Finite Group Databasebeta

← #27297861

Subgroups — #27297861

Order 13568.

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

normal subgroups only

This group has 364 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#20274341C211yesmaximal, characteristic, complemented
3#20272752C211yesmaximal, characteristic, complemented
4#20272691C212yesmaximal, complemented
5C53×#3292C211yesmaximal, characteristic
6#16811021C2211yescharacteristic, complemented
7#16810995—21no—
8#16810969—21no—
9#16810447—22no—
10#16810444C2212yescomplemented
11#16810433—22no—
12#16810344C2211yescharacteristic, complemented
13C53×#576—22no—
14C53×#493C2211yescharacteristic
15C53×#472—22no—
16C53×#453—21no—
17C53×#453—21no—
18C53×#443C2212yes—
19C2×#11719923—42no—
20#13133217—21no—
21#13133157—22no—
22#13133149—22no—
23#13133138—42no—
24C4×#88322—21no—
25H(5,𝔽2)×C53—21no—
26H(5,𝔽2)×C53—21no—
27AGL(1,ℤ/8)×C53—22no—
28AGL(1,ℤ/8)×C53—22no—
29AGL(1,ℤ/8)×C53—42no—
30AGL(1,ℤ/8)×C53—42no—
31#135×C53—42no—
32#135×C53D412yes—
33C2×D8×C53—22no—
34C2×D8×C53—42no—
35#131×C53—22no—
36#127×C53D411yescharacteristic
37#127×C53D411yescharacteristic
38#108×C53D412yes—
39#104×C53—22no—
40#100×C53—21no—
41#100×C53—21no—
42C4×C8×C53C2311yescharacteristic
43#41×C53—22no—
44#41×C53—42no—
45#41×C53—42no—
46C2×D4×C53—21no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C2×D4×C53—81no—
53C2×D4×C53—81no—
54C2×Dic106—41no—
55Q16×C53SD1612yescomplemented
56SD16×C53—22no—
57SD16×C53—42no—
58SD16×C53—42no—
59D8×C53—42no—
60D8×C53—42no—
61D8×C53—42no—
62D8×C53—42no—
63D8×C53SD1612yescomplemented
64M16×C53—22no—
65M16×C53—42no—
66C2×C8×C53—42no—
67C2×C8×C53C2×D412yes—
68C42×C53C2×D411yescharacteristic
69#11719923—84no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C2×#88322—22no—
73#11719908—42no—
74C23×C53—41no—
75C23×C53—41no—
76C23×C53—81no—
77C23×C53—81no—
78Q8×C53—22no—
79D4×C53—22no—
80D4×C53—42no—
81D4×C53—42no—
82D4×C53—42no—
83D4×C53—82no—
84D4×C53—82no—
85D4×C53—82no—
86D4×C53—82no—
87C2×C4×C53—21no—
88C2×C4×C53—42no—
89C2×C4×C53—81no—
90C2×C4×C53—81no—
91C2×C4×C53#12011yescharacteristic
92C2×Dic53—81no—
93Dic106—82no—
94C8×C53—42no—
95C8×C53C2×SD1612yes—
96C8×C53AGL(1,ℤ/8)12yes—
97#88322—42no—
98#12366—531nomaximal, Sylow
99C22×C53—42no—
100C22×C53—81no—
101C22×C53—81no—
102C22×C53—161no—
103C22×C53—161no—
104C22×C53—321no—
105C22×C53—321no—
106C22×C53#45311yescharacteristic
107C4×C53—22no—
108C4×C53—42no—
109C4×C53#45012yes—
110Dic53—161no—
111#3292D5311yescharacteristic
112#2230—532no—
113#1715—531no—
114#1357—531no—
115#1336—532no—
116C2×C53—21no—
117C2×C53—82no—
118C2×C53—161no—
119C2×C53—161no—
120C2×C53#162911yescharacteristic
121#576—22no—
122#498—531no—
123#493C2×D5311yescharacteristic
124#492—1061no—
125#486—1061no—
126#472—22no—
127#453—21no—
128#453—21no—
129#443C2×D5312yes—
130#365—1062no—
131#364—532no—
132#360—1062no—
133#322—531no—
134C53#1236611yescharacteristic, complemented, Sylow
135H(5,𝔽2)—21no—
136H(5,𝔽2)—21no—
137AGL(1,ℤ/8)—22no—
138AGL(1,ℤ/8)—22no—
139AGL(1,ℤ/8)—42no—
140AGL(1,ℤ/8)—42no—
141#135—42no—
142#135C22⋊D5312yes—
143C2×SD16—2122no—
144C2×D8—22no—
145C2×D8—42no—
146#131—22no—
147C4⋊Q8—1061no—
148#127C22⋊D5311yescharacteristic
149#127C22⋊D5311yescharacteristic
150M32—1062no—
151#108C22⋊D5312yes—
152#106—1062no—
153#104—22no—
154#102—2122no—
155#100—21no—
156#100—21no—
157C4×C8—1061no—
158C4×C8C22×D5311yescharacteristic
159#41—22no—
160#41—42no—
161#41—42no—
162C2×Q8—2121no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—81no—
170C2×D4—81no—
171Q16#1171992312yescomplemented
172SD16—22no—
173SD16—42no—
174SD16—42no—
175SD16—4244no—
176D8—42no—
177D8—42no—
178D8—42no—
179D8—42no—
180D8#1171992312yescomplemented
181M16—22no—
182M16—42no—
183C2×C8—42no—
184C2×C8—1062no—
185C2×C8C2×C22⋊D5312yes—
186C4⋊C4—2121no—
187C4⋊C4—2121no—
188C42C2×C22⋊D5311yescharacteristic
189C16—2122no—
190C23—41no—
191C23—41no—
192C23—81no—
193C23—81no—
194Q8—22no—
195Q8—4242no—
196D4—22no—
197D4—42no—
198D4—42no—
199D4—42no—
200D4—82no—
201D4—82no—
202D4—82no—
203D4—82no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—4241no—
209C2×C4#1313318611yescharacteristic
210C8—42no—
211C8—2122no—
212C8C2×#1171992312yes—
213C8#1313324312yes—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—161no—
218C22—161no—
219C22—321no—
220C22—321no—
221C22#1681079111yescharacteristic
222C4—22no—
223C4—42no—
224C4—8481no—
225C4#1681077412yes—
226C2—21no—
227C2—82no—
228C2—161no—
229C2—161no—
230C2#2027383111yescharacteristic
subgroupG/H
centerC2#20273831
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3292C2
socleC2×C53#1629
hypercenter#3292D53
hypocenterC53#12366
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3292

Complements

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

NH ≅ G/Ntype
1. #20276886C2semidirect
2. #20274341C2semidirect
3. #20272752C2semidirect
4. #20272691C2semidirect
6. #16811021C22semidirect
10. #16810444C22semidirect
12. #16810344C22semidirect
55. Q16×C53SD16semidirect
63. D8×C53SD16semidirect
134. C53#12366semidirect
171. Q16#11719923semidirect
180. D8#11719923semidirect

Maximal subgroups

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

subgroupindex
#202768862
#202743412
#202727522
#202726912
C53×#32922
#1236653