Finite Group Databasebeta

← #33241562

Subgroups — #33241562

Order 18688.

510 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620501C211yesmaximal, characteristic
2#22612214C211yesmaximal, characteristic, complemented
3#22612200C211yesmaximal, characteristic, complemented
4#22612176C211yesmaximal, characteristic, complemented
5#22612170C211yesmaximal, characteristic, complemented
6#22612046C212yesmaximal
7#22612037C212yesmaximal, complemented
8C4×#14556537C212yesmaximal, complemented
9C4×#14556534C212yesmaximal, complemented
10C4×#14556531C211yesmaximal, characteristic, complemented
11C73×#3170C211yesmaximal, characteristic, complemented
12#17989121—21no—
13C2×#14556537C2212yes—
14C2×#14556534C2212yes—
15C2×#14556531C2212yes—
16#17988989C2211yescharacteristic
17#17988971C2211yescharacteristic
18#17988942—21no—
19#17988828C2211yescharacteristic
20#17988823C2211yescharacteristic
21#17988816—21no—
22C4×Dic292C2211yescharacteristic
23C4×#11924017—21no—
24C4×D584C2211yescharacteristic
25#17988414C2212yes—
26#17988412C2212yes—
27#17988391C2212yes—
28#17988391C2212yes—
29#17988390C2212yes—
30#17988387C2212yes—
31#17988363C2211yescharacteristic
32#17988363C2211yescharacteristic
33C4×#11924003C2212yes—
34#533×C73—21no—
35#500×C73C2211yescharacteristic
36#494×C73C2211yescharacteristic
37#487×C73—21no—
38#467×C73—21no—
39#467×C73C2212yes—
40C4×Q16×C73—21no—
41C4×Q16×C73C2212yes—
42C2×C4×C8×C73C2211yescharacteristic
43C2×Dic292C2311yescharacteristic
44C2×#11924017—21no—
45C2×D584C2311yescharacteristic
46#14556585—41no—
47#14556583—21no—
48#14556579—21no—
49#14556574—41no—
50#14556570—21no—
51#14556565—21no—
52C4×Dic146—21no—
53C4×D292—21no—
54C4×C22⋊D73—41no—
55#14556548—41no—
56#14556537—24no—
57#14556534—24no—
58#14556531—24no—
59C2×#11924003C2312yes—
60C2×#11924003C2312yes—
61#14556525C2311yescharacteristic
62#14556525C2311yescharacteristic
63#14556523—21no—
64#14556516—21no—
65#14556516—21no—
66#14556513—21no—
67#14556513—21no—
68C2×Q16×C73—21no—
69C2×Q16×C73C2312yes—
70C22×C8×C73D411yescharacteristic
71C22×C8×C73C2311yescharacteristic
72C4⋊Q8×C73—21no—
73C4⋊Q8×C73—21no—
74#125×C73—41no—
75C22⋊Q8×C73—22no—
76C22⋊Q8×C73—41no—
77C4×Q8×C73—22no—
78C4×Q8×C73—41no—
79#117×C73—41no—
80C2×C42×C73D411yescharacteristic
81#107×C73—21no—
82#107×C73C2312yes—
83#103×C73—22no—
84#103×C73—22no—
85C4×C8×C73D411yescharacteristic
86C4×C8×C73D411yescharacteristic
87C4×C8×C73C2311yescharacteristic
88C4×C8×C73C2311yescharacteristic
89C2×Dic146—21no—
90C2×D292—21no—
91C2×C22⋊D73—41no—
92C2×C4×D73—41no—
93Dic292—22no—
94#11924017—42no—
95D584—22no—
96C4⋊Dic73—21no—
97C4⋊Dic73—21no—
98#11924007—41no—
99#11924007—41no—
100#11924007—41no—
101#11924007—41no—
102C22⋊Dic73—41no—
103#11924005—41no—
104#11924005—41no—
105#11924005—41no—
106#11924005—41no—
107C4×Dic73—41no—
108#11924003—22no—
109#11924003#4114yes—
110C2×Q8×C73—22no—
111C2×Q8×C73—41no—
112C22×C4×C73D811yescharacteristic
113C22×C4×C73D811yescharacteristic
114C22×C4×C73C2×D411yescharacteristic
115Q16×C73—22no—
116Q16×C73—22no—
117Q16×C73—42no—
118C2×C8×C73—22no—
