Finite Group Databasebeta

← C53×#11810 #27219322

Subgroups — C53×#11810 #27219322

Order 13568.

402 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2396C211yesmaximal, characteristic
2C53×#1820C211yesmaximal, characteristic
3C53×#1464C211yesmaximal, characteristic
4C53×#1352C212yesmaximal
5C53×#1351C212yesmaximal
6C2×C4×Q8×C53C411yescharacteristic, complemented
7C2×#117×C53C2211yescharacteristic
8C53×#432—21no—
9C53×#400C411yescharacteristic, complemented
10C53×#399C2211yescharacteristic
11C53×#398—21no—
12C53×#396—21no—
13C53×#395C411yescharacteristic, complemented
14C53×#393C2211yescharacteristic
15C53×#381—21no—
16C4×C4⋊C4×C53—21no—
17C4×C4⋊C4×C53C411yescharacteristic, complemented
18C4×C22⋊C4×C53—21no—
19C53×#342—21no—
20C53×#342C2212yes—
21C53×#324—21no—
22C53×#323C2212yes—
23C22×Q8×C53C2×C411yescharacteristic
24C23×C4×C53D411yescharacteristic
25C4×Q8×C53—41no—
26C4×Q8×C53—41no—
27#117×C53—41no—
28#117×C53—41no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C4⋊C4×C53C2×C411yescharacteristic
35C2×C4⋊C4×C53D411yescharacteristic
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—21no—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53C2311yescharacteristic
41C2×C42×C53—21no—
42C2×C42×C53—21no—
43C2×C42×C53C2×C411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C4⋊C8×C53—41no—
46C22⋊C8×C53—22no—
47#97×C53—41no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—41no—
53#95×C53—41no—
54#95×C53C2×C411yescharacteristic
55#95×C53C2×C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C2×Q8×C53—42no—
58C2×Q8×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—41no—
67C22×C4×C53—42no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C2×C8×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—42no—
93C42×C53—41no—
94C42×C53—41no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100C23×C53#13712yes—
101Q8×C53—42no—
102Q8×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—82no—
131C2×C4×C53—82no—
132C2×C4×C53—82no—
133C2×C4×C53#9911yescharacteristic
134C2×C4×C53#9911yescharacteristic
135C8×C53—82no—
136#11810C5311yesmaximal, characteristic, direct product factor, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—22no—
140C22×C53—42no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—81no—
144C22×C53C2×#9911yescharacteristic
145C22×C53#41911yescharacteristic
146C22×C53#42111yescharacteristic
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—42no—
150C4×C53—41no—
151C4×C53—42no—
152C4×C53—41no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—81no—
157C4×C53—81no—
158C4×C53—162no—
159#2396C2×C5311yescharacteristic
160#1820C2×C5311yescharacteristic
161#1464C2×C5311yescharacteristic
162#1352C2×C5312yes—
163#1351C2×C5312yes—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—42no—
167C2×C53#151111yescharacteristic
168C2×C53#212011yescharacteristic
169C2×C53#212611yescharacteristic
170C2×C4×Q8C4×C5311yescharacteristic, complemented
171C2×#117C22×C5311yescharacteristic
172#432—21no—
173#400C4×C5311yescharacteristic, complemented
174#399C22×C5311yescharacteristic
175#398—21no—
176#396—21no—
177#395C4×C5311yescharacteristic, complemented
178#393C22×C5311yescharacteristic
179#381—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4C4×C5311yescharacteristic, complemented
182C4×C22⋊C4—21no—
183#342—21no—
184#342C22×C5312yes—
185#324—21no—
186#323C22×C5312yes—
187C53#1181011yescharacteristic, direct product factor, Sylow
188C22×Q8C2×C4×C5311yescharacteristic
189C23×C4D4×C5311yescharacteristic
190C4×Q8—41no—
191C4×Q8—41no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×C4×C5311yescharacteristic
199C2×C4⋊C4C2×C4×C5311yescharacteristic
200C2×C4⋊C4D4×C5311yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4D4×C5311yescharacteristic
205C2×C22⋊C4C23×C5311yescharacteristic
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C2×C4×C5311yescharacteristic
209C2×C42D4×C5311yescharacteristic
210C4⋊C8—41no—
211C22⋊C8—22no—
212#97—41no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—41no—
218#95—41no—
219#95C2×C4×C5311yescharacteristic
220#95C2×C4×C5311yescharacteristic
221C24C2×D4×C5311yescharacteristic
222C2×Q8—42no—
223C2×Q8—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4C22⋊C4×C5311yescharacteristic
234C22×C4C22⋊C4×C5311yescharacteristic
235C22×C4C22⋊C4×C5311yescharacteristic
236C22×C4C22⋊C4×C5311yescharacteristic
237C22×C4C22×C4×C5311yescharacteristic
238C22×C4C2×D4×C5311yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—42no—
258C42—41no—
259C42—41no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊C4×C5311yescharacteristic
265C23#137×C5312yes—
266Q8—42no—
267Q8—42no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4#99×C5311yescharacteristic
299C2×C4#99×C5311yescharacteristic
300C8—82no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#99×C5311yescharacteristic
309C22C53×#41911yescharacteristic
310C22C53×#42111yescharacteristic
311C4—21no—
312C4—21no—
313C4—42no—
314C4—41no—
315C4—42no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—162no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2C53×#151111yescharacteristic
327C2C53×#212011yescharacteristic
328C2C53×#212611yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC24×C53C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×#117×C53 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C4×Q8×C53C4semidirect
9. C53×#400C4semidirect
13. C53×#395C4semidirect
17. C4×C4⋊C4×C53C4semidirect
136. #11810C53direct (N × H)
170. C2×C4×Q8C4×C53semidirect
173. #400C4×C53semidirect
177. #395C4×C53semidirect
181. C4×C4⋊C4C4×C53semidirect
187. C53#11810direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#23962
C53×#18202
C53×#14642
C53×#13522
C53×#13512
#1181053