Finite Group Databasebeta

← #27362272

Subgroups — #27362272

Order 13568.

928 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 179 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#20285863C212yesmaximal, complemented
2#20285175C211yesmaximal, characteristic
3#20273855C214yesmaximal, complemented
4#20273836C214yesmaximal, complemented
5#20273377C211yesmaximal, characteristic
6#20273136C212yesmaximal, complemented
7C53×#3550C211yesmaximal, characteristic
8C2×#13133247—22no—
9C2×#13133217C2211yescharacteristic
10#16811026—22no—
11#16811018—22no—
12#16811002C2212yes—
13#16810995C2212yes—
14#16810973—22no—
15#16810956—24no—
16#16810918C2218yes—
17#16810676C2212yes—
18C4×#11719917C2211yescharacteristic
19#16810372C2212yes—
20#16810367C2212yes—
21Q8⋊#88322C2218yes—
22C53×#563—22no—
23C53×#559C2212yes—
24Q82×C53C2212yes—
25C53×#556—22no—
26D4×Q8×C53—22no—
27C53×#542—24no—
28C53×#536—22no—
29C53×#533C2212yes—
30C53×#532C2211yescharacteristic
31C22×Dic106—21no—
32#13133247—48no—
33C2×#11719925—24no—
34C2×#11719923—24no—
35C2×#11719917—21no—
36#13133217—21no—
37#13133217—22no—
38#13133217C2312yes—
39C2×C4⋊Dic53—22no—
40#13133150—28no—
41C4⋊#88322C2314yes—
42#13133141—28no—
43C22⋊#88322—22no—
44#13133128—21no—
45C4×#88322—22no—
46C4×#88322C2312yes—
47C2×#41×C53D412yes—
48C22×Q8×C53—22no—
49C4⋊Q8×C53—24no—
50C4⋊Q8×C53—24no—
51C4⋊Q8×C53D412yes—
52C4⋊Q8×C53D412yes—
53C4⋊Q8×C53C2311yescharacteristic
54#127×C53C2311yescharacteristic
55#126×C53—28no—
56#125×C53—22no—
57#124×C53—24no—
58#124×C53—24no—
59#124×C53D412yes—
60#123×C53—24no—
61#123×C53—24no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—28no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53D414yes—
66#121×C53—22no—
67C4×Q8×C53—22no—
68C4×Q8×C53—24no—
69C4×Q8×C53C2314yes—
70C4×D4×C53—24no—
71C4×D4×C53—22no—
72#117×C53—22no—
73#117×C53—22no—
74C2×C42×C53C2311yescharacteristic
75#41×C53—44no—
76C2×Q8×C53—24no—
77C2×Q8×C53—22no—
78C2×Q8×C53—84no—
79C2×Q8×C53C2×D414yes—
80C2×Q8×C53C2×D412yes—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53C2×D412yes—
84C22×C4×C53—24no—
85C22×C4×C53—22no—
86C22×C4×C53C2×D412yes—
87C22×C4×C53C2×D411yescharacteristic
88C22×Dic53—41no—
89C2×Dic106—22no—
90C2×Dic106—44no—
91C2×Dic106—41no—
92C4⋊C4×C53—28no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—24no—
99C4⋊C4×C53C2×D414yes—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—28no—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—24no—
104C42×C53—24no—
105C42×C53—24no—
106C42×C53C2×D412yes—
107C42×C53C2×D411yescharacteristic
108C42×C53C2411yescharacteristic
109#11719925—48no—
110#11719923—48no—
111C4⋊Dic53—24no—
112C4⋊Dic53—42no—
113#11719917—24no—
114C2×#88322—22no—
115C2×#88322—24no—
116C23×C53—22no—
117C23×C53#12011yescharacteristic
118Q8×C53—28no—
119Q8×C53—42no—
120Q8×C53—44no—
121Q8×C53—84no—
122D4×C53—42no—
123D4×C53—44no—
124D4×C53—82no—
125C2×C4×C53—24no—
126C2×C4×C53—22no—
127C2×C4×C53—24no—
128C2×C4×C53—22no—
129C2×C4×C53—24no—
130C2×C4×C53—44no—
131C2×C4×C53—84no—
132C2×C4×C53#12012yes—
133C2×C4×C53#12011yescharacteristic
134C2×C4×C53C22×D411yescharacteristic
135C2×C4×C53C22×D412yes—
136C2×Dic53—42no—
137C2×Dic53—82no—
138Dic106—44no—
139Dic106—82no—
140#88322—24no—
141#88322—44no—
142#18719—531nomaximal, Sylow
143C22×C53—22no—
144C22×C53—21no—
145C22×C53—44no—
146C22×C53—42no—
147C22×C53C2×#12011yescharacteristic
148C4×C53—22no—
149C4×C53—21no—
150C4×C53—44no—
151C4×C53—42no—
152C4×C53—44no—
153C4×C53#45312yes—
