Finite Group Databasebeta

← #34002906

Subgroups — #34002906

Order 18688.

1226 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 172 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
1C2×C4×C22⋊#64408C212yesmaximal
2#22624811C214yesmaximal, complemented
3#407×D73C211yesmaximal, characteristic
4C22×C22⋊#64408—22no—
5C22×C4×#64408—22no—
6C4×C22⋊#64408—22no—
7C4×C22⋊#64408—22no—
8C4×C22⋊#64408C414yescomplemented
9C4×C22⋊#64408C2214yes—
10C2×(C2×#64408)⋊C4C414yescomplemented
11C23×C4×D73C2211yescharacteristic
12C2×M16×D73C412yescomplemented
13#17988551C411yescharacteristic, complemented
14#17988550C2212yes—
15#17988549—22no—
16C22⋊C8×D73—22no—
17#407×C73C411yescharacteristic, complemented
18C23×#64408—24no—
19C2×C22⋊#64408—216no—
20C2×C22⋊#64408—24no—
21C2×C22⋊#64408—24no—
22C2×C4×#64408—24no—
23C2×C4×#64408—24no—
24C2×C4×#64408—48no—
25(C2×#64408)⋊C4—24no—
26(C2×#64408)⋊C4C2×C418yescomplemented
27C24×D73D411yescharacteristic
28C23×Dic73Q811yescharacteristic
29C22×C4×D73—44no—
30C22×C4×D73C2×C412yes—
31C22×C4×D73C2×C412yes—
32C22×C4×D73C2×C411yescharacteristic
33C22×C4×D73D412yes—
34C22×C4×D73D412yes—
35C22×C4×D73D411yescharacteristic
36C22×C4×D73Q811yescharacteristic
37C22×C4×D73C2311yescharacteristic
38C2×#11924012C2×C412yescomplemented
39C2×#11924011—22no—
40M16×D73—44no—
41C2×C8×D73—22no—
42C22⋊#10596831—22no—
43#14556503—24no—
44C23×C4×C73C2×C411yescharacteristic
45C2×M16×C73C2×C412yescomplemented
46C22⋊C8×C73—22no—
47C22×#64408—216no—
48C22×#64408—28no—
49C22×#64408—48no—
50C22×#64408—48no—
51C22⋊#64408—216no—
52C22⋊#64408—216no—
53C4×#64408—48no—
54C4×#64408—48no—
55C23×D73—28no—
56C23×D73—42no—
57C23×D73C22⋊C412yes—
58C23×D73C22⋊C412yes—
59C23×D73C4⋊C411yescharacteristic
60C23×D73C2×D411yescharacteristic
61C22×Dic73—42no—
62C22×Dic73C4⋊C411yescharacteristic
63C22×Dic73C4⋊C412yes—
64C22×Dic73C4⋊C412yes—
65C22×Dic73C2×Q811yescharacteristic
66C2×C4×D73—24no—
67C2×C4×D73—28no—
68C2×C4×D73—44no—
69C2×C4×D73—42no—
70C2×C4×D73—44no—
71C2×C4×D73—42no—
72C2×C4×D73—44no—
73C2×C4×D73—42no—
74C2×C4×D73C4211yescharacteristic
75C2×C4×D73C4211yescharacteristic
76C2×C4×D73C22⋊C412yes—
77C2×C4×D73C22⋊C411yescharacteristic
78C2×C4×D73C22⋊C412yes—
79C2×C4×D73C22⋊C411yescharacteristic
80C2×C4×D73C4⋊C412yes—
81C2×C4×D73C4⋊C412yes—
82C2×C4×D73C22×C412yes—
83C2×C4×D73C2×D412yes—
84#11924012—42no—
85#11924011—44no—
86C2×#10596831—22no—
87C8×D73—44no—
88C24×C73C4⋊C411yescharacteristic
89C22×C4×C73—42no—
90C22×C4×C73C4212yes—
91C22×C4×C73C22⋊C412yes—
92C22×C4×C73C4⋊C411yescharacteristic
93C22×C4×C73C22×C411yescharacteristic
94M16×C73—42no—
95C2×C8×C73—22no—
96C23×C73—24no—
97C23×C73—41no—
98C23×C73#9512yes—
99C23×C73C2×C4⋊C411yescharacteristic
