Finite Group Databasebeta

← #34008335

Subgroups — #34008335

Order 18688.

1374 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 216 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#22626375C212yesmaximal
2C2×#17989455C214yesmaximal, complemented
3C22×M16×D73C211yesmaximal, characteristic
4C2×#14556670C2214yes—
5#17989530—22no—
6#17989529—24no—
7#17989528—22no—
8C4×C22⋊#64408C418yescomplemented
9#17989455—28no—
10C23×C4×D73C2211yescharacteristic
11C22×#11924012C411yescharacteristic, complemented
12C22×#11924011C2212yes—
13C2×M16×D73—24no—
14C2×M16×D73—28no—
15C22×C8×D73C412yescomplemented
16C22×M16×C73C411yescharacteristic, complemented
17#14556670—28no—
18#14556670—28no—
19C2×C4⋊#64408—24no—
20C2×C22⋊#64408—28no—
21C2×C22⋊#64408—24no—
22C2×C4×#64408C2×C418yes—
23(C2×#64408)⋊C4—28no—
24C24×D73D411yescharacteristic
25C23×Dic73Q811yescharacteristic
26C22×C4×D73—24no—
27C22×C4×D73—24no—
28C22×C4×D73C2×C414yes—
29C22×C4×D73C2×C411yescharacteristic
30C22×C4×D73D414yes—
31C22×C4×D73D411yescharacteristic
32C22×C4×D73Q811yescharacteristic
33C22×C4×D73C2311yescharacteristic
34C2×#11924012—24no—
35C2×#11924012—22no—
36C2×#11924011—24no—
37C2×#11924011—24no—
38C2×#11924011D418yes—
39M16×D73—232no—
40C22×#10596831C2×C412yes—
41C2×C8×D73—24no—
42C2×C8×D73—24no—
43C2×C8×D73—24no—
44C23×C4×C73C2×C411yescharacteristic
45C2×M16×C73—22no—
46C2×M16×C73—24no—
47C22×C8×C73C2×C412yes—
48C22×#64408—28no—
49C4⋊#64408—44no—
50C4⋊#64408—44no—
51C22⋊#64408—416no—
52C22⋊#64408—48no—
53C4×#64408—216no—
54C23×D73—24no—
55C23×D73—44no—
56C23×D73C22⋊C414yes—
57C23×D73C4⋊C411yescharacteristic
58C23×D73C2×D411yescharacteristic
59C22×Dic73—24no—
60C22×Dic73C4⋊C411yescharacteristic
61C22×Dic73C4⋊C414yes—
62C22×Dic73C2×Q811yescharacteristic
63C2×C4×D73—24no—
64C2×C4×D73—22no—
65C2×C4×D73—22no—
66C2×C4×D73—24no—
67C2×C4×D73—216no—
68C2×C4×D73—28no—
69C2×C4×D73—28no—
70C2×C4×D73C4212yes—
71C2×C4×D73C22⋊C414yes—
72C2×C4×D73C22⋊C412yes—
73C2×C4×D73C4⋊C414yes—
74C2×C4×D73C22×C412yes—
75C2×C4×D73C2×D414yes—
76C2×C4×D73C2×D412yes—
77C2×C4×D73#4114yes—
78#11924012—28no—
79#11924011—216no—
80C2×#10596831—22no—
81C2×#10596831—22no—
82C2×#10596831C22⋊C414yescomplemented
83C8×D73—216no—
84C24×C73C4⋊C411yescharacteristic
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73C4212yes—
88C22×C4×C73C22⋊C412yes—
89C22×C4×C73C4⋊C411yescharacteristic
90C22×C4×C73C22×C411yescharacteristic
91M16×C73—28no—
92C2×C8×C73—22no—
93C2×C8×C73—22no—
94C2×C8×C73C22⋊C414yescomplemented
95C23×C73—22no—
96C23×C73—42no—
97C23×C73#9512yes—
98C23×C73C2×C4⋊C411yescharacteristic
