Finite Group Databasebeta

← #33243697

Subgroups — #33243697

Order 18688.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622799C211yesmaximal, characteristic, complemented
2#22612172C212yesmaximal, complemented
3#22612154C211yesmaximal, characteristic, complemented
4#22612088C212yesmaximal, complemented
5#22612078C211yesmaximal, characteristic, complemented
6#22611963C212yesmaximal, complemented
7#22611963C211yesmaximal, characteristic, complemented
8#22611840C211yesmaximal, characteristic, complemented
9#22611805C212yesmaximal, complemented
10C4×#14556532C211yesmaximal, characteristic, complemented
11C73×#3391C211yesmaximal, characteristic, complemented
12#17989297—21no—
13C2×#14556532C2211yescharacteristic
14C2×#14556532C2212yes—
15#17988992C2211yescharacteristic
16#17988989C2212yes—
17#17988888—22no—
18#17988878—21no—
19C4×D584C2211yescharacteristic
20C4×D584C2212yes—
21#17988414C2212yes—
22#17988412C2211yescharacteristic
23C4⋊#11924003C2211yescharacteristic
24C4⋊#11924003C2212yes—
25#17988388C2211yescharacteristic
26#17988388C2212yes—
27#17988388C2212yes—
28#17988388C2212yes—
29#17988388C2212yes—
30#17988387C2211yescharacteristic
31#17988363C2212yes—
32C4×#11924003C2211yescharacteristic
33#550×C73—21no—
34#493×C73C2211yescharacteristic
35#493×C73C2212yes—
36#459×C73—21no—
37#459×C73—22no—
38C8×Q8×C73C2211yescharacteristic
39C4×D8×C73C2211yescharacteristic
40C4×D8×C73C2212yes—
41C2×D584C2311yescharacteristic
42#14556591—21no—
43#14556583—22no—
44#14556569—42no—
45#14556567—41no—
46C4×D292—21no—
47C4×D292—22no—
48C4⋊C4×D73—41no—
49#14556532—22no—
50#14556532—24no—
51C2×#11924003C2311yescharacteristic
52C2×#11924003C2311yescharacteristic
53C2×#11924003C2312yes—
54#14556525—21no—
55#14556525C2311yescharacteristic
56#14556525C2312yes—
57#14556523—22no—
58#14556513—22no—
59#14556513—21no—
60C2×#41×C73—41no—
61C2×D8×C73—22no—
62C2×D8×C73—21no—
63C2×D8×C73C2311yescharacteristic
64C2×D8×C73C2312yes—
65#127×C73—22no—
66#127×C73—21no—
67#121×C73—41no—
68#121×C73—42no—
69C4×Q8×C73D411yescharacteristic
70C4×D4×C73—22no—
71C4×D4×C73—21no—
72#107×C73C2311yescharacteristic
73C4⋊C8×C73D411yescharacteristic
74C4⋊C8×C73D412yes—
75#102×C73—22no—
76#102×C73—21no—
77C4×C8×C73C2311yescharacteristic
78C4×C8×C73C2312yes—
79C2×Dic146—41no—
80C2×D292—21no—
81C2×C4×D73—41no—
82C2×C4×D73—42no—
83D584—21no—
84D584#4112yescomplemented
85C4⋊Dic73—21no—
86C4⋊Dic73—22no—
87C4⋊Dic73—41no—
88#11924007—42no—
89#11924007—42no—
90#11924005—42no—
91#11924005—41no—
92#11924003—21no—
93#11924003—22no—
94#11924003—21no—
95#41×C73—84no—
96C2×Q8×C73D811yescharacteristic
97C2×D4×C73—22no—
98C2×D4×C73—21no—
99C2×D4×C73—42no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C2×D4×C73—42no—
103C22×C4×C73—42no—
104C22×C4×C73—41no—
105D8×C73—21no—