119C2×C8×C73—22no—
120C2×C8×C73C2×D411yescharacteristic
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73C2411yescharacteristic
124C4⋊C4×C73—22no—
125C4⋊C4×C73—41no—
126C4⋊C4×C73—41no—
127C4⋊C4×C73—42no—
128C4⋊C4×C73—42no—
129C4⋊C4×C73—41no—
130C22⋊C4×C73—42no—
131C42×C73—42no—
132C42×C73D811yescharacteristic
133C42×C73D811yescharacteristic
134C42×C73C2×D411yescharacteristic
135C42×C73C2×D411yescharacteristic
136C23×C73C2×D811yescharacteristic
137C22×D73—41no—
138Q8×C73—42no—
139Q8×C73—42no—
140Q8×C73—82no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—22no—
144C2×C4×C73—41no—
145C2×C4×C73—42no—
146C2×C4×C73—42no—
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73C2×D811yescharacteristic
149C2×C4×C73C2×D811yescharacteristic
150C2×C4×C73C2×D811yescharacteristic
151C2×C4×C73C2×D811yescharacteristic
152C2×C4×C73C22×D411yescharacteristic
153C22⋊D73—82no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157D292—41no—
158D292—41no—
159C4×D73—82no—
160Dic146—41no—
161Dic146—41no—
162C8×C73—21no—
163C8×C73—21no—
164C8×C73#13512yes—
165C8×C73C2×#4112yes—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73C22×D811yescharacteristic
170C2×D73—81no—
171C2×D73—81no—
172C2×D73—81no—
173C4×C73—21no—
174C4×C73—21no—
175C4×C73—21no—
176C4×C73—21no—
177C4×C73—82no—
178C4×C73—82no—
179C4×C73#51012yes—
180C4×C73#53211yescharacteristic
181C4×C73C2×#13511yescharacteristic
182Dic73—81no—
183Dic73—81no—
184Dic73—81no—
185#33738—731nomaximal, Sylow
186C2×C73—41no—
187C2×C73#316911yescharacteristic
188C2×C73C2×#51011yescharacteristic
189C2×C73#341411yescharacteristic
190D73—161no—
191#3170D7311yescharacteristic, complemented
192#3168—731no—
193#2255—731no—
194#2251—731no—
195#2245—731no—
196#2243—731no—
197#2220—732no—
198#2219—732no—
199C4×Q32—732no—
200C4×SD32—732no—
201C4×M32—731no—
202C73#3373811yescharacteristic, complemented, Sylow
203#533—21no—
204#529—1461no—
205C2×Q32—732no—
206C2×SD32—732no—
207C2×M32—732no—
208#500C2×D7311yescharacteristic
209#499—731no—
210#495—731no—
211#494C2×D7311yescharacteristic
212#488—1461no—
213#487—21no—
214#467—21no—
215#467—731no—
216#467C2×D7312yes—
217#466—731no—
218#465—1461no—
219C4×Q16—21no—
220C4×Q16—731no—
221C4×Q16C2×D7312yes—
222C4×SD16—1461no—
223C4×D8—731no—
224C2×C4×C8C2×D7311yescharacteristic
225#367—732no—
226#366—732no—
227#358—732no—
228#358—732no—
229#358—732no—
230#357—732no—
231#346—731no—
232#346—731no—
233C4×C16—732no—
234C2×Q16—21no—
235C2×Q16—731no—
236C2×Q16C22×D7312yes—
237C2×SD16—1461no—
238C2×D8—731no—
239C22×C8C22⋊D7311yescharacteristic
240C22×C8C22×D7311yescharacteristic
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243#126—2921no—
244#125—41no—
245#125—1461no—
246#124—1461no—
247#123—2921no—
248C22⋊Q8—22no—
249C22⋊Q8—41no—
250C22⋊Q8—1461no—
251#121—1461no—
252C4×Q8—22no—
253C4×Q8—41no—
254C4×Q8—1461no—
255C4×D4—2921no—
256C4×D4—1461no—
257#117—41no—
258#117—2921no—
259C2×C42C22⋊D7311yescharacteristic
260Q32—1464no—
261SD32—1464no—
262M32—1464no—
263C2×C16—732no—
264C2×C16—732no—
265#107—21no—
266#107—731no—
267#107—731no—
