Finite Group Databasebeta

← C53×#8456 #27215968

Subgroups — C53×#8456 #27215968

Order 13568.

1026 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 154 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×#3496C211yesmaximal, characteristic
2C53×#1955C211yesmaximal, characteristic, complemented
3C53×#1551C212yesmaximal, complemented
4C53×#1530C212yesmaximal, complemented
5C53×#1466C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C53C412yes—
7C53×#547—22no—
8C53×#546—24no—
9D42×C53—22no—
10C53×#537C412yes—
11C53×#534—22no—
12C53×#532C2212yes—
13C2×#121×C53—22no—
14C2×C4×D4×C53C2211yescharacteristic
15C53×#432C2212yescomplemented
16C2×C4⋊C8×C53C2211yescharacteristic, complemented
17C2×C22⋊C8×C53—22no—
18C2×#97×C53C2211yescharacteristic, complemented
19C53×#333—22no—
20C53×#331—22no—
21C53×#329—24no—
22C53×#323—22no—
23H(5,𝔽2)×C53—48no—
24C2×#41×C53—24no—
25C2×#41×C53C2×C414yes—
26C22×D4×C53—24no—
27C22×D4×C53—22no—
28C22×D4×C53—24no—
29C22×D4×C53C2×C411yescharacteristic
30C23×C4×C53D412yes—
31C22×C8×C53—21no—
32C22×C8×C53—21no—
33C4⋊Q8×C53D412yes—
34#127×C53D412yes—
35#124×C53D414yes—
36#123×C53—28no—
37C22⋊Q8×C53—24no—
38#121×C53—24no—
39#121×C53—24no—
40#121×C53—24no—
41#121×C53—24no—
42#121×C53—22no—
43#121×C53—22no—
44#120×C53—28no—
45C4×D4×C53—22no—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C4×D4×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—24no—
52C2×C22⋊C4×C53D412yes—
53C2×C42×C53C2311yescharacteristic
54C4⋊C8×C53—22no—
55C4⋊C8×C53—22no—
56C22⋊C8×C53—22no—
57C22⋊C8×C53—44no—
58#97×C53—22no—
59#97×C53—22no—
60C24×C53—24no—
61C24×C53C22⋊C412yes—
62#41×C53—48no—
63#41×C53—416no—
64C2×Q8×C53C22⋊C414yes—
65C2×D4×C53—24no—
66C2×D4×C53—28no—
67C2×D4×C53—24no—
68C2×D4×C53—24no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—416no—
73C2×D4×C53—48no—
74C2×D4×C53—48no—
75C2×D4×C53—82no—
76C2×D4×C53—82no—
77C2×D4×C53C22⋊C414yes—
78C22×C4×C53—24no—
79C22×C4×C53—24no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—42no—
83C22×C4×C53—42no—
84C22×C4×C53C22⋊C412yes—
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C2×C8×C53—22no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—42no—
92C2×C8×C53—42no—
93C2×C8×C53—41no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C22⋊C4×C53—28no—
99C22⋊C4×C53—24no—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—24no—
102C42×C53C2×D412yes—
103C42×C53C2×D412yes—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—24no—
107C23×C53—42no—
108C23×C53—42no—
109C23×C53—48no—
110C23×C53—84no—
111C23×C53—84no—
112C23×C53#10014yes—
113C23×C53C2×C22⋊C411yescharacteristic
114Q8×C53—44no—
115D4×C53—28no—
116D4×C53—44no—
117D4×C53—416no—
118D4×C53—48no—
119D4×C53—42no—
120D4×C53—42no—
121D4×C53—82no—
122D4×C53—84no—
123C2×C4×C53—22no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—24no—
127C2×C4×C53—22no—
128C2×C4×C53—41no—
129C2×C4×C53—48no—
130C2×C4×C53—48no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—82no—
135C2×C4×C53C2×C22⋊C412yes—
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12012yes—
138C2×C4×C53#12011yescharacteristic
139C8×C53—42no—
140C8×C53—82no—
141C8×C53—81no—
142#8456C5311yesmaximal, characteristic, direct product factor, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—24no—
146C22×C53—44no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—44no—
150C22×C53—44no—
151C22×C53—44no—
152C22×C53—82no—
153C22×C53—168no—
154C22×C53#37911yescharacteristic
155C22×C53C2×#10012yes—
156C4×C53—21no—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—42no—
160C4×C53—44no—
161C4×C53#45311yescharacteristic
162C4×C53#45511yescharacteristic
