Finite Group Databasebeta

← #33071260

Subgroups — #33071260

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620874C211yesmaximal, characteristic
2#22619736C211yesmaximal, characteristic, complemented
3C2×#17988787C211yesmaximal, characteristic
4#22609347C214yesmaximal, complemented
5#22609279C214yesmaximal, complemented
6#22608702C211yesmaximal, characteristic, complemented
7#22608598C211yesmaximal, characteristic
8#22608586C211yesmaximal, characteristic, complemented
9C73×#2982C211yesmaximal, characteristic, complemented
10C22×#11924015—21no—
11#17989144C2211yescharacteristic
12C2×#14556588C2211yescharacteristic
13#17988960C2212yes—
14#17988934C2212yes—
15#17988894—22no—
16#17988882C2212yescomplemented
17#17988872C2212yescomplemented
18#17988788—22no—
19#17988787—22no—
20#17988787C2214yes—
21#17988740—22no—
22#17988627—21no—
23#17988611C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25#17988596—21no—
26C2×#14556511—21no—
27#17988338C2214yescomplemented
28#17988335C2212yescomplemented
29#17988334C2212yescomplemented
30#17988330C2214yes—
31C2×C4×Q8×C73C2211yescharacteristic
32C8×Q8×C73—22no—
33C8×Q8×C73C2214yescomplemented
34#431×C73—22no—
35#431×C73C2211yescharacteristic
36#431×C73C2211yescharacteristic
37C22×D292—21no—
38C22×D292—21no—
39C22×D292—21no—
40C2×#11924017—24no—
41C2×#11924015—24no—
42C2×#11924015—44no—
43C2×#11924012—21no—
44C22×#10596831—21no—
45#14556591C2311yescharacteristic
46#14556591C2311yescharacteristic
47#14556588—21no—
48#14556588—21no—
49#14556588—21no—
50#14556588C2312yes—
51#14556583—21no—
52#14556570—21no—
53#14556570—21no—
54C4×Dic146—21no—
55C4×Dic146—21no—
56#14556550—21no—
57#14556524—22no—
58#14556522—22no—
59C4⋊#10596831—21no—
60C4⋊#10596831—21no—
61C4⋊#10596831C2312yes—
62C4⋊#10596831C2312yes—
63#14556515—22no—
64#14556515—22no—
65#14556513—24no—
66#14556511—24no—
67C22×Q8×C73D411yescharacteristic
68C4×Q8×C73—21no—
69C4×Q8×C73—21no—
70C4×Q8×C73D412yes—
71C4×Q8×C73C2312yes—
72C2×C4⋊C4×C73—21no—
73C2×C4⋊C4×C73D411yescharacteristic
74C2×C42×C73—21no—
75C2×C42×C73C2311yescharacteristic
76C4⋊C8×C73—24no—
77C4⋊C8×C73D412yes—
78C4⋊C8×C73C2312yes—
79C22⋊C8×C73—21no—
80C22⋊C8×C73D412yes—
81C4×C8×C73—24no—
82C4×C8×C73—21no—
83C4×C8×C73C2312yes—
84C23×D73—41no—
85C2×Dic146—21no—
86C2×Dic146—21no—
87C2×D292—22no—
88C2×D292—22no—
89C2×D292—22no—
90C2×D292—42no—
91C2×D292—42no—
92C2×D292—41no—
93C2×D292—41no—
94C2×D292—41no—
95C2×D292—82no—
96C2×D292—82no—
97#11924017—84no—
98#11924015—48no—
99#11924012—42no—
100C2×#10596831—22no—
101C2×#10596831—22no—
102C2×#10596831—42no—
103C4⋊Dic73—21no—
104C4⋊Dic73—21no—
105C4⋊Dic73—21no—
106C4⋊Dic73—21no—
107#11924007—41no—
108#11924007—41no—
109C22⋊Dic73—41no—
110C4×Dic73—41no—
111C2×Q8×C73—21no—
112C2×Q8×C73—22no—
113C2×Q8×C73SD1614yescomplemented
114C2×Q8×C73C2×D412yes—
115C22×C4×C73—21no—
116C22×C4×C73—21no—
117C22×C4×C73C2×D411yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73#4111yescharacteristic
120C2×C8×C73—22no—
121C2×C8×C73C2×D414yes—
122C4⋊C4×C73—22no—
123C4⋊C4×C73—21no—
124C4⋊C4×C73—22no—
125C4⋊C4×C73C2×D412yes—
