Finite Group Databasebeta

← #27288039

Subgroups — #27288039

Order 13568.

344 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 81 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#20285861C211yesmaximal, characteristic
2#20275704C211yesmaximal, characteristic
3#20273503C211yesmaximal, characteristic
4#20273452C211yesmaximal, characteristic
5#20273401C211yesmaximal, characteristic
6#20273137C211yesmaximal, characteristic
7C53×#1299C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9C2×#13133217C2211yescharacteristic
10#16811021—21no—
11#16810974—21no—
12#16810954—22no—
13#16810909—21no—
14#16810909—21no—
15#16810676C2211yescharacteristic
16#16810647C2211yescharacteristic
17#16810391—22no—
18#16810375—21no—
19#16810373—21no—
20#16810370—21no—
21#16810368—21no—
22#16810368—22no—
23#16810355—22no—
24C53×#431C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C4×M16×C53C2211yescharacteristic
27C22×Dic106—21no—
28C2×#11719917—21no—
29#13133217—21no—
30#13133217—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217D411yescharacteristic
34#13133217D411yescharacteristic
35#13133209—21no—
36#13133196—21no—
37#13133196—21no—
38C4×Dic106—21no—
39C4×Dic106—21no—
40#13133176—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53D412yes—
43#13133149—22no—
44#13133149—22no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322D412yes—
48#13133142—22no—
49#13133142—22no—
50C22⋊#88322—21no—
51C4×#88322—21no—
52C2×M16×C53—21no—
53C2×M16×C53—21no—
54C2×C42×C53C2311yescharacteristic
55C4⋊C8×C53—21no—
56C4⋊C8×C53—21no—
57C4⋊C8×C53D411yescharacteristic
58C4⋊C8×C53D411yescharacteristic
59C22⋊C8×C53—21no—
60#97×C53—21no—
61C4×C8×C53—21no—
62C4×C8×C53—21no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53#4111yescharacteristic
65C22×C4×C53#4111yescharacteristic
66C22×Dic53—22no—
67C2×Dic106—21no—
68C2×Dic106—21no—
69C2×Dic106—41no—
70C2×Dic106—41no—
71C2×Dic106—82no—
72M16×C53—22no—
73M16×C53—41no—
74M16×C53—81no—
75C2×C8×C53—21no—
76C2×C8×C53—21no—
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C42×C53C2×D411yescharacteristic
80C42×C53C2×D411yescharacteristic
81C42×C53C2×D411yescharacteristic
82C42×C53#4111yescharacteristic
83C22⋊Dic53—41no—
84C4⋊Dic53—22no—
85C4⋊Dic53—21no—
86C4⋊Dic53—22no—
87C4⋊Dic53—21no—
88C4⋊Dic53—21no—
89C4⋊Dic53—21no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92#11719919—41no—
93#11719919—41no—
94C4×Dic53—41no—
95#11719917—42no—
96C2×#88322—22no—
97C2×#88322—41no—
98C23×C53#12311yescharacteristic
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53#12011yescharacteristic
105C2×C4×C53#12111yescharacteristic
106C2×C4×C53#12111yescharacteristic
107C2×C4×C53#12111yescharacteristic
108C2×C4×C53#12311yescharacteristic
109C2×C4×C53#12411yescharacteristic
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—42no—
113C2×Dic53—82no—
114Dic106—41no—
115Dic106—41no—
116Dic106—41no—
117Dic106—41no—
118Dic106—82no—
119Dic106—82no—
120C8×C53—42no—
121C8×C53—41no—
122C8×C53—41no—
123C8×C53—81no—
124#88322—42no—
125#88322—81no—
126#10494—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53#39411yescharacteristic
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—21no—
134C4×C53#45411yescharacteristic
135C4×C53#45611yescharacteristic
136C4×C53#47111yescharacteristic
137C4×C53#47112yes—
138C4×C53#47311yescharacteristic
139Dic53—81no—
140Dic53—81no—
141Dic53—82no—
142#3229—531no—
143#1997—531no—
144#1574—531no—
145#1561—531no—
146#1549—531no—
