Finite Group Databasebeta

← #32983452

Subgroups — #32983452

Order 18688.

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

normal subgroups only

This group has 552 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#22615411C211yesmaximal, characteristic
2#22612534C211yesmaximal, characteristic, complemented
3#22608448C212yesmaximal
4#22607816C212yesmaximal, complemented
5C73×#1463C211yesmaximal, characteristic, complemented
6C2×#14556565C2211yescharacteristic
7#17988526C412yescomplemented
8#17988504C412yescomplemented
9#17988477—22no—
10#17988473—24no—
11#17988466—21no—
12#17988466—21no—
13#17988450—21no—
14#17988447—21no—
15C4×C22⋊Dic73C2211yescharacteristic
16C4×C22⋊Dic73C2212yes—
17#17988322—24no—
18C22×C4⋊C4×C73C2211yescharacteristic
19#407×C73C2212yes—
20#324×C73—22no—
21C22×D292C2×C411yescharacteristic
22C22×C22⋊D73—21no—
23#14556565—22no—
24#14556565—22no—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×C22⋊Dic73—22no—
28C2×C22⋊Dic73—21no—
29C2×#11924005—22no—
30C2×#11924005—22no—
31C2×#11924005—21no—
32C2×C4×Dic73—22no—
33C2×C4×Dic73—21no—
34#14556498—22no—
35#14556498—24no—
36#14556498—22no—
37#14556498—22no—
38#14556498C2×C412yes—
39#14556498D414yes—
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—
50C23×D73—21no—
51C2×D292—22no—
52C2×D292—81no—
53C2×C22⋊D73—44no—
54C2×C22⋊D73—41no—
55C2×C22⋊D73—41no—
56C22×Dic73—21no—
57C22×Dic73—22no—
58C22×Dic73—24no—
59C4⋊Dic73—22no—
60C4⋊Dic73—82no—
61C22⋊Dic73—44no—
62C22⋊Dic73—42no—
63#11924005—44no—
64#11924005—42no—
65#11924005—82no—
66C4×Dic73—81no—
67C4×Dic73—82no—
68C24×C73C2×D411yescharacteristic
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—42no—
74C22×C4×C73—81no—
75C22×C4×C73—81no—
76C22×C4×C73C22⋊C412yes—
77C22×C4×C73C22⋊C412yes—
78C22×C4×C73C22×C411yescharacteristic
79C22×C4×C73C2×D412yes—
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73#4112yes—
82M16×C73—82no—
83C2×C8×C73—42no—
84C4⋊C4×C73—24no—
85C4⋊C4×C73—24no—
86C23×C73—22no—
87C23×C73—22no—
88C23×C73—41no—
89C23×C73C2×C22⋊C411yescharacteristic
90C23×C73#12011yescharacteristic
91C23×C73#12311yescharacteristic
92C22×D73—42no—
93C22×D73—42no—
94C22×D73—41no—
95C22×D73—42no—
96C2×C4×C73—22no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—42no—
101C2×C4×C73—41no—
102C2×C4×C73—42no—
103C2×C4×C73—41no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—42no—
108C2×C4×C73—81no—
109C2×C4×C73C4×D412yes—
110C2×C4×C73#12112yes—
111C22⋊D73—82no—
112C22⋊D73—82no—
113C2×Dic73—42no—
114C2×Dic73—44no—
115C2×Dic73—42no—
116C2×Dic73—44no—
117C2×Dic73—42no—
118C2×Dic73—41no—
119C2×Dic73—42no—
120C2×Dic73—44no—
121D292—82no—
122C8×C73—82no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73—21no—
126C22×C73—21no—
127C22×C73—21no—
128C22×C73—21no—
129C22×C73—21no—
130C22×C73—44no—
131C22×C73#38611yescharacteristic
132C22×C73#42112yes—
133C22×C73#45412yes—
134C2×D73—84no—
135C2×D73—81no—
136C2×D73—81no—
137C2×D73—81no—
138C4×C73—22no—
139C4×C73—22no—
140C4×C73—42no—
141C4×C73—42no—
142Dic73—84no—
143Dic73—82no—
144Dic73—81no—
145#9414—731nomaximal, Sylow
146C2×C73—22no—
147C2×C73—22no—
148C2×C73—41no—
149C2×C73#188811yescharacteristic
150C2×C73#189711yescharacteristic