154C4×C53#57814yes—
155Dic53—82no—
156#3550D5311yescharacteristic
157#3230—532no—
158#3152—531no—
159#1633—534no—
160#1630—534no—
161#1543—531no—
162#1485—532no—
163C2×C53—41no—
164C2×C53—82no—
165C2×C53#301112yes—
166C2×C53C2×#45311yescharacteristic
167C2×#137—1062no—
168#563—22no—
169#559C2×D5312yes—
170Q82C2×D5312yes—
171#556—22no—
172D4×Q8—22no—
173#542—24no—
174#536—22no—
175#533C2×D5312yes—
176#532C2×D5311yescharacteristic
177C2×C4⋊Q8—531no—
178#500—1062no—
179#497—1062no—
180#494—532no—
181#492—532no—
182#487—1062no—
183#484—1064no—
184#477—538no—
185#431—532no—
186C4×M16—531no—
187#333—532no—
188#331—532no—
189Q8⋊C8—538no—
190C53#1871911yescharacteristic, complemented, Sylow
191C2×#41C22⋊D5312yes—
192C22×Q8—22no—
193C22×Q8—1061no—
194#137—2128no—
195C2×Q16—1064no—
196C2×SD16—1064no—
197C2×M16—1061no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—532no—
201C4⋊Q8—1062no—
202C4⋊Q8—1061no—
203C4⋊Q8C22⋊D5312yes—
204C4⋊Q8C22⋊D5312yes—
205C4⋊Q8C22×D5311yescharacteristic
206#127C22×D5311yescharacteristic
207#126—28no—
208#125—22no—
209#124—24no—
210#124—24no—
211#124C22⋊D5312yes—
212#123—24no—
213#123—24no—
214C22⋊Q8—24no—
215C22⋊Q8—28no—
216C22⋊Q8—22no—
217C22⋊Q8C22⋊D5314yes—
218#121—22no—
219C4×Q8—24no—
220C4×Q8—22no—
221C4×Q8C22×D5314yes—
222C4×D4—24no—
223C4×D4—22no—
224#117—22no—
225#117—22no—
226C2×C4⋊C4—1062no—
227C2×C42C22×D5311yescharacteristic
228#107—1068no—
229C4⋊C8—534no—
230#103—1068no—
231C22⋊C8—1062no—
232#97—1061no—
233C4×C8—532no—
234C4×C8—1062no—
235#41—44no—
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—84no—
239C2×Q8—1062no—
240C2×Q8—2124no—
241C2×Q8—2121no—
242C2×Q8C2×C22⋊D5314yes—
243C2×Q8C2×C22⋊D5312yes—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4C2×C22⋊D5312yes—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—2121no—
250C22×C4C2×C22⋊D5312yes—
251C22×C4C2×C22⋊D5311yescharacteristic
252Q16—2128no—
253SD16—2128no—
254M16—1064no—
255C2×C8—1062no—
256C2×C8—1064no—
257C4⋊C4—28no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—1064no—
265C4⋊C4—2122no—
266C4⋊C4C2×C22⋊D5314yes—
267C22⋊C4—24no—
268C22⋊C4—28no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C42—24no—
272C42—24no—
273C42C2×C22⋊D5312yes—
274C42C2×C22⋊D5311yescharacteristic
275C42C23×D5311yescharacteristic
276C23—22no—
277C23#1313318611yescharacteristic
278Q8—28no—
279Q8—42no—
280Q8—44no—
281Q8—84no—
282Q8—4242no—
283Q8—2124no—
284D4—42no—
285D4—44no—
286D4—82no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—44no—
293C2×C4—84no—
294C2×C4—4242no—
295C2×C4—2122no—
296C2×C4#1313318612yes—
297C2×C4#1313318611yescharacteristic
298C2×C4C22×C22⋊D5311yescharacteristic
299C2×C4C22×C22⋊D5312yes—
300C8—1064no—
301C8—2124no—
302C22—22no—
303C22—21no—
304C22—44no—
305C22—42no—
306C22C2×#1313318611yescharacteristic
307C4—22no—
308C4—21no—
309C4—44no—
310C4—42no—
311C4—44no—
312C4—4242no—
313C4#1681079112yes—
314C4#1681143814yes—
315C2—41no—
316C2—82no—
317C2#2028401012yes—
318C2C2×#1681079111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3550C2
socleC22×C53C2×#120
prime radicalC53×#532C22
hypercenter#3550D53
hypocenterC53#18719
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #20285863C2semidirect
3. #20273855C2semidirect
4. #20273836C2semidirect
6. #20273136C2semidirect
190. C53#18719semidirect

Maximal subgroups

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

subgroupindex
#202858632
#202851752
#202738552
#202738362
#202733772
#202731362
C53×#35502
#1871953