Finite Group Databasebeta

← #32983816

Subgroups — #32983816

Order 18688.

518 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 99 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#22614218C211yesmaximal, characteristic
2#22612534C211yesmaximal, characteristic
3#22610028C212yesmaximal
4#22607785C212yesmaximal
5C73×#1463C211yesmaximal, characteristic
6C2×#14556550C2211yescharacteristic
7#17988526C412yescomplemented
8#17988517C412yescomplemented
9#17988473—24no—
10#17988468—22no—
11#17988460—21no—
12#17988450—21no—
13#17988450—21no—
14#17988450C2211yescharacteristic
15#17988447—21no—
16C4×C22⋊Dic73C2212yes—
17#17988348—24no—
18C22×C4⋊C4×C73C2211yescharacteristic
19#407×C73C2212yes—
20#324×C73—22no—
21#14556550—22no—
22#14556550—22no—
23C2×C4⋊Dic73—22no—
24C2×C4⋊Dic73C2×C411yescharacteristic
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×#11924007—22no—
27C2×C22⋊Dic73—22no—
28C2×C22⋊Dic73—21no—
29C2×C22⋊Dic73—21no—
30C2×C4×Dic73—21no—
31C2×C4×Dic73D414yes—
32#14556498—22no—
33#14556498—24no—
34#14556498—22no—
35#14556498—22no—
36#14556498—22no—
37#14556498—22no—
38#14556498—21no—
39#14556498C2×C412yes—
40C23×C4×C73D411yescharacteristic
41C23×C4×C73D411yescharacteristic
42C23×C4×C73C2311yescharacteristic
43C2×M16×C73—22no—
44C2×C4⋊C4×C73—22no—
45C2×C4⋊C4×C73—22no—
46C2×C4⋊C4×C73C2×C412yes—
47C2×C4⋊C4×C73D412yes—
48C22⋊C8×C73—22no—
49C22⋊C8×C73—22no—
50C22×Dic73—22no—
51C22×Dic73—21no—
52C22×Dic73—21no—
53C22×Dic73—24no—
54C4⋊Dic73—22no—
55C4⋊Dic73—22no—
56C4⋊Dic73—82no—
57#11924007—44no—
58C22⋊Dic73—44no—
59C22⋊Dic73—42no—
60C22⋊Dic73—41no—
61C22⋊Dic73—41no—
62C4×Dic73—42no—
63C4×Dic73—44no—
64C24×C73C2×D411yescharacteristic
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73—21no—
69C22×C4×C73—42no—
70C22×C4×C73—42no—
71C22×C4×C73—42no—
72C22×C4×C73C22⋊C412yes—
73C22×C4×C73C22⋊C412yes—
74C22×C4×C73C22×C411yescharacteristic
75C22×C4×C73C2×D412yes—
76C22×C4×C73C2×D411yescharacteristic
77C22×C4×C73#4112yes—
78M16×C73—44no—
79C2×C8×C73—42no—
80C4⋊C4×C73—24no—
81C4⋊C4×C73—24no—
82C23×C73—22no—
83C23×C73—22no—
84C23×C73—41no—
85C23×C73C2×C22⋊C411yescharacteristic
86C23×C73#12011yescharacteristic
87C23×C73#12311yescharacteristic
88C2×C4×C73—24no—
89C2×C4×C73—22no—
90C2×C4×C73—22no—
91C2×C4×C73—22no—
92C2×C4×C73—22no—
93C2×C4×C73—41no—
94C2×C4×C73—41no—
95C2×C4×C73—42no—
96C2×C4×C73—42no—
97C2×C4×C73—42no—
98C2×C4×C73—42no—
99C2×C4×C73—42no—
100C2×C4×C73—81no—
101C2×C4×C73C4×D412yes—
102C2×C4×C73#12112yes—
103C2×Dic73—42no—
104C2×Dic73—44no—
105C2×Dic73—42no—
106C2×Dic73—44no—
107C2×Dic73—42no—
108C2×Dic73—41no—
109C2×Dic73—41no—
110C2×Dic73—42no—
111C2×Dic73—42no—
112C2×Dic73—44no—
113C8×C73—82no—
114C22×C73—21no—
115C22×C73—21no—
116C22×C73—21no—
117C22×C73—21no—
118C22×C73—21no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—44no—
122C22×C73#38611yescharacteristic
123C22×C73#42112yes—
124C22×C73#45812yes—
125C4×C73—22no—
126C4×C73—22no—
127C4×C73—42no—
128C4×C73—42no—
129Dic73—84no—
130Dic73—82no—
131Dic73—81no—
132Dic73—81no—
133#9482—731nomaximal, Sylow
