Finite Group Databasebeta

← #33071280

Subgroups — #33071280

Order 18688.

740 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620846C211yesmaximal, characteristic
2C2×#17988787C211yesmaximal, characteristic, complemented
3#22619012C211yesmaximal, characteristic, complemented
4#22609377C214yesmaximal, complemented
5#22609289C214yesmaximal, complemented
6#22608707C211yesmaximal, characteristic, complemented
7#22608600C211yesmaximal, characteristic, complemented
8#22608580C211yesmaximal, characteristic, complemented
9C73×#3037C211yesmaximal, characteristic, complemented
10C22×#11924015—21no—
11C2×#14556588C2211yescharacteristic
12#17988963C2212yes—
13#17988935C2212yes—
14#17988787—22no—
15#17988787C2214yes—
16#17988782—22no—
17#17988720—22no—
18Q8×#10596831—22no—
19Q8×#10596831C2214yescomplemented
20#17988640—22no—
21#17988640C2211yescharacteristic
22#17988640C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556511—21no—
25#17988339C2214yescomplemented
26#17988333C2212yescomplemented
27#17988331C2212yescomplemented
28#17988330C2214yes—
29C2×#137×C73—21no—
30#532×C73C2211yescharacteristic
31C2×C4×Q8×C73C2211yescharacteristic
32#464×C73—22no—
33#462×C73C2212yescomplemented
34#460×C73C2212yescomplemented
35#423×C73C2211yescharacteristic
36#417×C73—21no—
37C22×D292—21no—
38C22×D292—21no—
39C22×D292—21no—
40C2×#11924015—24no—
41C2×#11924015—44no—
42C2×#11924013—24no—
43C22×#10596831—21no—
44#14556588—21no—
45#14556588—21no—
46#14556588—21no—
47#14556588C2312yes—
48C4⋊#10596831—24no—
49C4⋊#10596831—21no—
50C4⋊#10596831D412yes—
51C4⋊#10596831C2312yes—
52C4⋊#10596831C2312yes—
53#14556511—24no—
54#14556511—24no—
55C22⋊#10596831—21no—
56C22⋊#10596831D412yes—
57C4×#10596831—24no—
58C4×#10596831—21no—
59C4×#10596831C2312yes—
60C2×#41×C73—21no—
61C22×Q8×C73D411yescharacteristic
62#137×C73—44no—
63C2×Q16×C73—22no—
64C2×SD16×C73—22no—
65C2×M16×C73—21no—
66C4⋊Q8×C73C2311yescharacteristic
67#127×C73C2311yescharacteristic
68#124×C73—21no—
69#121×C73—21no—
70#121×C73—21no—
71C4×Q8×C73—21no—
72C4×Q8×C73—21no—
73C4×Q8×C73D412yes—
74C4×Q8×C73C2312yes—
75C4×D4×C73—21no—
76C4×D4×C73—21no—
77C2×C4⋊C4×C73—21no—
78C2×C4⋊C4×C73D411yescharacteristic
79C2×C42×C73—21no—
80C2×C42×C73C2311yescharacteristic
81C4⋊C8×C73—21no—
82C4⋊C8×C73C2312yes—
83#103×C73—22no—
84#102×C73—22no—
85C23×D73—41no—
86C2×D292—22no—
87C2×D292—22no—
88C2×D292—22no—
89C2×D292—42no—
90C2×D292—42no—
91C2×D292—41no—
92C2×D292—41no—
93C2×D292—41no—
94C2×D292—82no—
95C2×D292—82no—
96#11924015—48no—
97#11924013—84no—
98C2×#10596831—22no—
99C2×#10596831—22no—
100C2×#10596831—42no—
101C2×#10596831C2×D414yes—
102#41×C73—42no—
103#41×C73—42no—
104C2×Q8×C73—21no—
105C2×Q8×C73—22no—
106C2×Q8×C73—21no—
107C2×Q8×C73D814yescomplemented
108C2×Q8×C73C2×D412yes—
109C2×D4×C73—21no—
110C2×D4×C73—21no—
111C2×D4×C73—21no—
112C2×D4×C73—41no—
113C22×C4×C73—21no—
114C22×C4×C73—21no—
115C22×C4×C73—41no—
116C22×C4×C73C2×D411yescharacteristic
117C22×C4×C73C2×D411yescharacteristic
118C22×C4×C73#4111yescharacteristic
119Q16×C73—44no—
120SD16×C73—44no—
121M16×C73—42no—
122C2×C8×C73—22no—
123C4⋊C4×C73—22no—
124C4⋊C4×C73—21no—
125C4⋊C4×C73—22no—
126C4⋊C4×C73—21no—
127C4⋊C4×C73—21no—
128C4⋊C4×C73C2×D412yes—
129C22⋊C4×C73—41no—
