Finite Group Databasebeta

← #33241320

Subgroups — #33241320

Order 18688.

602 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#22620471C211yesmaximal, characteristic, complemented
2#22620418C211yesmaximal, characteristic
3#22612171C212yesmaximal, complemented
4#22612169C212yesmaximal, complemented
5#22612164C212yesmaximal, complemented
6#22612153C212yesmaximal, complemented
7#22611646C211yesmaximal, characteristic, complemented
8#22611643C211yesmaximal, characteristic, complemented
9C2×#17988389C211yesmaximal, characteristic, complemented
10#22611527C211yesmaximal, characteristic, complemented
11C73×#2138C211yesmaximal, characteristic, complemented
12C2×#14556616—21no—
13C22×D584C2211yescharacteristic
14#17989215—21no—
15#17989151—21no—
16C2×#14556531C2211yescharacteristic
17#17988994—22no—
18#17988991C2212yescomplemented
19#17988988C2212yescomplemented
20#17988973—22no—
21#17988972C2212yes—
22#17988962C2212yes—
23#17988630—21no—
24C2×#14556525C2211yescharacteristic
25#17988542C2211yescharacteristic
26#17988391C2212yescomplemented
27#17988389C2212yes—
28#17988389C2212yes—
29#17988388C2212yescomplemented
30#17988387C2212yes—
31#17988387C2212yes—
32#17988362C2212yescomplemented
33C16×Dic73C2212yescomplemented
34C2×#135×C73C2211yescharacteristic
35C22×C16×C73C2211yescharacteristic
36#427×C73C2211yescharacteristic
37#358×C73C2211yescharacteristic, complemented
38#358×C73C2211yescharacteristic, complemented
39#357×C73C2211yescharacteristic, complemented
40#357×C73C2211yescharacteristic, complemented
41C22×D292—21no—
42#14556616—44no—
43C2×#11924015—22no—
44C2×D584—21no—
45C2×D584—22no—
46C2×D584—22no—
47C2×D584C2312yes—
48C2×#11924013—22no—
49C2×#11924012D411yescharacteristic
50#14556591—41no—
51#14556590—22no—
52#14556587—22no—
53#14556583—41no—
54#14556582—22no—
55#14556580—22no—
56#14556568—42no—
57#14556565—41no—
58#14556565—41no—
59#14556559—42no—
60#14556550D411yescharacteristic
61C2×C4⋊Dic73—21no—
62#14556531—22no—
63C2×#11924003C2312yes—
64#14556525—21no—
65#14556525C2312yes—
66#14556524—22no—
67#14556522—22no—
68#14556501D412yes—
69C8×Dic73C2312yes—
70C2×#41×C73—21no—
71#135×C73—22no—
72#135×C73—22no—
73C2×Q16×C73C2311yescharacteristic
74C2×SD16×C73—21no—
75C2×D8×C73C2311yescharacteristic
76C22×C8×C73C2311yescharacteristic
77#117×C73—21no—
78C2×C16×C73—22no—
79C2×C16×C73C2311yescharacteristic
80C2×C16×C73C2311yescharacteristic
81#107×C73C2311yescharacteristic
82#107×C73C2311yescharacteristic
83#106×C73—21no—
84C23×D73—41no—
85C2×Dic146—42no—
86C2×D292—22no—
87C2×D292—42no—
88C2×D292—42no—
89C2×D292—41no—
90C2×C22⋊D73—42no—
91C2×C22⋊D73—42no—
92C22×Dic73—41no—
93#11924015—44no—
94D584—22no—
95D584—22no—
96D584—42no—
97#11924013—44no—
98#11924012SD1614yes—
99C2×#10596831C2×D412yes—
100C4⋊Dic73—22no—
101C4⋊Dic73—41no—
102C4⋊Dic73—42no—
103C4⋊Dic73D811yescharacteristic
104C4⋊Dic73D811yescharacteristic
105#11924007—42no—
106#11924007—42no—
107C22⋊Dic73D812yes—
108#11924005—42no—
109#11924005—42no—
110C4×Dic73C2×D412yes—
111#11924003—22no—
112#41×C73—42no—
113#41×C73—42no—
114C2×Q8×C73—21no—
115C2×D4×C73—21no—
116C2×D4×C73—41no—
117C22×C4×C73—41no—
118C22×C4×C73C2×D411yescharacteristic