106D8×C73—22no—
107D8×C73—21no—
108D8×C73—22no—
109D8×C73—82no—
110D8×C73—81no—
111C2×C8×C73C2×D412yes—
112C2×C8×C73C2×D411yescharacteristic
113C2×C8×C73C2411yescharacteristic
114C4⋊C4×C73—21no—
115C4⋊C4×C73D811yescharacteristic
116C4⋊C4×C73D812yes—
117C22⋊C4×C73—41no—
118C22⋊C4×C73—42no—
119C42×C73C2×D412yes—
120C42×C73C2×D411yescharacteristic
121C23×C73—42no—
122C23×C73—41no—
123C22×D73—41no—
124Q8×C73—22no—
125D4×C73—41no—
126D4×C73—42no—
127D4×C73—41no—
128D4×C73—42no—
129D4×C73—84no—
130D4×C73—82no—
131D4×C73—161no—
132D4×C73—162no—
133C2×C4×C73—41no—
134C2×C4×C73—84no—
135C2×C4×C73—82no—
136C2×C4×C73C2×D811yescharacteristic
137C2×C4×C73C2×D812yes—
138C2×C4×C73C2×D812yes—
139C2×C4×C73C2×D811yescharacteristic
140C2×C4×C73C22×D411yescharacteristic
141C2×Dic73—41no—
142C2×Dic73—42no—
143D292—22no—
144D292—41no—
145C4×D73—82no—
146C4×D73—162no—
147Dic146—161no—
148C8×C73—22no—
149C8×C73—21no—
150C8×C73C2×#4111yescharacteristic
151C8×C732−1+411yescharacteristic
152C22×C73—82no—
153C22×C73—81no—
154C22×C73—82no—
155C22×C73—81no—
156C22×C73—82no—
157C22×C73—81no—
158C22×C73C22×D811yescharacteristic
159C2×D73—42no—
160C2×D73—81no—
161C2×D73—81no—
162C4×C73—22no—
163C4×C73—21no—
164C4×C73—22no—
165C4×C73—81no—
166C4×C73#50912yes—
167C4×C73#54811yescharacteristic
168C4×C73#57711yescharacteristic
169Dic73—81no—
170Dic73—82no—
171#33933—731nomaximal, Sylow
172C2×C73—162no—
173C2×C73—161no—
174C2×C73#335811yescharacteristic
175C2×C73C2×#50911yescharacteristic
176C2×C73#341511yescharacteristic
177D73—82no—
178#3391D7311yescharacteristic, complemented
179#3380—731no—
180#2244—732no—
181#2240—731no—
182#2228—732no—
183#2226—731no—
184#2206—732no—
185#2206—731no—
186#2183—731no—
187#2177—732no—
188C4×D16—731no—
189C73#3393311yescharacteristic, complemented, Sylow
190#554—1461no—
191#550—21no—
192C2×D16—731no—
193C2×D16—732no—
194#500—731no—
195#499—732no—
196#493C2×D7311yescharacteristic
197#493C2×D7312yes—
198#478—1462no—
199#476—1461no—
200#459—21no—
201#459—22no—
202C8×Q8C2×D7311yescharacteristic
203C4×D8—731no—
204C4×D8—732no—
205C4×D8C2×D7311yescharacteristic
206C4×D8C2×D7312yes—
207#367—732no—
208#366—731no—
209C4⋊C16—731no—
210C4⋊C16—732no—
211#357—731no—
212#357—732no—
213#357—732no—
214#357—732no—
215#357—732no—
216#357—731no—
217#346—732no—
218C4×C16—731no—
219C2×#41—41no—
220C2×D8—22no—
221C2×D8—21no—
222C2×D8—731no—
223C2×D8C22×D7311yescharacteristic
224C2×D8C22×D7312yes—
225C4⋊Q8—1461no—
226#127—22no—
227#127—21no—
228#125—1462no—
229C22⋊Q8—2921no—
230C22⋊Q8—2922no—
231#121—41no—
232#121—42no—
233C4×Q8C22⋊D7311yescharacteristic
234C4×D4—22no—
235C4×D4—21no—
236C4×D4—1461no—
237C4×D4—1462no—
238C2×C4⋊C4—2921no—
239D16—1462no—
240D16—1464no—
241C2×C16—731no—
242C2×C16—731no—
243C2×C16—732no—
244#107—731no—
