Finite Group Databasebeta

← #33241558

Subgroups — #33241558

Order 18688.

554 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#22620510C211yesmaximal, characteristic, complemented
2#22612218C211yesmaximal, characteristic, complemented
3#22612198C211yesmaximal, characteristic, complemented
4#22612174C211yesmaximal, characteristic, complemented
5#22612167C211yesmaximal, characteristic, complemented
6#22612046C212yesmaximal, complemented
7#22612032C212yesmaximal, complemented
8C4×#14556534C212yesmaximal, complemented
9C4×#14556532C212yesmaximal, complemented
10C4×#14556531C211yesmaximal, characteristic, complemented
11C73×#3168C211yesmaximal, characteristic, complemented
12#17989137—21no—
13C2×#14556534C2212yes—
14C2×#14556532C2212yes—
15C2×#14556531C2212yes—
16#17988992C2211yescharacteristic
17#17988964C2211yescharacteristic
18#17988934—21no—
19#17988823—21no—
20#17988823C2212yes—
21C4×D584—21no—
22C4×D584C2212yes—
23#17988414C2212yes—
24#17988412C2212yes—
25#17988391C2212yes—
26#17988388C2212yes—
27#17988387C2212yes—
28#17988387C2212yes—
29#17988363C2211yescharacteristic
30#17988363C2211yescharacteristic
31C4×#11924003C2212yes—
32#529×C73—21no—
33#499×C73C2211yescharacteristic
34#495×C73C2211yescharacteristic
35#488×C73—21no—
36#467×C73C2211yescharacteristic
37#466×C73C2211yescharacteristic
38#465×C73—21no—
39C4×Q16×C73C2211yescharacteristic
40C4×SD16×C73—21no—
41C4×D8×C73C2211yescharacteristic
42C2×C4×C8×C73C2211yescharacteristic
43C2×#11924028—41no—
44C2×D584—21no—
45C2×D584C2312yes—
46#14556591—21no—
47#14556588—21no—
48#14556579—41no—
49#14556565—22no—
50#14556565—41no—
51C4×D292—22no—
52C4×D292—41no—
53#14556534—24no—
54#14556532—24no—
55#14556531—24no—
56C2×#11924003C2312yes—
57C2×#11924003C2312yes—
58#14556525—21no—
59#14556525C2312yes—
60#14556513—22no—
61#14556513—22no—
62C2×Q16×C73C2311yescharacteristic
63C2×SD16×C73—21no—
64C2×D8×C73C2311yescharacteristic
65C22×C8×C73D411yescharacteristic
66C22×C8×C73C2311yescharacteristic
67#126×C73—41no—
68#125×C73—21no—
69#124×C73—21no—
70#123×C73—41no—
71C22⋊Q8×C73—21no—
72#121×C73—21no—
73C4×Q8×C73—21no—
74C4×D4×C73—21no—
75C4×D4×C73—41no—
76#117×C73—41no—
77C2×C42×C73D411yescharacteristic
78#107×C73C2311yescharacteristic
79#107×C73C2311yescharacteristic
80#106×C73—21no—
81#103×C73—21no—
82#103×C73—21no—
83#102×C73—21no—
84#102×C73—21no—
85C4×C8×C73D411yescharacteristic
86C4×C8×C73D411yescharacteristic
87C4×C8×C73C2311yescharacteristic
88C4×C8×C73C2311yescharacteristic
89#11924028—84no—
90C2×Dic146—41no—
91C2×D292—22no—
92C2×D292—41no—
93C2×D292—41no—
94C2×C22⋊D73—42no—
95C2×C4×D73—42no—
96D584—22no—
97D584—22no—
98D584—42no—
99C4⋊Dic73—22no—
100C4⋊Dic73—41no—
101#11924005—42no—
102#11924005—42no—
103#11924003—22no—
104#11924003#4114yescomplemented
105C2×Q8×C73—21no—
106C2×D4×C73—21no—
107C2×D4×C73—41no—
108C22×C4×C73—41no—
109C22×C4×C73D811yescharacteristic
110C22×C4×C73D811yescharacteristic
111C22×C4×C73C2×D411yescharacteristic
112Q16×C73—22no—
113SD16×C73—42no—
114D8×C73—22no—
115C2×C8×C73—22no—
116C2×C8×C73—22no—
117C2×C8×C73C2×D411yescharacteristic