99C22×D73—22no—
100C22×D73—24no—
101C22×D73—28no—
102C22×D73—28no—
103C22×D73—44no—
104C22×D73—42no—
105C22×D73—416no—
106C22×D73#9512yes—
107C22×D73C2×C22⋊C412yes—
108C22×D73#12114yes—
109C2×#64408—416no—
110C2×#64408—48no—
111C2×C4×C73—22no—
112C2×C4×C73—24no—
113C2×C4×C73—22no—
114C2×C4×C73—22no—
115C2×C4×C73#9512yes—
116C2×C4×C73C2×C4211yescharacteristic
117C2×C4×C73C2×C22⋊C412yes—
118C2×C4×C73C2×C22⋊C411yescharacteristic
119C2×C4×C73#11712yes—
120C2×Dic73—22no—
121C2×Dic73—28no—
122C2×Dic73#9512yes—
123C2×Dic73C2×C4⋊C412yes—
124C2×Dic73C22⋊Q814yes—
125C4×D73—28no—
126C4×D73—24no—
127C4×D73—22no—
128C4×D73—22no—
129C4×D73—28no—
130C4×D73C4×D418yes—
131C4×D73#12014yes—
132C4×D73#12314yes—
133C8×C73—24no—
134#10596831—24no—
135C22×C73—22no—
136C22×C73—22no—
137C22×C73—24no—
138C22×C73—44no—
139C22×C73C2×#9511yescharacteristic
140C22×C73#38012yes—
141C2×D73—28no—
142C2×D73—22no—
143C2×D73—24no—
144C2×D73—22no—
145C2×D73—44no—
146C2×D73—416no—
147C2×D73H(3,ℤ/4)14yescomplemented
148C2×D73#38614yes—
149#64408—88no—
150C4×C73—22no—
151C4×C73C4×C22⋊C412yes—
152C4×C73#37911yescharacteristic
153C4×C73#38111yescharacteristic
154Dic73—22no—
155Dic73#38514yes—
156#10717—731nomaximal, Sylow
157C2×C73—22no—
158C2×C73—42no—
159C2×C73C22⋊#9511yescharacteristic
160C2×C73C2×H(3,ℤ/4)12yes—
161D73—24no—
162D73—24no—
163C23×M16—731no—
164#2280—732no—
165C2×H(3,ℤ/4)—734no—
166C73#1071711yescharacteristic, complemented, Sylow
167C24×C4—731no—
168C22×M16—731no—
169C22×M16—732no—
170C22×M16—1468no—
171C22×M16—1464no—
172C22×M16#6440811yescharacteristic, complemented
173C23×C8—732no—
174C2×#117—734no—
175#381—1462no—
176#380—1464no—
177#379—1462no—
178C4×C22⋊C4—738no—
179H(3,ℤ/4)—1468no—
180C25—731no—
181C23×C4—731no—
182C23×C4—731no—
183C23×C4—734no—
184C23×C4—734no—
185C23×C4—731no—
186C23×C4—731no—
187C23×C4—731no—
188C23×C4—1464no—
189C23×C4—1464no—
190C23×C4C2×#6440811yescharacteristic
191C2×M16—22no—
192C2×M16—24no—
193C2×M16—738no—
194C2×M16—1464no—
195C2×M16—1464no—
196C2×M16—1464no—
197C2×M16—14632no—
198C2×M16—1462no—
199C22×C8—732no—
200C22×C8—1464no—
201C22×C8—1464no—
202C22×C8—1464no—
203C22×C8C2×#6440812yes—
204#117—1468no—
205#117—1468no—
206C2×C4⋊C4—1464no—
207C2×C22⋊C4—1464no—
208C2×C22⋊C4—1468no—
209C2×C42—738no—
210#95—1468no—
211C24—2924no—
212C24—731no—
213C24—734no—
214C24—731no—
215C24—1464no—
216C24C4⋊#6440811yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—731no—
220C22×C4—732no—
221C22×C4—732no—
222C22×C4—734no—
223C22×C4—734no—
224C22×C4—734no—
225C22×C4—731no—
226C22×C4—734no—
