Finite Group Databasebeta

← #33062104

Subgroups — #33062104

Order 18688.

982 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 179 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#22620305C211yesmaximal, characteristic
2#22620061C212yesmaximal, complemented
3#22609117C218yesmaximal, complemented
4#22608645C211yesmaximal, characteristic
5#22608378C212yesmaximal, complemented
6C2×#559×C73C211yesmaximal, characteristic
7#17989144C2211yescharacteristic
8#17988951—22no—
9#17988933C2214yes—
10#17988917—22no—
11#17988872C2214yescomplemented
12#17988797C2214yescomplemented
13#17988743—22no—
14#17988692—24no—
15#17988644C2212yes—
16#17988592—22no—
17C4×#11924012C2211yescharacteristic
18#17988331C2214yes—
19Q8⋊#10596831C2218yescomplemented
20#559×C73—22no—
21#559×C73—24no—
22#559×C73C2214yes—
23C2×#127×C73C2211yescharacteristic
24C2×#124×C73—22no—
25C2×#123×C73—22no—
26C2×C4×Q8×C73C2212yes—
27C2×C4×D4×C73—22no—
28C2×#11924018—24no—
29C2×#11924012—21no—
30#14556591C2312yes—
31#14556583—21no—
32#14556570—22no—
33C4×Dic146—22no—
34#14556550—21no—
35#14556522—24no—
36C4⋊#10596831C2314yes—
37#14556515—24no—
38#14556514—24no—
39#14556512—28no—
40C22⋊#10596831—22no—
41#14556502—22no—
42#14556502—21no—
43C4×#10596831C2312yes—
44C22×Q8×C73D412yes—
45C22×D4×C73—22no—
46C22×D4×C73D412yes—
47C23×C4×C73—22no—
48#127×C73—22no—
49#127×C73D412yes—
50#127×C73C2312yes—
51#124×C73—28no—
52#124×C73—28no—
53#123×C73—28no—
54#123×C73—28no—
55C4×Q8×C73—22no—
56C4×Q8×C73—22no—
57C4×Q8×C73D414yes—
58C4×Q8×C73C2314yes—
59C4×D4×C73—28no—
60C4×D4×C73—28no—
61C2×C4⋊C4×C73—22no—
62C2×C4⋊C4×C73D412yes—
63C2×C22⋊C4×C73—24no—
64C2×C22⋊C4×C73—22no—
65C2×C42×C73—22no—
66C2×C42×C73C2311yescharacteristic
67C2×Dic146—22no—
68#11924018—48no—
69#11924012—42no—
70C2×#10596831—22no—
71C2×#10596831—24no—
72C4⋊Dic73—22no—
73C4⋊Dic73—22no—
74#11924007—42no—
75C22⋊Dic73—41no—
76C4×Dic73—41no—
77C24×C73—22no—
78C2×Q8×C73—22no—
79C2×Q8×C73—44no—
80C2×Q8×C73C2×D414yes—
81C2×D4×C73—24no—
82C2×D4×C73—22no—
83C2×D4×C73—24no—
84C2×D4×C73—48no—
85C2×D4×C73—44no—
86C2×D4×C73C2×D414yes—
87C22×C4×C73—22no—
88C22×C4×C73—22no—
89C22×C4×C73—28no—
90C22×C4×C73—84no—
91C22×C4×C73C2×D412yes—
92C22×C4×C73C2×D411yescharacteristic
93C4⋊C4×C73—24no—
94C4⋊C4×C73—22no—
95C4⋊C4×C73—24no—
96C4⋊C4×C73C2×D414yes—
97C22⋊C4×C73—216no—
98C22⋊C4×C73—28no—
99C42×C73—24no—
100C42×C73—24no—
101C42×C73C2×D411yescharacteristic
102C42×C73C2×D412yes—
103C42×C73C2411yescharacteristic
104C23×C73—24no—
105C23×C73—48no—
106C23×C73—44no—
107C23×C73#12011yescharacteristic
108Q8×C73—24no—
109Q8×C73—24no—
110Q8×C73—44no—
111D4×C73—44no—
112D4×C73—48no—
113D4×C73—48no—
114D4×C73—82no—
115C2×C4×C73—22no—
116C2×C4×C73—24no—
117C2×C4×C73—24no—
118C2×C4×C73—24no—
119C2×C4×C73—44no—
120C2×C4×C73—44no—
121C2×C4×C73—88no—
122C2×C4×C73#12012yes—
123C2×C4×C73#12011yescharacteristic
124C2×C4×C73C22×D411yescharacteristic
125C2×C4×C73C22×D412yes—
126C2×Dic73—42no—
127Dic146—44no—
128#10596831—42no—
129#10596831—44no—
130C22×C73—22no—
131C22×C73—48no—
132C22×C73—44no—
133C22×C73—88no—
134C22×C73#45512yes—
135C22×C73C2×#12011yescharacteristic
136C4×C73—22no—
137C4×C73—21no—
138C4×C73—21no—
139C4×C73—44no—
140C4×C73—44no—
141C4×C73#57514yes—
142Dic73—82no—
