Finite Group Databasebeta

← #27270714

Subgroups — #27270714

Order 13568.

248 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 53 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#20275508C211yesmaximal, characteristic
2#20272655C211yesmaximal, characteristic, complemented
3C53×#1322C211yesmaximal, characteristic, complemented
4C2×C4×D212C411yescharacteristic
5C2×#13133129—21no—
6C2×#13133129—21no—
7C2×C4×#88322C411yescharacteristic
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—
21C4×#88322—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—21no—
33C2×D212—41no—
34C2×D212—41no—
35C2×D212C1612yescomplemented
36C2×C4×D53—41no—
37C2×C4×D53—41no—
38C2×C4×D53—41no—
39C2×C4×D53—41no—
40C2×C4×D53—41no—
41C2×C4×D53—41no—
42C2×C8×C53—21no—
43C2×C8×C53—21no—
44C2×C8×C53—41no—
45C2×C8×C53—41no—
46C42×C53—21no—
47C42×C53—21no—
48#11719921—42no—
49C4⋊Dic53—41no—
50C2×#88322—22no—
51C2×#88322—41no—
52C2×#88322—41no—
53C16×C53—81no—
54#11719908—81no—
55C23×C53C22⋊C811yescharacteristic
56C22×D53—22no—
57C22×D53—41no—
58C22×D53—41no—
59C22×D53—41no—
60C22×D53—41no—
61C22×D53—41no—
62C22×D53—41no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53#9911yescharacteristic
70C2×C4×C53#10011yescharacteristic
71C2×C4×C53C2×C1611yescharacteristic
72C2×C4×C53M3211yescharacteristic
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76D212—42no—
77D212—42no—
78C4×D53—81no—
79C4×D53—81no—
80C4×D53—81no—
81C4×D53—81no—
82C8×C53—41no—
83C8×C53—41no—
84#88322—42no—
85#7056—531nomaximal, Sylow
86C22×C53—21no—
87C22×C53—21no—
88C22×C53#32311yescharacteristic
89C22×C53C22⋊C1611yescharacteristic
90C22×C53#34911yescharacteristic
91C2×D53—42no—
92C2×D53—42no—
93C2×D53—42no—
94C2×D53—81no—
95C2×D53—81no—
96C2×D53—81no—
97C2×D53—81no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—41no—
102C4×C53#35311yescharacteristic
103C4×C53#35511yescharacteristic
104Dic53—81no—
105#1954—531no—
106#1322—531no—
107#1322D5311yescharacteristic, complemented
108C2×C53—21no—
109C2×C53—21no—
110C2×C53#131411yescharacteristic
111C2×C53#131811yescharacteristic
112C2×C53#132111yescharacteristic
113D53—82no—
114C2×C4×D4—531no—
115C2×C4⋊C8C2×D5311yescharacteristic
116C2×C22⋊C8—1061no—
117C2×C22⋊C8—1061no—
118C2×C4×C8—531no—
119C22⋊C16—21no—
120C22⋊C16—1061no—
121C53#705611yescharacteristic, complemented, Sylow
122C22×D4—531no—
123C23×C4—1061no—
124C22×C8—1061no—
125C22×C8D21211yescharacteristic
126C22×C8C22⋊D5311yescharacteristic
127C4×D4—2121no—
128C4×D4—2121no—
129C2×C4⋊C4—531no—
130C2×C22⋊C4—1061no—
131C2×C42C4×D5311yescharacteristic
132C2×C16—41no—
133C2×C16—2121no—
134C4⋊C8—41no—
135C22⋊C8—2122no—
136C22⋊C8—2122no—
137C4×C8—2121no—
138C24—1061no—
139C2×D4—532no—
140C2×D4—1061no—
141C2×D4—2121no—
142C2×D4—2121no—
143C22×C4—1061no—
144C22×C4—2121no—
145C22×C4—2121no—
146C22×C4—2121no—
147C22×C4—2121no—
148C22×C4—2121no—
149C22×C4—2121no—
150C22×C4C8×D5311yescharacteristic
151C22×C4#1171991211yescharacteristic
152C22×C4#1171992111yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—1062no—
158C2×C8—2121no—
159C2×C8—2121no—
160C4⋊C4—2121no—
161C22⋊C4—2122no—
162C42—21no—
163C42—21no—
164C16—81no—
165C16—4241no—
166C23—1062no—
167C23—2121no—
168C23—2121no—
169C23—2121no—
170C23—2121no—
171C23—2121no—
172C23—2121no—
173C23#1313312911yescharacteristic
174D4—2122no—
175D4—2122no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—4241no—
183C2×C4—4241no—
184C2×C4—4241no—
185C2×C4—4241no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4—2121no—
189C2×C4#1313313111yescharacteristic
190C2×C4#1313313311yescharacteristic
191C2×C4C16×D5311yescharacteristic
192C2×C4#1313315611yescharacteristic
193C8—41no—
194C8—41no—
195C8—2122no—
196C22—21no—
197C22—21no—
198C22—4241no—
199C22—4241no—
200C22—4241no—
201C22—4241no—
202C22—2122no—
203C22—2122no—
204C22—2122no—
205C22#1681034611yescharacteristic
206C22#1681040211yescharacteristic
207C22#1681040411yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—41no—
212C4—4241no—
213C4#1681041511yescharacteristic
214C4#1681041911yescharacteristic
215C2—21no—
216C2—21no—
217C2—4242no—
218C2#2027263811yescharacteristic
219C2#2027264611yescharacteristic
220C2#2027265211yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC2×C4×C53C2×C16
FrattiniC2×C4⋊C8C2×D53
FittingC53×#1322C2
socleC22×C53#323
prime radicalC22×D212C8
cosocleC2×C4⋊C8×C53C22
hypercenter#1322D53
hypocenterC53#7056
LCSG ▷ C2×C4×C53 ▷ C22×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
35. C2×D212C16semidirect
107. #1322D53semidirect
121. C53#7056semidirect

Maximal subgroups

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

subgroupindex
#202755082
#202726552
C53×#13222
#705653