Finite Group Databasebeta

← #33240645

Subgroups — #33240645

Order 18688.

534 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620444C211yesmaximal, characteristic
2C4×Q16×D73C211yesmaximal, characteristic
3#22612096C211yesmaximal, characteristic, complemented
4#22612095C211yesmaximal, characteristic, complemented
5#22612092C211yesmaximal, characteristic, complemented
6#22612082C211yesmaximal, characteristic, complemented
7#22612081C211yesmaximal, characteristic, complemented
8#22612072C211yesmaximal, characteristic, complemented
9#22612071C211yesmaximal, characteristic, complemented
10#22611718C211yesmaximal, characteristic
11#22611716C211yesmaximal, characteristic
12#22611685C211yesmaximal, characteristic
13#22611649C211yesmaximal, characteristic
14#22611649C211yesmaximal, characteristic
15C73×#2229C211yesmaximal, characteristic, complemented
16C2×Q16×D73C2211yescharacteristic
17#17989143—21no—
18C4×Q8×D73—21no—
19C2×#14556530C2211yescharacteristic
20C2×#14556530C2211yescharacteristic
21#17988991C2211yescharacteristic
22#17988991C2211yescharacteristic
23#17988989C2211yescharacteristic
24#17988986—21no—
25#17988984—21no—
26C4×Dic292C2211yescharacteristic
27Q16×Dic73C2211yescharacteristic
28#17988681C2211yescharacteristic
29C4×#11924018—21no—
30#17988679—21no—
31#17988624C2211yescharacteristic
32#17988624C2211yescharacteristic
33#17988622—21no—
34#107×D73C2211yescharacteristic
35#103×D73—21no—
36C4×C8×D73C2211yescharacteristic
37#17988415C2211yescharacteristic
38#17988414C2211yescharacteristic
39#17988414C2211yescharacteristic
40#17988413C2211yescharacteristic
41#17988412C2211yescharacteristic
42#17988412C2211yescharacteristic
43C4⋊#11924003C2211yescharacteristic
44#17988405C2211yescharacteristic
45#17988405C2211yescharacteristic
46#17988392C2211yescharacteristic
47#17988392C2211yescharacteristic
48#17988391C2211yescharacteristic
49#17988391C2211yescharacteristic
50#17988390C2211yescharacteristic
51#17988390C2211yescharacteristic
52#499×C73C2211yescharacteristic
53C4×Q16×C73C2211yescharacteristic
54#367×C73C2211yescharacteristic
55#366×C73C2211yescharacteristic
56C4⋊C16×C73C2211yescharacteristic
57#358×C73C2211yescharacteristic
58#358×C73C2211yescharacteristic
59C2×Q8×D73—21no—
60C2×Dic292C2311yescharacteristic
61C2×#11924018—21no—
62Q16×D73—22no—
63Q16×D73Q814yescomplemented
64C2×C8×D73D411yescharacteristic
65C2×C8×D73C2311yescharacteristic
66#14556590—21no—
67#14556590—21no—
68#14556583—21no—
69#14556569—41no—
70#14556569—41no—
71Q8×Dic73—21no—
72C4×Dic146—21no—
73C4×Dic146—41no—
74#14556556—21no—
75#14556556—41no—
76#14556556—41no—
77#14556556—41no—
78#14556549—21no—
79#14556549—21no—
80C4⋊C4×D73—21no—
81C4⋊C4×D73—41no—
82C42×D73—41no—
83C42×D73D411yescharacteristic
84#14556530—22no—
85#14556530—22no—
86C2×#11924003C2311yescharacteristic
87C2×#11924003C2311yescharacteristic
88#14556525C2311yescharacteristic
89#14556525C2311yescharacteristic
90#14556525C2311yescharacteristic
91#14556524—21no—
92#14556524—21no—
93#14556524—21no—
94#14556523—21no—
95#14556522—21no—
96#14556522—21no—
97#14556516—21no—
98#14556515—21no—
99#14556514—21no—
100C4×#10596831D411yescharacteristic
101C8×Dic73D411yescharacteristic
102C8×Dic73C2311yescharacteristic
103C2×Q16×C73C2311yescharacteristic
104#125×C73—21no—
105C4×Q8×C73—21no—
106C2×C16×C73C2311yescharacteristic
107C2×C16×C73C2311yescharacteristic
108#107×C73C2311yescharacteristic
109#107×C73C2311yescharacteristic
110#107×C73C2311yescharacteristic
111#106×C73—21no—
112#103×C73—21no—
113C4×C8×C73C2311yescharacteristic
114C2×Dic146—21no—
115C2×Dic146—41no—
116C2×Dic146—41no—
117C2×Dic146—41no—
118Q8×D73—24no—
119Q8×D73—42no—
