Finite Group Databasebeta

← #27297980

Subgroups — #27297980

Order 13568.

294 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 65 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#20287701C211yesmaximal, characteristic
2#20276943C211yesmaximal, characteristic, complemented
3#20276852C211yesmaximal, characteristic
4#20274110C211yesmaximal, characteristic, complemented
5#20272763C211yesmaximal, characteristic, complemented
6#20272695C211yesmaximal, characteristic, complemented
7C53×#1331C211yesmaximal, characteristic
8#16811343—21no—
9#16811028C2211yescharacteristic
10#16811006—21no—
11#16811003C2211yescharacteristic
12#16810844—22no—
13#16810834—21no—
14#16810828—21no—
15C8×Dic106C2211yescharacteristic
16#16810717—21no—
17#16810448—21no—
18C4⋊#11719908C2211yescharacteristic
19#16810428—22no—
20#16810424—21no—
21#16810370—21no—
22C4×D8×C53C2211yescharacteristic
23C4×C16×C53C2211yescharacteristic
24C53×#335C2211yescharacteristic
25C2×Dic212—21no—
26C2×Dic212—41no—
27C2×Dic212D412yes—
28C2×#11719923—22no—
29#13133217—21no—
30#13133217—21no—
31#13133210—42no—
32#13133204—22no—
33#13133196—41no—
34C4×Dic106D411yescharacteristic
35#13133182—22no—
36#13133176—41no—
37C2×#11719908—21no—
38#13133151—21no—
39#13133151—21no—
40C4⋊#88322D411yescharacteristic
41#13133146—21no—
42#13133142—41no—
43#13133140—22no—
44C8×Dic53—21no—
45C2×D8×C53D411yescharacteristic
46C4×D4×C53—21no—
47C2×C16×C53—21no—
48#107×C53D411yescharacteristic
49C4⋊C8×C53—21no—
50#102×C53—21no—
51C4×C8×C53C2311yescharacteristic
52C2×D4×C53—21no—
53C22×C4×C53—41no—
54C2×Dic106—22no—
55C2×Dic106—41no—
56C2×Dic106D811yescharacteristic
57D8×C53—21no—
58D8×C53—21no—
59C2×C8×C53—41no—
60C2×C8×C53C2×D411yescharacteristic
61C2×C8×C53C2×D411yescharacteristic
62C4⋊C4×C53—21no—
63C22⋊C4×C53—41no—
64C42×C53C2×D411yescharacteristic
65C22⋊Dic53—42no—
66C22⋊Dic53—42no—
67#11719923—82no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C4⋊Dic53D811yescharacteristic
73#11719919—21no—
74#11719919—42no—
75C4×Dic53—21no—
76C4×Dic53—42no—
77C2×#88322—21no—
78Dic212—42no—
79Dic212—41no—
80Dic212—41no—
81Dic212—82no—
82C16×C53—41no—
83#11719908—41no—
84C23×C53—41no—
85D4×C53—41no—
86D4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—81no—
89C2×C4×C53—81no—
90C2×C4×C53#12011yescharacteristic
91C2×C4×C53C2×D811yescharacteristic
92C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—42no—
96C2×Dic53—81no—
97Dic106—21no—
98Dic106—21no—
99Dic106—42no—
100Dic106—42no—
101Dic106—81no—
102Dic106—81no—
103C8×C53—21no—
104C8×C53—81no—
105C8×C53#13511yescharacteristic
106C8×C53#13711yescharacteristic
107#88322—41no—
108#12383—531nomaximal, Sylow
109C22×C53—81no—
110C22×C53—81no—
111C22×C53—81no—
112C22×C53#44711yescharacteristic
113C4×C53—21no—
114C4×C53—81no—
115C4×C53#45211yescharacteristic
116C4×C53#45511yescharacteristic
117C4×C53#50811yescharacteristic
118C4×C53#50911yescharacteristic
119Dic53—41no—
120Dic53—41no—
121Dic53—82no—
122Dic53—161no—
123#3393—531no—
124#2241—531no—
