Finite Group Databasebeta

← #34003232

Subgroups — #34003232

Order 18688.

562 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626610C211yesmaximal, characteristic
2#22626378C211yesmaximal, characteristic
3#22624831C214yesmaximal
4#22618563C211yesmaximal, characteristic
5#17989610—21no—
6#17989607C412yescomplemented
7#17989605—21no—
8#17989604C412yescomplemented
9#17989592—21no—
10#17989558C2212yes—
11C4×#11924038—21no—
12#17989532—21no—
13#17989531—21no—
14#17989530C412yes—
15#17989529C2212yes—
16C4×C4⋊#64408—21no—
17C4×C22⋊#64408C412yes—
18#17989523—21no—
19C2×C42×D73C2211yescharacteristic
20#17988643—21no—
21#17988643—21no—
22#17988639—21no—
23#17988639—21no—
24#17988611C411yescharacteristic, complemented
25#17988550C2212yes—
26M16×Dic73C412yescomplemented
27#423×C73C411yescharacteristic, complemented
28C2×#11924038—22no—
29C2×C4⋊#64408—22no—
30C2×C22⋊#64408C2×C414yes—
31C2×C4×#64408—22no—
32#14556658—22no—
33#14556658—22no—
34#14556648—22no—
35C22⋊#10596833C2×C414yes—
36#14556646—22no—
37C4×#10596833—22no—
38(C2×#64408)⋊C4—22no—
39(C2×#64408)⋊C4—22no—
40(C2×#64408)⋊C4—22no—
41(C2×#64408)⋊C4—22no—
42C22×C4×D73C2×C411yescharacteristic
43C22×C4×D73C2×C411yescharacteristic
44C22×C4×D73C2311yescharacteristic
45C2×#11924012C2×C412yes—
46C2×C4×Dic73C2×C411yescharacteristic
47C2×C4×Dic73C2×C411yescharacteristic
48C2×C4×Dic73C2×C411yescharacteristic
49C42×D73D411yescharacteristic
50C42×D73D411yescharacteristic
51C42×D73D411yescharacteristic
52C42×D73D411yescharacteristic
53C42×D73D411yescharacteristic
54C42×D73D411yescharacteristic
55C42×D73Q811yescharacteristic
56C42×D73Q811yescharacteristic
57C4⋊#10596831—21no—
58C4⋊#10596831—21no—
59#14556503—22no—
60#14556503—22no—
61#14556501—22no—
62C8×Dic73—22no—
63C2×M16×C73C2×C412yes—
64C2×C42×C73C2×C411yescharacteristic
65C4⋊C8×C73—21no—
66C4⋊C8×C73—21no—
67C22×#64408—24no—
68#11924038—42no—
69#11924038—42no—
70C2×#10596833—24no—
71C4⋊#64408—42no—
72C4⋊#64408—42no—
73C22⋊#64408—44no—
74C4×#64408—44no—
75C23×D73C22×C411yescharacteristic
76C22×Dic73C4211yescharacteristic
77C22×Dic73C4211yescharacteristic
78C22×Dic73C22×C411yescharacteristic
79C2×C4×D73—42no—
80C2×C4×D73—41no—
81C2×C4×D73—41no—
82C2×C4×D73—42no—
83C2×C4×D73C22⋊C411yescharacteristic
84C2×C4×D73C22⋊C411yescharacteristic
85C2×C4×D73C22⋊C411yescharacteristic
86C2×C4×D73C22⋊C411yescharacteristic
87C2×C4×D73C4⋊C411yescharacteristic
88C2×C4×D73C4⋊C411yescharacteristic
89C2×C4×D73C4⋊C411yescharacteristic
90C2×C4×D73C4⋊C411yescharacteristic
91C2×C4×D73C2×D411yescharacteristic
92C2×C4×D73C2×D411yescharacteristic
93C2×C4×D73C2×D411yescharacteristic
94C2×C4×D73C2×Q811yescharacteristic
95#11924012—42no—
96C2×#10596831—22no—
97C4×Dic73C22⋊C411yescharacteristic
98C4×Dic73C22⋊C411yescharacteristic
99C4×Dic73C22⋊C411yescharacteristic
100C4×Dic73C22⋊C411yescharacteristic
101C4×Dic73C22⋊C411yescharacteristic
102C4×Dic73C22⋊C411yescharacteristic
103C4×Dic73C4⋊C411yescharacteristic
104C4×Dic73C4⋊C411yescharacteristic
105C4×Dic73C4⋊C411yescharacteristic
106C4×Dic73C4⋊C411yescharacteristic
107C4×Dic73C4⋊C411yescharacteristic
108C4×Dic73C4⋊C411yescharacteristic
