Finite Group Databasebeta

← #27304745

Subgroups — #27304745

Order 13568.

298 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×#16811056C211yesmaximal, characteristic, complemented
2C2×#16810460C211yesmaximal, characteristic, complemented
3C2×D1696C211yesmaximal, characteristic, complemented
4#20276802C211yesmaximal, characteristic, complemented
5#20272892C211yesmaximal, characteristic, complemented
6#20272852C211yesmaximal, characteristic, complemented
7C53×#1416C211yesmaximal, characteristic, complemented
8C2×#13133240—21no—
9#16811056—22no—
10#16811056—22no—
11C2×#13133160—21no—
12C2×D848C2211yescharacteristic
13C2×D848C2211yescharacteristic
14C2×C16×D53C2211yescharacteristic
15#16810860—21no—
16#16810460—22no—
17D1696—22no—
18C2×#13133123C2211yescharacteristic
19#16810423—21no—
20C2×Q32×C53C2211yescharacteristic
21C2×C32×C53C2211yescharacteristic
22C53×#366C2211yescharacteristic
23C2×#11719949—41no—
24#13133240—24no—
25#13133240—42no—
26C2×#11719924—41no—
27C2×D424—21no—
28C2×D424—21no—
29C2×C8×D53D411yescharacteristic
30#13133192—41no—
31#13133160—24no—
32D848—21no—
33D848—21no—
34D848—21no—
35D848D412yescomplemented
36C2×#11719908D411yescharacteristic
37C16×D53—21no—
38C16×D53—21no—
39#13133137—41no—
40#13133123—21no—
41C2×Q16×C53—21no—
42Q32×C53—21no—
43Q32×C53D412yescomplemented
44C2×C16×C53C2311yescharacteristic
45#107×C53—21no—
46C32×C53—21no—
47C2×Q8×C53—41no—
48#11719949—44no—
49#11719949—82no—
50C2×D212—41no—
51C2×D212—41no—
52C2×D212—81no—
53C2×D212—81no—
54C2×C4×D53—81no—
55C2×C4×D53D811yescharacteristic
56Q16×C53—22no—
57Q16×C53—41no—
58C2×C8×C53C2×D411yescharacteristic
59C4⋊C4×C53—41no—
60#11719924—44no—
61#11719921—81no—
62C2×#88322D811yescharacteristic
63D424—22no—
64D424—41no—
65D424—41no—
66D424—41no—
67C8×D53—21no—
68C8×D53—21no—
69C16×C53C2×D411yescharacteristic
70C16×C53C2×D411yescharacteristic
71#11719908—21no—
72C22×D53—81no—
73C22×D53—81no—
74C22×D53D1611yescharacteristic
75Q8×C53—42no—
76Q8×C53—81no—
77C2×C4×C53—81no—
78C2×C4×C53—81no—
79C2×C4×C53C2×D811yescharacteristic
80C2×Dic53D1611yescharacteristic
81D212—42no—
82D212—84no—
83D212—81no—
84D212—81no—
85D212—81no—
86D212—162no—
87C4×D53—21no—
88C4×D53—21no—
89C4×D53—84no—
90C8×C53#12011yescharacteristic
91C8×C53AGL(1,ℤ/8)11yescharacteristic
92#88322—21no—
93#13663—531nomaximal, Sylow
94C22×C53C2×D1611yescharacteristic
95C2×D53—21no—
96C2×D53—21no—
97C2×D53—21no—
98C2×D53—82no—
99C2×D53—161no—
100C2×D53—161no—
101C2×D53—161no—
102C2×D53—161no—
103C2×D53—161no—
104C4×C53—82no—
105C4×C53—161no—
106C4×C53#44711yescharacteristic
107C4×C53#50911yescharacteristic
108Dic53—21no—
109C2×#508—531no—
110C2×SD64—531no—
111C2×D32—531no—
112#2215—531no—
113#1416D5311yescharacteristic, complemented
114#1415—531no—
115C22⋊C32—531no—
116C2×C53#218411yescharacteristic
117C2×C53#226211yescharacteristic
