Finite Group Databasebeta

← #27297689

Subgroups — #27297689

Order 13568.

306 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 67 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#20287146C211yesmaximal, characteristic, complemented
2#20276938C211yesmaximal, characteristic, complemented
3#20276752C211yesmaximal, characteristic, complemented
4#20272771C211yesmaximal, characteristic, complemented
5#20272678C211yesmaximal, characteristic, complemented
6D4⋊#11719908C211yesmaximal, characteristic, complemented
7C53×#1667C211yesmaximal, characteristic
8#16811343—21no—
9C2×#13133161—21no—
10#16811028C2211yescharacteristic
11#16810965—22no—
12#16810909—21no—
13#16810864—21no—
14C4×Dic212C2211yescharacteristic
15#16810661—21no—
16C4⋊#11719908C2211yescharacteristic
17#16810424—21no—
18#16810424—22no—
19C4×#11719908C2211yescharacteristic
20C53×#493C2211yescharacteristic
21C53×#459—21no—
22C8×D4×C53C2211yescharacteristic
23C53×#335C2211yescharacteristic
24C53×#331—21no—
25C2×Dic212D411yescharacteristic
26#13133217—21no—
27#13133210—42no—
28#13133204—41no—
29#13133196—22no—
30C4×Dic106—21no—
31#13133182—41no—
32#13133176—22no—
33#13133161—42no—
34C2×#11719908—21no—
35C2×#11719908—21no—
36#13133151—21no—
37#13133151D412yes—
38#13133151D411yescharacteristic
39#13133149—22no—
40#13133142—22no—
41#13133142—21no—
42C2×D8×C53—21no—
43C2×D8×C53—21no—
44C2×D8×C53—41no—
45C22×C8×C53—21no—
46#127×C53—21no—
47C4×D4×C53D411yescharacteristic
48C4⋊C8×C53—21no—
49C4⋊C8×C53D411yescharacteristic
50#102×C53—41no—
51C22⋊C8×C53—21no—
52C4×C8×C53C2311yescharacteristic
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53D811yescharacteristic
57C22×C4×C53—21no—
58C2×Dic106—21no—
59C2×Dic106—41no—
60D8×C53—42no—
61D8×C53—41no—
62D8×C53—41no—
63D8×C53—82no—
64C2×C8×C53—21no—
65C2×C8×C53—41no—
66C2×C8×C53—41no—
67C2×C8×C53—41no—
68C2×C8×C53C2×D411yescharacteristic
69C2×C8×C53C2×D411yescharacteristic
70C4⋊C4×C53D811yescharacteristic
71C22⋊C4×C53—21no—
72C42×C53C2×D411yescharacteristic
73C22⋊Dic53—42no—
74C22⋊Dic53—42no—
75C4⋊Dic53—21no—
76C4⋊Dic53—22no—
77C4⋊Dic53—41no—
78#11719919—42no—
79#11719919—41no—
80C4×Dic53—42no—
81C4×Dic53—41no—
82Dic212—21no—
83Dic212—21no—
84#11719908—41no—
85#11719908—41no—
86C23×C53—21no—
87C23×C53—81no—
88D4×C53—21no—
89D4×C53—21no—
90D4×C53—81no—
91D4×C53—81no—
92D4×C53—81no—
93D4×C53—81no—
94D4×C53—161no—
95C2×C4×C53—21no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53#12011yescharacteristic
99C2×C4×C53C2×D811yescharacteristic
100C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
101C2×Dic53—41no—
102C2×Dic53—42no—
103C2×Dic53—41no—
104Dic106—41no—
105Dic106—41no—
106Dic106—81no—
107Dic106—81no—
108C8×C53—21no—
109C8×C53—41no—
110C8×C53—81no—
111C8×C53#13511yescharacteristic
112C8×C53AGL(1,ℤ/8)11yescharacteristic
113#12341—531nomaximal, Sylow
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—41no—
117C22×C53—161no—
118C22×C53—161no—
119C22×C53—161no—
120C22×C53#44711yescharacteristic
121C4×C53—21no—
122C4×C53—41no—
123C4×C53#44911yescharacteristic
124C4×C53#45411yescharacteristic
125C4×C53#50811yescharacteristic
126C4×C53#50911yescharacteristic
127Dic53—81no—
128Dic53—82no—
129Dic53—81no—
130#3342—531no—
131#2240—531no—
132#2207—531no—
