Finite Group Databasebeta

← #33069260

Subgroups — #33069260

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621079C211yesmaximal, characteristic, complemented
2#22620869C211yesmaximal, characteristic
3C2×#17988934C211yesmaximal, characteristic
4#22609471C214yesmaximal, complemented
5#22609347C214yesmaximal, complemented
6#22608679C211yesmaximal, characteristic, complemented
7#22608613C211yesmaximal, characteristic, complemented
8#22608600C211yesmaximal, characteristic
9C73×#2960C211yesmaximal, characteristic, complemented
10C2×#14556591C2211yescharacteristic
11C2×#14556588C2211yescharacteristic
12#17988990—21no—
13#17988961—21no—
14#17988937—21no—
15#17988935—21no—
16#17988934—21no—
17#17988934—21no—
18#17988934C2212yes—
19#17988934C2212yes—
20#17988920—22no—
21#17988882C2214yescomplemented
22#17988787C2214yes—
23#17988737—22no—
24#17988640C2211yescharacteristic
25#17988640C2211yescharacteristic
26C2×#14556513—21no—
27C2×#14556513—21no—
28#17988340C2212yescomplemented
29#17988338C2212yescomplemented
30#17988335C2214yescomplemented
31#17988330C2214yes—
32#533×C73C2211yescharacteristic
33#446×C73—21no—
34#446×C73C2212yescomplemented
35C8×Q8×C73—21no—
36C8×Q8×C73C2212yescomplemented
37#424×C73—21no—
38#423×C73C2211yescharacteristic
39#405×C73—21no—
40C2×C4×C8×C73C2211yescharacteristic
41C22×Dic146—21no—
42C22×D292—21no—
43C22×D292—21no—
44C22×D292—21no—
45C2×#11924015—24no—
46#14556591—21no—
47#14556591—21no—
48#14556591—21no—
49#14556591C2312yes—
50#14556588—21no—
51#14556588—21no—
52#14556588—21no—
53#14556588C2312yes—
54C2×C4⋊Dic73—21no—
55C2×C4⋊Dic73—21no—
56#14556524—24no—
57C4⋊#10596831C2312yes—
58C4⋊#10596831C2312yes—
59#14556515—24no—
60#14556513—22no—
61#14556513—22no—
62#14556513—22no—
63#14556513—22no—
64#14556511—24no—
65C22⋊#10596831—21no—
66C22⋊#10596831—21no—
67C4×#10596831—21no—
68C4×#10596831—21no—
69C2×M16×C73—21no—
70C22×C8×C73D411yescharacteristic
71C22×C8×C73D411yescharacteristic
72C4⋊Q8×C73D411yescharacteristic
73C4⋊Q8×C73D411yescharacteristic
74#125×C73—21no—
75C22⋊Q8×C73—21no—
76C22⋊Q8×C73D412yes—
77C4×Q8×C73—21no—
78C4×Q8×C73C2312yes—
79#117×C73—21no—
80C2×C42×C73C2311yescharacteristic
81C4⋊C8×C73—22no—
82C4⋊C8×C73—22no—
83C4⋊C8×C73—21no—
84C4⋊C8×C73C2312yes—
85#97×C73—22no—
86C4×C8×C73—22no—
87C4×C8×C73D411yescharacteristic
88C4×C8×C73D411yescharacteristic
89C4×C8×C73C2311yescharacteristic
90C4×C8×C73C2311yescharacteristic
91C23×D73—41no—
92C2×Dic146—22no—
93C2×Dic146—41no—
94C2×Dic146—82no—
95C2×D292—22no—
96C2×D292—22no—
97C2×D292—22no—
98C2×D292—42no—
99C2×D292—42no—
100C2×D292—41no—
101C2×D292—41no—
102C2×D292—41no—
103C2×D292—82no—
104C2×D292—82no—
105C22×Dic73—41no—
106#11924015—84no—
107C2×#10596831—22no—
108C2×#10596831—22no—
109C4⋊Dic73—22no—
110C4⋊Dic73—22no—
111C4⋊Dic73—41no—
112C4⋊Dic73—41no—
113C2×Q8×C73—21no—
114C2×Q8×C73C2×D412yes—
115C22×C4×C73C2×D411yescharacteristic
116C22×C4×C73C2×D411yescharacteristic
117C22×C4×C73#4111yescharacteristic
118M16×C73—42no—
119C2×C8×C73—22no—
120C2×C8×C73—22no—
121C2×C8×C73—22no—
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73C2×D411yescharacteristic
124C2×C8×C73C2×D411yescharacteristic
125C2×C8×C73C2×D411yescharacteristic
126C4⋊C4×C73—21no—
