented
7C53×#480C211yesmaximal, characteristic, complemented
8C53×#478C211yesmaximal, characteristic, complemented
9C53×#476C211yesmaximal, characteristic, complemented
10C53×#469C211yesmaximal, characteristic, complemented
11C53×#466C211yesmaximal, characteristic, complemented
12C53×#442C211yesmaximal, characteristic, complemented
13C53×#435C211yesmaximal, characteristic, complemented
14C53×#428C211yesmaximal, characteristic, complemented
15C2×#106×C53C211yesmaximal, characteristic, complemented
16C2×#41×C53—21no—
17C2×D8×C53C2211yescharacteristic
18C2×M16×C53C2211yescharacteristic
19C22×C8×C53C2211yescharacteristic
20C4⋊Q8×C53C2211yescharacteristic
21#126×C53—21no—
22#125×C53C2211yescharacteristic
23#123×C53—21no—
24#123×C53—21no—
25#123×C53—21no—
26C22⋊Q8×C53—21no—
27#121×C53—21no—
28#121×C53C2211yescharacteristic
29#121×C53C2211yescharacteristic
30#121×C53C2211yescharacteristic
31#121×C53C2211yescharacteristic
32C4×D4×C53—21no—
33C4×D4×C53C2211yescharacteristic
34C4×D4×C53C2211yescharacteristic
35C4×D4×C53C2211yescharacteristic
36#117×C53C2211yescharacteristic
37C2×C4⋊C4×C53C2211yescharacteristic
38C2×C4⋊C4×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic
40#107×C53C2211yescharacteristic
41#107×C53C2211yescharacteristic
42#107×C53C2211yescharacteristic
43#106×C53C2211yescharacteristic
44#106×C53C2211yescharacteristic
45#106×C53C2211yescharacteristic
46#106×C53C2211yescharacteristic
47#106×C53C2211yescharacteristic
48#106×C53C2211yescharacteristic
49C4⋊C8×C53C2211yescharacteristic
50#102×C53C2211yescharacteristic
51#102×C53C2211yescharacteristic
52#102×C53C2211yescharacteristic
53#102×C53C2211yescharacteristic
54#102×C53C2211yescharacteristic
55#102×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59#41×C53—44no—
60C2×Q8×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53D411yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C2×D4×C53C2311yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75D8×C53—22no—
76M16×C53—22no—
77C2×C8×C53—22no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53D411yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C4⋊C4×C53C2311yescharacteristic
93C4⋊C4×C53C2311yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C22⋊C4×C53—21no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53—21no—
101C22⋊C4×C53—21no—
102C22⋊C4×C53D411yescharacteristic
103C22⋊C4×C53D411yescharacteristic
104C42×C53—21no—
105C42×C53C2311yescharacteristic
106C23×C53—21no—
107C23×C53—21no—
108C23×C53C2×D411yescharacteristic
109C23×C53C2×D411yescharacteristic
110Q8×C53—42no—
111D4×C53—22no—
112D4×C53—21no—
113D4×C53—21no—
114D4×C53—21no—
115D4×C53—21no—
116D4×C53—42no—
117D4×C53—42no—
118D4×C53—42no—
119D4×C53—81no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2411yescharacteristic
140C8×C53—21no—
141C8×C53—21no—
142C8×C53—21no—
143C8×C53#4112yescomplemented
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53AGL(1,ℤ/8)12yes—
156C22×C53C22×D411yescharacteristic
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53C2×#4111yescharacteristic
168C4×C532−1+411yescharacteristic
169#3345C5311yesmaximal, characteristic, direct product factor, Sylow
170C2×C53—22no—
171C2×C53—41no—
172C2×C53—81no—
173C2×C53—81no—
174C2×C53#54711yescharacteristic
175C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C53#57711yescharacteristic
177#553C2×C5311yescharacteristic, complemented
178#547C2×C5311yescharacteristic, complemented
179#501C2×C5311yescharacteristic, complemented
180#482C2×C5311yescharacteristic, complemented
181#481C2×C5311yescharacteristic, complemented
182#481C2×C5311yescharacteristic, complemented
183#480C2×C5311yescharacteristic, complemented
184#478C2×C5311yescharacteristic, complemented
185#476C2×C5311yescharacteristic, complemented
186#469C2×C5311yescharacteristic, complemented
187#466C2×C5311yescharacteristic, complemented
188#442C2×C5311yescharacteristic, complemented
189#435C2×C5311yescharacteristic, complemented
190#428C2×C5311yescharacteristic, complemented
191C2×#106C2×C5311yescharacteristic, complemented
192C53#334511yescharacteristic, direct product factor, Sylow
193C2×#41—21no—
194C2×D8C22×C5311yescharacteristic
195C2×M16C22×C5311yescharacteristic
