Finite Group Databasebeta

← #27281449

Subgroups — #27281449

Order 13568.

486 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 117 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#20283795C211yesmaximal, characteristic
2#20279924C211yesmaximal, characteristic
3#20274708C211yesmaximal, characteristic
4#20273396C211yesmaximal, characteristic
5#20272617C211yesmaximal, characteristic
6#20272616C211yesmaximal, characteristic
7C53×#1548C211yesmaximal, characteristic
8C2×C4×Dic106C2211yescharacteristic
9C8×Dic106—22no—
10#16810747—22no—
11#16810676C2211yescharacteristic
12#16810673—22no—
13#16810647C2211yescharacteristic
14C2×#13133135—21no—
15#16810542C414yes—
16#16810528—21no—
17#16810519—22no—
18#16810492—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810391—21no—
21#16810372—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C53×#431C2211yescharacteristic
24C53×#423C2211yescharacteristic
25C53×#333—22no—
26C22×Dic106D411yescharacteristic
27C2×#11719917—21no—
28C4×Dic106—21no—
29C4×Dic106—21no—
30C4×Dic106—21no—
31C4×Dic106—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×#11719919—24no—
35C2×#11719919C2×C412yes—
36C2×C4×Dic53—21no—
37C2×C4×Dic53D412yes—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322D412yes—
41#13133146—24no—
42#13133135—42no—
43C22⋊#88322—21no—
44C4×#88322—21no—
45C8×Dic53—24no—
46#13133124—24no—
47#13133124C2×C412yes—
48C22×Q8×C53C2×C412yes—
49C2×M16×C53—21no—
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53D412yes—
60C22⋊C8×C53—21no—
61C4×C8×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—22no—
64C2×Q8×C53—82no—
65C22×C4×C53—22no—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—22no—
70C22×Dic53C22⋊C412yes—
71C22×Dic53C22⋊C412yes—
72C2×Dic106—21no—
73C2×Dic106—21no—
74C2×Dic106—21no—
75C2×Dic106—21no—
76C2×Dic106SD1612yes—
77C2×Dic106Q1612yes—
78M16×C53—41no—
79M16×C53—41no—
80C2×C8×C53—22no—
81C2×C8×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C42×C53C2×D411yescharacteristic
87C42×C53C2×D411yescharacteristic
88C42×C53#4111yescharacteristic
89C42×C53#4111yescharacteristic
90C4⋊Dic53—24no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93#11719919—22no—
94#11719919—22no—
95#11719919—84no—
96C4×Dic53—22no—
97C4×Dic53—22no—
98C4×Dic53—41no—
99C4×Dic53—41no—
100#11719917—41no—
101#11719917—41no—
102C2×#88322—22no—
103C2×#88322—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105Q8×C53—82no—
106Q8×C53—82no—
107C2×C4×C53—22no—
108C2×C4×C53—41no—
109C2×C4×C53—44no—
110C2×C4×C53—42no—
111C2×C4×C53C4×D411yescharacteristic
112C2×C4×C53C4×D411yescharacteristic
113C2×C4×C53#12011yescharacteristic
114C2×C4×C53#12111yescharacteristic
115C2×C4×C53#12111yescharacteristic
116C2×C4×C53#12311yescharacteristic
117C2×C4×C53C2×SD1611yescharacteristic
118C2×C4×C53C2×Q1611yescharacteristic
119C2×C4×C53#13511yescharacteristic
120C2×C4×C53#13511yescharacteristic
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—24no—
124C2×Dic53—22no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53#10314yes—
129Dic106—22no—
130Dic106—22no—
131Dic106—22no—
132Dic106—22no—
133C8×C53—42no—
134C8×C53—81no—
135#88322—42no—
136#88322—81no—
137#9392—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53#38611yescharacteristic
141C22×C53C2×#10311yescharacteristic
142C22×C53#41611yescharacteristic
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—82no—
147C4×C53#45612yes—
148C4×C53#46511yescharacteristic
149C4×C53#46511yescharacteristic
150C4×C53#46711yescharacteristic
151C4×C53#46711yescharacteristic
152Dic53—24no—
153Dic53—44no—
154Dic53—42no—
155#2982—531no—
156#2567—531no—
157#1801—531no—
158#1548—531no—