147#1485—531no—
148#1299D5311yescharacteristic
149C2×C53—41no—
150C2×C53#203811yescharacteristic
151C2×C53#204611yescharacteristic
152C2×C53#205211yescharacteristic
153#533—531no—
154C2×C4⋊Q8—531no—
155#498—1061no—
156#487—1061no—
157#483—1062no—
158#475—1061no—
159#475—1061no—
160#431—531no—
161#431C2×D5311yescharacteristic
162#423—531no—
163#423C2×D5311yescharacteristic
164C4×M16C2×D5311yescharacteristic
165#343—1062no—
166#335—1061no—
167#334—1061no—
168#332—1062no—
169#332—1061no—
170#332—1061no—
171Q8⋊C8—1062no—
172C53#1049411yescharacteristic, complemented, Sylow
173C22×Q8—1061no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—1061no—
177C4⋊Q8—531no—
178C4⋊Q8—531no—
179C4⋊Q8—1061no—
180C4⋊Q8—1061no—
181C4⋊Q8—1061no—
182C4⋊Q8—1061no—
183#125—1061no—
184C22⋊Q8—1061no—
185C22⋊Q8—1061no—
186C4×Q8—1061no—
187C4×Q8—1061no—
188#117—1061no—
189C2×C4⋊C4—532no—
190C2×C4⋊C4—1061no—
191C2×C42C22×D5311yescharacteristic
192#106—1062no—
193#106—1062no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—532no—
197C4⋊C8—1061no—
198C4⋊C8—1061no—
199C4⋊C8D21211yescharacteristic
200C4⋊C8D21211yescharacteristic
201#103—1062no—
202#103—1062no—
203C22⋊C8—21no—
204C22⋊C8—1061no—
205#97—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—1061no—
209C2×Q8—1061no—
210C2×Q8—1061no—
211C2×Q8—4242no—
212C2×Q8—2121no—
213C2×Q8—2121no—
214C22×C4—1062no—
215C22×C4#1171994511yescharacteristic
216C22×C4D4×D5311yescharacteristic
217C22×C4#1171994911yescharacteristic
218M16—22no—
219M16—41no—
220M16—81no—
221M16—2122no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—1062no—
227C2×C8—2121no—
228C4⋊C4—1062no—
229C4⋊C4—1061no—
230C4⋊C4—1062no—
231C4⋊C4—1061no—
232C4⋊C4—1061no—
233C4⋊C4—1061no—
234C4⋊C4—2121no—
235C4⋊C4—2121no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C22⋊C4—2121no—
239C42—2121no—
240C42C2×D21211yescharacteristic
241C42#1171994511yescharacteristic
242C42D4×D5311yescharacteristic
243C42D4×D5311yescharacteristic
244C23#1313320211yescharacteristic
245Q8—4242no—
246Q8—4242no—
247Q8—2121no—
248Q8—2121no—
249Q8—2121no—
250Q8—2121no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—4242no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4—2122no—
260C2×C4#1313318711yescharacteristic
261C2×C4#1313319011yescharacteristic
262C2×C4#1313319011yescharacteristic
263C2×C4#1313319211yescharacteristic
264C2×C4#1313320211yescharacteristic
265C2×C4#1313320511yescharacteristic
266C8—42no—
267C8—41no—
268C8—41no—
269C8—81no—
270C8—4241no—
271C8—2122no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22#1681054111yescharacteristic
276C4—21no—
277C4—21no—
278C4—21no—
279C4—4242no—
280C4—4241no—
281C4—4241no—
282C4#1681079711yescharacteristic
283C4#1681080711yescharacteristic
284C4#1681088511yescharacteristic
285C4#1681088712yes—
286C4#1681089711yescharacteristic
287C2—41no—
288C2#2027593511yescharacteristic
289C2#2027597911yescharacteristic
290C2#2027601711yescharacteristic
subgroupG/H
centerC22#16810541
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1299C2
socleC22×C53#394
prime radicalC23×C53#123
hypercenter#1299D53
hypocenterC53#10494
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1299

Complements

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

NH ≅ G/Ntype
172. C53#10494semidirect

Maximal subgroups

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

subgroupindex
#202858612
#202757042
#202735032
#202734522
#202734012
#202731372
C53×#12992
#1049453