126C42×C73—22no—
127C42×C73—22no—
128C42×C73C2×D411yescharacteristic
129C42×C73C2×D411yescharacteristic
130C42×C73#4111yescharacteristic
131C42×C73C2411yescharacteristic
132C23×C73#12111yescharacteristic
133C22×D73—42no—
134C22×D73—82no—
135C22×D73—82no—
136C22×D73—82no—
137Q8×C73—22no—
138Q8×C73—24no—
139Q8×C73C2×SD1614yes—
140C2×C4×C73—22no—
141C2×C4×C73—24no—
142C2×C4×C73—22no—
143C2×C4×C73—22no—
144C2×C4×C73—41no—
145C2×C4×C73—81no—
146C2×C4×C73#12111yescharacteristic
147C2×C4×C73#12111yescharacteristic
148C2×C4×C73#12111yescharacteristic
149C2×C4×C73C2×SD1612yes—
150C2×C4×C73#13512yes—
151C2×C4×C73C22×D411yescharacteristic
152C2×C4×C73C22×D411yescharacteristic
153C2×C4×C73C2×#4111yescharacteristic
154C2×Dic73—41no—
155C2×Dic73—41no—
156D292—42no—
157D292—42no—
158D292—44no—
159D292—82no—
160D292—82no—
161D292—82no—
162D292—82no—
163Dic146—42no—
164Dic146—81no—
165C8×C73—24no—
166C8×C73—42no—
167#10596831—42no—
168#10596831—42no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73#46512yes—
172C22×C73C2×#12111yescharacteristic
173C2×D73—82no—
174C2×D73—82no—
175C2×D73—82no—
176C2×D73—164no—
177C4×C73—21no—
178C4×C73—24no—
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—42no—
182C4×C73#46512yes—
183C4×C73C22×SD1611yescharacteristic
184C4×C73C2×#13511yescharacteristic
185C4×C73#57712yes—
186Dic73—81no—
187Dic73—81no—
188#19396—731nomaximal, Sylow
189C2×C73—22no—
190C2×C73#304311yescharacteristic
191C2×C73C2×#46511yescharacteristic
192C2×C73#306811yescharacteristic
193D73—162no—
194#3208—731no—
195#3078—731no—
196C2×#460—731no—
197#2982D7311yescharacteristic, complemented
198#1678—734no—
199#1666—734no—
200#1570—731no—
201#1545—731no—
202#1542—731no—
203C73#1939611yescharacteristic, complemented, Sylow
204C22×SD16—1461no—
205#533—731no—
206C2×#127—731no—
207C2×C4×Q8C2×D7311yescharacteristic
208#492—732no—
209#486—732no—
210#479—1462no—
211#477—732no—
212#475—732no—
213#460—734no—
214#460—1462no—
215#460—1462no—
216#450—1462no—
217C8×Q8—22no—
218C8×Q8C2×D7314yescomplemented
219#431—22no—
220#431C2×D7311yescharacteristic
221#431C2×D7311yescharacteristic
222#428—1461no—
223#423—731no—
224C2×C4⋊C8—731no—
225#419—1461no—
226C2×#102—1461no—
227#334—734no—
228#333—732no—
229#332—732no—
230#331—734no—
231C22×Q8D29211yescharacteristic
232C22×D4—1461no—
233C22×D4—1461no—
234C22×D4—1461no—
235C2×SD16—2924no—
236C2×SD16—1464no—
237C2×SD16—1464no—
238C2×M16—1461no—
239C22×C8—1461no—
240C4⋊Q8—731no—
241C4⋊Q8—731no—
242#127—732no—
243#127—1461no—
244#127—1461no—
245#127—1461no—
246#125—1461no—
247C22⋊Q8—1461no—
248C22⋊Q8—1461no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—1461no—
252C4×Q8—1461no—
253C4×Q8D29212yes—
254C4×Q8C22×D7312yes—
255#117—1461no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D29211yescharacteristic
258C2×C42—21no—
259C2×C42C22×D7311yescharacteristic
260#107—1462no—
261#106—1462no—
262C4⋊C8—24no—
263C4⋊C8—732no—
264C4⋊C8—732no—
265C4⋊C8—1461no—
266C4⋊C8—1461no—
267C4⋊C8C22⋊D7312yes—
268C4⋊C8C22×D7312yes—
269#103—1462no—
270#103—1462no—
271#102—1464no—
272#102—1464no—