100C22×D73—24no—
101C22×D73—24no—
102C22×D73—24no—
103C22×D73—24no—
104C22×D73—22no—
105C22×D73—24no—
106C22×D73—24no—
107C22×D73—42no—
108C22×D73—41no—
109C22×D73—416no—
110C22×D73#9511yescharacteristic
111C22×D73#9511yescharacteristic
112C22×D73#10418yes—
113C22×D73C2×C22⋊C412yes—
114C2×#64408—216no—
115C2×#64408—48no—
116C2×#64408—432no—
117C2×#64408—48no—
118C2×C4×C73—22no—
119C2×C4×C73—24no—
120C2×C4×C73—42no—
121C2×C4×C73—41no—
122C2×C4×C73#9512yes—
123C2×C4×C73C2×C4211yescharacteristic
124C2×C4×C73C2×C22⋊C411yescharacteristic
125C2×Dic73—22no—
126C2×Dic73—24no—
127C2×Dic73—42no—
128C2×Dic73—41no—
129C2×Dic73#9511yescharacteristic
130C2×Dic73#9511yescharacteristic
131C2×Dic73C2×C4⋊C412yes—
132C4×D73—22no—
133C4×D73—24no—
134C4×D73—22no—
135C4×D73—24no—
136C4×D73—22no—
137C4×D73—24no—
138C4×D73—42no—
139C4×D73—42no—
140C4×D73—42no—
141C4×D73#9914yes—
142C8×C73—42no—
143#10596831—42no—
144C22×C73—22no—
145C22×C73—22no—
146C22×C73—22no—
147C22×C73—21no—
148C22×C73—44no—
149C22×C73#33914yes—
150C22×C73C2×#9511yescharacteristic
151C2×D73—22no—
152C2×D73—22no—
153C2×D73—216no—
154C2×D73—22no—
155C2×D73—21no—
156C2×D73—44no—
157C2×D73—48no—
158C2×D73—44no—
159C2×D73#33914yes—
160C2×D73#34212yes—
161C2×D73C2×#10414yes—
162#64408—416no—
163C4×C73—22no—
164C4×C73—21no—
165C4×C73C2×#9912yes—
166Dic73—21no—
167Dic73—22no—
168Dic73#34212yes—
169C2×#1297—731nomaximal, Sylow
170C2×C73—24no—
171C2×C73—41no—
172C2×C73C2×#33912yes—
173C2×C73C2×#34211yescharacteristic
174D73—24no—
175D73—41no—
176D73#129714yescomplemented
177C2×#407—731no—
178C2×C4×C22⋊C4—732no—
179#1297—734no—
180C73C2×#129711yescharacteristic, complemented, Sylow
181C24×C4—731no—
182C22×M16—732no—
183C22×C22⋊C4—1462no—
184C22×C42—1462no—
185#407—731no—
186#407—732no—
187#407—1462no—
188#407#6440811yescharacteristic, complemented
189C2×C22⋊C8—1462no—
190C4×C22⋊C4—734no—
191C4×C22⋊C4—734no—
192C4×C22⋊C4—1462no—
193C4×C22⋊C4—1462no—
194C2×#95—734no—
195C25—731no—
196C23×C4—2924no—
197C23×C4—731no—
198C23×C4—732no—
199C23×C4—732no—
200C23×C4—732no—
201C23×C4—731no—
202C23×C4—731no—
203C23×C4—731no—
204C23×C4—732no—
205C23×C4—731no—
206C23×C4—1464no—
207C23×C4C2×#6440811yescharacteristic
208C2×M16—2924no—
209C2×M16—732no—
210C2×M16—1462no—
211C2×M16C2×#6440812yescomplemented
212C22×C8—1462no—
213C2×C22⋊C4—1464no—
214C2×C22⋊C4—1464no—
215C2×C22⋊C4—14616no—
216C2×C42—2928no—
217C2×C42—1464no—
218C2×C42—1464no—
219C22⋊C8—22no—
220C22⋊C8—1464no—
221C22⋊C8—1462no—
222#95—738no—
223#95—1464no—
224C24—2922no—
225C24—732no—
226C24—731no—
227C24—732no—
228C24—731no—
229C24—1468no—
230C24C4⋊#6440811yescharacteristic
231C22×C4—42no—
232C22×C4—2922no—
233C22×C4—2924no—
234C22×C4—2924no—
235C22×C4—2922no—