127C4⋊C4×C73—21no—
128C4⋊C4×C73—21no—
129C4⋊C4×C73—22no—
130C4⋊C4×C73—22no—
131C4⋊C4×C73C2×D412yes—
132C22⋊C4×C73—22no—
133C42×C73—22no—
134C42×C73C2×D411yescharacteristic
135C42×C73C2×D411yescharacteristic
136C42×C73#4111yescharacteristic
137C42×C73C2411yescharacteristic
138C23×C73#12111yescharacteristic
139C22×D73—42no—
140C22×D73—82no—
141C22×D73—82no—
142C22×D73—82no—
143Q8×C73—24no—
144Q8×C73—42no—
145C2×C4×C73—22no—
146C2×C4×C73—22no—
147C2×C4×C73—22no—
148C2×C4×C73—22no—
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73#12111yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73#13512yes—
153C2×C4×C73#13712yes—
154C2×C4×C73C22×D411yescharacteristic
155C2×C4×C73C22×D411yescharacteristic
156C2×C4×C73C2×#4111yescharacteristic
157C2×Dic73—42no—
158C2×Dic73—82no—
159D292—42no—
160D292—42no—
161D292—44no—
162D292—82no—
163D292—82no—
164D292—82no—
165D292—82no—
166Dic146—82no—
167Dic146—82no—
168C8×C73—21no—
169C8×C73—21no—
170C8×C73—21no—
171C8×C73—21no—
172C8×C73—42no—
173#10596831—42no—
174#10596831—42no—
175C22×C73—21no—
176C22×C73—21no—
177C22×C73#46412yes—
178C22×C73C2×#12111yescharacteristic
179C2×D73—82no—
180C2×D73—82no—
181C2×D73—82no—
182C2×D73—164no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—42no—
186C4×C73—42no—
187C4×C73#46412yes—
188C4×C73#47212yes—
189C4×C73C2×#13511yescharacteristic
190C4×C73C2×#13711yescharacteristic
191C4×C73#57712yes—
192Dic73—82no—
193#19212—731nomaximal, Sylow
194C2×C73—22no—
195C2×C73C2×#46411yescharacteristic
196C2×C73#305711yescharacteristic
197C2×C73C2×#47211yescharacteristic
198D73—162no—
199#3230—731no—
200#3208—731no—
201C2×#486—731no—
202#2960D7311yescharacteristic, complemented
203#1698—734no—
204#1678—734no—
205#1564—731no—
206#1548—731no—
207#1545—731no—
208C73#1921211yescharacteristic, complemented, Sylow
209#533C2×D7311yescharacteristic
210C2×C4⋊Q8—731no—
211C2×#127—731no—
212#500—1461no—
213#492—1461no—
214#487—1461no—
215#486—732no—
216#486—732no—
217#486—1461no—
218#486—1461no—
219#486—1461no—
220#484—1462no—
221#477—734no—
222#460—734no—
223#450—1462no—
224#446—21no—
225#446C2×D7312yescomplemented
226C8×Q8—21no—
227C8×Q8C2×D7312yescomplemented
228#431—731no—
229#431—731no—
230#424—21no—
231#423C2×D7311yescharacteristic
232C2×#102—1461no—
233C2×#102—1461no—
234#405—21no—
235C2×C4×C8C2×D7311yescharacteristic
236#335—732no—
237#334—732no—
238#333—734no—
239#331—734no—
240C22×Q8—1461no—
241C22×D4—1461no—
242C22×D4—1461no—
243C22×D4—1461no—
244C2×SD16—1464no—
245C2×M16—21no—
246C22×C8C22⋊D7311yescharacteristic
247C22×C8C22⋊D7311yescharacteristic
248C4⋊Q8—732no—
249C4⋊Q8—1461no—
250C4⋊Q8—1461no—
251C4⋊Q8—1461no—
252C4⋊Q8D29211yescharacteristic
253C4⋊Q8D29211yescharacteristic
254#127—732no—
255#127—1461no—
256#127—1461no—
257#127—1461no—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8D29212yes—
261C4×Q8—21no—
262C4×Q8C22×D7312yes—
263#117—21no—
264C2×C4⋊C4—1461no—
265C2×C4⋊C4—1461no—
266C2×C42C22×D7311yescharacteristic
267#107—1464no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—21no—
271C4⋊C8—732no—
272C4⋊C8—732no—
273C4⋊C8C22×D7312yes—
274#103—1464no—
275#102—1462no—
276#102—1462no—
277#102—1462no—
278#102—1462no—
279#102—1464no—