151C2×C73#191311yescharacteristic
152D73—161no—
153#2591—731no—
154#2313—731no—
155#1507—732no—
156#1463D7311yescharacteristic, complemented
157#1297—732no—
158C73#941411yescharacteristic, complemented, Sylow
159C2×#121—731no—
160C22×C4⋊C4C2×D7311yescharacteristic
161#407C2×D7312yes—
162#400—732no—
163#394—732no—
164#388—1462no—
165#387—1464no—
166#386—1461no—
167#386—1461no—
168#381—1461no—
169#380—1461no—
170C4×C22⋊C4—732no—
171C4×C22⋊C4—731no—
172#327—1464no—
173#324—22no—
174C22×D4—731no—
175C22×D4—1461no—
176C23×C4D29211yescharacteristic
177C23×C4D29211yescharacteristic
178C23×C4C22×D7311yescharacteristic
179C2×M16—22no—
180#121—1462no—
181#121—1462no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—731no—
185C2×C4⋊C4—1462no—
186C2×C4⋊C4C4×D7312yes—
187C2×C4⋊C4C22⋊D7312yes—
188C2×C22⋊C4—1462no—
189C2×C22⋊C4—1461no—
190C2×C22⋊C4—1462no—
191C2×C22⋊C4—1461no—
192C2×C22⋊C4—1462no—
193C2×C42—1462no—
194C2×C42—1461no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197#95—732no—
198#95—734no—
199#95—1462no—
200#95—1464no—
201#95—1462no—
202#95—1462no—
203C24—1461no—
204C24C2×D29211yescharacteristic
205C2×D4—2924no—
206C2×D4—2921no—
207C2×D4—2921no—
208C2×D4—5841no—
209C2×D4—1462no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—1462no—
218C22×C4—1461no—
219C22×C4—1464no—
220C22×C4#1192400512yes—
221C22×C4#1192400512yes—
222C22×C4C2×C4×D7311yescharacteristic
223C22×C4D4×D7312yes—
224C22×C4C2×C22⋊D7311yescharacteristic
225C22×C4#1192402712yes—
226M16—82no—
227C2×C8—42no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—5842no—
231C4⋊C4—1462no—
232C22⋊C4—2924no—
233C22⋊C4—2922no—
234C22⋊C4—2924no—
235C22⋊C4—2922no—
236C22⋊C4—5842no—
237C42—5842no—
238C42—5841no—
239C23—22no—
240C23—22no—
241C23—41no—
242C23—2922no—
243C23—2922no—
244C23—2921no—
245C23—2922no—
246C23C2×#1192400511yescharacteristic
247C23#1455656111yescharacteristic
248C23#1455657511yescharacteristic
249D4—5842no—
250D4—5842no—
251D4—5842no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—2922no—
266C2×C4—2924no—
267C2×C4—2922no—
268C2×C4—2924no—
269C2×C4—2922no—
270C2×C4—2921no—
271C2×C4—2924no—
272C2×C4—2922no—
273C2×C4#1455655112yes—
274C2×C4#1455656312yes—
275C8—82no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—5844no—
285C22—5841no—
286C22—5841no—
287C22—5841no—
288C22#1798846411yescharacteristic
289C22#1798860312yes—
290C22#1798876212yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—5844no—
296C4—5842no—
297C4—5841no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—11681no—
302C2#2261033211yescharacteristic
303C2#2261037711yescharacteristic
304C2#2261047111yescharacteristic
subgroupG/H
centerC22#17988464
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#1463C2
socleC22×C73#386
prime radicalC2×#14556565C22
cosocleC23×C4×C73C23
hypercenter#1463D73
hypocenterC73#9414
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
2. #22612534C2semidirect
4. #22607816C2semidirect
5. C73×#1463C2semidirect
7. #17988526C4semidirect
8. #17988504C4semidirect
156. #1463D73semidirect
158. C73#9414semidirect

Maximal subgroups

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

subgroupindex
#226154112
#226125342
#226084482
#226078162
C73×#14632
#941473