236C22×C4—2924no—
237C22×C4—2922no—
238C22×C4—2928no—
239C22×C4—2928no—
240C22×C4—2922no—
241C22×C4—732no—
242C22×C4—731no—
243C22×C4—732no—
244C22×C4—732no—
245C22×C4—732no—
246C22×C4—731no—
247C22×C4—732no—
248C22×C4—731no—
249C22×C4—732no—
250C22×C4—732no—
251C22×C4—731no—
252C22×C4—731no—
253C22×C4—731no—
254C22×C4—732no—
255C22×C4—1468no—
256C22×C4—1464no—
257C22×C4—1468no—
258C22×C4—14616no—
259C22×C4C4×#6440812yes—
260C22×C4C22⋊#6440812yes—
261C22×C4C4⋊#6440811yescharacteristic
262C22×C4C22×#6440811yescharacteristic
263M16—42no—
264M16—2924no—
265M16—2922no—
266C2×C8—22no—
267C2×C8—2924no—
268C2×C8—1462no—
269C22⋊C4—14616no—
270C22⋊C4—14616no—
271C42—2928no—
272C42—2928no—
273C23—24no—
274C23—41no—
275C23—2922no—
276C23—2921no—
277C23—29216no—
278C23—731no—
279C23—738no—
280C23—732no—
281C23—731no—
282C23—1464no—
283C23—1462no—
284C23—1464no—
285C23—1464no—
286C23—1464no—
287C23—1464no—
288C23—1464no—
289C23(C2×#64408)⋊C412yes—
290C23C2×C4⋊#6440811yescharacteristic
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—2922no—
296C2×C4—2922no—
297C2×C4—2922no—
298C2×C4—2928no—
299C2×C4—2928no—
300C2×C4—29232no—
301C2×C4—2922no—
302C2×C4—2921no—
303C2×C4—731no—
304C2×C4—734no—
305C2×C4—731no—
306C2×C4—732no—
307C2×C4—1464no—
308C2×C4—1462no—
309C2×C4—1464no—
310C2×C4—1462no—
311C2×C4—14616no—
312C2×C4—1464no—
313C2×C4—1462no—
314C2×C4—1464no—
315C2×C4—1462no—
316C2×C4(C2×#64408)⋊C412yes—
317C2×C4C2×C4×#6440811yescharacteristic
318C2×C4C2×C22⋊#6440811yescharacteristic
319C8—42no—
320C8—2922no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—44no—
326C22—2928no—
327C22—2924no—
328C22—2924no—
329C22—734no—
330C22—732no—
331C22—734no—
332C22—14616no—
333C22—1462no—
334C22—1461no—
335C22—1462no—
336C22—1462no—
337C22#1798945914yes—
338C22C2×(C2×#64408)⋊C411yescharacteristic
339C4—22no—
340C4—21no—
341C4—29216no—
342C4—732no—
343C4—1461no—
344C4—1462no—
345C4D73⋊#9912yes—
346C2—24no—
347C2—41no—
348C2—2921no—
349C2—734no—
350C2—1464no—
351C2C2×#1798945912yes—
352C2#2262562811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#407×C73C4
socleC22×C73C2×#95
prime radicalC24×D73D4
cosocleC22×C4×D73C23
hypercenter#407#64408
hypocenterC73C2×#1297
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #22624811C2semidirect
8. C4×C22⋊#64408C4semidirect
10. C2×(C2×#64408)⋊C4C4semidirect
12. C2×M16×D73C4semidirect
13. #17988551C4semidirect
17. #407×C73C4semidirect
26. (C2×#64408)⋊C4C2×C4semidirect
38. C2×#11924012C2×C4semidirect
45. C2×M16×C73C2×C4semidirect
176. D73#1297semidirect
180. C73C2×#1297semidirect
188. #407#64408semidirect
211. C2×M16C2×#64408semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊#644082
#226248112
#407×D732
C2×#129773