280C22⋊C8—1461no—
281C22⋊C8—1461no—
282#97—22no—
283C4×C8—22no—
284C4×C8—1461no—
285C4×C8—1461no—
286C4×C8C22⋊D7311yescharacteristic
287C4×C8C22⋊D7311yescharacteristic
288C4×C8C22×D7311yescharacteristic
289C4×C8C22×D7311yescharacteristic
290C24—2921no—
291C2×Q8—21no—
292C2×Q8—2921no—
293C2×Q8—5842no—
294C2×Q8—1462no—
295C2×Q8C2×D29212yes—
296C2×D4—2921no—
297C2×D4—2921no—
298C2×D4—2921no—
299C2×D4—2922no—
300C2×D4—2922no—
301C2×D4—5842no—
302C2×D4—5842no—
303C2×D4—1462no—
304C2×D4—1462no—
305C2×D4—1462no—
306C22×C4—2921no—
307C22×C4C2×C22⋊D7311yescharacteristic
308C22×C4C2×D29211yescharacteristic
309C22×C4#1192402811yescharacteristic
310SD16—5844no—
311M16—42no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—1462no—
316C2×C8—1462no—
317C2×C8C2×C22⋊D7311yescharacteristic
318C2×C8C2×C22⋊D7311yescharacteristic
319C2×C8C2×C22⋊D7311yescharacteristic
320C2×C8C2×C22⋊D7311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—2921no—
327C4⋊C4—2921no—
328C4⋊C4—1462no—
329C4⋊C4—1462no—
330C4⋊C4C2×D29212yes—
331C22⋊C4—22no—
332C42—22no—
333C42C2×C22⋊D7311yescharacteristic
334C42C2×D29211yescharacteristic
335C42#1192402811yescharacteristic
336C42C23×D7311yescharacteristic
337C23—2922no—
338C23—5842no—
339C23—5842no—
340C23—5842no—
341C23#1455656511yescharacteristic
342Q8—24no—
343Q8—42no—
344Q8—5842no—
345Q8—5842no—
346D4—2922no—
347D4—2924no—
348D4—2922no—
349D4—5842no—
350D4—5842no—
351D4—5842no—
352D4—5842no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—2922no—
358C2×C4—5842no—
359C2×C4#1455656511yescharacteristic
360C2×C4#1455656511yescharacteristic
361C2×C4#1455656511yescharacteristic
362C2×C4#1455661112yes—
363C2×C4#1455662212yes—
364C2×C4C22×C22⋊D7311yescharacteristic
365C2×C4C22×D29211yescharacteristic
366C2×C4C2×#1192402811yescharacteristic
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—42no—
372C8—2922no—
373C8—2922no—
374C22—21no—
375C22—21no—
376C22—5842no—
377C22—5842no—
378C22—5842no—
379C22—11684no—
380C22#1798880912yes—
381C22C2×#1455656511yescharacteristic
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—5842no—
387C4#1798880912yes—
388C4#1798885712yes—
389C4C2×#1455661111yescharacteristic
390C4C2×#1455662211yescharacteristic
391C4#1798939912yes—
392C2—22no—
393C2—11682no—
394C2C2×#1798880911yescharacteristic
395C2#2261957011yescharacteristic
396C2C2×#1798885711yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2960C2
socleC22×C73C2×#121
prime radicalC2×#14556588C22
hypercenter#2960D73
hypocenterC73#19212
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960

Complements

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

NH ≅ G/Ntype
1. #22621079C2semidirect
4. #22609471C2semidirect
5. #22609347C2semidirect
6. #22608679C2semidirect
7. #22608613C2semidirect
9. C73×#2960C2semidirect
21. #17988882C22semidirect
28. #17988340C22semidirect
29. #17988338C22semidirect
30. #17988335C22semidirect
34. #446×C73C22semidirect
36. C8×Q8×C73C22semidirect
202. #2960D73semidirect
208. C73#19212semidirect
225. #446C2×D73semidirect
227. C8×Q8C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226210792
#226208692
C2×#179889342
#226094712
#226093472
#226086792
#226086132
#226086002
C73×#29602
#1921273