227C22×C4—734no—
228C22×C4—732no—
229C22×C4—732no—
230C22×C4—1468no—
231C22×C4—1464no—
232C22×C4—14616no—
233C22×C4—1468no—
234C22×C4—1464no—
235C22×C4—1468no—
236C22×C4—1464no—
237C22×C4—1462no—
238C22×C4—1462no—
239C22×C4C4×#6440812yes—
240C22×C4C22⋊#6440812yes—
241C22×C4C4⋊#6440811yescharacteristic
242C22×C4C22×#6440811yescharacteristic
243M16—28no—
244M16—14616no—
245M16—1468no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—734no—
249C2×C8—14616no—
250C2×C8—1462no—
251C2×C8—1462no—
252C2×C8C22⋊#6440814yescomplemented
253C4⋊C4—2924no—
254C4⋊C4—2924no—
255C22⋊C4—2928no—
256C22⋊C4—29216no—
257C42—14616no—
258C23—22no—
259C23—42no—
260C23—2922no—
261C23—2924no—
262C23—29216no—
263C23—734no—
264C23—732no—
265C23—732no—
266C23—1468no—
267C23—1468no—
268C23—1464no—
269C23—1462no—
270C23(C2×#64408)⋊C412yes—
271C23C2×C4⋊#6440811yescharacteristic
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—2928no—
277C2×C4—29216no—
278C2×C4—732no—
279C2×C4—734no—
280C2×C4—738no—
281C2×C4—734no—
282C2×C4—734no—
283C2×C4—732no—
284C2×C4—1468no—
285C2×C4—1468no—
286C2×C4—1468no—
287C2×C4—1462no—
288C2×C4—1464no—
289C2×C4—1462no—
290C2×C4—1462no—
291C2×C4(C2×#64408)⋊C412yes—
292C2×C4C2×C4×#6440811yescharacteristic
293C2×C4C2×C22⋊#6440812yes—
294C2×C4C2×C22⋊#6440811yescharacteristic
295C2×C4#1455667012yes—
296C8—24no—
297C8—1464no—
298C22—22no—
299C22—24no—
300C22—22no—
301C22—44no—
302C22—2924no—
303C22—29216no—
304C22—734no—
305C22—734no—
306C22—1468no—
307C22—1464no—
308C22—1462no—
309C22—1462no—
310C22C2×(C2×#64408)⋊C411yescharacteristic
311C22#1798952912yes—
312C4—22no—
313C4—5848no—
314C4—734no—
315C4—1462no—
316C4C4×C22⋊#6440812yes—
317C4#1798952811yescharacteristic
318C4#1798953011yescharacteristic
319C2—22no—
320C2—42no—
321C2—1464no—
322C2—1464no—
323C2#2262510511yescharacteristic
324C2#2262560312yes—
subgroupG/H
centerC2×C4C2×C4×#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
FittingC22×M16×C73C4
socleC22×C73C2×#95
prime radicalC24×D73D4
cosocleC22×C4×D73C23
hypercenterC22×M16#64408
hypocenterC73#10717
LCSG ▷ C2×C4×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#17989455C2semidirect
8. C4×C22⋊#64408C4semidirect
11. C22×#11924012C4semidirect
15. C22×C8×D73C4semidirect
16. C22×M16×C73C4semidirect
82. C2×#10596831C22⋊C4semidirect
94. C2×C8×C73C22⋊C4semidirect
147. C2×D73H(3,ℤ/4)semidirect
166. C73#10717semidirect
172. C22×M16#64408semidirect
252. C2×C8C22⋊#64408semidirect

Maximal subgroups

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

subgroupindex
#226263752
C2×#179894552
C22×M16×D732
#1071773