130C22⋊C4×C73—41no—
131C42×C73—22no—
132C42×C73—22no—
133C42×C73C2×D411yescharacteristic
134C42×C73C2×D411yescharacteristic
135C42×C73#4111yescharacteristic
136C42×C73C2411yescharacteristic
137C23×C73—41no—
138C23×C73#12111yescharacteristic
139C22×D73—42no—
140C22×D73—82no—
141C22×D73—82no—
142C22×D73—82no—
143Q8×C73—22no—
144Q8×C73—24no—
145Q8×C73—41no—
146Q8×C73—41no—
147Q8×C73C2×D814yes—
148D4×C73—41no—
149D4×C73—41no—
150D4×C73—42no—
151D4×C73—81no—
152D4×C73—82no—
153C2×C4×C73—22no—
154C2×C4×C73—24no—
155C2×C4×C73—22no—
156C2×C4×C73—22no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—81no—
160C2×C4×C73—82no—
161C2×C4×C73#12111yescharacteristic
162C2×C4×C73#12111yescharacteristic
163C2×C4×C73#12111yescharacteristic
164C2×C4×C73C2×D812yes—
165C2×C4×C73#13512yes—
166C2×C4×C73C22×D411yescharacteristic
167C2×C4×C73C22×D411yescharacteristic
168C2×C4×C73C2×#4111yescharacteristic
169D292—42no—
170D292—42no—
171D292—44no—
172D292—82no—
173D292—82no—
174D292—82no—
175D292—82no—
176C8×C73—42no—
177#10596831—24no—
178#10596831—42no—
179#10596831—42no—
180C22×C73—21no—
181C22×C73—21no—
182C22×C73—81no—
183C22×C73—81no—
184C22×C73—81no—
185C22×C73#46612yes—
186C22×C73C2×#12111yescharacteristic
187C2×D73—82no—
188C2×D73—82no—
189C2×D73—82no—
190C2×D73—164no—
191C4×C73—21no—
192C4×C73—24no—
193C4×C73—21no—
194C4×C73—21no—
195C4×C73—42no—
196C4×C73—81no—
197C4×C73#46612yes—
198C4×C73C22×D811yescharacteristic
199C4×C73C2×#13511yescharacteristic
200C4×C73#57712yes—
201#19398—731nomaximal, Sylow
202C2×C73—22no—
203C2×C73—161no—
204C2×C73#304311yescharacteristic
205C2×C73C2×#46611yescharacteristic
206C2×C73#306811yescharacteristic
207D73—162no—
208#3205—731no—
209#3037D7311yescharacteristic, complemented
210C2×#460—731no—
211#2982—731no—
212#1682—734no—
213#1668—734no—
214#1571—731no—
215#1545—731no—
216#1540—731no—
217C73#1939811yescharacteristic, complemented, Sylow
218C2×#137—21no—
219C22×SD16—1461no—
220#532C2×D7311yescharacteristic
221C2×#127—731no—
222C2×C4×Q8C2×D7311yescharacteristic
223#493—732no—
224#486—732no—
225#464—22no—
226#462C2×D7312yescomplemented
227#460—734no—
228#460—1462no—
229#460C2×D7312yescomplemented
230#459—1462no—
231#447—1462no—
232C8×Q8—734no—
233C8×Q8—1462no—
234#431—731no—
235#431—731no—
236#431—1462no—
237#423C2×D7311yescharacteristic
238C2×C4⋊C8—731no—
239#417—21no—
240C2×#102—1461no—
241#335—734no—
242#332—732no—
243#331—732no—
244#331—734no—
245C2×#41—21no—
246C22×Q8C22⋊D7311yescharacteristic
247C22×D4—1461no—
248C22×D4—1461no—
249C22×D4—1461no—
250#137—44no—
251C2×Q16—22no—
252C2×SD16—22no—
253C2×SD16—2924no—
254C2×SD16—1464no—
255C2×D8—1464no—
256C2×M16—21no—
257C22×C8—1461no—
258C4⋊Q8C22×D7311yescharacteristic
259#127—732no—
260#127—1461no—
261#127—1461no—
262#127—1461no—
263#127C22×D7311yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C22⋊D7312yes—
270C4×Q8C22×D7312yes—
271C4×D4—21no—
272C4×D4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C22⋊D7311yescharacteristic
275C2×C42—21no—
276C2×C42C22×D7311yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—732no—
279C4⋊C8—732no—
280C4⋊C8—732no—
281C4⋊C8—1464no—
282C4⋊C8—1461no—
283C4⋊C8C22×D7312yes—
284#103—22no—
285#102—22no—
286#102—1464no—