133#1667D5311yescharacteristic
134#1364—531no—
135#1332—531no—
136D4⋊C16—531no—
137C2×C53—81no—
138C2×C53—321no—
139C2×C53#162511yescharacteristic
140C2×C53#218611yescharacteristic
141C2×C53#219311yescharacteristic
142#556—1061no—
143C2×SD32—1061no—
144#500—531no—
145#493C2×D5311yescharacteristic
146#485—1062no—
147#475—1061no—
148#467—1061no—
149#459—21no—
150C4×Q16—531no—
151C8×D4C2×D5311yescharacteristic
152#427—1061no—
153C4⋊C16—531no—
154#357—1061no—
155#357—1062no—
156C4×C16—531no—
157#335C2×D5311yescharacteristic
158#331—21no—
159C53#1234111yescharacteristic, complemented, Sylow
160C2×Q16—531no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—41no—
164C22×C8—21no—
165C4⋊Q8—1061no—
166#127—21no—
167#126—2122no—
168#124—2121no—
169C22⋊Q8—1062no—
170C4×Q8—1061no—
171C4×Q8—2121no—
172C4×D4C22⋊D5311yescharacteristic
173#117—1062no—
174SD32—2122no—
175C2×C16—1061no—
176C2×C16—1061no—
177#107—531no—
178#107—532no—
179#107—1061no—
180#106—1062no—
181C4⋊C8—21no—
182C4⋊C8C22⋊D5311yescharacteristic
183#103—1062no—
184#103—1061no—
185#102—41no—
186C22⋊C8—21no—
187C4×C8C22×D5311yescharacteristic
188C2×Q8—1061no—
189C2×Q8—2121no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4#1171992211yescharacteristic
194C22×C4—21no—
195Q16—1061no—
196Q16—1061no—
197D8—42no—
198D8—41no—
199D8—41no—
200D8—82no—
201C2×C8—21no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8D4×D5311yescharacteristic
206C2×C8D4×D5311yescharacteristic
207C4⋊C4—1061no—
208C4⋊C4—1062no—
209C4⋊C4—2122no—
210C4⋊C4—2121no—
211C4⋊C4—2121no—
212C4⋊C4#1171992211yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—2122no—
215C22⋊C4—2122no—
216C42—2122no—
217C42—2121no—
218C42C2×C22⋊D5311yescharacteristic
219C16—2121no—
220C16—2121no—
221C23—21no—
222C23—81no—
223Q8—4241no—
224Q8—4241no—
225Q8—2121no—
226Q8—2121no—
227D4—21no—
228D4—21no—
229D4—81no—
230D4—81no—
231D4—81no—
232D4—81no—
233D4—161no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—2121no—
238C2×C4—2122no—
239C2×C4—2121no—
240C2×C4#1313318511yescharacteristic
241C2×C4C2×#1171992211yescharacteristic
242C2×C4#1313324211yescharacteristic
243C8—21no—
244C8—41no—
245C8—81no—
246C8#1313323611yescharacteristic
247C8#1313324111yescharacteristic
248C22—41no—
249C22—41no—
250C22—41no—
251C22—161no—
252C22—161no—
253C22—161no—
254C22#1681075611yescharacteristic
255C4—21no—
256C4—41no—
257C4—4241no—
258C4—4242no—
259C4—4241no—
260C4#1681076511yescharacteristic
261C4#1681079411yescharacteristic
262C4#1681105311yescharacteristic
263C4#1681105911yescharacteristic
264C2—81no—
265C2—321no—
266C2#2027380211yescharacteristic
267C2#2027663611yescharacteristic
268C2#2027667711yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#1667C2
socleC22×C53#447
hypercenter#1667D53
hypocenterC53#12341
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1667

Complements

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

NH ≅ G/Ntype
1. #20287146C2semidirect
2. #20276938C2semidirect
3. #20276752C2semidirect
4. #20272771C2semidirect
5. #20272678C2semidirect
6. D4⋊#11719908C2semidirect
159. C53#12341semidirect

Maximal subgroups

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

subgroupindex
#202871462
#202769382
#202767522
#202727712
#202726782
D4⋊#117199082
C53×#16672
#1234153