Finite Group Databasebeta

← #27276267

Subgroups — #27276267

Order 13568.

1014 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#20274633C211yesmaximal, characteristic, complemented
2#20274627C211yesmaximal, characteristic, complemented
3#20273271C214yesmaximal, complemented
4C53×#3485C211yesmaximal, characteristic
5C2×#13133132—22no—
6C2×C22⋊#88322—22no—
7C2×C22⋊#88322C2212yescomplemented
8#16810486C2212yescomplemented
9#16810360—24no—
10#16810357—24no—
11C2×H(5,𝔽2)×C53C411yescharacteristic, complemented
12C22×#41×C53C2211yescharacteristic
13C53×#547—24no—
14C53×#546—24no—
15C53×#536C412yescomplemented
16C53×#534—22no—
17C53×#525C2212yes—
18C2×#123×C53—22no—
19C53×#518C411yescharacteristic, complemented
20C22×#88322—22no—
21C2×C22⋊Dic53—22no—
22#13133132—44no—
23C22⋊#88322—24no—
24C22⋊#88322—44no—
25#13133124—24no—
26H(5,𝔽2)×C53—44no—
27C2×#41×C53—22no—
28C2×#41×C53—24no—
29C2×#41×C53—44no—
30C2×#41×C53C2×C412yes—
31C22×Q8×C53C2×C411yescharacteristic
32C22×D4×C53—22no—
33C22×D4×C53—22no—
34C22×D4×C53C2×C411yescharacteristic
35C22×D4×C53D412yes—
36C23×C4×C53D412yes—
37C23×C4×C53C2311yescharacteristic
38C4⋊Q8×C53—22no—
39#124×C53—22no—
40#123×C53—28no—
41#123×C53—24no—
42#123×C53—24no—
43#123×C53—22no—
44C22⋊Q8×C53—24no—
45C22⋊Q8×C53D414yes—
46#121×C53—28no—
47#121×C53D414yes—
48#120×C53—24no—
49#120×C53—22no—
50C4×D4×C53—24no—
51C4×D4×C53—24no—
52#117×C53C2×C412yes—
53C2×C4⋊C4×C53—24no—
54C2×C22⋊C4×C53—24no—
55C2×C22⋊C4×C53—22no—
56C24×C53—22no—
57C24×C53C22⋊C412yes—
58C24×C53C2×D411yescharacteristic
59#41×C53—44no—
60#41×C53—416no—
61#41×C53—48no—
62C2×Q8×C53—22no—
63C2×Q8×C53—42no—
64C2×Q8×C53C22⋊C412yes—
65C2×D4×C53—24no—
66C2×D4×C53—24no—
67C2×D4×C53—24no—
68C2×D4×C53—22no—
69C2×D4×C53—24no—
70C2×D4×C53—24no—
71C2×D4×C53—22no—
72C2×D4×C53—42no—
73C2×D4×C53—48no—
74C2×D4×C53—44no—
75C2×D4×C53—44no—
76C2×D4×C53C22⋊C412yes—
77C22×C4×C53—22no—
78C22×C4×C53—24no—
79C22×C4×C53—24no—
80C22×C4×C53—22no—
81C22×C4×C53—24no—
82C22×C4×C53—42no—
83C22×C4×C53—42no—
84C22×C4×C53C22⋊C412yes—
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D414yes—
88C22×Dic53—42no—
89C4⋊C4×C53—28no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53C22⋊C414yes—
92C22⋊C4×C53—24no—
93C22⋊C4×C53—28no—
94C22⋊C4×C53—22no—
95C22⋊C4×C53—24no—
96C22⋊C4×C53—24no—
97C42×C53—22no—
98C22⋊Dic53—44no—
99C2×#88322—44no—
100C2×#88322—42no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—42no—
106C23×C53—44no—
107C23×C53—44no—
108C23×C53—84no—
109C23×C53#9914yes—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53#12012yes—
112Q8×C53—44no—
113Q8×C53—42no—
114D4×C53—28no—
115D4×C53—42no—
116D4×C53—48no—
117D4×C53—48no—
118D4×C53—44no—
119D4×C53—84no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—24no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—44no—
128C2×C4×C53—48no—
129C2×C4×C53—44no—
130C2×C4×C53—44no—
131C2×C4×C53—84no—
132C2×C4×C53C2×C22⋊C412yes—
133C2×C4×C53#12012yes—
134C2×Dic53—84no—
135C2×Dic53—82no—
136#88322—82no—
137#8462—531nomaximal, Sylow
138C22×C53—22no—
139C22×C53—24no—
140C22×C53—22no—
141C22×C53—24no—
142C22×C53—44no—
143C22×C53—44no—
144C22×C53—44no—
145C22×C53—42no—
146C22×C53—82no—
147C22×C53—82no—
148C22×C53—164no—
149C22×C53#37911yescharacteristic
150C22×C53C2×#9912yes—
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—44no—
154C4×C53—42no—
155C4×C53—42no—
156Dic53—162no—
157#3485D5311yescharacteristic
158#1782—531no—
