Finite Group Databasebeta

← #27270722

Subgroups — #27270722

Order 13568.

240 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 47 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#20275512C211yesmaximal, characteristic
2#20272655C211yesmaximal, characteristic, complemented
3C53×#1322C211yesmaximal, characteristic, complemented
4C2×C4×D212C411yescharacteristic
5C2×#13133129—21no—
6C2×#13133129—21no—
7C2×#13133128C411yescharacteristic
8C22⋊#11719908—21no—
9C2×C4⋊C8×C53C2211yescharacteristic
10C53×C22⋊C16—21no—
11C22×D212C811yescharacteristic
12C22×C4×D53—21no—
13C22×#88322—21no—
14C4×D212—41no—
15C4×D212—41no—
16C2×C4⋊Dic53C811yescharacteristic
17C2×#11719921—21no—
18C2×#11719908—41no—
19#13133129—42no—
20#13133129—42no—
21#13133128—41no—
22C22×C8×C53D411yescharacteristic
23C22×C8×C53D411yescharacteristic
24C2×C42×C53C2×C411yescharacteristic
25C2×C16×C53—41no—
26C4⋊C8×C53—41no—
27C23×D53—21no—
28C22×C4×C53C22⋊C411yescharacteristic
29C22×C4×C53C2×C811yescharacteristic
30C22×C4×C53M1611yescharacteristic
31C22×Dic53—21no—
32C2×D212—41no—
33C2×D212—42no—
34C2×C4×D53—41no—
35C2×C4×D53—41no—
36C2×C4×D53—41no—
37C2×C4×D53—41no—
38C2×C4×D53—41no—
39C2×C4×D53—41no—
40C2×C8×C53—21no—
41C2×C8×C53—21no—
42C2×C8×C53—41no—
43C2×C8×C53—41no—
44C42×C53—21no—
45C42×C53—21no—
46#11719921—42no—
47C4⋊Dic53—41no—
48C2×#88322—22no—
49C2×#88322—41no—
50C2×#88322—41no—
51C16×C53—81no—
52#11719908—81no—
53C23×C53C22⋊C811yescharacteristic
54C22×D53—22no—
55C22×D53—41no—
56C22×D53—41no—
57C22×D53—41no—
58C22×D53—41no—
59C22×D53—41no—
60C22×D53—41no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—41no—
68C2×C4×C53#9911yescharacteristic
69C2×C4×C53#10011yescharacteristic
70C2×Dic53—41no—
71C2×Dic53—41no—
72C2×Dic53—41no—
73D212—42no—
74D212—42no—
75C4×D53—81no—
76C4×D53—81no—
77C4×D53—81no—
78C4×D53—81no—
79C8×C53—41no—
80C8×C53—41no—
81#88322—42no—
82#7060—531nomaximal, Sylow
83C22×C53—21no—
84C22×C53—21no—
85C22×C53#32311yescharacteristic
86C22×C53#35011yescharacteristic
87C22×C53#35011yescharacteristic
88C2×D53—42no—
89C2×D53—42no—
90C2×D53—42no—
91C2×D53—81no—
92C2×D53—81no—
93C2×D53—81no—
94C2×D53—81no—
95C4×C53—21no—
96C4×C53—41no—
97C4×C53—41no—
98C4×C53#35311yescharacteristic
99C4×C53#35511yescharacteristic
100Dic53—81no—
101#1955—531no—
102#1322—531no—
103#1322D5311yescharacteristic, complemented
104C2×C53—21no—
105C2×C53—21no—
106C2×C53#131511yescharacteristic
107C2×C53#131811yescharacteristic
108C2×C53#132011yescharacteristic
109D53—82no—
110C2×C4×D4—531no—
111C2×C4⋊C8C2×D5311yescharacteristic
112C2×C22⋊C8—1061no—
113C2×C22⋊C8—1061no—
114C2×#97—531no—
115C22⋊C16—21no—
116C22⋊C16—1061no—
117C53#706011yescharacteristic, complemented, Sylow
118C22×D4—531no—
119C23×C4—1061no—
120C22×C8—1061no—
121C22×C8D21211yescharacteristic
122C22×C8C22⋊D5311yescharacteristic
123C4×D4—2121no—
124C4×D4—2121no—
125C2×C4⋊C4—531no—
126C2×C22⋊C4—1061no—
127C2×C42C4×D5311yescharacteristic
128C2×C16—41no—
129C2×C16—2121no—
130C4⋊C8—41no—
131C22⋊C8—2122no—
132C22⋊C8—2122no—
133#97—2121no—
134C24—1061no—
135C2×D4—2121no—
136C2×D4—2122no—
137C22×C4—1061no—
138C22×C4—2121no—
139C22×C4—2121no—
140C22×C4—2121no—
141C22×C4—2121no—
142C22×C4—2121no—
143C22×C4—2121no—
144C22×C4C8×D5311yescharacteristic
145C22×C4#1171991211yescharacteristic
146C22×C4#1171992111yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—1062no—
152C2×C8—2121no—
153C2×C8—2121no—
154C4⋊C4—2121no—
155C22⋊C4—2122no—
156C42—21no—
157C42—21no—
158C16—81no—
159C16—4241no—
160C23—1062no—
161C23—2121no—
162C23—2121no—
163C23—2121no—
164C23—2121no—
165C23—2121no—
166C23—2121no—
167C23#1313312911yescharacteristic
168D4—2122no—
169D4—2122no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—4241no—
178C2×C4—4241no—
179C2×C4—4241no—
180C2×C4—4241no—
181C2×C4—2121no—
182C2×C4—2121no—
183C2×C4—2121no—
184C2×C4#1313313111yescharacteristic
185C2×C4#1313313311yescharacteristic
186C8—41no—
187C8—41no—
188C8—2122no—
189C22—21no—
190C22—21no—
191C22—4241no—
192C22—4241no—
193C22—4241no—
194C22—4241no—
195C22—2122no—
196C22—2122no—
197C22—2122no—
198C22#1681034611yescharacteristic
199C22#1681040611yescharacteristic
200C22#1681040811yescharacteristic
201C4—21no—
202C4—41no—
203C4—41no—
204C4—4241no—
205C4#1681041511yescharacteristic
206C4#1681041911yescharacteristic
207C2—21no—
208C2—21no—
209C2—4242no—
210C2#2027264011yescharacteristic
211C2#2027264611yescharacteristic
212C2#2027265011yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC22×C4×C53C2×C8
FrattiniC2×C4⋊C8C2×D53
FittingC53×#1322C2
socleC22×C53#323
prime radicalC22×D212C8
cosocleC2×C4⋊C8×C53C22
hypercenter#1322D53
hypocenterC53#7060
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #20272655C2semidirect
3. C53×#1322C2semidirect
103. #1322D53semidirect
117. C53#7060semidirect

Maximal subgroups

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

subgroupindex
#202755122
#202726552
C53×#13222
#706053