Finite Group Databasebeta

← #27294939

Subgroups — #27294939

Order 13568.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 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#20276527C211yesmaximal, characteristic, complemented
2#20274864C211yesmaximal, characteristic, complemented
3#20272706C212yesmaximal, complemented
4#20272700C212yesmaximal, complemented
5C53×#3184C211yesmaximal, characteristic
6#16810444—21no—
7#16810405C2212yes—
8#16810401—21no—
9#16810394—21no—
10#16810383—21no—
11#16810381C2211yescharacteristic
12#16810380C2211yescharacteristic
13C53×#532—21no—
14C53×#498C411yescharacteristic, complemented
15C53×#493C411yescharacteristic, complemented
16C53×#486—21no—
17C53×#469—21no—
18C53×#469C2212yes—
19C53×#442—21no—
20C53×#442C412yescomplemented
21C4×M16×C53C2211yescharacteristic
22C2×#11719917—21no—
23#13133176—21no—
24#13133157—22no—
25C2×#41×C53—41no—
26C2×D8×C53—21no—
27C2×D8×C53C2×C412yes—
28C2×M16×C53D411yescharacteristic
29C2×M16×C53C2311yescharacteristic
30C4⋊Q8×C53—21no—
31#127×C53—21no—
32#124×C53—41no—
33#121×C53—22no—
34#121×C53—41no—
35C4×D4×C53—22no—
36C4×D4×C53—41no—
37C2×C42×C53D411yescharacteristic
38#106×C53—21no—
39#106×C53C2×C412yes—
40#102×C53—22no—
41#102×C53—22no—
42#97×C53C2×C411yescharacteristic
43#97×C53D411yescharacteristic
44C4×C8×C53C2×C411yescharacteristic
45C4×C8×C53D411yescharacteristic
46#41×C53—84no—
47C2×Q8×C53—41no—
48C2×D4×C53—22no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53—42no—
52C22×C4×C53—21no—
53C22×C4×C53—42no—
54C22×C4×C53C2×D411yescharacteristic
55D8×C53—42no—
56D8×C53—42no—
57M16×C53—41no—
58M16×C53—41no—
59C2×C8×C53C22⋊C411yescharacteristic
60C2×C8×C53C22⋊C411yescharacteristic
61C2×C8×C53C22×C411yescharacteristic
62C2×C8×C53C2×D411yescharacteristic
63C4⋊C4×C53—22no—
64C4⋊C4×C53—41no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C42×C53—21no—
68C42×C53C22⋊C411yescharacteristic
69C42×C53C22⋊C411yescharacteristic
70C22⋊Dic53—41no—
71C4⋊Dic53—41no—
72C4×Dic53—41no—
73#11719917—41no—
74#11719917—41no—
75C2×#88322—41no—
76#11719908—42no—
77C23×C53—42no—
78C23×C53AGL(1,ℤ/8)11yescharacteristic
79Q8×C53—82no—
80D4×C53—42no—
81D4×C53—42no—
82D4×C53—82no—
83D4×C53—82no—
84D4×C53—84no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—84no—
92C2×C4×C53C2×C22⋊C411yescharacteristic
93C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
94C2×Dic53—81no—
95C8×C53—21no—
96C8×C53—21no—
97C8×C53—21no—
98C8×C53—21no—
99#88322—81no—
100#11837—531nomaximal, Sylow
101C22×C53—21no—
102C22×C53—41no—
103C22×C53—82no—
104C22×C53—82no—
105C22×C53—82no—
106C22×C53#41811yescharacteristic
107C4×C53—22no—
108C4×C53—21no—
109C4×C53—41no—
110C4×C53—82no—
111C4×C53#41011yescharacteristic
112C4×C53#42111yescharacteristic
113Dic53—161no—
114#3184D5311yescharacteristic
115#2165—531no—
116#1841—531no—
117#1341—532no—
118#1339—532no—
119C2×C53—21no—
120C2×C53—41no—
121C2×C53—162no—
122C2×C53#151411yescharacteristic
123#532—21no—
124#498Dic5311yescharacteristic, complemented
125#493Dic5311yescharacteristic, complemented
126#486—21no—
127#469—21no—
128#469C2×D5312yes—
129#442—21no—
130#442Dic5312yescomplemented
131C4×M16C2×D5311yescharacteristic
132#364—1061no—
133#349—532no—
134#347—1061no—
135#344—1061no—
136#339—1061no—
137#338—531no—
138H(3,ℤ/4)—531no—
139C53#1183711yescharacteristic, complemented, Sylow
140C2×#41—41no—
141C2×D8—21no—
142C2×D8C2×Dic5312yes—
143C2×M16—1061no—
144C2×M16C22⋊D5311yescharacteristic
145C2×M16C22×D5311yescharacteristic
146C4⋊Q8—21no—
147#127—21no—
148#124—41no—
149#121—22no—
150#121—41no—
151C4×D4—22no—
152C4×D4—41no—
153#117—1061no—
154C2×C42C22⋊D5311yescharacteristic
155M32—1062no—
156#106—21no—
157#106C2×Dic5312yes—
158#102—22no—
159#102—22no—
160#97C2×Dic5311yescharacteristic
161#97C22⋊D5311yescharacteristic
162C4×C8C2×Dic5311yescharacteristic
163C4×C8C22⋊D5311yescharacteristic
164#41—84no—
165C2×Q8—41no—
166C2×D4—22no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—42no—
170C22×C4—21no—
171C22×C4—42no—
172C22×C4C2×C22⋊D5311yescharacteristic
173D8—42no—
174D8—42no—
175M16—41no—
176M16—41no—
177M16—2121no—
178M16—2121no—
179C2×C8—2121no—
180C2×C8C22⋊Dic5311yescharacteristic
181C2×C8C22⋊Dic5311yescharacteristic
182C2×C8C22×Dic5311yescharacteristic
183C2×C8C2×C22⋊D5311yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—41no—
186C4⋊C4—2121no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—2121no—
190C42—21no—
191C42—2121no—
192C42C22⋊Dic5311yescharacteristic
193C42C22⋊Dic5311yescharacteristic
194C16—2122no—
195C23—42no—
196C23#1313324311yescharacteristic
197Q8—82no—
198D4—42no—
199D4—42no—
200D4—82no—
201D4—82no—
202D4—84no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—84no—
210C2×C4—4241no—
211C2×C4C2×C22⋊Dic5311yescharacteristic
212C2×C4#1313324311yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—4241no—
218C22—21no—
219C22—41no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22#1681062811yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—82no—
228C4—8481no—
229C4#1681059711yescharacteristic
230C4#1681064011yescharacteristic
231C2—21no—
232C2—41no—
233C2—162no—
234C2#2027326211yescharacteristic
subgroupG/H
centerC2#20273262
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
Fitting, prime radicalC53×#3184C2
socleC2×C53#1514
cosocleC2×M16×C53C23
hypercenter#3184D53
hypocenterC53#11837
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3184

Complements

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

NH ≅ G/Ntype
1. #20276527C2semidirect
2. #20274864C2semidirect
3. #20272706C2semidirect
4. #20272700C2semidirect
14. C53×#498C4semidirect
15. C53×#493C4semidirect
20. C53×#442C4semidirect
124. #498Dic53semidirect
125. #493Dic53semidirect
130. #442Dic53semidirect
139. C53#11837semidirect

Maximal subgroups

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

subgroupindex
#202765272
#202748642
#202727062
#202727002
C53×#31842
#1183753