Finite Group Databasebeta

← #28298842

Subgroups — #28298842

Order 13568.

256 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 52 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#20290887C211yesmaximal, characteristic
2#20290582C211yesmaximal, characteristic
3#20273057C211yesmaximal, characteristic
4#16811626—21no—
5C2×#13133284C411yescharacteristic
6C2×#13133278C412yes—
7C2×#13133272C411yescharacteristic
8#16811503—21no—
9#16811489—21no—
10#16811489—21no—
11C2×#13133182C2211yescharacteristic
12#16810677—21no—
13#16810349C411yescharacteristic
14#16810348—21no—
15C53×#324C411yescharacteristic
16C2×#11719940—22no—
17C22×#88324—21no—
18C22×#88324—21no—
19#13133284—41no—
20#13133284—41no—
21C4⋊#88324—41no—
22#13133278—42no—
23#13133272—41no—
24C22×Dic106C2×C411yescharacteristic
25#13133182—41no—
26#13133182—41no—
27C2×#11719919D411yescharacteristic
28C2×#11719919Q811yescharacteristic
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53D411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32#13133146—41no—
33C22⋊#88322—21no—
34#13133127—41no—
35C2×C4⋊C4×C53C2×C411yescharacteristic
36C22⋊C8×C53—21no—
37C22×C4×C53C4211yescharacteristic
38C22×C4×C53C22⋊C411yescharacteristic
39C22×C4×C53C4⋊C411yescharacteristic
40C22×Dic53C22⋊C411yescharacteristic
41C22×Dic53C22⋊C411yescharacteristic
42C22×Dic53C4⋊C411yescharacteristic
43C22×Dic53C4⋊C411yescharacteristic
44C2×Dic106—21no—
45C2×Dic106—21no—
46C2×Dic106—41no—
47C2×Dic106—41no—
48#11719940—42no—
49#11719940—42no—
50C2×#88324—22no—
51C2×#88324—41no—
52C2×#88324—41no—
53C2×#88324—42no—
54C2×#88324—42no—
55C2×#88324—41no—
56C2×C8×C53—41no—
57C4⋊C4×C53—41no—
58#11719919—41no—
59#11719919—41no—
60C4×Dic53—21no—
61C4×Dic53—21no—
62C4×Dic53—21no—
63C4×Dic53—21no—
64C4×Dic53—21no—
65C4×Dic53—21no—
66C2×#88322—41no—
67C23×C53#9511yescharacteristic
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—41no—
84C2×Dic53#9911yescharacteristic
85C2×Dic53#9911yescharacteristic
86C2×Dic53#10011yescharacteristic
87C2×Dic53#10111yescharacteristic
88Dic106—42no—
89Dic106—42no—
90#88324—42no—
91#88324—82no—
92#88324—81no—
93C8×C53—81no—
94#88322—81no—
95#7323—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—21no—
98C22×C53#34211yescharacteristic
99C22×C53#34311yescharacteristic
100C22×C53#34411yescharacteristic
101C4×C53—41no—
102C4×C53—41no—
103C4×C53—41no—
104Dic53—21no—
105Dic53—21no—
106Dic53—21no—
107Dic53—41no—
108Dic53—41no—
109Dic53#35511yescharacteristic
110Dic53#35611yescharacteristic
111#1985—531no—
112#1801—531no—
113#1467—531no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53#130511yescharacteristic
117C2×C53#139411yescharacteristic
118C2×C53#139511yescharacteristic
119C2×C4×Q8—531no—
120#432—1061no—
121#423—1061no—
122C2×C4⋊C8—531no—
123C2×#101—532no—
124C2×#97—531no—
125#343—1061no—
126#336—1061no—
127#336—1061no—
128#324—531no—
129#324—1061no—
130#324#615711yescharacteristic
131C53#732311yescharacteristic, complemented, Sylow
132C22×Q8—531no—
133C2×M16—1062no—
134C22×C8—1061no—
135C22×C8—1061no—
136C4×Q8—2121no—
137C4×Q8—2121no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—531no—
140C2×C4⋊C4C2×#615711yescharacteristic
141C2×C42—531no—
142C2×C42—531no—
143C2×C42—531no—
144C4⋊C8—2121no—
145C4⋊C8—2121no—
146C4⋊C8—2121no—
147C4⋊C8—2121no—
148#101—2122no—
149C22⋊C8—21no—
150C22⋊C8—1061no—
151#97—2121no—
152#97—2121no—
153C2×Q8—1061no—
154C2×Q8—1061no—
155C2×Q8—2121no—
156C2×Q8—2121no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4C4×#615711yescharacteristic
162C22×C4C4⋊#615711yescharacteristic
163C22×C4C22⋊#615711yescharacteristic
164M16—2122no—
165M16—2122no—
166C2×C8—41no—
167C2×C8—1062no—
168C2×C8—2121no—
169C2×C8—2121no—
170C2×C8—2122no—
171C2×C8—2122no—
172C2×C8—2121no—
173C2×C8—2121no—
174C4⋊C4—41no—
175C4⋊C4—2121no—
176C4⋊C4—2121no—
177C42—1061no—
178C42—1061no—
179C42—1061no—
180C42—1061no—
181C42—1061no—
182C42—1061no—
183C23(C2×#6157)⋊C411yescharacteristic
184Q8—2122no—
185Q8—2122no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—531no—
194C2×C4—531no—
195C2×C4—531no—
196C2×C4—531no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—2121no—
206C8—81no—
207C8—4241no—
208C8—4241no—
209C8—4242no—
210C8—2122no—
211C22—21no—
212C22—21no—
213C22#1681150011yescharacteristic
214C22#1681150211yescharacteristic
215C22#1681150511yescharacteristic
216C4—41no—
217C4—41no—
218C4—41no—
219C4—531no—
220C4—531no—
221C4—1061no—
222C4—1061no—
223C4—1061no—
224C4—2121no—
225C4—2121no—
226C2—21no—
227C2—21no—
228C2#2028961711yescharacteristic
229C2#2028981111yescharacteristic
230C2#2028981411yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC22×C4×C53C42
FrattiniC2×C4⋊C4C2×#6157
FittingC53×#324C4
socleC22×C53#342
prime radicalC23×C53#95
cosocleC2×#13133182C22
hypercenter#324#6157
hypocenterC53#7323
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
131. C53#7323semidirect

Maximal subgroups

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

subgroupindex
#202908872
#202905822
#202730572
#732353