Finite Group Databasebeta

← #28308098

Subgroups — #28308098

Order 13568.

392 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 90 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#20289958C212yesmaximal
2#20289656C212yesmaximal
3#20289646C212yesmaximal
4#20277433C211yesmaximal, characteristic
5#16811639—22no—
6#16811468—22no—
7#16811468C2214yes—
8C2×#13133215C411yescharacteristic
9C2×Q8×Dic53C2211yescharacteristic
10C2×#13133182C411yescharacteristic
11#16810529C2211yescharacteristic
12#16810510—22no—
13#16810505—22no—
14#16810488C2211yescharacteristic
15#16810469C411yescharacteristic
16C53×#391C411yescharacteristic
17C4⋊#88324—42no—
18C22⋊#88324—24no—
19C4×#88324—42no—
20C22×Dic106C2×C411yescharacteristic
21#13133215—22no—
22#13133215C814yescomplemented
23Q8×Dic53—42no—
24#13133182—42no—
25C2×C4⋊Dic53D412yes—
26C2×C4⋊Dic53C2311yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C4×Dic53—21no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D412yes—
33#13133124—21no—
34#13133124—21no—
35#13133124—22no—
36#13133124—22no—
37C22×Q8×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40#95×C53—22no—
41C2×Q8×C53—41no—
42C2×Q8×C53—41no—
43C2×Q8×C53C2×C812yes—
44C2×Q8×C53M1612yes—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C22⋊C412yes—
48C22×C4×C53C22×C411yescharacteristic
49C22×Dic53—21no—
50C22×Dic53—21no—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C22⋊C411yescharacteristic
53C22×Dic53C2×D412yes—
54C2×Dic106—21no—
55C2×Dic106—41no—
56C2×Dic106—41no—
57C2×Dic106C2×C812yes—
58C2×#88324—44no—
59C4⋊C4×C53—41no—
60C42×C53—41no—
61C4⋊Dic53—42no—
62C4⋊Dic53—41no—
63#11719919—28no—
64#11719919—42no—
65C4×Dic53—21no—
66C4×Dic53—22no—
67C4×Dic53—22no—
68C4×Dic53—22no—
69C4×Dic53—42no—
70C4×Dic53C2×C812yes—
71C23×C53C2×C22⋊C411yescharacteristic
72Q8×C53—42no—
73Q8×C53—42no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—42no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53C22⋊C814yes—
84C2×C4×C53C22×C811yescharacteristic
85C2×C4×C53C2×M1611yescharacteristic
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53—41no—
91C2×Dic53—41no—
92C2×Dic53—41no—
93C2×Dic53—41no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C2×Dic53—42no—
97C2×Dic53#9912yes—
98Dic106—42no—
99Dic106—42no—
100#88324—84no—
101#11482—531nomaximal, Sylow
102C22×C53—21no—
103C22×C53—21no—
104C22×C53C2×C22⋊C811yescharacteristic
105C22×C53C2×#9911yescharacteristic
106C22×C53#41311yescharacteristic
107C4×C53—21no—
108C4×C53—21no—
109C4×C53—24no—
110C4×C53—81no—
111C4×C53—81no—
112Dic53—22no—
113Dic53—42no—
114Dic53—42no—
115Dic53—41no—
116Dic53—42no—
117#2327—531no—
118#1461—532no—
119#1324—532no—
120#1319—532no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53#145911yescharacteristic
124C2×C53#212911yescharacteristic
125C2×C53#213311yescharacteristic
126C2×C4⋊Q8—531no—
127C2×C4×Q8—531no—
128C2×C4×Q8—531no—
129#431—1062no—
130#391—531no—
131#391#615711yescharacteristic
132#387—1062no—
133#387—1062no—
134#382—531no—
135#376—531no—
136#324—534no—
137#324—1062no—
138C53#1148211yescharacteristic, complemented, Sylow
139C22×Q8—531no—
140C22×Q8C2×#615711yescharacteristic
141C4⋊Q8—534no—
142C4⋊Q8—1062no—
143C4×Q8—2122no—
144C4×Q8—2122no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—531no—
147C2×C4⋊C4—1062no—
148C2×C4⋊C4—1061no—
149C2×C4⋊C4C2×#615711yescharacteristic
150C2×C42—531no—
151C2×C42—532no—
152C2×C42—531no—
153C2×C42—1061no—
154C2×C42C2×#615711yescharacteristic
155C4⋊C8—2122no—
156C22⋊C8—1064no—
157C4×C8—2122no—
158#95—22no—
159#95—1061no—
160#95—1061no—
161#95—1062no—
162#95—1062no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—532no—
166C2×Q8—1061no—
167C2×Q8—2121no—
168C2×Q8—2121no—
169C2×Q8D53⋊C812yes—
170C2×Q8#1171993612yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—531no—
174C22×C4—531no—
175C22×C4—532no—
176C22×C4—1061no—
177C22×C4—1061no—
178C22×C4C22⋊#615712yes—
179C22×C4C22×#615711yescharacteristic
180C2×C8—2124no—
181C4⋊C4—41no—
182C4⋊C4—1068no—
183C4⋊C4—2121no—
184C4⋊C4—2122no—
185C4⋊C4—2122no—
186C42—41no—
187C42—532no—
188C42—1062no—
189C42—1062no—
190C42—1062no—
191C42—1061no—
192C42—2122no—
193C23C2×C22⋊#615711yescharacteristic
194Q8—42no—
195Q8—42no—
196Q8—2122no—
197Q8—2122no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—532no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1061no—
211C2×C4—1062no—
212C2×C4—2121no—
213C2×C4—2121no—
214C2×C4—2121no—
215C2×C4—2121no—
216C2×C4—2121no—
217C2×C4—2122no—
218C2×C4—2121no—
219C2×C4#1313327414yes—
220C2×C4C2×D53⋊C811yescharacteristic
221C2×C4C2×#1171993611yescharacteristic
222C8—4244no—
223C22—21no—
224C22—21no—
225C22C2×#1313327411yescharacteristic
226C22D53⋊#9911yescharacteristic
227C22#1681161011yescharacteristic
228C4—24no—
229C4—21no—
230C4—21no—
231C4—81no—
232C4—81no—
233C4—1062no—
234C4—2122no—
235C4—2122no—
236C4—2122no—
237C4—2121no—
238C2—21no—
239C2—21no—
240C2#2028995311yescharacteristic
241C2#2029099811yescharacteristic
242C2#2029101111yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenter#391#6157
hypocenterC53#11482
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
22. #13133215C8semidirect
138. C53#11482semidirect

Maximal subgroups

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

subgroupindex
#202899582
#202896562
#202896462
#202774332
#1148253