Finite Group Databasebeta

← #34010793

Subgroups — #34010793

Order 18688.

448 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 88 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#22625630C211yesmaximal, characteristic
2#22625164C211yesmaximal, characteristic
3#22624926C214yesmaximal
4#22615753C211yesmaximal, characteristic
5C2×#14556670—21no—
6#17989603—21no—
7#17989465—21no—
8#17989465C2212yes—
9#17989433—21no—
10#17989431C2212yes—
11C2×#14556548C2211yescharacteristic
12#17988525—21no—
13#17988522—21no—
14#17988506C412yescomplemented
15#17988498C2212yes—
16#17988483—21no—
17#17988453C411yescharacteristic, complemented
18#17988444—21no—
19C2×C4⋊Q8×C73C411yescharacteristic, complemented
20#14556670—24no—
21#14556670—42no—
22C2×C4⋊#64408—41no—
23C2×C22⋊#64408—22no—
24C2×C4×#64408—41no—
25C4⋊#10596833—41no—
26C22⋊#10596833—22no—
27C4×#10596833—41no—
28C22×C4×D73D411yescharacteristic
29#14556548—41no—
30#14556548—41no—
31C2×C4⋊Dic73—21no—
32C2×#11924007—22no—
33C2×#11924007C2×C411yescharacteristic
34C2×#11924007D411yescharacteristic
35C2×#11924005—22no—
36C2×#11924005D411yescharacteristic
37C2×#11924005C2311yescharacteristic
38C2×C4×Dic73D411yescharacteristic
39#14556498—22no—
40#14556498—22no—
41#14556498—21no—
42#14556498C2×C412yes—
43C22×Q8×C73C2×C412yes—
44C4⋊Q8×C73—41no—
45C4⋊Q8×C73—41no—
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73—21no—
48C2×C42×C73C2×C411yescharacteristic
49C22×#64408—42no—
50C2×#10596833—42no—
51C4⋊#64408—44no—
52C22⋊#64408—24no—
53C22⋊#64408—42no—
54C4×#64408—44no—
55C23×D73C2×D411yescharacteristic
56C22×Dic73—22no—
57C22×Dic73C22⋊C411yescharacteristic
58C22×Dic73C22⋊C411yescharacteristic
59C22×Dic73C2×D411yescharacteristic
60C2×C4×D73—21no—
61C2×C4×D73—21no—
62C2×C4×D73—22no—
63C2×C4×D73SD1612yes—
64C2×C4×D73Q1612yes—
65C4⋊Dic73—42no—
66#11924007—41no—
67#11924007—41no—
68#11924007—82no—
69#11924005—41no—
70#11924005—41no—
71#11924005—82no—
72C4×Dic73—21no—
73C4×Dic73—21no—
74C2×Q8×C73—42no—
75C2×Q8×C73—82no—
76C22×C4×C73—22no—
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22×C411yescharacteristic
80C4⋊C4×C73—42no—
81C4⋊C4×C73—41no—
82C4⋊C4×C73—41no—
83C42×C73—21no—
84C42×C73—21no—
85C23×C73C2×C22⋊C411yescharacteristic
86C22×D73—21no—
87C22×D73—21no—
88C22×D73—41no—
89C22×D73—41no—
90C22×D73C2×SD1611yescharacteristic
91C22×D73C2×Q1611yescharacteristic
92C2×#64408—44no—
93C2×#64408—82no—
94C2×#64408—82no—
95Q8×C73—84no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—41no—
99C2×C4×C73—41no—
100C2×C4×C73—42no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73#9911yescharacteristic
104C2×C4×C73#9911yescharacteristic
105C2×C4×C73#10312yes—
106C2×C4×C73#10412yes—
107C2×Dic73—21no—
108C2×Dic73—21no—
109C2×Dic73—21no—
110C2×Dic73—21no—
111C2×Dic73—41no—
112C2×Dic73—41no—
113C2×Dic73—42no—
114C2×Dic73—42no—
115C2×Dic73—42no—
116C2×Dic73#10312yes—
117C4×D73—22no—
118C4×D73—22no—
119C4×D73—22no—
120C4×D73—22no—
121#10596833—82no—
122C22×C73—21no—
123C22×C73—21no—
124C22×C73C2×#9911yescharacteristic
125C22×C73C2×#10311yescharacteristic
126C22×C73C2×#10411yescharacteristic
127C2×D73—21no—
128C2×D73—21no—
129C2×D73—41no—
130C2×D73—41no—
131C2×D73—41no—
132C2×D73—41no—
133C2×D73—81no—
134#64408—84no—
135C4×C73—41no—
136C4×C73—41no—
137C4×C73—82no—
138C4×C73#32812yes—
139C4×C73#32812yes—
140Dic73—22no—
141Dic73—41no—
142Dic73—41no—
143Dic73—82no—
144#11717—731nomaximal, Sylow
145C2×C73—21no—