120C2×C4×D73—41no—
121C2×C4×D73—41no—
122C2×C4×D73D811yescharacteristic
123C2×C4×D73D811yescharacteristic
124C2×C4×D73C2×D411yescharacteristic
125Dic292—21no—
126Dic292C2×Q812yes—
127#11924018—24no—
128C2×#10596831C2×D411yescharacteristic
129C2×#10596831C2×D411yescharacteristic
130C8×D73—22no—
131C8×D73C2×Q812yes—
132C8×D73#4112yes—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—41no—
137#11924007—41no—
138#11924007—41no—
139#11924007—41no—
140#11924007—41no—
141#11924007—41no—
142#11924007—41no—
143#11924007—41no—
144#11924005—41no—
145#11924005—41no—
146C4×Dic73—41no—
147C4×Dic73—41no—
148C4×Dic73—41no—
149C4×Dic73—41no—
150C4×Dic73—41no—
151C4×Dic73D811yescharacteristic
152C4×Dic73D811yescharacteristic
153C4×Dic73C2×D411yescharacteristic
154#11924003—21no—
155#11924003—21no—
156C2×Q8×C73—21no—
157Q16×C73—21no—
158Q16×C73C2×Q812yes—
159C2×C8×C73C2×D411yescharacteristic
160C2×C8×C73C2411yescharacteristic
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—41no—
165C4⋊C4×C73—41no—
166C4⋊C4×C73—41no—
167C42×C73—41no—
168C42×C73C2×D411yescharacteristic
169C16×C73—21no—
170C16×C73—21no—
171C22×D73C2×D811yescharacteristic
172Q8×C73—22no—
173Q8×C73—41no—
174C2×C4×C73—41no—
175C2×C4×C73—41no—
176C2×C4×C73—41no—
177C2×C4×C73—41no—
178C2×C4×C73C2×D811yescharacteristic
179C2×C4×C73C2×D811yescharacteristic
180C2×C4×C73C22×D411yescharacteristic
181C2×Dic73—41no—
182C2×Dic73—41no—
183C2×Dic73—41no—
184C2×Dic73—41no—
185C2×Dic73C2×D811yescharacteristic
186C2×Dic73C2×D811yescharacteristic
187C2×Dic73C2×D811yescharacteristic
188C4×D73—22no—
189C4×D73—41no—
190C4×D73—44no—
191C4×D73—82no—
192C4×D73C22⋊Q812yes—
193C4×D73#13712yes—
194Dic146—22no—
195Dic146—41no—
196Dic146—44no—
197Dic146—82no—
198Dic146—82no—
199C8×C73—21no—
200C8×C73C22×Q811yescharacteristic
201C8×C73C2×#4111yescharacteristic
202#10596831—21no—
203#10596831C22⋊Q812yes—
204C22×C73C22×D811yescharacteristic
205C2×D73—21no—
206C2×D73—21no—
207C2×D73#47412yes—
208C4×C73—21no—
209C4×C73—21no—
210C4×C73—42no—
211C4×C73—81no—
212C4×C73—81no—
213C4×C73—81no—
214C4×C73C2×C22⋊Q811yescharacteristic
215C4×C73C2×#13711yescharacteristic
216Dic73—21no—
217Dic73—42no—
218Dic73—81no—
219Dic73—81no—
220Dic73—81no—
221Dic73#47412yes—
222Dic73#51012yes—
223#33628—731nomaximal, Sylow
224C2×C73C2×#47411yescharacteristic
225C2×C73C2×#51011yescharacteristic
226C2×C73#341411yescharacteristic
227D73—22no—
228#3160—731no—
229C2×C4×Q16—731no—
230#2229—731no—
231#2229—731no—
232#2229—731no—
233#2229D7311yescharacteristic, complemented
234#2227—731no—
235#2227—731no—
236#2225—731no—
237#2225—731no—
238#2159—731no—
239#2159—731no—
240#2150—731no—
241#2141—731no—
242#2141—731no—
243C73#3362811yescharacteristic, complemented, Sylow
244C22×Q16—731no—
245#533—1461no—
246C2×C4×Q8—1461no—
247C2×M32—731no—
248C2×M32—731no—
249#500—731no—
250#500—731no—
251#499—731no—
252#499—1461no—
253#499C2×D7311yescharacteristic
254#498—1461no—
255C4×Q16—731no—
256C4×Q16—731no—
257C4×Q16—731no—
258C4×Q16—1461no—
259C4×Q16—1461no—
260C4×Q16C2×D7311yescharacteristic
261#427—731no—
262#427—731no—
263#427—1461no—
264C2×#107—731no—
265C2×#103—1461no—
266C2×C4×C8—731no—
267#367—731no—
268#367—731no—
269#367—731no—
270#367C2×D7311yescharacteristic
271#366—731no—
272#366—731no—
273#366—731no—
274#366C2×D7311yescharacteristic
275C4⋊C16—731no—
276C4⋊C16—731no—
277C4⋊C16—731no—
278C4⋊C16C2×D7311yescharacteristic
279#358—731no—
280#358—731no—
281#358—731no—
282#358—731no—
283#358—731no—
284#358—731no—
285#358C2×D7311yescharacteristic
286#358C2×D7311yescharacteristic
287C22×Q8—1461no—
288C2×Q16—734no—
289C2×Q16—731no—
290C2×Q16—1462no—
291C2×Q16—1461no—
292C2×Q16C22×D7311yescharacteristic