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73C2411yescharacteristic
121C4⋊C4×C73—21no—
122C4⋊C4×C73—21no—
123C4⋊C4×C73—41no—
124C4⋊C4×C73—41no—
125C4⋊C4×C73—41no—
126C4⋊C4×C73—41no—
127C22⋊C4×C73—41no—
128C22⋊C4×C73—41no—
129C22⋊C4×C73—41no—
130C22⋊C4×C73—41no—
131C22⋊C4×C73—41no—
132C42×C73—41no—
133C42×C73D811yescharacteristic
134C42×C73D811yescharacteristic
135C42×C73C2×D411yescharacteristic
136C42×C73C2×D411yescharacteristic
137C23×C73—41no—
138C23×C73C2×D811yescharacteristic
139C22×D73—42no—
140Q8×C73—41no—
141Q8×C73—41no—
142D4×C73—41no—
143D4×C73—41no—
144D4×C73—82no—
145C2×C4×C73—21no—
146C2×C4×C73—21no—
147C2×C4×C73—22no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—82no—
153C2×C4×C73C2×D811yescharacteristic
154C2×C4×C73C2×D811yescharacteristic
155C2×C4×C73C2×D811yescharacteristic
156C2×C4×C73C2×D811yescharacteristic
157C2×C4×C73C2×D811yescharacteristic
158C2×C4×C73C22×D411yescharacteristic
159C22⋊D73—84no—
160C2×Dic73—42no—
161D292—42no—
162D292—42no—
163D292—82no—
164D292—82no—
165C4×D73—84no—
166Dic146—82no—
167C8×C73—21no—
168C8×C73—21no—
169C8×C73#13512yes—
170C8×C73C2×#4112yes—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—81no—
175C22×C73—81no—
176C22×C73—81no—
177C22×C73C22×D811yescharacteristic
178C2×D73—82no—
179C2×D73—82no—
180C2×D73—82no—
181C4×C73—21no—
182C4×C73—21no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—81no—
186C4×C73—81no—
187C4×C73—81no—
188C4×C73#50912yes—
189C4×C73#53211yescharacteristic
190C4×C73C2×#13511yescharacteristic
191Dic73—82no—
192#33737—731nomaximal, Sylow
193C2×C73—41no—
194C2×C73—161no—
195C2×C73#316911yescharacteristic
196C2×C73C2×#50911yescharacteristic
197C2×C73#341411yescharacteristic
198D73—162no—
199#3169—731no—
200#3168D7311yescharacteristic, complemented
201#2255—731no—
202#2250—731no—
203#2244—731no—
204#2243—731no—
205#2220—732no—
206#2218—732no—
207C4×SD32—732no—
208C4×D16—732no—
209C4×M32—731no—
210C73#3373711yescharacteristic, complemented, Sylow
211#532—1461no—
212#529—21no—
213C2×SD32—732no—
214C2×D16—732no—
215C2×M32—732no—
216#500—731no—
217#499C2×D7311yescharacteristic
218#495C2×D7311yescharacteristic
219#493—731no—
220#488—21no—
221#486—1461no—
222#467C2×D7311yescharacteristic
223#466—732no—
224#466—1461no—
225#466C2×D7311yescharacteristic
226#465—21no—
227C4×Q16C2×D7311yescharacteristic
228C4×SD16—21no—
229C4×D8—732no—
230C4×D8—1461no—
231C4×D8C2×D7311yescharacteristic
232C2×C4×C8C2×D7311yescharacteristic
233#367—732no—
234#366—732no—
235#358—732no—
236#357—732no—
237#357—732no—
238#357—732no—
239#346—731no—
240#346—731no—
241C4×C16—732no—
242C2×#41—2921no—
243C2×Q16C22×D7311yescharacteristic
244C2×SD16—21no—
245C2×D8—732no—
246C2×D8—1461no—
247C2×D8C22×D7311yescharacteristic
248C22×C8C22⋊D7311yescharacteristic
249C22×C8C22×D7311yescharacteristic
250C4⋊Q8—1461no—
251#127—1461no—
252#126—41no—
253#125—21no—
254#124—21no—
255#124—2921no—
256#123—41no—
257C22⋊Q8—21no—
258#121—21no—
259#121—2921no—
260#121—1462no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—41no—
264C4×D4—2921no—
265C4×D4—1462no—
266#117—41no—
267C2×C42C22⋊D7311yescharacteristic
268SD32—1464no—