245#107—732no—
246#107—1461no—
247#107C22×D7311yescharacteristic
248#106—1462no—
249C4⋊C8C22⋊D7311yescharacteristic
250C4⋊C8C22⋊D7312yes—
251#102—22no—
252#102—21no—
253#102—1462no—
254#102—1461no—
255C4×C8C22×D7311yescharacteristic
256C4×C8C22×D7312yes—
257#41—84no—
258C2×Q8—2921no—
259C2×Q8#1192401311yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—1461no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—2921no—
270C22×C4—2922no—
271D8—21no—
272D8—22no—
273D8—21no—
274D8—22no—
275D8—82no—
276D8—81no—
277D8—732no—
278D8—1461no—
279C2×C8C2×C22⋊D7312yes—
280C2×C8C2×C22⋊D7311yescharacteristic
281C2×C8C23×D7311yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—2922no—
284C4⋊C4—2922no—
285C4⋊C4—2921no—
286C4⋊C4—1461no—
287C4⋊C4—1462no—
288C4⋊C4#1192401311yescharacteristic
289C4⋊C4#1192401312yes—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—2921no—
293C22⋊C4—2922no—
294C42C2×C22⋊D7312yes—
295C42C2×C22⋊D7311yescharacteristic
296C16—1461no—
297C16—1462no—
298C16—1461no—
299C23—42no—
300C23—41no—
301C23—2921no—
302Q8—22no—
303Q8—11681no—
304D4—41no—
305D4—42no—
306D4—41no—
307D4—42no—
308D4—84no—
309D4—82no—
310D4—162no—
311D4—161no—
312D4—2921no—
313D4—1462no—
314C2×C4—41no—
315C2×C4—84no—
316C2×C4—82no—
317C2×C4—2921no—
318C2×C4—2922no—
319C2×C4—5842no—
320C2×C4—11682no—
321C2×C4C2×#1192401311yescharacteristic
322C2×C4C2×#1192401312yes—
323C2×C4C2×#1192401312yes—
324C2×C4C2×#1192401311yescharacteristic
325C2×C4C22×C22⋊D7311yescharacteristic
326C8—22no—
327C8—21no—
328C8#41×D7311yescharacteristic
329C8#1455663811yescharacteristic
330C22—82no—
331C22—81no—
332C22—81no—
333C22—82no—
334C22—82no—
335C22—81no—
336C22—2922no—
337C22—5841no—
338C22—5841no—
339C22C22×#1192401311yescharacteristic
340C4—22no—
341C4—22no—
342C4—21no—
343C4—81no—
344C4—5841no—
345C4—5842no—
346C4#1798902312yes—
347C4#1798924411yescharacteristic
348C4#1798939811yescharacteristic
349C2—162no—
350C2—161no—
351C2—5842no—
352C2#2262256211yescharacteristic
353C2C2×#1798902311yescharacteristic
354C2#2262307011yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3391C2
socleC22×C73C22×D8
hypercenter#3391D73
hypocenterC73#33933
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. #22622799C2semidirect
2. #22612172C2semidirect
3. #22612154C2semidirect
4. #22612088C2semidirect
5. #22612078C2semidirect
6. #22611963C2semidirect
7. #22611963C2semidirect
8. #22611840C2semidirect
9. #22611805C2semidirect
10. C4×#14556532C2semidirect
11. C73×#3391C2semidirect
84. D584#41semidirect
178. #3391D73semidirect
189. C73#33933semidirect

Maximal subgroups

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

subgroupindex
#226227992
#226121722
#226121542
#226120882
#226120782
#226119632
#226118402
#226118052
C4×#145565322
C73×#33912
#3393373