119Q16×C73—21no—
120Q16×C73—21no—
121SD16×C73—42no—
122D8×C73—21no—
123D8×C73—21no—
124C2×C8×C73C2×D411yescharacteristic
125C2×C8×C73#4112yes—
126C2×C8×C73#4112yes—
127C2×C8×C73C2411yescharacteristic
128C4⋊C4×C73—21no—
129C4⋊C4×C73—21no—
130C22⋊C4×C73—41no—
131C42×C73—41no—
132C16×C73—21no—
133C16×C73—21no—
134C23×C73—41no—
135C23×C73C2×D811yescharacteristic
136C22×D73—42no—
137C22×D73—82no—
138C22×D73—161no—
139C22×D73—161no—
140Q8×C73—41no—
141Q8×C73—41no—
142D4×C73—41no—
143D4×C73—41no—
144D4×C73—82no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—82no—
149C2×C4×C73#12412yes—
150C2×C4×C73C2×D811yescharacteristic
151C2×C4×C73C2×SD1612yes—
152C2×C4×C73C22×D411yescharacteristic
153C22⋊D73—162no—
154C22⋊D73—162no—
155C2×Dic73—42no—
156C2×Dic73—161no—
157C2×Dic73C2×D812yes—
158C2×Dic73C2×D812yes—
159D292—42no—
160D292—42no—
161D292—82no—
162Dic146—162no—
163C8×C73#12412yes—
164C8×C73C2×#4111yescharacteristic
165C8×C73C2×#4111yescharacteristic
166#10596831C2×SD1614yes—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73—81no—
172C22×C73#48612yes—
173C22×C73C22×D811yescharacteristic
174C2×D73—82no—
175C2×D73—82no—
176C2×D73—82no—
177C2×D73—164no—
178C4×C73—81no—
179C4×C73—81no—
180C4×C73—81no—
181C4×C73#48612yes—
182C4×C73C2×#12411yescharacteristic
183C4×C73C22×SD1611yescharacteristic
184Dic73—22no—
185Dic73—22no—
186Dic73—82no—
187#33702—731nomaximal, Sylow
188C2×C73—22no—
189C2×C73—161no—
190C2×C73C2×#48611yescharacteristic
191C2×C73#341511yescharacteristic
192C2×C73#341611yescharacteristic
193D73—162no—
194#3164—731no—
195#3155—731no—
196#2244—732no—
197#2243—732no—
198#2242—732no—
199#2240—732no—
200#2140—731no—
201#2139—731no—
202#2138D7311yescharacteristic, complemented
203C2×#357—731no—
204#2108—731no—
205C73#3370211yescharacteristic, complemented, Sylow
206C2×AGL(1,ℤ/8)—1461no—
207C2×#135C2×D7311yescharacteristic
208C22×D8—731no—
209#544—1461no—
210#535—1461no—
211C2×M32—731no—
212C22×C16C2×D7311yescharacteristic
213#501—1462no—
214#500—732no—
215#499—732no—
216#496—1462no—
217#495—732no—
218#493—732no—
219#428—1461no—
220#427C2×D7311yescharacteristic
221C2×#107—731no—
222#405—731no—
223#358—732no—
224#358C2×D7311yescharacteristic, complemented
225#358C2×D7311yescharacteristic, complemented
226#357—732no—
227#357—732no—
228#357—732no—
229#357—732no—
230#357—732no—
231#357C2×D7311yescharacteristic, complemented
232#357C2×D7311yescharacteristic, complemented
233#346—732no—
234C4×C16—732no—
235C2×#41—21no—
236C22×D4—1461no—
237AGL(1,ℤ/8)—2924no—
238#135—22no—
239#135—22no—
240C2×Q16C22×D7311yescharacteristic
241C2×SD16—21no—
242C2×SD16—1462no—
243C2×D8—732no—
244C2×D8—1462no—
245C2×D8—1461no—
246C2×D8—1462no—
247C2×D8—1462no—
248C2×D8C22×D7311yescharacteristic
249C2×M16—731no—
250C22×C8C22×D7311yescharacteristic
251C4⋊Q8—2921no—
252C4⋊Q8—1462no—
253#127—1462no—
254#125—2921no—
255#125—1462no—
256#124—1462no—
257C22⋊Q8—2922no—
258#121—2921no—
259#121—2921no—
260#120—2922no—
261#117—21no—
262#117—731no—
263C2×C4⋊C4—1461no—
264M32—1462no—
265C2×C16—22no—
266C2×C16—732no—
267C2×C16C22×D7311yescharacteristic
268C2×C16C22×D7311yescharacteristic
269#107—732no—