287#102—1464no—
288C22⋊C8—732no—
289C22⋊C8—1461no—
290C4×C8—732no—
291C4×C8—1464no—
292C4×C8—1461no—
293C24—2921no—
294#41—42no—
295#41—42no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—21no—
299C2×Q8#1192401314yescomplemented
300C2×Q8C2×C22⋊D7312yes—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—2921no—
306C2×D4—2921no—
307C2×D4—2921no—
308C2×D4—2922no—
309C2×D4—2922no—
310C2×D4—5842no—
311C2×D4—5842no—
312C2×D4—1462no—
313C2×D4—1462no—
314C2×D4—1462no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4D4×D7311yescharacteristic
319C22×C4C2×C22⋊D7311yescharacteristic
320C22×C4#1192402711yescharacteristic
321Q16—44no—
322SD16—44no—
323SD16—2928no—
324D8—5844no—
325M16—42no—
326C2×C8—22no—
327C2×C8—2922no—
328C2×C8—734no—
329C2×C8—1462no—
330C2×C8—1462no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4C2×C22⋊D7312yes—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C42—22no—
340C42—22no—
341C42D4×D7311yescharacteristic
342C42C2×C22⋊D7311yescharacteristic
343C42#1192402711yescharacteristic
344C42C23×D7311yescharacteristic
345C23—41no—
346C23—2922no—
347C23—5842no—
348C23—5842no—
349C23—5842no—
350C23#1455656311yescharacteristic
351Q8—24no—
352Q8—22no—
353Q8—41no—
354Q8—41no—
355Q8C2×#1192401314yes—
356D4—42no—
357D4—41no—
358D4—41no—
359D4—81no—
360D4—82no—
361D4—2924no—
362D4—2922no—
363D4—2922no—
364D4—5842no—
365D4—5842no—
366D4—5842no—
367D4—5842no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4#1455656311yescharacteristic
377C2×C4#1455656311yescharacteristic
378C2×C4#1455656311yescharacteristic
379C2×C4C2×#1192401312yes—
380C2×C4#1455661112yes—
381C2×C4C2×D4×D7311yescharacteristic
382C2×C4C22×C22⋊D7311yescharacteristic
383C2×C4C2×#1192402711yescharacteristic
384C8—42no—
385C8—2922no—
386C8—2922no—
387C8—1464no—
388C22—21no—
389C22—21no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—5842no—
394C22—5842no—
395C22—5842no—
396C22—11684no—
397C22#1798882112yes—
398C22C2×#1455656311yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—24no—
403C4—42no—
404C4—81no—
405C4#1798882112yes—
406C4C22×#1192401311yescharacteristic
407C4C2×#1455661111yescharacteristic
408C4#1798939612yes—
409C2—22no—
410C2—161no—
411C2—11682no—
412C2#2261945411yescharacteristic
413C2C2×#1798882111yescharacteristic
414C2#2261966111yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3037C2
socleC22×C73C2×#121
prime radical#22620846C2
hypercenter#3037D73
hypocenterC73#19398
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
2. C2×#17988787C2semidirect
3. #22619012C2semidirect
4. #22609377C2semidirect
5. #22609289C2semidirect
6. #22608707C2semidirect
7. #22608600C2semidirect
8. #22608580C2semidirect
9. C73×#3037C2semidirect
19. Q8×#10596831C22semidirect
25. #17988339C22semidirect
26. #17988333C22semidirect
27. #17988331C22semidirect
33. #462×C73C22semidirect
34. #460×C73C22semidirect
107. C2×Q8×C73D8semidirect
209. #3037D73semidirect
217. C73#19398semidirect
226. #462C2×D73semidirect
229. #460C2×D73semidirect
299. C2×Q8#11924013semidirect

Maximal subgroups

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

subgroupindex
#226208462
C2×#179887872
#226190122
#226093772
#226092892
#226087072
#226086002
#226085802
C73×#30372
#1939873