146C2×C73—21no—
147C2×C73C2×#32811yescharacteristic
148C2×C73#212211yescharacteristic
149C2×C73#212911yescharacteristic
150D73—41no—
151D73—41no—
152#2628—731no—
153#1740—731no—
154#1458—731no—
155#1351—734no—
156C73#1171711yescharacteristic, complemented, Sylow
157C2×C4⋊Q8#6440811yescharacteristic, complemented
158C2×#117—731no—
159C2×#117—1461no—
160#431—1461no—
161#400—1461no—
162#399—1461no—
163#395—732no—
164#393—732no—
165#389—1461no—
166#382—731no—
167#380—1461no—
168#342—732no—
169#342—1461no—
170#324—1461no—
171#323—732no—
172C22×Q8C2×#6440812yes—
173C23×C4—731no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—2921no—
177#117—2921no—
178#117—2922no—
179#117—1464no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—2921no—
183C2×C4⋊C4—731no—
184C2×C4⋊C4—731no—
185C2×C4⋊C4—1461no—
186C2×C4⋊C4—1462no—
187C2×C22⋊C4—731no—
188C2×C22⋊C4—731no—
189C2×C22⋊C4—1462no—
190C2×C22⋊C4—1462no—
191C2×C42—2921no—
192C2×C42—731no—
193C2×C42C2×#6440811yescharacteristic
194C4⋊C8—2921no—
195C22⋊C8—1462no—
196C4×C8—2921no—
197#95—732no—
198#95—1462no—
199#95—1462no—
200#95—1461no—
201C24—731no—
202C2×Q8—42no—
203C2×Q8—82no—
204C22×C4—22no—
205C22×C4—2922no—
206C22×C4—731no—
207C22×C4—731no—
208C22×C4—731no—
209C22×C4—732no—
210C22×C4—732no—
211C22×C4—1462no—
212C22×C4—1461no—
213C22×C4—1461no—
214C22×C4—1462no—
215C22×C4C22⋊#6440811yescharacteristic
216C22×C4C22⋊#6440811yescharacteristic
217C22×C4C22×#6440811yescharacteristic
218C2×C8—2922no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—2921no—
223C4⋊C4—2921no—
224C4⋊C4—2924no—
225C4⋊C4—2922no—
226C4⋊C4—5842no—
227C22⋊C4—2921no—
228C22⋊C4—2921no—
229C22⋊C4—2922no—
230C22⋊C4—5842no—
231C22⋊C4—1464no—
232C42—21no—
233C42—21no—
234C42—2924no—
235C42—1461no—
236C42—1461no—
237C23—2921no—
238C23—2921no—
239C23—731no—
240C23—731no—
241C23—1461no—
242C23—1461no—
243C23C2×C22⋊#6440811yescharacteristic
244Q8—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—2921no—
253C2×C4—2921no—
254C2×C4—2922no—
255C2×C4—2922no—
256C2×C4—2924no—
257C2×C4—2922no—
258C2×C4—5842no—
259C2×C4—5842no—
260C2×C4—732no—
261C2×C4—1461no—
262C2×C4—1461no—
263C2×C4—1461no—
264C2×C4—1461no—
265C2×C4—1462no—
266C2×C4—1462no—
267C2×C4—1462no—
268C2×C4—1462no—
269C2×C4#1455664911yescharacteristic
270C2×C4#1455664911yescharacteristic
271C2×C4#1455665412yes—
272C2×C4#1455665612yes—
273C8—5842no—
274C22—21no—
275C22—21no—
276C22—2921no—
277C22—2921no—
278C22—2921no—
279C22—2921no—
280C22—5841no—
281C22—1461no—
282C22—1461no—
283C22C2×#1455664911yescharacteristic
284C22C2×#1455665411yescharacteristic
285C22C2×#1455665611yescharacteristic
286C4—41no—
287C4—41no—
288C4—82no—
289C4—2921no—
290C4—2921no—
291C4—5842no—
292C4—5844no—
293C4—1462no—
294C4#1798944112yes—
295C4#1798944112yes—
296C2—21no—
297C2—21no—
298C2—2921no—
299C2—2921no—
300C2C2×#1798944111yescharacteristic
301C2#2262619011yescharacteristic
302C2#2262621311yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC22×C73C2×#99
prime radicalC23×D73C2×D4
cosocleC2×#11924005C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#11717
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
14. #17988506C4semidirect
17. #17988453C4semidirect
19. C2×C4⋊Q8×C73C4semidirect
156. C73#11717semidirect
157. C2×C4⋊Q8#64408semidirect

Maximal subgroups

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

subgroupindex
#226256302
#226251642
#226249262
#226157532
#1171773