125#2224—531no—
126#1674—531no—
127#1362—531no—
128Q8⋊C16—531no—
129#1331D5311yescharacteristic
130C2×C53—161no—
131C2×C53#164111yescharacteristic
132C2×C53#218611yescharacteristic
133C2×C53#219511yescharacteristic
134#556—1061no—
135#500—531no—
136#495—1061no—
137#494—531no—
138#464—1062no—
139#462—1061no—
140#461—1061no—
141C8×Q8—531no—
142C4×Q16—1061no—
143C4×D8C2×D5311yescharacteristic
144#366—1061no—
145C4⋊C16—531no—
146#358—1062no—
147#357—1061no—
148C4×C16C2×D5311yescharacteristic
149#335C2×D5311yescharacteristic
150#332—1061no—
151C53#1238311yescharacteristic, complemented, Sylow
152C2×Q16—532no—
153C2×Q16—1061no—
154C2×Q16—2121no—
155C2×SD16—1062no—
156C2×D8D21211yescharacteristic
157C4⋊Q8—1061no—
158C4⋊Q8—1061no—
159#126—2122no—
160#124—1062no—
161C22⋊Q8—2121no—
162C4×Q8—531no—
163C4×Q8—1062no—
164C4×D4—21no—
165#117—2121no—
166C2×C16—21no—
167C2×C16—1061no—
168#107—1061no—
169#107—1061no—
170#107D21211yescharacteristic
171C4⋊C8—21no—
172C4⋊C8—531no—
173C4⋊C8—1061no—
174#103—1062no—
175#103—2121no—
176#102—21no—
177C4×C8—1061no—
178C4×C8C22×D5311yescharacteristic
179C2×Q8—531no—
180C2×Q8—1062no—
181C2×Q8—2121no—
182C2×D4—21no—
183C22×C4—41no—
184Q16—4242no—
185Q16—2122no—
186Q16—2121no—
187Q16—2121no—
188SD16—4242no—
189D8—21no—
190D8—21no—
191C2×C8—41no—
192C2×C8—1061no—
193C2×C8C2×D21211yescharacteristic
194C2×C8D4×D5311yescharacteristic
195C4⋊C4—21no—
196C4⋊C4—531no—
197C4⋊C4—1061no—
198C4⋊C4—2122no—
199C4⋊C4—2121no—
200C4⋊C4—2121no—
201C4⋊C4—2121no—
202C4⋊C4—2121no—
203C22⋊C4—41no—
204C22⋊C4—2122no—
205C22⋊C4—2122no—
206C42—1061no—
207C42—2122no—
208C42D4×D5311yescharacteristic
209C16—41no—
210C16—2121no—
211C23—41no—
212Q8—1061no—
213Q8—1061no—
214Q8—4241no—
215Q8—4241no—
216Q8—2122no—
217Q8—2122no—
218D4—41no—
219D4—41no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—4241no—
226C2×C4—2122no—
227C2×C4#1313318711yescharacteristic
228C2×C4D8×D5311yescharacteristic
229C2×C4#1313324411yescharacteristic
230C8—21no—
231C8—81no—
232C8—2121no—
233C8#1313323911yescharacteristic
234C8#1313325011yescharacteristic
235C22—81no—
236C22—81no—
237C22—81no—
238C22#1681075811yescharacteristic
239C4—21no—
240C4—81no—
241C4—8481no—
242C4—4242no—
243C4—2121no—
244C4—2121no—
245C4#1681078711yescharacteristic
246C4#1681080211yescharacteristic
247C4#1681105411yescharacteristic
248C4#1681105711yescharacteristic
249C2—161no—
250C2#2027391111yescharacteristic
251C2#2027663911yescharacteristic
252C2#2027669411yescharacteristic
subgroupG/H
centerC22#16810758
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1331C2
socleC22×C53#447
prime radicalC2×D8×C53D4
hypercenter#1331D53
hypocenterC53#12383
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
2. #20276943C2semidirect
4. #20274110C2semidirect
5. #20272763C2semidirect
6. #20272695C2semidirect
151. C53#12383semidirect

Maximal subgroups

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

subgroupindex
#202877012
#202769432
#202768522
#202741102
#202727632
#202726952
C53×#13312
#1238353