Finite Group Databasebeta

← #27295624

Subgroups — #27295624

Order 13568.

412 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 87 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#20276527C212yesmaximal, complemented
2#20272756C212yesmaximal, complemented
3#20272752C212yesmaximal, complemented
4C53×#3151C211yesmaximal, characteristic
5#16810444C2214yescomplemented
6#16810405—22no—
7#16810401—22no—
8C2×AGL(1,ℤ/8)×C53—22no—
9C53×#532C411yescharacteristic
10C2×#127×C53C411yescharacteristic
11C53×#496—22no—
12C53×#493C411yescharacteristic
13C53×#493C2211yescharacteristic
14C53×#492C411yescharacteristic
15C53×#492C2211yescharacteristic
16C4×M16×C53C2211yescharacteristic
17#13133157—24no—
18C2×#41×C53—21no—
19C22×D4×C53—21no—
20C22×D4×C53—41no—
21AGL(1,ℤ/8)×C53—48no—
22C2×SD16×C53—22no—
23C2×SD16×C53—22no—
24C2×D8×C53—22no—
25C2×D8×C53—22no—
26C2×M16×C53D412yes—
27C4⋊Q8×C53C2×C411yescharacteristic
28#127×C53—21no—
29#127×C53—41no—
30#127×C53C2×C411yescharacteristic
31#127×C53C2×C411yescharacteristic
32#127×C53C2×C411yescharacteristic
33#124×C53—21no—
34#121×C53—41no—
35C4×D4×C53—41no—
36C2×C42×C53C2×C411yescharacteristic
37#97×C53D412yes—
38C4×C8×C53C2×C411yescharacteristic
39C4×C8×C53C2311yescharacteristic
40C24×C53—41no—
41#41×C53—44no—
42C2×Q8×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53—44no—
47C2×D4×C53—42no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53—84no—
54C22×C4×C53—21no—
55C22×C4×C53—41no—
56C22×C4×C53C22⋊C411yescharacteristic
57SD16×C53—44no—
58SD16×C53—44no—
59D8×C53—44no—
60D8×C53—44no—
61M16×C53—24no—
62C2×C8×C53C22⋊C412yes—
63C2×C8×C53C2×D412yes—
64C4⋊C4×C53—41no—
65C22⋊C4×C53—42no—
66C42×C53—21no—
67C42×C53C22⋊C411yescharacteristic
68C42×C53C22×C411yescharacteristic
69#11719908—44no—
70C23×C53—41no—
71C23×C53—41no—
72C23×C53—41no—
73C23×C53—82no—
74C23×C53—84no—
75C23×C53#10011yescharacteristic
76Q8×C53—42no—
77D4×C53—42no—
78D4×C53—42no—
79D4×C53—42no—
80D4×C53—82no—
81D4×C53—82no—
82D4×C53—82no—
83D4×C53—82no—
84D4×C53—84no—
85D4×C53—82no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—82no—
92C2×C4×C53#10011yescharacteristic
93C2×C4×C53C2×C22⋊C411yescharacteristic
94C8×C53—22no—
95C8×C53AGL(1,ℤ/8)14yes—
96#11992—531nomaximal, Sylow
97C22×C53—21no—
98C22×C53—41no—
99C22×C53—82no—
100C22×C53—82no—
101C22×C53—82no—
102C22×C53—81no—
103C22×C53—81no—
104C22×C53—81no—
105C22×C53—164no—
106C22×C53C2×#10011yescharacteristic
107C4×C53—22no—
108C4×C53—21no—
109C4×C53—41no—
110C4×C53—81no—
111C4×C53#41812yes—
112#3151D5311yescharacteristic
113#2165—532no—
114#1359—532no—
115#1357—532no—
116C2×C53—21no—
117C2×C53—41no—
118C2×C53—161no—
119C2×C53—161no—
120C2×C53—161no—
121C2×C53#155511yescharacteristic
122C2×AGL(1,ℤ/8)—22no—
123#532Dic5311yescharacteristic
124C2×#127Dic5311yescharacteristic
125#496—22no—
126#493Dic5311yescharacteristic
127#493C2×D5311yescharacteristic
128#492Dic5311yescharacteristic
129#492C2×D5311yescharacteristic
130C4×M16C2×D5311yescharacteristic
131#364—534no—
132#349—1062no—
133#347—1062no—
134C53#1199211yescharacteristic, complemented, Sylow
135C2×#41—21no—
136C22×D4—21no—
137C22×D4—41no—
138AGL(1,ℤ/8)—48no—
139C2×SD16—22no—
140C2×SD16—22no—
141C2×D8—22no—
142C2×D8—22no—
143C2×M16C22⋊D5312yes—
144C4⋊Q8C2×Dic5311yescharacteristic
145#127—21no—
146#127—41no—
147#127C2×Dic5311yescharacteristic
148#127C2×Dic5311yescharacteristic
149#127C2×Dic5311yescharacteristic
150#124—21no—
151#121—41no—
152C4×D4—41no—
153C2×C42C2×Dic5311yescharacteristic
154M32—1064no—
155#97C22⋊D5312yes—
156C4×C8C2×Dic5311yescharacteristic
157C4×C8C22×D5311yescharacteristic
158C24—41no—
159#41—44no—
160C2×Q8—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—44no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—84no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4C22⋊Dic5311yescharacteristic
175SD16—44no—
176SD16—44no—
177D8—44no—
178D8—44no—
179M16—24no—
180C2×C8C22⋊Dic5312yes—
181C2×C8C2×C22⋊D5312yes—
182C4⋊C4—41no—
183C22⋊C4—42no—
184C42—21no—
185C42C22⋊Dic5311yescharacteristic
186C42C22×Dic5311yescharacteristic
187C16—2124no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—82no—
192C23—84no—
193C23#1313313411yescharacteristic
194Q8—42no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—82no—
199D4—82no—
200D4—82no—
201D4—82no—
202D4—84no—
203D4—82no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—82no—
210C2×C4#1313313411yescharacteristic
211C2×C4C2×C22⋊Dic5311yescharacteristic
212C8—22no—
213C8#1313324314yes—
214C22—21no—
215C22—41no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—164no—
223C22C2×#1313313411yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—81no—
228C4#1681062812yes—
229C2—21no—
230C2—41no—
231C2—161no—
232C2—161no—
233C2—161no—
234C2#2027342911yescharacteristic
subgroupG/H
centerC2#20273429
commutatorC42×C53C22×C4
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3151C2
socleC2×C53#1555
cosocleC4×C8×C53C23
hypercenter#3151D53
hypocenterC53#11992
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #20276527C2semidirect
2. #20272756C2semidirect
3. #20272752C2semidirect
5. #16810444C22semidirect
134. C53#11992semidirect

Maximal subgroups

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

subgroupindex
#202765272
#202727562
#202727522
C53×#31512
#1199253