268#107C22×D7312yes—
269#106—1461no—
270#103—22no—
271#103—22no—
272#103—1461no—
273#103—1461no—
274#102—1461no—
275#102—1461no—
276C4×C8C22⋊D7311yescharacteristic
277C4×C8C22⋊D7311yescharacteristic
278C4×C8C22×D7311yescharacteristic
279C4×C8C22×D7311yescharacteristic
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—1461no—
283C2×D4—2921no—
284C2×D4—1461no—
285C22×C4—2921no—
286C22×C4#1192401311yescharacteristic
287C22×C4#1192401311yescharacteristic
288C22×C4C2×C22⋊D7311yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—42no—
292Q16—1462no—
293SD16—2922no—
294D8—1462no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×C22⋊D7311yescharacteristic
298C2×C8C2×C22⋊D7311yescharacteristic
299C2×C8C2×C22⋊D7311yescharacteristic
300C2×C8C23×D7311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—2921no—
308C4⋊C4—2921no—
309C4⋊C4—2921no—
310C4⋊C4—2921no—
311C4⋊C4—1461no—
312C4⋊C4—1461no—
313C22⋊C4—42no—
314C22⋊C4—2921no—
315C22⋊C4—2921no—
316C22⋊C4—2921no—
317C22⋊C4—2921no—
318C22⋊C4—2921no—
319C42—42no—
320C42—2921no—
321C42#1192401311yescharacteristic
322C42#1192401311yescharacteristic
323C42C2×C22⋊D7311yescharacteristic
324C42C2×C22⋊D7311yescharacteristic
325C16—734no—
326C16—1462no—
327C23—2921no—
328C23C2×#1192401311yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—82no—
332Q8—2921no—
333Q8—2921no—
334D4—2921no—
335D4—2921no—
336D4—5842no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—2921no—
344C2×C4—2921no—
345C2×C4—2921no—
346C2×C4—5842no—
347C2×C4C2×#1192401311yescharacteristic
348C2×C4C2×#1192401311yescharacteristic
349C2×C4C2×#1192401311yescharacteristic
350C2×C4C2×#1192401311yescharacteristic
351C2×C4C2×#1192401311yescharacteristic
352C2×C4C22×C22⋊D7311yescharacteristic
353C8—21no—
354C8—21no—
355C8#1455661112yes—
356C8#41×D7312yes—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—5841no—
361C22—5841no—
362C22—5841no—
363C22C22×#1192401311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—82no—
370C4—5841no—
371C4—5841no—
372C4—5841no—
373C4#1798902812yes—
374C4#1798913311yescharacteristic
375C4C2×#1455661111yescharacteristic
376C2—41no—
377C2—11681no—
378C2#2262050611yescharacteristic
379C2C2×#1798902811yescharacteristic
380C2#2262306511yescharacteristic
subgroupG/H
centerC2×C4C2×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3170C2
socleC22×C73C22×D8
prime radical#17988823C22
hypercenter#3170D73
hypocenterC73#33738
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3170

Complements

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

NH ≅ G/Ntype
2. #22612214C2semidirect
3. #22612200C2semidirect
4. #22612176C2semidirect
5. #22612170C2semidirect
7. #22612037C2semidirect
8. C4×#14556537C2semidirect
9. C4×#14556534C2semidirect
10. C4×#14556531C2semidirect
11. C73×#3170C2semidirect
191. #3170D73semidirect
202. C73#33738semidirect

Maximal subgroups

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

subgroupindex
#226205012
#226122142
#226122002
#226121762
#226121702
#226120462
#226120372
C4×#145565372
C4×#145565342
C4×#145565312
C73×#31702
#3373873