270#107—1462no—
271#107—1461no—
272#107C22×D7311yescharacteristic
273#107C22×D7311yescharacteristic
274#106—21no—
275#106—1462no—
276#97—732no—
277C4×C8—732no—
278C24—2921no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—2922no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—2922no—
286C2×D4—2922no—
287C2×D4—2922no—
288C2×D4—2921no—
289C2×D4—2922no—
290C2×D4—1462no—
291C22×C4—41no—
292C22×C4—2921no—
293C22×C4D4×D7311yescharacteristic
294Q16—21no—
295Q16—21no—
296SD16—42no—
297SD16—2924no—
298D8—21no—
299D8—21no—
300D8—2924no—
301D8—2922no—
302D8—1462no—
303D8—1462no—
304M16—734no—
305C2×C8—732no—
306C2×C8D4×D7311yescharacteristic
307C2×C8#1192402712yes—
308C2×C8#1192402812yes—
309C2×C8C23×D7311yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—2922no—
313C4⋊C4—2922no—
314C4⋊C4—2922no—
315C4⋊C4—2921no—
316C4⋊C4—731no—
317C4⋊C4—731no—
318C4⋊C4—1462no—
319C22⋊C4—41no—
320C22⋊C4—2922no—
321C22⋊C4—2922no—
322C22⋊C4—732no—
323C42—41no—
324C42—732no—
325C16—21no—
326C16—21no—
327C16—1462no—
328C23—41no—
329C23—2922no—
330C23—5842no—
331C23—11681no—
332C23—11681no—
333C23D8×D7311yescharacteristic
334Q8—41no—
335Q8—41no—
336Q8—11682no—
337D4—41no—
338D4—41no—
339D4—82no—
340D4—2922no—
341D4—2922no—
342D4—5842no—
343D4—11682no—
344D4—11682no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—2922no—
350C2×C4—732no—
351C2×C4—732no—
352C2×C4—11681no—
353C2×C4#1455657712yes—
354C2×C4D8×D7311yescharacteristic
355C2×C4SD16×D7312yes—
356C2×C4C2×D4×D7311yescharacteristic
357C8—734no—
358C8#1455657712yes—
359C8C2×#1192402711yescharacteristic
360C8C2×#1192402811yescharacteristic
361C22—21no—
362C22—21no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—5842no—
367C22—5842no—
368C22—5842no—
369C22—11684no—
370C22#1798893212yes—
371C22C2×D8×D7311yescharacteristic
372C4—81no—
373C4—81no—
374C4—81no—
375C4—5842no—
376C4—1462no—
377C4—1462no—
378C4#1798893212yes—
379C4C2×#1455657711yescharacteristic
380C4C2×SD16×D7311yescharacteristic
381C2—22no—
382C2—161no—
383C2—11682no—
384C2C2×#1798893211yescharacteristic
385C2#2262306911yescharacteristic
386C2#2262307411yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2138C2
socleC22×C73C22×D8
prime radical#22620418C2
hypercenter#2138D73
hypocenterC73#33702
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #22620471C2semidirect
3. #22612171C2semidirect
4. #22612169C2semidirect
5. #22612164C2semidirect
6. #22612153C2semidirect
7. #22611646C2semidirect
8. #22611643C2semidirect
9. C2×#17988389C2semidirect
10. #22611527C2semidirect
11. C73×#2138C2semidirect
18. #17988991C22semidirect
19. #17988988C22semidirect
26. #17988391C22semidirect
29. #17988388C22semidirect
32. #17988362C22semidirect
33. C16×Dic73C22semidirect
37. #358×C73C22semidirect
38. #358×C73C22semidirect
39. #357×C73C22semidirect
40. #357×C73C22semidirect
202. #2138D73semidirect
205. C73#33702semidirect
224. #358C2×D73semidirect
225. #358C2×D73semidirect
231. #357C2×D73semidirect
232. #357C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226204712
#226204182
#226121712
#226121692
#226121642
#226121532
#226116462
#226116432
C2×#179883892
#226115272
C73×#21382
#3370273