159#1780—531no—
160#1516—534no—
161C2×C53—22no—
162C2×C53—42no—
163C2×C53—44no—
164C2×C53—82no—
165C2×C53#178111yescharacteristic
166C2×C53#179412yes—
167C2×H(5,𝔽2)Dic5311yescharacteristic, complemented
168C22×#41C2×D5311yescharacteristic
169#547—24no—
170#546—24no—
171#536Dic5312yescomplemented
172#534—22no—
173#525C2×D5312yes—
174C2×#123—22no—
175#518Dic5311yescharacteristic, complemented
176C2×#99—1062no—
177C2×C22⋊C8—532no—
178C2×C22⋊C8—1062no—
179#381—532no—
180#328—1064no—
181#327—1064no—
182C53#846211yescharacteristic, complemented, Sylow
183H(5,𝔽2)—44no—
184C2×#41—22no—
185C2×#41—24no—
186C2×#41—44no—
187C2×#41C2×Dic5312yes—
188C22×Q8C2×Dic5311yescharacteristic
189C22×D4—22no—
190C22×D4—22no—
191C22×D4C2×Dic5311yescharacteristic
192C22×D4C22⋊D5312yes—
193C23×C4C22⋊D5312yes—
194C23×C4C22×D5311yescharacteristic
195C22×C8—1062no—
196C4⋊Q8—22no—
197#124—22no—
198#123—28no—
199#123—24no—
200#123—22no—
201#123—24no—
202C22⋊Q8—24no—
203C22⋊Q8C22⋊D5314yes—
204#121—28no—
205#121C22⋊D5314yes—
206#120—24no—
207#120—22no—
208C4×D4—24no—
209C4×D4—24no—
210#117C2×Dic5312yes—
211C2×C4⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—1062no—
215#99—2124no—
216C22⋊C8—1064no—
217C22⋊C8—2124no—
218#95—1064no—
219C24—22no—
220C24C22⋊Dic5312yes—
221C24C2×C22⋊D5311yescharacteristic
222#41—44no—
223#41—416no—
224#41—48no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×Q8C22⋊Dic5312yes—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—48no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4C22⋊Dic5312yes—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—2122no—
248C22×C4C22⋊Dic5312yes—
249C22×C4C22×Dic5311yescharacteristic
250C22×C4C2×C22⋊D5311yescharacteristic
251C22×C4C2×C22⋊D5314yes—
252C2×C8—2124no—
253C2×C8—2122no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4C22⋊Dic5314yes—
257C22⋊C4—24no—
258C22⋊C4—28no—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—2124no—
263C42—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—42no—
269C23—44no—
270C23—44no—
271C23—84no—
272C23#1313313214yes—
273C23C2×C22⋊Dic5311yescharacteristic
274C23#1313318612yes—
275Q8—42no—
276Q8—44no—
277D4—28no—
278D4—42no—
279D4—48no—
280D4—48no—
281D4—44no—
282D4—84no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—84no—
295C2×C4—4242no—
296C2×C4—4244no—
297C2×C4C2×C22⋊Dic5312yes—
298C2×C4#1313318612yes—
299C8—4242no—
300C22—22no—
301C22—24no—
302C22—22no—
303C22—24no—
304C22—44no—
305C22—44no—
306C22—42no—
307C22—44no—
308C22—82no—
309C22—82no—
310C22—164no—
311C22#1681047911yescharacteristic
312C22C2×#1313313212yes—
313C4—22no—
314C4—22no—
315C4—42no—
316C4—42no—
317C4—44no—
318C4—8482no—
319C2—22no—
320C2—42no—
321C2—44no—
322C2—82no—
323C2#2027463011yescharacteristic
324C2#2027467912yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3485C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenter#3485D53
hypocenterC53#8462
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. #20274633C2semidirect
2. #20274627C2semidirect
3. #20273271C2semidirect
7. C2×C22⋊#88322C22semidirect
8. #16810486C22semidirect
11. C2×H(5,𝔽2)×C53C4semidirect
15. C53×#536C4semidirect
19. C53×#518C4semidirect
167. C2×H(5,𝔽2)Dic53semidirect
171. #536Dic53semidirect
175. #518Dic53semidirect
182. C53#8462semidirect

Maximal subgroups

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

subgroupindex
#202746332
#202746272
#202732712
C53×#34852
#846253