Finite Group Databasebeta

← #28298797

Subgroups — #28298797

Order 13568.

484 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 54 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#20290544C212yesmaximal
2#20274626C211yesmaximal, characteristic
3C2×#13133276—22no—
4C2×#13133275C414yescomplemented
5#16811566—22no—
6C22×#11719945C2211yescharacteristic
7C2×C22⋊#88322C411yescharacteristic, complemented
8C2×#13133129—22no—
9C2×C22⋊C8×C53C411yescharacteristic, complemented
10C2×C22⋊#6157—24no—
11#13133276—44no—
12#13133275—44no—
13(C2×#6157)⋊C4—42no—
14(C2×#6157)⋊C4—42no—
15C2×#11719945—41no—
16C2×#11719945—41no—
17C2×#11719945—44no—
18C22×Dic106Q811yescharacteristic
19C22×D212D411yescharacteristic
20C22×C22⋊D53C2×C412yes—
21C22×C4×D53D412yes—
22C22×#88322—21no—
23C22⋊#88322—41no—
24#13133129—44no—
25C23×C4×C53C2×C411yescharacteristic
26C22×C8×C53—21no—
27C22⋊C8×C53—41no—
28C24×C53C4211yescharacteristic
29C23×D53C22⋊C412yes—
30C22×#6157—44no—
31C22×C4×C53—21no—
32C22×C4×C53—21no—
33C22×C4×C53—42no—
34C22×C4×C53C22⋊C411yescharacteristic
35C22×C4×C53C4⋊C411yescharacteristic
36C2×C22⋊D53—24no—
37C2×C22⋊D53—42no—
38C2×C22⋊D53—42no—
39C22×Dic53C4⋊C412yes—
40#11719945—48no—
41#11719945—48no—
42C2×D212—24no—
43C2×D212—41no—
44C2×C4×D53—24no—
45C2×C4×D53—22no—
46C2×C4×D53—22no—
47C2×C4×D53—42no—
48C2×Dic106—41no—
49C2×Dic106—42no—
50C22⋊#6157—44no—
51C22⋊#6157—44no—
52C2×C8×C53—42no—
53C2×C8×C53—41no—
54C2×#88322—42no—
55C2×#88322—41no—
56C23×C53—21no—
57C23×C53—22no—
58C23×C53—41no—
59C23×C53—41no—
60C23×C53#9511yescharacteristic
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—42no—
64C22×D53—42no—
65C22×D53#9914yes—
66C2×#6157—84no—
67C2×#6157—84no—
68C2×#6157—84no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—21no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53#9911yescharacteristic
75C2×C4×C53#9911yescharacteristic
76C22⋊D53—48no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80D212—44no—
81C4×D53—44no—
82C4×D53—44no—
83Dic106—44no—
84C8×C53—81no—
85#88322—81no—
86#7305—531nomaximal, Sylow
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—82no—
93C22×C53#34212yes—
94C22×C53#34211yescharacteristic
95C2×D53—24no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—44no—
99C2×D53—82no—
100#6157—164no—
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—41no—
104Dic53—42no—
105#1782—532no—
106#1780—531no—
107C2×C53—22no—
108C2×C53—81no—
109C2×C53#130411yescharacteristic
110C2×C53#139111yescharacteristic
111C2×C53#139111yescharacteristic
112D53—44no—
113C22×#41—531no—
114C2×#99—534no—
115C2×#99—1062no—
116C2×C22⋊C8—531no—
117C2×C22⋊C8—1062no—
118C2×C22⋊C8#615711yescharacteristic, complemented
119#381—1062no—
120C53#730511yescharacteristic, complemented, Sylow
121C2×#41—2121no—
122C2×#41—2121no—
123C2×#41—2124no—
124C22×Q8—531no—
125C22×D4—532no—
126C22×D4—531no—
127C23×C4—532no—
128C23×C4C2×#615711yescharacteristic
129C22×C8—21no—
130C22×C8—1061no—
131C2×C22⋊C4—1064no—
132#99—2124no—
133#99—2124no—
134C22⋊C8—41no—
135C22⋊C8—2124no—
136C22⋊C8—2121no—
137#95—2122no—
138#95—2122no—
139C24—532no—
140C24C4×#615711yescharacteristic
141#41—2128no—
142#41—2128no—
143C2×Q8—2122no—
144C2×Q8—2121no—
145C2×D4—1064no—
146C2×D4—1064no—
147C2×D4—2121no—
148C2×D4—2122no—
149C2×D4—2122no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—42no—
153C22×C4—532no—
154C22×C4—1064no—
155C22×C4—1062no—
156C22×C4—1062no—
157C22×C4—2122no—
158C22×C4—2124no—
159C22×C4C4⋊#615711yescharacteristic
160C22×C4C22⋊#615711yescharacteristic
161C2×C8—42no—
162C2×C8—41no—
163C2×C8—2122no—
164C2×C8—2121no—
165C22⋊C4—2124no—
166C22⋊C4—2124no—
167C23—21no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—534no—
172C23—1062no—
173C23—1062no—
174C23—2122no—
175C23—2122no—
176C23(C2×#6157)⋊C411yescharacteristic
177Q8—2124no—
178D4—2124no—
179D4—2128no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—4244no—
189C2×C4—4244no—
190C2×C4—4244no—
191C2×C4—2124no—
192C2×C4—2124no—
193C2×C4#1313327511yescharacteristic
194C2×C4#1313327611yescharacteristic
195C8—81no—
196C8—4241no—
197C22—21no—
198C22—21no—
199C22—41no—
200C22—41no—
201C22—41no—
202C22—82no—
203C22—1064no—
204C22—4242no—
205C22—2124no—
206C22—2122no—
207C22—2122no—
208C22#1681150012yes—
209C22#1681150111yescharacteristic
210C4—21no—
211C4—21no—
212C4—41no—
213C4—8484no—
214C4—2122no—
215C2—22no—
216C2—81no—
217C2—2124no—
218C2#2028961411yescharacteristic
219C2#2028980311yescharacteristic
220C2#2028980411yescharacteristic
subgroupG/H
centerC22#16811501
commutatorC24×C53C42
FrattiniC23×C4C2×#6157
FittingC2×C22⋊C8×C53C4
socleC22×C53#342
prime radical, cosocleC22×#11719945C22
hypercenterC2×C22⋊C8#6157
hypocenterC53#7305
LCSG ▷ C24×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#13133275C4semidirect
7. C2×C22⋊#88322C4semidirect
9. C2×C22⋊C8×C53C4semidirect
118. C2×C22⋊C8#6157semidirect
120. C53#7305semidirect

Maximal subgroups

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

subgroupindex
#202905442
#202746262
#730553