Finite Group Databasebeta

← #27297851

Subgroups — #27297851

Order 13568.

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

normal subgroups only

This group has 358 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#20276886C212yesmaximal, complemented
2#20274301C211yesmaximal, characteristic, complemented
3#20272756C211yesmaximal, characteristic, complemented
4#20272688C212yesmaximal, complemented
5C53×#3291C211yesmaximal, characteristic
6#16811021C2211yescharacteristic, complemented
7#16811019—21no—
8#16810995—21no—
9#16810447—22no—
10#16810444C2212yescomplemented
11#16810433—22no—
12C8×#88322C2211yescharacteristic, complemented
13C53×#577—22no—
14C53×#492C2211yescharacteristic
15C53×#471—22no—
16C53×#456—21no—
17C53×#453—21no—
18C53×#443C2212yes—
19C2×#11719923—42no—
20#13133217—21no—
21#13133157—22no—
22#13133149—22no—
23#13133148—42no—
24C4×#88322—21no—
252−1+4×C53—21no—
26H(5,𝔽2)×C53—21no—
27#137×C53—22no—
28#137×C53—42no—
29AGL(1,ℤ/8)×C53—22no—
30AGL(1,ℤ/8)×C53—42no—
31#135×C53—42no—
32#135×C53D412yes—
33C2×SD16×C53—22no—
34C2×SD16×C53—42no—
35#131×C53—22no—
36C4⋊Q8×C53D411yescharacteristic
37#127×C53D411yescharacteristic
38#108×C53D412yes—
39#104×C53—22no—
40#101×C53—21no—
41#100×C53—21no—
42C4×C8×C53C2311yescharacteristic
43#41×C53—22no—
44#41×C53—42no—
45#41×C53—42no—
46#41×C53—81no—
47C2×Q8×C53—21no—
48C2×Q8×C53—42no—
49C2×D4×C53—21no—
50C2×D4×C53—41no—
51C2×D4×C53—42no—
52C2×D4×C53—81no—
53C2×Dic106—41no—
54Q16×C53—42no—
55Q16×C53SD1612yescomplemented
56SD16×C53—22no—
57SD16×C53—42no—
58SD16×C53—42no—
59SD16×C53—42no—
60SD16×C53—42no—
61D8×C53—42no—
62D8×C53SD1612yescomplemented
63M16×C53—22no—
64M16×C53—42no—
65C2×C8×C53—42no—
66C2×C8×C53C2×D412yes—
67C4⋊C4×C53—41no—
68C42×C53C2×D411yescharacteristic
69#11719923—84no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C2×#88322—22no—
73#11719908—42no—
74C23×C53—41no—
75C23×C53—81no—
76Q8×C53—22no—
77Q8×C53—42no—
78Q8×C53—81no—
79D4×C53—22no—
80D4×C53—42no—
81D4×C53—42no—
82D4×C53—82no—
83D4×C53—82no—
84D4×C53—81no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—81no—
89C2×C4×C53—82no—
90C2×C4×C53#12011yescharacteristic
91C2×Dic53—81no—
92Dic106—82no—
93C8×C53—42no—
94C8×C53C2×SD1612yes—
95C8×C53AGL(1,ℤ/8)12yes—
96#88322—24no—
97#12364—531nomaximal, Sylow
98C22×C53—42no—
99C22×C53—81no—
100C22×C53—161no—
101C22×C53—321no—
102C22×C53#45311yescharacteristic
103C4×C53—22no—
104C4×C53—42no—
105C4×C53—81no—
106C4×C53#45012yes—
107Dic53—161no—
108#3291D5311yescharacteristic
109#2230—532no—
110#1705—531no—
111#1359—531no—
112#1335—532no—
113C2×C53—21no—
114C2×C53—82no—
115C2×C53—161no—
116C2×C53#162911yescharacteristic
117#577—22no—
118#498—531no—
119#498—1061no—
120#492—1061no—
121#492C2×D5311yescharacteristic
122#471—22no—
123#456—21no—
124#453—21no—
125#443C2×D5312yes—
126#365—1062no—
127#364—532no—
128#360—1062no—
129C82—531no—
130C53#1236411yescharacteristic, complemented, Sylow
1312−1+4—21no—
132H(5,𝔽2)—21no—
133#137—22no—
134#137—42no—
135AGL(1,ℤ/8)—22no—
136AGL(1,ℤ/8)—42no—
137#135—42no—
138#135C22⋊D5312yes—
139C2×SD16—22no—
140C2×SD16—42no—
141C2×SD16—2122no—
142#131—22no—
143C4⋊Q8—1061no—
144C4⋊Q8C22⋊D5311yescharacteristic
145#127C22⋊D5311yescharacteristic
146M32—1062no—
147#108C22⋊D5312yes—
148#106—1062no—
149#106—2122no—
150#104—22no—
151#101—21no—
152#100—21no—
153C4×C8—1061no—
154C4×C8C22×D5311yescharacteristic
155#41—22no—
156#41—42no—
157#41—42no—
158#41—81no—
159C2×Q8—21no—
160C2×Q8—42no—
161C2×Q8—2121no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—81no—
166Q16—42no—
167Q16#1171992312yescomplemented
168SD16—22no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172SD16—42no—
173SD16—4244no—
174D8—42no—
175D8#1171992312yescomplemented
176M16—22no—
177M16—42no—
178C2×C8—42no—
179C2×C8—1062no—
180C2×C8C2×C22⋊D5312yes—
181C4⋊C4—41no—
182C4⋊C4—2121no—
183C4⋊C4—2121no—
184C42C2×C22⋊D5311yescharacteristic
185C16—2122no—
186C23—41no—
187C23—81no—
188Q8—22no—
189Q8—42no—
190Q8—81no—
191Q8—4242no—
192D4—22no—
193D4—42no—
194D4—42no—
195D4—82no—
196D4—82no—
197D4—81no—
198C2×C4—21no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4—4241no—
204C2×C4#1313318611yescharacteristic
205C8—42no—
206C8—1064no—
207C8C2×#1171992312yes—
208C8#1313324312yes—
209C22—42no—
210C22—81no—
211C22—161no—
212C22—321no—
213C22#1681079111yescharacteristic
214C4—22no—
215C4—42no—
216C4—81no—
217C4—8481no—
218C4#1681077412yes—
219C2—21no—
220C2—82no—
221C2—161no—
222C2#2027383111yescharacteristic
subgroupG/H
centerC2#20273831
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3291C2
socleC2×C53#1629
hypercenter#3291D53
hypocenterC53#12364
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3291

Complements

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

NH ≅ G/Ntype
1. #20276886C2semidirect
2. #20274301C2semidirect
3. #20272756C2semidirect
4. #20272688C2semidirect
6. #16811021C22semidirect
10. #16810444C22semidirect
12. C8×#88322C22semidirect
55. Q16×C53SD16semidirect
62. D8×C53SD16semidirect
130. C53#12364semidirect
167. Q16#11719923semidirect
175. D8#11719923semidirect

Maximal subgroups

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

subgroupindex
#202768862
#202743012
#202727562
#202726882
C53×#32912
#1236453