109C22×C4×C73C4211yescharacteristic
110C22×C4×C73C4211yescharacteristic
111C22×C4×C73C22×C411yescharacteristic
112M16×C73—42no—
113C2×C8×C73—22no—
114C42×C73C22⋊C411yescharacteristic
115C42×C73C22⋊C411yescharacteristic
116C42×C73C4⋊C411yescharacteristic
117C42×C73C4⋊C411yescharacteristic
118C23×C73C2×C4211yescharacteristic
119C22×D73—22no—
120C22×D73—41no—
121C22×D73—41no—
122C22×D73C2×C22⋊C411yescharacteristic
123C22×D73C2×C4⋊C411yescharacteristic
124C2×#64408—44no—
125C2×#64408—44no—
126C2×#64408—44no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73#9511yescharacteristic
132C2×C4×C73#9511yescharacteristic
133C2×C4×C73#9511yescharacteristic
134C2×C4×C73#9511yescharacteristic
135C2×C4×C73C2×C22⋊C411yescharacteristic
136C2×C4×C73C2×C4⋊C411yescharacteristic
137C2×Dic73—22no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140C2×Dic73#9511yescharacteristic
141C2×Dic73#9511yescharacteristic
142C2×Dic73#9511yescharacteristic
143C2×Dic73#9511yescharacteristic
144C2×Dic73C2×C22⋊C411yescharacteristic
145C2×Dic73C2×C4⋊C411yescharacteristic
146C4×D73—22no—
147C4×D73—22no—
148C4×D73—21no—
149C4×D73—21no—
150C4×D73—21no—
151C4×D73—21no—
152C4×D73—22no—
153C4×D73—22no—
154C4×D73—41no—
155C4×D73—41no—
156C4×D73—41no—
157C4×D73—41no—
158C4×D73—41no—
159C4×D73—41no—
160#10596833—44no—
161C8×C73—42no—
162#10596831—42no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73C2×#9511yescharacteristic
167C2×D73—21no—
168C2×D73—21no—
169C2×D73—21no—
170C2×D73—21no—
171C2×D73—21no—
172C2×D73—44no—
173C2×D73#41312yes—
174#64408—84no—
175C4×C73—21no—
176C4×C73—21no—
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—21no—
180C4×C73#41011yescharacteristic
181C4×C73#41011yescharacteristic
182Dic73—21no—
183Dic73—21no—
184Dic73—21no—
185Dic73—21no—
186Dic73—21no—
187Dic73#34412yes—
188#8106—731nomaximal, Sylow
189C2×C73—41no—
190C2×C73#174011yescharacteristic
191C2×C73#174211yescharacteristic
192C2×C73C2×#34411yescharacteristic
193D73—22no—
194D73—41no—
195#2920—731no—
196#2920—731no—
197#2283—731no—
198#1305—734no—
199C73#810611yescharacteristic, complemented, Sylow
200C22×C42—731no—
201#432—732no—
202#432—1461no—
203#432—1461no—
204#432—1461no—
205#431—732no—
206#431—1461no—
207#431—1461no—
208#431—1461no—
209#423—731no—
210#423—1461no—
211#423#6440811yescharacteristic, complemented
212#407—732no—
213#407—732no—
214C4×M16—732no—
215C4×M16—1461no—
216#383—1461no—
217#382—1461no—
218#381—732no—
219#380—732no—
220C4×C4⋊C4—1461no—
221C4×C22⋊C4—732no—
222#376—1461no—
223C23×C4—731no—
224C23×C4—731no—
225C23×C4—731no—
226C2×M16—732no—
227C2×M16—1462no—
228C2×M16C2×#6440812yes—
229C2×C4⋊C4—1462no—
230C2×C22⋊C4—734no—
231C2×C42—731no—
232C2×C42—731no—
233C2×C42—731no—
234C2×C42—731no—
235C2×C42—731no—
236C2×C42—731no—
237C2×C42—731no—
238C2×C42—731no—
239C2×C42—731no—
240C2×C42—731no—
241C2×C42—731no—
242C2×C42—1462no—
243C2×C42C2×#6440811yescharacteristic
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—1462no—
247C4⋊C8—1462no—
248C4⋊C8—1461no—
249C4⋊C8—1461no—
250C22⋊C8—734no—
251C22⋊C8—1462no—
252C22⋊C8—1462no—
253C22⋊C8—1462no—