134C2×C73—22no—
135C2×C73—22no—
136C2×C73—41no—
137C2×C73#189011yescharacteristic
138C2×C73#189711yescharacteristic
139C2×C73#191211yescharacteristic
140#2484—731no—
141#2313—731no—
142#1826—732no—
143#1463D7311yescharacteristic
144#1284—732no—
145C73#948211yescharacteristic, complemented, Sylow
146C2×#117—731no—
147C22×C4⋊C4C2×D7311yescharacteristic
148#407C2×D7312yes—
149#400—732no—
150#398—732no—
151#387—1464no—
152#387—1462no—
153#385—1461no—
154#381—731no—
155#381—1461no—
156#381—1461no—
157#380—1461no—
158C4×C22⋊C4—732no—
159#339—1464no—
160#324—22no—
161C23×C4D29211yescharacteristic
162C23×C4D29211yescharacteristic
163C23×C4C22×D7311yescharacteristic
164C2×M16—22no—
165#117—1462no—
166#117—1462no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—731no—
170C2×C4⋊C4—731no—
171C2×C4⋊C4—1462no—
172C2×C4⋊C4—1462no—
173C2×C4⋊C4C4×D7312yes—
174C2×C4⋊C4C22⋊D7312yes—
175C2×C22⋊C4—1461no—
176C2×C22⋊C4—1462no—
177C2×C22⋊C4—1461no—
178C2×C42—734no—
179C2×C42—1461no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182#95—732no—
183#95—1462no—
184#95—1464no—
185#95—1462no—
186#95—1462no—
187#95—1462no—
188#95—1462no—
189#95—1461no—
190C24C2×D29211yescharacteristic
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—1462no—
199C22×C4—1461no—
200C22×C4—1461no—
201C22×C4—1464no—
202C22×C4#1192400512yes—
203C22×C4#1192400512yes—
204C22×C4C2×C4×D7311yescharacteristic
205C22×C4D4×D7312yes—
206C22×C4C2×C22⋊D7311yescharacteristic
207C22×C4#1192402712yes—
208M16—44no—
209C2×C8—42no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—2924no—
213C4⋊C4—5842no—
214C4⋊C4—1462no—
215C4⋊C4—1462no—
216C22⋊C4—2922no—
217C22⋊C4—2924no—
218C22⋊C4—2921no—
219C22⋊C4—2921no—
220C42—2924no—
221C42—2922no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23C2×#1192400511yescharacteristic
226C23#1455656111yescharacteristic
227C23#1455657511yescharacteristic
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—81no—
241C2×C4—2922no—
242C2×C4—2924no—
243C2×C4—2922no—
244C2×C4—2924no—
245C2×C4—2922no—
246C2×C4—2921no—
247C2×C4—2921no—
248C2×C4—2924no—
249C2×C4—2922no—
250C2×C4—2922no—
251C2×C4#1455655112yes—
252C2×C4#1455656312yes—
253C8—82no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22#1798846411yescharacteristic
263C22#1798860312yes—
264C22#1798877812yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—42no—
269C4—5844no—
270C4—5842no—
271C4—5841no—
272C4—5841no—
273C2—22no—
274C2—22no—
275C2—41no—
276C2#2261034211yescharacteristic
277C2#2261037711yescharacteristic
278C2#2261046611yescharacteristic
subgroupG/H
centerC22#17988464
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#1463C2
socleC22×C73#386
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#1463D73
hypocenterC73#9482
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
7. #17988526C4semidirect
8. #17988517C4semidirect
145. C73#9482semidirect

Maximal subgroups

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

subgroupindex
#226142182
#226125342
#226100282
#226077852
C73×#14632
#948273