Finite Group Databasebeta

← #28303731

Subgroups — #28303731

Order 13568.

348 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 82 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#20290888C211yesmaximal, characteristic
3#20290288C212yesmaximal
4#20289610C212yesmaximal
5#20273037C211yesmaximal, characteristic
6#16811673—22no—
7#16811673—22no—
8#16811644—21no—
9#16811644C2212yes—
10#16811639—21no—
11C2×C4⋊#88324C2211yescharacteristic
12C2×#13133272C2211yescharacteristic
13#16811489—21no—
14#16811489—21no—
15#16811487—22no—
16#16811484—22no—
17C2×#13133182C2211yescharacteristic
18#16810673C2212yes—
19#16810349C411yescharacteristic
20#16810348C412yes—
21C53×#324C411yescharacteristic
22C22×#88324—21no—
23C22×#88324—21no—
24C4⋊#88324—21no—
25C4⋊#88324—21no—
26C4⋊#88324—21no—
27C4⋊#88324—21no—
28C4⋊#88324—22no—
29C4⋊#88324D412yes—
30C4⋊#88324D412yes—
31C22⋊#88324—22no—
32#13133272—22no—
33#13133272—22no—
34#13133272—22no—
35C4×#88324—22no—
36C22×Dic106Q811yescharacteristic
37#13133182—22no—
38#13133182—22no—
39C2×#11719919C2×C411yescharacteristic
40C2×#11719919Q811yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53C2311yescharacteristic
44#13133146—22no—
45C22⋊#88322C2×C412yes—
46C8×Dic53—22no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C22⋊C8×C53C2×C412yes—
49C22×C4×C53C4⋊C411yescharacteristic
50C22×C4×C53C4⋊C411yescharacteristic
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53C4⋊C411yescharacteristic
53C22×Dic53C4⋊C411yescharacteristic
54C22×Dic53C2×D411yescharacteristic
55C22×Dic53C2×Q811yescharacteristic
56C2×Dic106—22no—
57C2×Dic106—22no—
58C2×Dic106—41no—
59C2×#88324—22no—
60C2×#88324—22no—
61C2×#88324—22no—
62C2×#88324—41no—
63C2×#88324—41no—
64C2×#88324—42no—
65C2×#88324—41no—
66C2×C8×C53—22no—
67C4⋊C4×C53—22no—
68#11719919—22no—
69#11719919—22no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C4×Dic53C2×D412yes—
75C4×Dic53#4112yes—
76C2×#88322—22no—
77C23×C53C2×C4⋊C411yescharacteristic
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53#10812yes—
84C2×C4×C53C4×D412yes—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—41no—
94C2×Dic53#12011yescharacteristic
95C2×Dic53C22⋊Q812yes—
96C2×Dic53#12311yescharacteristic
97Dic106—44no—
98#88324—42no—
99#88324—42no—
100#88324—42no—
101#88324—81no—
102C8×C53—42no—
103#88322—42no—
104#8865—531nomaximal, Sylow
105C22×C53—21no—
106C22×C53—21no—
107C22×C53#38511yescharacteristic
108C22×C53C2×#10811yescharacteristic
109C22×C53#43011yescharacteristic
110C4×C53—22no—
111C4×C53—22no—
112C4×C53—41no—
113Dic53—22no—
114Dic53—21no—
115Dic53—41no—
116Dic53—41no—
117Dic53#45411yescharacteristic
118Dic53#45611yescharacteristic
119#2983—531no—
120#1985—531no—
121#1573—532no—
122#1461—531no—
123#1302—532no—
124C2×C53—21no—
125C2×C53—21no—
126C2×C53#185511yescharacteristic
127C2×C53#185711yescharacteristic
128C2×C53#186811yescharacteristic
129C2×C4×Q8—531no—
130#446—1062no—
131#446—1062no—
132#432—532no—
133#432—1061no—
134#431—532no—
135#431—1061no—
136C2×C4⋊C8—531no—
137C2×#97—531no—
138#336—1061no—
139#336—1061no—
140#335—1062no—
141#334—1062no—
142#324—531no—
143#324—532no—
144#324#615711yescharacteristic
145C53#886511yescharacteristic, complemented, Sylow
146C22×Q8—531no—
147C22×C8—1061no—
148C22×C8—1061no—
149C4×Q8—1062no—
150C4×Q8—1062no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4C2×#615711yescharacteristic
154C2×C42—531no—
155C2×C42—531no—
156C2×C42—531no—
157C4⋊C8—532no—
158C4⋊C8—532no—
159C4⋊C8—1062no—
160C4⋊C8—1062no—
161C4⋊C8—1061no—
162C4⋊C8—1061no—
163C4⋊C8—1061no—
164C4⋊C8—1061no—
165C22⋊C8—532no—
166C22⋊C8—1062no—
167C22⋊C8C2×#615712yes—
168#97—1062no—
169#97—1062no—
170#97—1062no—
171C4×C8—1062no—
172C4×C8—1062no—
173C2×Q8—1062no—
174C2×Q8—1062no—
175C2×Q8—2121no—
176C22×C4—531no—
177C22×C4—531no—
178C22×C4—531no—
179C22×C4—531no—
180C22×C4C4⋊#615711yescharacteristic
181C22×C4C4⋊#615711yescharacteristic
182C22×C4C22×#615711yescharacteristic
183C2×C8—22no—
184C2×C8—1062no—
185C2×C8—1062no—
186C2×C8—1062no—
187C2×C8—1062no—
188C2×C8—2121no—
189C2×C8—2122no—
190C2×C8—2121no—
191C2×C8—2121no—
192C4⋊C4—22no—
193C4⋊C4—1062no—
194C4⋊C4—1062no—
195C42—532no—
196C42—532no—
197C42—1061no—
198C42—1061no—
199C42—1061no—
200C42—1061no—
201C23C2×C4⋊#615711yescharacteristic
202Q8—2124no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—531no—
209C2×C4—532no—
210C2×C4—531no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1061no—
215C2×C4—1061no—
216C2×C4—1061no—
217C2×C4—1061no—
218C2×C4—1061no—
219C2×C4—2121no—
220C2×C4#1313328712yes—
221C2×C4D4×#615712yes—
222C8—42no—
223C8—4241no—
224C8—2122no—
225C8—2122no—
226C8—2122no—
227C8—2122no—
228C22—21no—
229C22—21no—
230C22#1681157011yescharacteristic
231C22C2×#1313328711yescharacteristic
232C22#1681163811yescharacteristic
233C4—22no—
234C4—22no—
235C4—41no—
236C4—531no—
237C4—531no—
238C4—1062no—
239C4—1061no—
240C4—2121no—
241C4—2121no—
242C2—21no—
243C2—21no—
244C2#2029066811yescharacteristic
245C2#2029067111yescharacteristic
246C2#2029068711yescharacteristic
subgroupG/H
centerC22#16811570
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#324C4
socleC22×C53#385
prime radicalC23×C53C2×C4⋊C4
cosocleC2×C4×Dic53C23
hypercenter#324#6157
hypocenterC53#8865
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
145. C53#8865semidirect

Maximal subgroups

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

subgroupindex
#202916132
#202908882
#202902882
#202896102
#202730372
#886553