293C22×C8—731no—
294C22×C8—731no—
295C4⋊Q8—1461no—
296C4⋊Q8—1461no—
297#125—21no—
298#125—1461no—
299C22⋊Q8—2921no—
300C22⋊Q8—2921no—
301C4×Q8—21no—
302C4×Q8—2921no—
303C4×Q8—2921no—
304C4×Q8—2921no—
305C4×Q8—2921no—
306C4×Q8—1461no—
307C4×Q8—1461no—
308C4×Q8—1461no—
309#117—1461no—
310#117—1461no—
311C2×C4⋊C4—2921no—
312C2×C4⋊C4—1461no—
313C2×C42—2921no—
314C2×C42—731no—
315M32—1462no—
316M32—1462no—
317C2×C16—731no—
318C2×C16—731no—
319C2×C16C22×D7311yescharacteristic
320C2×C16C22×D7311yescharacteristic
321#107—731no—
322#107—731no—
323#107—731no—
324#107—1461no—
325#107—1461no—
326#107—1461no—
327#107C22×D7311yescharacteristic
328#107C22×D7311yescharacteristic
329#107C22×D7311yescharacteristic
330#106—21no—
331#106—1461no—
332#106—1461no—
333#106—1461no—
334#103—21no—
335#103—1461no—
336#103—1461no—
337#103—1461no—
338C4×C8—731no—
339C4×C8—731no—
340C4×C8—731no—
341C4×C8C22×D7311yescharacteristic
342C2×Q8—21no—
343C2×Q8—2921no—
344C2×Q8—2921no—
345C2×Q8—2921no—
346C2×Q8—2922no—
347C2×Q8—1464no—
348C2×Q8—1461no—
349C22×C4—2921no—
350C22×C4—2921no—
351C22×C4—731no—
352C22×C4—731no—
353C22×C4—731no—
354Q16—21no—
355Q16—732no—
356Q16—1464no—
357Q16—1461no—
358Q16Q8×D7312yes—
359C2×C8—731no—
360C2×C8—732no—
361C2×C8—732no—
362C2×C8—731no—
363C2×C8—1462no—
364C2×C8D4×D7311yescharacteristic
365C2×C8C23×D7311yescharacteristic
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—2921no—
373C4⋊C4—2921no—
374C4⋊C4—2921no—
375C4⋊C4—2921no—
376C4⋊C4—2921no—
377C4⋊C4—2921no—
378C4⋊C4—2921no—
379C4⋊C4—2921no—
380C4⋊C4—1461no—
381C4⋊C4—1461no—
382C4⋊C4—1461no—
383C22⋊C4—2921no—
384C22⋊C4—2921no—
385C42—41no—
386C42—2921no—
387C42—2921no—
388C42—2921no—
389C42—2921no—
390C42—2921no—
391C42—731no—
392C42—731no—
393C42—731no—
394C42D4×D7311yescharacteristic
395C16—21no—
396C16—21no—
397C16—1461no—
398C16—1461no—
399C23—731no—
400Q8—22no—
401Q8—41no—
402Q8—2924no—
403Q8—2921no—
404Q8—5842no—
405Q8—5842no—
406Q8—1462no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—2921no—
412C2×C4—2924no—
413C2×C4—2921no—
414C2×C4—2921no—
415C2×C4—2921no—
416C2×C4—2921no—
417C2×C4—5842no—
418C2×C4—731no—
419C2×C4—731no—
420C2×C4—732no—
421C2×C4—732no—
422C2×C4—731no—
423C2×C4—1462no—
424C2×C4D8×D7311yescharacteristic
425C2×C4D8×D7311yescharacteristic
426C2×C4C2×D4×D7311yescharacteristic
427C8—21no—
428C8—732no—
429C8—1461no—
430C8C2×Q8×D7311yescharacteristic
431C8#41×D7311yescharacteristic
432C22—732no—
433C22—1461no—
434C22—1461no—
435C22C2×D8×D7311yescharacteristic
436C4—21no—
437C4—21no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—2922no—
443C4—5841no—
444C4—5841no—
445C4—5841no—
446C4—732no—
447C4—732no—
448C4—1461no—
449C4C22⋊Q8×D7311yescharacteristic
450C4#137×D7311yescharacteristic
451C2—1462no—
452C2#474×D7311yescharacteristic
453C2#510×D7311yescharacteristic
454C2#2262306411yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2229C2
socleC22×C73C22×D8
prime radicalC22×D73C2×D8
hypercenter#2229D73
hypocenterC73#33628
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2229

Complements

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

NH ≅ G/Ntype
3. #22612096C2semidirect
4. #22612095C2semidirect
5. #22612092C2semidirect
6. #22612082C2semidirect
7. #22612081C2semidirect
8. #22612072C2semidirect
9. #22612071C2semidirect
15. C73×#2229C2semidirect
63. Q16×D73Q8semidirect
233. #2229D73semidirect
243. C73#33628semidirect

Maximal subgroups

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

subgroupindex
#226204442
C4×Q16×D732
#226120962
#226120952
#226120922
#226120822
#226120812
#226120722
#226120712
#226117182
#226117162
#226116852
#226116492
C73×#22292
#3362873