163#3496C2×C5311yescharacteristic
164#1955C2×C5311yescharacteristic, complemented
165#1551C2×C5312yescomplemented
166#1530C2×C5312yescomplemented
167#1466C2×C5311yescharacteristic, complemented
168C2×C53—22no—
169C2×C53—44no—
170C2×C53—84no—
171C2×C53#178311yescharacteristic
172C2×C53#179211yescharacteristic
173C2×C53#179411yescharacteristic
174C2×H(5,𝔽2)C4×C5312yes—
175#547—22no—
176#546—24no—
177D42—22no—
178#537C4×C5312yes—
179#534—22no—
180#532C22×C5312yes—
181C2×#121—22no—
182C2×C4×D4C22×C5311yescharacteristic
183#432C22×C5312yescomplemented
184C2×C4⋊C8C22×C5311yescharacteristic, complemented
185C2×C22⋊C8—22no—
186C2×#97C22×C5311yescharacteristic, complemented
187#333—22no—
188#331—22no—
189#329—24no—
190#323—22no—
191C53#845611yescharacteristic, direct product factor, Sylow
192H(5,𝔽2)—48no—
193C2×#41—24no—
194C2×#41C2×C4×C5314yes—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4C2×C4×C5311yescharacteristic
199C23×C4D4×C5312yes—
200C22×C8—21no—
201C22×C8—21no—
202C4⋊Q8D4×C5312yes—
203#127D4×C5312yes—
204#124D4×C5314yes—
205#123—28no—
206C22⋊Q8—24no—
207#121—24no—
208#121—24no—
209#121—24no—
210#121—22no—
211#121—22no—
212#121—24no—
213#120—28no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4C2×C4×C5311yescharacteristic
220C2×C22⋊C4—24no—
221C2×C22⋊C4D4×C5312yes—
222C2×C42C23×C5311yescharacteristic
223C4⋊C8—22no—
224C4⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—44no—
227#97—22no—
228#97—22no—
229C24—24no—
230C24C22⋊C4×C5312yes—
231#41—48no—
232#41—416no—
233C2×Q8C22⋊C4×C5314yes—
234C2×D4—24no—
235C2×D4—28no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—416no—
242C2×D4—48no—
243C2×D4—48no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4C22⋊C4×C5314yes—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C22⋊C4×C5312yes—
254C22×C4C22×C4×C5311yescharacteristic
255C22×C4C2×D4×C5311yescharacteristic
256C22×C4C2×D4×C5311yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—41no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C22⋊C4—28no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C42C2×D4×C5312yes—
272C42C2×D4×C5312yes—
273C23—22no—
274C23—22no—
275C23—24no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—84no—
280C23—84no—
281C23#100×C5314yes—
282C23C2×C22⋊C4×C5311yescharacteristic
283Q8—44no—
284D4—28no—
285D4—44no—
286D4—416no—
287D4—48no—
288D4—42no—
289D4—42no—
290D4—82no—
291D4—84no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4C2×C22⋊C4×C5312yes—
305C2×C4#120×C5312yes—
306C2×C4#120×C5311yescharacteristic
307C2×C4#120×C5311yescharacteristic
308C8—42no—
309C8—82no—
310C8—81no—
311C22—21no—
312C22—21no—
313C22—24no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—82no—
321C22—168no—
322C22C53×#37911yescharacteristic
323C22C2×#100×C5312yes—
324C4—22no—
325C4—21no—
326C4—22no—
327C4—44no—
328C4—42no—
329C4C53×#45311yescharacteristic
330C4C53×#45511yescharacteristic
331C2—22no—
332C2—44no—
333C2—84no—
334C2C53×#178311yescharacteristic
335C2C53×#179211yescharacteristic
336C2C53×#179411yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1955C2semidirect
3. C53×#1551C2semidirect
4. C53×#1530C2semidirect
5. C53×#1466C2semidirect
15. C53×#432C22semidirect
16. C2×C4⋊C8×C53C22semidirect
18. C2×#97×C53C22semidirect
142. #8456C53direct (N × H)
164. #1955C2×C53semidirect
165. #1551C2×C53semidirect
166. #1530C2×C53semidirect
167. #1466C2×C53semidirect
183. #432C22×C53semidirect
184. C2×C4⋊C8C22×C53semidirect
186. C2×#97C22×C53semidirect
191. C53#8456direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34962
C53×#19552
C53×#15512
C53×#15302
C53×#14662
#845653