159#1548D5311yescharacteristic
160#1303—531no—
161#1303—531no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53#188611yescharacteristic
165C2×C53#189211yescharacteristic
166C2×C53#191411yescharacteristic
167C2×C4⋊Q8C2×D5311yescharacteristic
168C2×C4×Q8—531no—
169C8×Q8—1062no—
170C8×Q8—1062no—
171#431—531no—
172#431—1062no—
173#431C2×D5311yescharacteristic
174#423—531no—
175#423C2×D5311yescharacteristic
176C2×#101—1061no—
177#395—534no—
178#391—1061no—
179#389—1062no—
180#384—1061no—
181C4×C4⋊C4—531no—
182#343—1061no—
183#333—22no—
184#333—1062no—
185C53#939211yescharacteristic, complemented, Sylow
186C22×Q8—531no—
187C22×Q8C4×D5312yes—
188C2×M16—21no—
189C2×M16—1061no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4×Q8—1061no—
195C4×Q8—1061no—
196C4×Q8—1061no—
197C4×Q8—1061no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—532no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1064no—
204C2×C42—532no—
205C2×C42—1061no—
206C2×C42C22×D5311yescharacteristic
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—532no—
210C4⋊C8—1064no—
211C4⋊C8—1061no—
212C4⋊C8—1061no—
213C4⋊C8C22⋊D5312yes—
214#101—2122no—
215C22⋊C8—21no—
216C22⋊C8—1061no—
217C4×C8—21no—
218C4×C8—1064no—
219C4×C8—1061no—
220#95—532no—
221#95—1064no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—82no—
225C2×Q8—532no—
226C2×Q8—532no—
227C2×Q8—1061no—
228C2×Q8—1061no—
229C2×Q8—1061no—
230C2×Q8—1061no—
231C22×C4—22no—
232C22×C4—532no—
233C22×C4—532no—
234C22×C4—1062no—
235C22×C4C2×C4×D5311yescharacteristic
236C22×C4D4×D5311yescharacteristic
237C22×C4D4×D5311yescharacteristic
238M16—41no—
239M16—41no—
240M16—2121no—
241M16—2121no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—1062no—
245C2×C8—2121no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—1064no—
251C4⋊C4—1062no—
252C4⋊C4—1062no—
253C4⋊C4—1061no—
254C4⋊C4—1061no—
255C4⋊C4—4244no—
256C42—1062no—
257C42—1062no—
258C42—2121no—
259C42—2121no—
260C42#1171994511yescharacteristic
261C42D4×D5311yescharacteristic
262C42#1171994711yescharacteristic
263C42C2×C22⋊D5311yescharacteristic
264C23C22⋊C4×D5311yescharacteristic
265Q8—82no—
266Q8—82no—
267Q8—1062no—
268Q8—1062no—
269Q8—1062no—
270Q8—1062no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—534no—
276C2×C4—1062no—
277C2×C4—1062no—
278C2×C4—1064no—
279C2×C4—1062no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4#1313317711yescharacteristic
284C2×C4C4×C22⋊D5311yescharacteristic
285C2×C4#1313318511yescharacteristic
286C2×C4#1313318811yescharacteristic
287C2×C4#1313319011yescharacteristic
288C2×C4#1313319911yescharacteristic
289C2×C4SD16×D5311yescharacteristic
290C2×C4Q16×D5311yescharacteristic
291C2×C4#1313323811yescharacteristic
292C2×C4#1313324011yescharacteristic
293C8—42no—
294C8—81no—
295C8—4241no—
296C8—2122no—
297C22—21no—
298C22—21no—
299C22#1681049911yescharacteristic
300C22#103×D5311yescharacteristic
301C22#1681061811yescharacteristic
302C4—21no—
303C4—21no—
304C4—21no—
305C4—82no—
306C4—1064no—
307C4—2124no—
308C4—2122no—
309C4#1681080412yes—
310C4#1681084911yescharacteristic
311C4#1681085511yescharacteristic
312C4#1681086111yescharacteristic
313C4#1681086311yescharacteristic
314C2—21no—
315C2—21no—
316C2#2027510511yescharacteristic
317C2#2027513511yescharacteristic
318C2#2027525911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1548C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1548D53
hypocenterC53#9392
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
185. C53#9392semidirect

Maximal subgroups

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

subgroupindex
#202837952
#202799242
#202747082
#202733962
#202726172
#202726162
C53×#15482
#939253