143#18531—731nomaximal, Sylow
144C2×C73—22no—
145C2×C73—84no—
146C2×C73#300512yes—
147C2×C73C2×#45511yescharacteristic
148C2×#559D7311yescharacteristic
149#3138—731no—
150#3113—732no—
151#1640—738no—
152#1555—731no—
153#1491—732no—
154C73#1853111yescharacteristic, complemented, Sylow
155#559—22no—
156#559—24no—
157#559C2×D7314yes—
158#533—731no—
159C2×#127C2×D7311yescharacteristic
160C2×#124—22no—
161C2×#123—22no—
162C2×C4×Q8C2×D7312yes—
163C2×C4×D4—22no—
164#490—1462no—
165#486—734no—
166#483—1462no—
167#475—734no—
168#462—734no—
169#451—1462no—
170#441—1464no—
171#432—732no—
172#418—1462no—
173C4×M16—731no—
174#331—734no—
175Q8⋊C8—738no—
176C22×Q8C22⋊D7312yes—
177C22×D4—22no—
178C22×D4C22⋊D7312yes—
179C23×C4—22no—
180C2×Q16—1464no—
181C2×M16—1461no—
182C4⋊Q8—732no—
183#127—22no—
184#127C22⋊D7312yes—
185#127C22×D7312yes—
186#125—1461no—
187#124—28no—
188#124—28no—
189#123—28no—
190#123—28no—
191C22⋊Q8—1462no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—1462no—
195C4×Q8C22⋊D7314yes—
196C4×Q8C22×D7314yes—
197C4×D4—28no—
198C4×D4—28no—
199#117—1461no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22⋊D7312yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42C22×D7311yescharacteristic
206#106—1464no—
207C4⋊C8—734no—
208#103—1464no—
209#103—1464no—
210#102—1468no—
211C22⋊C8—1462no—
212#97—1461no—
213#97—1462no—
214C4×C8—732no—
215C24—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—1462no—
219C2×Q8C2×C22⋊D7314yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4C2×C22⋊D7314yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—84no—
230C22×C4C2×C22⋊D7312yes—
231C22×C4C2×C22⋊D7311yescharacteristic
232Q16—2928no—
233M16—2922no—
234C2×C8—1462no—
235C2×C8—1464no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—2922no—
240C4⋊C4—1462no—
241C4⋊C4—1462no—
242C4⋊C4C2×C22⋊D7314yes—
243C22⋊C4—216no—
244C22⋊C4—28no—
245C22⋊C4—2921no—
246C42—24no—
247C42—24no—
248C42—2921no—
249C42C2×C22⋊D7311yescharacteristic
250C42C2×C22⋊D7312yes—
251C42C23×D7311yescharacteristic
252C23—24no—
253C23—48no—
254C23—44no—
255C23#1455656011yescharacteristic
256Q8—24no—
257Q8—24no—
258Q8—44no—
259Q8—2924no—
260D4—44no—
261D4—48no—
262D4—48no—
263D4—82no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—88no—
271C2×C4—2922no—
272C2×C4#1455656012yes—
273C2×C4#1455656011yescharacteristic
274C2×C4C22×C22⋊D7311yescharacteristic
275C2×C4C22×C22⋊D7312yes—
276C8—2922no—
277C8—2924no—
278C22—22no—
279C22—48no—
280C22—44no—
281C22—88no—
282C22#1798876512yes—
283C22C2×#1455656011yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—44no—
288C4—44no—
289C4—5842no—
290C4#1798938514yes—
291C2—22no—
292C2—84no—
293C2#2261917312yes—
294C2C2×#1798876511yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#559×C73C2
socleC22×C73C2×#120
prime radicalC2×#127×C73C22
hypercenterC2×#559D73
hypocenterC73#18531
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #22620061C2semidirect
3. #22609117C2semidirect
5. #22608378C2semidirect
11. #17988872C22semidirect
12. #17988797C22semidirect
19. Q8⋊#10596831C22semidirect
154. C73#18531semidirect

Maximal subgroups

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

subgroupindex
#226203052
#226200612
#226091172
#226086452
#226083782
C2×#559×C732
#1853173