196C22×C8C22×C5311yescharacteristic
197C4⋊Q8C22×C5311yescharacteristic
198#126—21no—
199#125C22×C5311yescharacteristic
200#123—21no—
201#123—21no—
202#123—21no—
203C22⋊Q8—21no—
204#121—21no—
205#121C22×C5311yescharacteristic
206#121C22×C5311yescharacteristic
207#121C22×C5311yescharacteristic
208#121C22×C5311yescharacteristic
209C4×D4—21no—
210C4×D4C22×C5311yescharacteristic
211C4×D4C22×C5311yescharacteristic
212C4×D4C22×C5311yescharacteristic
213#117C22×C5311yescharacteristic
214C2×C4⋊C4C22×C5311yescharacteristic
215C2×C4⋊C4C22×C5311yescharacteristic
216C2×C4⋊C4C22×C5311yescharacteristic
217#107C22×C5311yescharacteristic
218#107C22×C5311yescharacteristic
219#107C22×C5311yescharacteristic
220#106C22×C5311yescharacteristic
221#106C22×C5311yescharacteristic
222#106C22×C5311yescharacteristic
223#106C22×C5311yescharacteristic
224#106C22×C5311yescharacteristic
225#106C22×C5311yescharacteristic
226C4⋊C8C22×C5311yescharacteristic
227#102C22×C5311yescharacteristic
228#102C22×C5311yescharacteristic
229#102C22×C5311yescharacteristic
230#102C22×C5311yescharacteristic
231#102C22×C5311yescharacteristic
232#102C22×C5311yescharacteristic
233C22⋊C8C22×C5311yescharacteristic
234C22⋊C8C22×C5311yescharacteristic
235#97C22×C5311yescharacteristic
236#41—44no—
237C2×Q8—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4D4×C5311yescharacteristic
243C2×D4C23×C5311yescharacteristic
244C2×D4C23×C5311yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4C23×C5311yescharacteristic
251C22×C4C23×C5311yescharacteristic
252D8—22no—
253M16—22no—
254C2×C8—22no—
255C2×C8C23×C5311yescharacteristic
256C2×C8C23×C5311yescharacteristic
257C2×C8C23×C5311yescharacteristic
258C2×C8C23×C5311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4D4×C5311yescharacteristic
266C4⋊C4C23×C5311yescharacteristic
267C4⋊C4C23×C5311yescharacteristic
268C4⋊C4C23×C5311yescharacteristic
269C4⋊C4C23×C5311yescharacteristic
270C4⋊C4C23×C5311yescharacteristic
271C4⋊C4C23×C5311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4D4×C5311yescharacteristic
280C22⋊C4D4×C5311yescharacteristic
281C42—21no—
282C42C23×C5311yescharacteristic
283C23—21no—
284C23—21no—
285C23C2×D4×C5311yescharacteristic
286C23C2×D4×C5311yescharacteristic
287Q8—42no—
288D4—22no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—81no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4C2×D4×C5311yescharacteristic
313C2×C4C2×D4×C5311yescharacteristic
314C2×C4C2×D4×C5311yescharacteristic
315C2×C4C2×D4×C5311yescharacteristic
316C2×C4C24×C5311yescharacteristic
317C8—21no—
318C8—21no—
319C8—21no—
320C8#41×C5312yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22AGL(1,ℤ/8)×C5312yes—
333C22C22×D4×C5311yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4C2×#41×C5311yescharacteristic
345C42−1+4×C5311yescharacteristic
346C2—22no—
347C2—41no—
348C2—81no—
349C2—81no—
350C2C53×#54711yescharacteristic
351C2C2×AGL(1,ℤ/8)×C5311yescharacteristic
352C2C53×#57711yescharacteristic
subgroupG/H
center, socleC22×C53C22×D4
commutator, Frattini, cosocleC2×C4C24×C53
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C4×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#553C2semidirect
2. C53×#547C2semidirect
3. C53×#501C2semidirect
4. C53×#482C2semidirect
5. C53×#481C2semidirect
6. C53×#481C2semidirect
7. C53×#480C2semidirect
8. C53×#478C2semidirect
9. C53×#476C2semidirect
10. C53×#469C2semidirect
11. C53×#466C2semidirect
12. C53×#442C2semidirect
13. C53×#435C2semidirect
14. C53×#428C2semidirect
15. C2×#106×C53C2semidirect
143. C8×C53#41semidirect
169. #3345C53direct (N × H)
177. #553C2×C53semidirect
178. #547C2×C53semidirect
179. #501C2×C53semidirect
180. #482C2×C53semidirect
181. #481C2×C53semidirect
182. #481C2×C53semidirect
183. #480C2×C53semidirect
184. #478C2×C53semidirect
185. #476C2×C53semidirect
186. #469C2×C53semidirect
187. #466C2×C53semidirect
188. #442C2×C53semidirect
189. #435C2×C53semidirect
190. #428C2×C53semidirect
191. C2×#106C2×C53semidirect
192. C53#3345direct (N × H)
320. C8#41×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#5532
C53×#5472
C53×#5012
C53×#4822
C53×#4812
C53×#4802
C53×#4782
C53×#4762
C53×#4692
C53×#4662
C53×#4422
C53×#4352
C53×#4282
C2×#106×C532
#334553