269D16—1464no—
270M32—1464no—
271C2×C16—732no—
272C2×C16—732no—
273#107—732no—
274#107—1461no—
275#107C22×D7311yescharacteristic
276#107C22×D7311yescharacteristic
277#106—21no—
278#103—21no—
279#103—21no—
280#102—21no—
281#102—21no—
282#102—1462no—
283#102—1462no—
284C4×C8C22⋊D7311yescharacteristic
285C4×C8C22⋊D7311yescharacteristic
286C4×C8C22×D7311yescharacteristic
287C4×C8C22×D7311yescharacteristic
288#41—5844no—
289C2×Q8—21no—
290C2×Q8—2921no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—2922no—
294C2×D4—2921no—
295C2×D4—2921no—
296C2×D4—1462no—
297C22×C4—41no—
298C22×C4—2922no—
299C22×C4#1192401311yescharacteristic
300C22×C4#1192401311yescharacteristic
301C22×C4C2×C22⋊D7311yescharacteristic
302Q16—22no—
303SD16—42no—
304D8—22no—
305D8—2922no—
306D8—1462no—
307D8—1462no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8C2×C22⋊D7311yescharacteristic
311C2×C8C2×C22⋊D7311yescharacteristic
312C2×C8C2×C22⋊D7311yescharacteristic
313C2×C8C23×D7311yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—2921no—
321C4⋊C4—1462no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—2922no—
328C22⋊C4—2922no—
329C42—41no—
330C42#1192401311yescharacteristic
331C42#1192401311yescharacteristic
332C42C2×C22⋊D7311yescharacteristic
333C42C2×C22⋊D7311yescharacteristic
334C16—734no—
335C16—1462no—
336C23—41no—
337C23—2922no—
338C23C2×#1192401311yescharacteristic
339Q8—41no—
340Q8—41no—
341Q8—5842no—
342D4—41no—
343D4—41no—
344D4—82no—
345D4—2922no—
346D4—2922no—
347D4—5842no—
348D4—5842no—
349D4—5844no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—2922no—
359C2×C4—5844no—
360C2×C4C2×#1192401311yescharacteristic
361C2×C4C2×#1192401311yescharacteristic
362C2×C4C2×#1192401311yescharacteristic
363C2×C4C2×#1192401311yescharacteristic
364C2×C4C2×#1192401311yescharacteristic
365C2×C4C22×C22⋊D7311yescharacteristic
366C8—21no—
367C8—21no—
368C8#1455661112yes—
369C8#41×D7312yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—5842no—
377C22—5842no—
378C22—5842no—
379C22C22×#1192401311yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—5842no—
388C4#1798902212yes—
389C4#1798913311yescharacteristic
390C4C2×#1455661111yescharacteristic
391C2—41no—
392C2—161no—
393C2—11682no—
394C2#2262050611yescharacteristic
395C2C2×#1798902211yescharacteristic
396C2#2262306511yescharacteristic
subgroupG/H
centerC2×C4C2×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3168C2
socleC22×C73C22×D8
hypercenter#3168D73
hypocenterC73#33737
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #22620510C2semidirect
2. #22612218C2semidirect
3. #22612198C2semidirect
4. #22612174C2semidirect
5. #22612167C2semidirect
6. #22612046C2semidirect
7. #22612032C2semidirect
8. C4×#14556534C2semidirect
9. C4×#14556532C2semidirect
10. C4×#14556531C2semidirect
11. C73×#3168C2semidirect
104. #11924003#41semidirect
200. #3168D73semidirect
210. C73#33737semidirect

Maximal subgroups

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

subgroupindex
#226205102
#226122182
#226121982
#226121742
#226121672
#226120462
#226120322
C4×#145565342
C4×#145565322
C4×#145565312
C73×#31682
#3373773