118C2×C53#226311yescharacteristic
119D53—41no—
120D53—162no—
121D53—321no—
122C2×#135—1061no—
123#508—1062no—
124#508—1062no—
125C2×Q32C2×D5311yescharacteristic
126C2×SD32—1061no—
127C2×D16—531no—
128C2×D16—531no—
129C22×C16—531no—
130#466—1061no—
131SD64—1062no—
132D32—1062no—
133C2×C32—531no—
134C2×C32C2×D5311yescharacteristic
135#366C2×D5311yescharacteristic
136#357—1061no—
137C53#1366311yescharacteristic, complemented, Sylow
138C2×#41—2121no—
139#135—1064no—
140#135—2122no—
141C2×Q16—21no—
142C2×SD16—2121no—
143C2×D8—1061no—
144C2×D8—1061no—
145C22×C8—531no—
146#121—2121no—
147Q32—21no—
148Q32D21212yescomplemented
149SD32—1064no—
150D16—532no—
151D16—1061no—
152D16—1061no—
153D16—1061no—
154C2×C16—531no—
155C2×C16—1061no—
156C2×C16—1061no—
157C2×C16C22×D5311yescharacteristic
158#107—21no—
159#102—2121no—
160C32—21no—
161C32—1061no—
162#41—4242no—
163#41—2124no—
164C2×Q8—41no—
165C2×D4—4241no—
166C2×D4—4241no—
167C2×D4—2121no—
168C2×D4—2121no—
169C22×C4—531no—
170C22×C4—4241no—
171Q16—22no—
172Q16—41no—
173SD16—2124no—
174D8—1062no—
175D8—2121no—
176D8—2121no—
177D8—2121no—
178C2×C8—531no—
179C2×C8—1061no—
180C2×C8—1061no—
181C2×C8D4×D5311yescharacteristic
182C4⋊C4—41no—
183C22⋊C4—4241no—
184C16—1061no—
185C16C2×D21211yescharacteristic
186C16D4×D5311yescharacteristic
187C23—531no—
188C23—4241no—
189C23—4241no—
190Q8—42no—
191Q8—81no—
192D4—8482no—
193D4—4244no—
194D4—4241no—
195D4—4241no—
196D4—4241no—
197D4—2122no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—531no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—4244no—
204C2×C4D8×D5311yescharacteristic
205C8—1061no—
206C8#1313318711yescharacteristic
207C8#1313324411yescharacteristic
208C22—8481no—
209C22—8481no—
210C22—8481no—
211C22—8481no—
212C22—8481no—
213C22—1061no—
214C22—1061no—
215C22—1061no—
216C22—4242no—
217C22D16×D5311yescharacteristic
218C4—82no—
219C4—161no—
220C4—1061no—
221C4#1681075811yescharacteristic
222C4#1681106011yescharacteristic
223C2—8482no—
224C2—16961no—
225C2—2121no—
226C2#2027662811yescharacteristic
227C2#2027702511yescharacteristic
228C2#2027702911yescharacteristic
subgroupG/H
centerC22D16×D53
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#1416C2
socleC22×C53C2×D16
hypercenter#1416D53
hypocenterC53#13663
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1416

Complements

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

NH ≅ G/Ntype
1. C2×#16811056C2semidirect
2. C2×#16810460C2semidirect
3. C2×D1696C2semidirect
4. #20276802C2semidirect
5. #20272892C2semidirect
6. #20272852C2semidirect
7. C53×#1416C2semidirect
35. D848D4semidirect
43. Q32×C53D4semidirect
113. #1416D53semidirect
137. C53#13663semidirect
148. Q32D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168110562
C2×#168104602
C2×D16962
#202768022
#202728922
#202728522
C53×#14162
#1366353