Finite Group Databasebeta

← #27279124

Subgroups — #27279124

Order 13568.

572 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#16810562C211yesmaximal, characteristic
2#20277318C211yesmaximal, characteristic
3#20274806C212yesmaximal
4#20272595C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7#16810562—22no—
8#16810562C414yescomplemented
9#16810509—24no—
10#16810486—21no—
11#16810483C2212yes—
12C4×C22⋊Dic53—21no—
13C2×#13133124—21no—
14C2×#13133124—21no—
15C2×#13133124C2211yescharacteristic
16#16810386—24no—
17C22×C4⋊C4×C53C2211yescharacteristic
18C2×C22⋊C8×C53C2212yes—
19C53×#324—22no—
20C23×Dic53—21no—
21C23×Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53C2×C412yes—
26C2×C4⋊Dic53D414yes—
27C2×C22⋊Dic53—22no—
28C2×C4×Dic53—22no—
29#13133124—22no—
30#13133124—24no—
31#13133124—24no—
32#13133124—24no—
33#13133124—22no—
34#13133124—21no—
35#13133124C2×C412yes—
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39C22×C8×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C4⋊C4×C53D412yes—
44C22⋊C8×C53—22no—
45C22⋊C8×C53—22no—
46C24×C53C2×D411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—21no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22⋊C412yes—
57C22×C4×C53C22×C411yescharacteristic
58C22×C4×C53C2×D412yes—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53#4112yes—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—24no—
64C22×Dic53—42no—
65C22×Dic53—42no—
66C22×Dic53—44no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53—42no—
70C2×C8×C53—44no—
71C2×C8×C53—42no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—24no—
74C22⋊Dic53—42no—
75C22⋊Dic53—42no—
76C4⋊Dic53—24no—
77C4⋊Dic53—42no—
78C4⋊Dic53—44no—
79C4×Dic53—82no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53C2×C22⋊C411yescharacteristic
83C23×C53#12011yescharacteristic
84C23×C53#12311yescharacteristic
85C23×C53#13512yes—
86C23×C53#13712yes—
87C2×C4×C53—24no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—81no—
100C2×C4×C53C4×D412yes—
101C2×C4×C53#12112yes—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—44no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—44no—
108C2×Dic53—44no—
109C2×Dic53—44no—
110C2×Dic53—84no—
111C2×Dic53—82no—
112C2×Dic53—82no—
113C8×C53—82no—
114#9022—531nomaximal, Sylow
115C22×C53—24no—
116C22×C53—24no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53#38611yescharacteristic
122C22×C53#41611yescharacteristic
123C22×C53#41911yescharacteristic
124C22×C53#45212yes—
125C22×C53#45412yes—
126C22×C53#45812yes—
127C22×C53#48512yes—
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—42no—
131C4×C53—42no—
132Dic53—82no—
133Dic53—84no—
134Dic53—82no—
135C2×#400—531no—
136#2313—531no—
137#1824—532no—
138C2×#324D5311yescharacteristic
139#1294—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—22no—
143C2×C53#161514yes—
144C2×C53#187311yescharacteristic
145C2×C53#188811yescharacteristic
146C2×C53#189011yescharacteristic
147C22×C4⋊C4—531no—
148C22×C4⋊C4C2×D5311yescharacteristic
149C2×C22⋊C8C2×D5312yes—
150#400—534no—
151#400—1062no—
152#387—1064no—
153#381—1061no—
154#380—532no—
155C4×C22⋊C4—1061no—
156C2×#95—531no—
157C2×#95—1061no—
158C2×#95—1061no—
159#340—1064no—
160#324—22no—
161C53#902211yescharacteristic, complemented, Sylow
162C23×C4—1061no—
163C23×C4—1061no—
164C23×C4D21211yescharacteristic
165C23×C4D21211yescharacteristic
166C23×C4C22×D5311yescharacteristic
167C22×C8—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—534no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4—1062no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4C4×D5312yes—
176C2×C4⋊C4C22⋊D5312yes—
177C2×C22⋊C4—1062no—
178C2×C42—1062no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181#95—532no—
182#95—1062no—
183#95—1064no—
184#95—1062no—
185#95—1061no—
186#95—1064no—
187#95—1064no—
188C24C2×D21211yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—1062no—
198C22×C4—1062no—
199C22×C4—1064no—
200C22×C4—2122no—
201C22×C4—2122no—
202C22×C4—2124no—
203C22×C4—2121no—
204C22×C4—2121no—
205C22×C4—2122no—
206C22×C4#1171992112yes—
207C22×C4#1171992112yes—
208C22×C4C2×C4×D5311yescharacteristic
209C22×C4D4×D5312yes—
210C22×C4#1171994712yes—
211C22×C4C2×C22⋊D5311yescharacteristic
212C2×C8—44no—
213C2×C8—42no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—1064no—
217C4⋊C4—2124no—
218C4⋊C4—2122no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C42—4242no—
222C23—22no—
223C23—22no—
224C23C2×#1171992111yescharacteristic
225C23#1313318711yescharacteristic
226C23#1313320111yescharacteristic
227C23#1313323912yes—
228C23#1313325012yes—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—81no—
242C2×C4—4244no—
243C2×C4—4242no—
244C2×C4—4242no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4—2124no—
248C2×C4—2122no—
249C2×C4—2124no—
250C2×C4—2124no—
251C2×C4—2124no—
252C2×C4—2122no—
253C2×C4#1313317712yes—
254C2×C4#1313318912yes—
255C8—82no—
256C22—24no—
257C22—24no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22#1681050011yescharacteristic
263C22#1681062011yescharacteristic
264C22#1681063311yescharacteristic
265C22#1681078712yes—
266C22#1681079812yes—
267C22#1681081412yes—
268C22#1681096512yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—4242no—
274C4—4244no—
275C4—4242no—
276C2—21no—
277C2—21no—
278C2—22no—
279C2#2027374814yes—
280C2#2027502511yescharacteristic
281C2#2027511611yescharacteristic
282C2#2027512611yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#9022
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
8. #16810562C4semidirect
161. C53#9022semidirect

Maximal subgroups

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

subgroupindex
C2×#168105622
#202773182
#202748062
#202725952
C2×C53×#3242
#902253