273C22⋊C8—21no—
274C22⋊C8C22⋊D7312yes—
275C4×C8—24no—
276C4×C8—21no—
277C4×C8C22×D7312yes—
278C24—2921no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—1461no—
282C2×Q8—1461no—
283C2×Q8#1192401714yescomplemented
284C2×Q8C2×D29212yes—
285C2×D4—2921no—
286C2×D4—2921no—
287C2×D4—2921no—
288C2×D4—2922no—
289C2×D4—2922no—
290C2×D4—5842no—
291C2×D4—5842no—
292C2×D4—1462no—
293C2×D4—1462no—
294C2×D4—1462no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4C2×C22⋊D7311yescharacteristic
298C22×C4C2×D29211yescharacteristic
299C22×C4#1192402811yescharacteristic
300SD16—2928no—
301SD16—5844no—
302M16—2922no—
303C2×C8—22no—
304C2×C8—2922no—
305C2×C8—1462no—
306C2×C8—1462no—
307C2×C8C2×C22⋊D7314yes—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—2921no—
312C4⋊C4—2921no—
313C4⋊C4—1461no—
314C4⋊C4—1461no—
315C4⋊C4—1461no—
316C4⋊C4—1461no—
317C4⋊C4C2×D29212yes—
318C22⋊C4—2921no—
319C42—22no—
320C42—22no—
321C42—2921no—
322C42C2×C22⋊D7311yescharacteristic
323C42C2×D29211yescharacteristic
324C42#1192402811yescharacteristic
325C42C23×D7311yescharacteristic
326C23—2922no—
327C23—5842no—
328C23—5842no—
329C23—5842no—
330C23#1455656511yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—2922no—
334Q8—5841no—
335Q8C2×#1192401714yes—
336D4—2924no—
337D4—2922no—
338D4—2922no—
339D4—5842no—
340D4—5842no—
341D4—5842no—
342D4—5842no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—2921no—
350C2×C4—2921no—
351C2×C4#1455656511yescharacteristic
352C2×C4#1455656511yescharacteristic
353C2×C4#1455656511yescharacteristic
354C2×C4C2×#1192401712yes—
355C2×C4#1455661312yes—
356C2×C4C22×C22⋊D7311yescharacteristic
357C2×C4C22×D29211yescharacteristic
358C2×C4C2×#1192402811yescharacteristic
359C8—24no—
360C8—42no—
361C8—2922no—
362C8—2922no—
363C22—21no—
364C22—21no—
365C22—5842no—
366C22—5842no—
367C22—5842no—
368C22—11684no—
369C22#1798881612yes—
370C22C2×#1455656511yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—24no—
375C4—42no—
376C4—5841no—
377C4—5841no—
378C4#1798881612yes—
379C4C22×#1192401711yescharacteristic
380C4C2×#1455661311yescharacteristic
381C4#1798939512yes—
382C2—22no—
383C2—11682no—
384C2#2261945211yescharacteristic
385C2C2×#1798881611yescharacteristic
386C2#2261965911yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2982C2
socleC22×C73C2×#121
prime radicalC2×#14556588C22
hypercenter#2982D73
hypocenterC73#19396
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #22619736C2semidirect
4. #22609347C2semidirect
5. #22609279C2semidirect
6. #22608702C2semidirect
8. #22608586C2semidirect
9. C73×#2982C2semidirect
16. #17988882C22semidirect
17. #17988872C22semidirect
27. #17988338C22semidirect
28. #17988335C22semidirect
29. #17988334C22semidirect
33. C8×Q8×C73C22semidirect
113. C2×Q8×C73SD16semidirect
197. #2982D73semidirect
203. C73#19396semidirect
218. C8×Q8C2×D73semidirect
283. C2×Q8#11924017semidirect

Maximal subgroups

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

subgroupindex
#226208742
#226197362
C2×#179887872
#226093472
#226092792
#226087022
#226085982
#226085862
C73×#29822
#1939673