254#97—1462no—
255#97—1462no—
256C4×C8—1462no—
257C4×C8—1462no—
258#95—1462no—
259#95—1462no—
260#95—1462no—
261#95—1462no—
262C24—731no—
263C22×C4—2921no—
264C22×C4—2921no—
265C22×C4—2922no—
266C22×C4—2922no—
267C22×C4—731no—
268C22×C4—731no—
269C22×C4—731no—
270C22×C4—731no—
271C22×C4—731no—
272C22×C4—731no—
273C22×C4—731no—
274C22×C4—731no—
275C22×C4—731no—
276C22×C4—731no—
277C22×C4—731no—
278C22×C4—731no—
279C22×C4—731no—
280C22×C4—731no—
281C22×C4—731no—
282C22×C4—1464no—
283C22×C4C4×#6440811yescharacteristic
284C22×C4C4×#6440811yescharacteristic
285C22×C4C22×#6440811yescharacteristic
286M16—42no—
287M16—2922no—
288M16—2922no—
289M16—2922no—
290C2×C8—22no—
291C2×C8—1464no—
292C2×C8—1462no—
293C4⋊C4—2922no—
294C4⋊C4—2922no—
295C22⋊C4—2924no—
296C42—2924no—
297C42—731no—
298C42—731no—
299C42—731no—
300C42—731no—
301C42—731no—
302C42—731no—
303C42—731no—
304C42—731no—
305C42—731no—
306C42—731no—
307C42—731no—
308C42—731no—
309C42C22⋊#6440811yescharacteristic
310C42C22⋊#6440811yescharacteristic
311C42C4⋊#6440811yescharacteristic
312C42C4⋊#6440811yescharacteristic
313C23—2921no—
314C23—2921no—
315C23—731no—
316C23—731no—
317C23—1462no—
318C23C2×C4×#6440811yescharacteristic
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—2921no—
324C2×C4—2921no—
325C2×C4—2921no—
326C2×C4—2921no—
327C2×C4—2921no—
328C2×C4—2921no—
329C2×C4—2924no—
330C2×C4—2924no—
331C2×C4—2924no—
332C2×C4—2921no—
333C2×C4—2921no—
334C2×C4—731no—
335C2×C4—731no—
336C2×C4—731no—
337C2×C4—731no—
338C2×C4—731no—
339C2×C4—731no—
340C2×C4—1462no—
341C2×C4—1462no—
342C2×C4—1462no—
343C2×C4—1462no—
344C2×C4—1462no—
345C2×C4—1461no—
346C2×C4—1461no—
347C2×C4—1461no—
348C2×C4—1461no—
349C2×C4(C2×#64408)⋊C411yescharacteristic
350C2×C4(C2×#64408)⋊C411yescharacteristic
351C2×C4(C2×#64408)⋊C411yescharacteristic
352C2×C4(C2×#64408)⋊C411yescharacteristic
353C2×C4C2×C22⋊#6440811yescharacteristic
354C2×C4C2×C4⋊#6440811yescharacteristic
355C8—42no—
356C8—2924no—
357C8—2922no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—2924no—
362C22—732no—
363C22—1461no—
364C22—1461no—
365C22—1461no—
366C22—1461no—
367C22—1461no—
368C22C2×(C2×#64408)⋊C411yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—5844no—
375C4—732no—
376C4—1461no—
377C4—1461no—
378C4—1461no—
379C4—1461no—
380C4—1461no—
381C4#1798956711yescharacteristic
382C4#1798956811yescharacteristic
383C2—41no—
384C2—2921no—
385C2—1462no—
386C2#2262563111yescharacteristic
387C2#2262563511yescharacteristic
388C2#2262564311yescharacteristic
subgroupG/H
centerC22C2×(C2×#64408)⋊C4
commutatorC23×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#423×C73C4
socleC22×C73C2×#95
prime radicalC23×D73C22×C4
cosocleC22×C4×D73C23
hypercenter#423#64408
hypocenterC73#8106
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
6. #17989607C4semidirect
8. #17989604C4semidirect
24. #17988611C4semidirect
26. M16×Dic73C4semidirect
27. #423×C73C4semidirect
199. C73#8106semidirect
211. #423#64408semidirect

Maximal subgroups

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

subgroupindex
#226266102
#226263782
#226248312
#226185632
#810673