Finite Group Databasebeta

← #27298376

Subgroups — #27298376

Order 13568.

228 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 71 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#20276971C211yesmaximal, characteristic
2#20274085C211yesmaximal, characteristic, complemented
3#20274075C211yesmaximal, characteristic
4#20272787C211yesmaximal, characteristic, complemented
5#20272765C211yesmaximal, characteristic, complemented
6#20272762C211yesmaximal, characteristic
7C53×#2183C211yesmaximal, characteristic, complemented
8C2×#13133162—21no—
9C2×#13133162—21no—
10#16811003C2211yescharacteristic
11#16811000C2211yescharacteristic
12#16810908—21no—
13#16810823—21no—
14#16810376C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810370—21no—
17#16810366—21no—
18C8×Q8×C53C2211yescharacteristic
19C53×C4⋊C16—21no—
20C53×C4⋊C16C2211yescharacteristic
21C53×#346—21no—
22C4×C16×C53C2211yescharacteristic
23C2×Dic212—21no—
24C2×Dic212—21no—
25C2×#11719924—41no—
26C2×D424—21no—
27C2×D424—21no—
28#13133217—21no—
29#13133214—21no—
30#13133162—42no—
31#13133162—42no—
32#13133148—41no—
33C4⋊#88322—21no—
34C4⋊#88322—21no—
35#13133141—41no—
36#13133137—41no—
37C4×Q8×C53D411yescharacteristic
38C2×C16×C53—21no—
39C2×C16×C53D411yescharacteristic
40C2×C16×C53D411yescharacteristic
41C4⋊C8×C53—21no—
42C4⋊C8×C53D411yescharacteristic
43C4×C8×C53—21no—
44C4×C8×C53C2311yescharacteristic
45C2×Q8×C53D811yescharacteristic
46C2×D212—41no—
47C2×D212—41no—
48C2×D212—41no—
49C2×Dic106—41no—
50C2×C8×C53—21no—
51C2×C8×C53C2×D411yescharacteristic
52C2×C8×C53#4111yescharacteristic
53C4⋊C4×C53—21no—
54C4⋊C4×C53D811yescharacteristic
55C42×C53—21no—
56C42×C53C2×D411yescharacteristic
57#11719924—82no—
58C4⋊Dic53—41no—
59C4⋊Dic53—41no—
60C2×#88322—41no—
61C2×#88322—41no—
62Dic212—41no—
63Dic212—41no—
64Dic212—81no—
65D424—41no—
66D424—41no—
67D424—81no—
68C16×C53—21no—
69C16×C53—41no—
70C16×C53SD1612yescomplemented
71C22×D53—81no—
72Q8×C53—21no—
73Q8×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53#12111yescharacteristic
77C2×C4×C53C2×D811yescharacteristic
78C2×C4×C53#13511yescharacteristic
79C2×Dic53—81no—
80D212—81no—
81D212—81no—
82D212—81no—
83D212—81no—
84D212—161no—
85Dic106—81no—
86Dic106—81no—
87C8×C53—21no—
88C8×C53—41no—
89C8×C53C2×SD1611yescharacteristic
90C8×C53AGL(1,ℤ/8)11yescharacteristic
91#88322—81no—
92#88322—81no—
93#12444—531nomaximal, Sylow
94C22×C53#46611yescharacteristic
95C2×D53—161no—
96C2×D53—161no—
97C2×D53—161no—
98C4×C53—21no—
99C4×C53—41no—
100C4×C53—41no—
101C4×C53#46011yescharacteristic
102C4×C53#47211yescharacteristic
103C4×C53#50911yescharacteristic
104C4×C53#51011yescharacteristic
105Dic53—161no—
106#2248—531no—
107#2183D5311yescharacteristic, complemented
108#1670—531no—
109#1668—531no—
110#1371—531no—
111#1362—531no—
112#1361—531no—
113C2×C53#166811yescharacteristic
114C2×C53#222011yescharacteristic
115C2×C53#222311yescharacteristic
116D53—321no—
117C2×SD32—1061no—
118C2×SD32—1061no—
119#494—531no—
120#493—531no—
121#475—1061no—
122#460—1061no—
123C8×Q8C2×D5311yescharacteristic
124C4⋊C16—21no—
125C4⋊C16C2×D5311yescharacteristic
126#346—21no—
127C4×C16C2×D5311yescharacteristic
128#335—531no—
129#335—531no—
130#332—1061no—
131#331—1061no—
132C53#1244411yescharacteristic, complemented, Sylow
133C2×Q16—1061no—
134C2×Q16—1061no—
135C2×SD16—2121no—
136C2×D8—1061no—
137C2×D8—1061no—
138C4⋊Q8—1061no—
139#127—1061no—
140C4×Q8D21211yescharacteristic
141SD32—2122no—
142SD32—2122no—
143C2×C16—21no—
144C2×C16C22⋊D5311yescharacteristic
145C2×C16C22⋊D5311yescharacteristic
146#106—2121no—
147C4⋊C8—21no—
148C4⋊C8—1061no—
149C4⋊C8—1061no—
150C4⋊C8D21211yescharacteristic
151#103—2121no—
152#102—2121no—
153C4×C8—21no—
154C4×C8C22×D5311yescharacteristic
155C2×Q8—2121no—
156C2×Q8D42411yescharacteristic
157C2×D4—2121no—
158C2×D4—2121no—
159C2×D4—2121no—
160Q16—4241no—
161Q16—2121no—
162Q16—2121no—
163SD16—4242no—
164D8—4241no—
165D8—2121no—
166D8—2121no—
167C2×C8—21no—
168C2×C8—2121no—
169C2×C8—2121no—
170C2×C8#1171994511yescharacteristic
171C2×C8C2×C22⋊D5311yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—2121no—
174C4⋊C4—2121no—
175C4⋊C4D42411yescharacteristic
176C42—21no—
177C42C2×D21211yescharacteristic
178C16—21no—
179C16—41no—
180C16#1171992412yescomplemented
181C23—4241no—
182Q8—21no—
183Q8—21no—
184Q8—4241no—
185Q8—4241no—
186D4—8481no—
187D4—4241no—
188D4—4241no—
189D4—4241no—
190D4—4241no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—4241no—
194C2×C4#1313319111yescharacteristic
195C2×C4C2×D42411yescharacteristic
196C2×C4#1313323911yescharacteristic
197C8—21no—
198C8—41no—
199C8—4241no—
200C8—4241no—
201C8C2×#1171992411yescharacteristic
202C8#1313324211yescharacteristic
203C22—8481no—
204C22—8481no—
205C22—8481no—
206C22#1681085911yescharacteristic
207C4—21no—
208C4—41no—
209C4—41no—
210C4—8481no—
211C4#1681082311yescharacteristic
212C4#1681089311yescharacteristic
213C4#1681106111yescharacteristic
214C4#1681106611yescharacteristic
215C2—16961no—
216C2#2027407511yescharacteristic
217C2#2027682811yescharacteristic
218C2#2027684511yescharacteristic
subgroupG/H
centerC22#16810859
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2183C2
socleC22×C53#466
prime radical#16811000C22
hypercenter#2183D53
hypocenterC53#12444
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #2183

Complements

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

NH ≅ G/Ntype
2. #20274085C2semidirect
4. #20272787C2semidirect
5. #20272765C2semidirect
7. C53×#2183C2semidirect
70. C16×C53SD16semidirect
107. #2183D53semidirect
132. C53#12444semidirect
180. C16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202769712
#202740852
#202740752
#202727872
#202727652
#202727622
C53×#21832
#1244453