Finite Group Databasebeta

← #28303651

Subgroups — #28303651

Order 13568.

386 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 102 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#20291613C211yesmaximal, characteristic
2#20290885C211yesmaximal, characteristic
3#20290266C212yesmaximal
4#20289548C212yesmaximal
5#20274510C211yesmaximal, characteristic
6#16811673—22no—
7#16811673—22no—
8#16811644—22no—
9#16811639C2212yes—
10C2×C4⋊#88324C2211yescharacteristic
11C2×C4×#88324C2211yescharacteristic
12#16811489—22no—
13#16811486—22no—
14#16811484—22no—
15C2×#13133182C2211yescharacteristic
16C2×#13133135C412yes—
17M16×Dic53C2212yes—
18#16810392C411yescharacteristic
19C53×#343C411yescharacteristic
20C22×#88324—21no—
21C22×#88324—21no—
22C4⋊#88324—22no—
23C4⋊#88324—22no—
24C4⋊#88324—22no—
25C4⋊#88324D412yes—
26C4⋊#88324D412yes—
27C22⋊#88324—22no—
28#13133272—24no—
29C4×#88324—22no—
30C4×#88324—22no—
31C22×Dic106C2×C411yescharacteristic
32C2×#11719917C2×C412yes—
33#13133182—22no—
34#13133182—22no—
35C2×#11719919Q812yes—
36C2×C4×Dic53D412yes—
37C2×C4×Dic53C2311yescharacteristic
38#13133135—24no—
39#13133127—22no—
40C8×Dic53—22no—
41C2×M16×C53C2×C412yes—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C22×C4×C53C4⋊C412yes—
44C22×C4×C53C22×C411yescharacteristic
45C22×Dic53C4⋊C412yes—
46C22×Dic53C2×D411yescharacteristic
47C22×Dic53C2×Q811yescharacteristic
48C2×Dic106—22no—
49C2×Dic106—22no—
50C2×Dic106—21no—
51C2×Dic106—21no—
52C2×#88324—22no—
53C2×#88324—22no—
54C2×#88324—22no—
55C2×#88324—41no—
56C2×#88324—41no—
57C2×#88324—42no—
58C2×#88324—41no—
59M16×C53—22no—
60M16×C53M1614yes—
61C2×C8×C53—22no—
62C4⋊C4×C53—22no—
63#11719919—24no—
64C4×Dic53—22no—
65C4×Dic53—22no—
66C4×Dic53C2×D412yes—
67C4×Dic53#4112yes—
68#11719917—22no—
69#11719917—22no—
70C2×#88322—22no—
71C23×C53C2×C4⋊C411yescharacteristic
72C2×C4×C53—22no—
73C2×C4×C53—42no—
74C2×C4×C53C4×D412yes—
75C2×C4×C53C2×M1612yes—
76C2×C4×C53#13111yescharacteristic
77C2×C4×C53#13111yescharacteristic
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—41no—
84C2×Dic53#12011yescharacteristic
85C2×Dic53C22⋊Q812yes—
86C2×Dic53#12311yescharacteristic
87Dic106—42no—
88Dic106—42no—
89#88324—42no—
90#88324—42no—
91#88324—42no—
92#88324—81no—
93C8×C53—24no—
94#88322—42no—
95#8838—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—21no—
98C22×C53#38511yescharacteristic
99C22×C53#42311yescharacteristic
100C22×C53#42411yescharacteristic
101C4×C53—42no—
102C4×C53#43512yes—
103C4×C53#43612yes—
104Dic53—21no—
105Dic53—22no—
106Dic53—42no—
107Dic53#45411yescharacteristic
108Dic53#45611yescharacteristic
109#2983—531no—
110#1984—531no—
111C4×#101—531no—
112#1565—532no—
113#1278—532no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53#185111yescharacteristic
117C2×C53#185711yescharacteristic
118C2×C53#185811yescharacteristic
119C2×C4×Q8—531no—
120#446—1062no—
121#446—1062no—
122#432—1062no—
123#431—532no—
124C2×C4⋊C8—531no—
125C2×#101—532no—
126C4×M16—532no—
127C2×C4×C8—531no—
128#343—531no—
129#343#615711yescharacteristic
130#336—1062no—
131#335—1062no—
132#334—1062no—
133C53#883811yescharacteristic, complemented, Sylow
134C22×Q8—531no—
135C2×M16—532no—
136C2×M16C2×#615712yes—
137C22×C8—1061no—
138C22×C8—1061no—
139C4×Q8—1062no—
140C4×Q8—1062no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4C2×#615711yescharacteristic
143C2×C42—531no—
144C2×C42—532no—
145C4⋊C8—532no—
146C4⋊C8—532no—
147C4⋊C8—1062no—
148C4⋊C8—1062no—
149C4⋊C8—1062no—
150#101—1064no—
151C22⋊C8—1062no—
152#97—1062no—
153#97—1064no—
154C4×C8—1062no—
155C4×C8—1062no—
156C4×C8—1062no—
157C2×Q8—1062no—
158C2×Q8—1062no—
159C2×Q8—1061no—
160C2×Q8—1061no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—532no—
164C22×C4C4⋊#615712yes—
165C22×C4C22×#615711yescharacteristic
166M16—22no—
167M16—1062no—
168M16—1062no—
169M16#1171993614yes—
170C2×C8—22no—
171C2×C8—1062no—
172C2×C8—1062no—
173C2×C8—1062no—
174C2×C8—1062no—
175C2×C8—2121no—
176C2×C8—2122no—
177C2×C8—2121no—
178C2×C8—2121no—
179C4⋊C4—22no—
180C4⋊C4—1064no—
181C42—532no—
182C42—532no—
183C42—1062no—
184C42—1062no—
185C23C2×C4⋊#615711yescharacteristic
186Q8—2122no—
187Q8—2122no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—531no—
191C2×C4—532no—
192C2×C4—531no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—2121no—
199C2×C4D4×#615712yes—
200C2×C4C2×#1171993612yes—
201C2×C4#1313330411yescharacteristic
202C2×C4#1313330511yescharacteristic
203C8—24no—
204C8—4241no—
205C8—2122no—
206C8—2122no—
207C8—2122no—
208C8—2122no—
209C22—21no—
210C22—21no—
211C22#1681157011yescharacteristic
212C22#1681162711yescharacteristic
213C22#1681163011yescharacteristic
214C4—42no—
215C4—531no—
216C4—531no—
217C4—1061no—
218C4—1062no—
219C4—2122no—
220C4#1681165212yes—
221C4#1681165712yes—
222C2—21no—
223C2—21no—
224C2#2029066111yescharacteristic
225C2#2029067111yescharacteristic
226C2#2029067211yescharacteristic
subgroupG/H
centerC22#16811570
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#343C4
socleC22×C53#385
prime radicalC23×C53C2×C4⋊C4
cosocleC2×C4×Dic53C23
hypercenter#343#6157
hypocenterC53#8838
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
133. C53#8838semidirect

Maximal subgroups

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

subgroupindex
#202916132
#202908852
#202902662
#202895482
#202745102
#883853