Finite Group Databasebeta

← #33092891

Subgroups — #33092891

Order 18688.

874 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988859C211yesmaximal, characteristic
2C4×#14556622C212yesmaximal
3#22618356C211yesmaximal, characteristic
4#22610847C211yesmaximal, characteristic, complemented
5#22610709C211yesmaximal, characteristic, complemented
6#22610583C212yesmaximal, complemented
7#22610449C212yesmaximal
8#22609991C211yesmaximal, characteristic
9#22609837C211yesmaximal, characteristic, complemented
10C4×#14556510C212yesmaximal, complemented
11C2×#441×C73C211yesmaximal, characteristic, complemented
12C2×#14556622C2212yes—
13C2×#14556597C2211yescharacteristic
14C2×#14556597C2211yescharacteristic
15C4×#11924028C2211yescharacteristic
16#17988859C418yescomplemented
17#17988692—22no—
18#17988685—22no—
19C4×#11924018—22no—
20C4×#11924015—22no—
21C2×#14556527C2211yescharacteristic
22C2×#14556527C2211yescharacteristic
23#17988627C2212yes—
24#17988596C2212yes—
25#17988586C2212yes—
26C2×#14556510C2212yes—
27C2×#14556510C2212yes—
28#17988544—21no—
29#17988544—21no—
30#17988542C2211yescharacteristic
31#17988542C2211yescharacteristic
32C4×#11924012C2211yescharacteristic
33C4×#11924012C2212yes—
34#17988539—21no—
35#97×D73—21no—
36#17988356C2212yes—
37#17988356C2212yes—
38#17988352C2212yes—
39C22×Q16×C73C2211yescharacteristic
40C2×C4×Q8×C73C2212yes—
41#441×C73—21no—
42#441×C73—21no—
43#441×C73—21no—
44#441×C73—21no—
45C2×#106×C73C2211yescharacteristic
46C2×#103×C73C2212yes—
47C2×#97×C73C2211yescharacteristic
48C2×#11924028C2311yescharacteristic
49#14556622—24no—
50#14556622C2×C418yes—
51C2×#11924018—22no—
52C2×#11924015—22no—
53#14556597—22no—
54#14556597—22no—
55#14556597—22no—
56#14556597C2×C414yes—
57C2×#11924012C2312yes—
58C2×#11924012C2312yes—
59C2×#11924012C2311yescharacteristic
60C2×#11924012C2311yescharacteristic
61C2×#11924011—21no—
62C2×#11924011—21no—
63C2×C8×D73—21no—
64C2×C8×D73—21no—
65C4×Dic146D412yes—
66C4×D292D412yes—
67C4×C22⋊D73—22no—
68#14556550C2311yescharacteristic
69#14556548—21no—
70C42×D73—21no—
71#14556527—22no—
72#14556527C2×C414yescomplemented
73#14556524—22no—
74#14556522—22no—
75#14556515—22no—
76#14556515—22no—
77#14556514—22no—
78#14556511—22no—
79#14556510—24no—
80#14556510C2×C418yescomplemented
81#14556502—22no—
82#14556502—21no—
83#14556501—21no—
84#14556501—21no—
85C4×#10596831—22no—
86C4×#10596831—21no—
87C8×Dic73—21no—
88C8×Dic73—21no—
89C22×Q8×C73C2312yes—
90C2×Q16×C73—21no—
91C2×Q16×C73—21no—
92C2×Q16×C73—22no—
93C2×Q16×C73C2×C414yescomplemented
94C22×C8×C73C2311yescharacteristic
95C22×C8×C73C2311yescharacteristic
96C4×Q8×C73—22no—
97C4×Q8×C73—22no—
98C4×Q8×C73—22no—
99C4×Q8×C73—22no—
100C2×C4⋊C4×C73—22no—
101C2×C4⋊C4×C73C2312yes—
102C2×C42×C73—22no—
103C2×C42×C73C2311yescharacteristic
104#106×C73—21no—
105#106×C73—21no—
106#103×C73—22no—
107#103×C73—22no—
108#97×C73D411yescharacteristic
109#97×C73D411yescharacteristic
110#97×C73D411yescharacteristic
111#97×C73D411yescharacteristic
112#11924028—21no—
113#11924028—21no—
114#11924028C22×C412yes—
115#11924028#4112yes—
116C2×Dic146C2×D411yescharacteristic
117C2×D292C2×D411yescharacteristic
118C2×C22⋊D73—21no—
119C2×C4×D73—21no—
120C2×C4×D73—22no—
121#11924018—28no—
122#11924015—28no—
123#11924012—21no—
124#11924012—21no—
125#11924012—22no—
126#11924012—22no—
127#11924012—22no—
128#11924012C22×C412yes—
129#11924012C22×C414yes—
130#11924012#4112yes—
131#11924011—24no—
132#11924011—24no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C8×D73—24no—
138C8×D73—24no—
139C4⋊Dic73C2×D412yes—
140#11924007—22no—
141C22⋊Dic73—21no—
142#11924005—22no—
143C4×Dic73—22no—
144C4×Dic73—21no—
145C2×Q8×C73—22no—
146C2×Q8×C73—22no—
147C2×Q8×C73—22no—
148C2×Q8×C73—22no—
149C2×Q8×C73—22no—
150C2×Q8×C73C22×C414yes—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73#4112yes—
154C22×C4×C73C2411yescharacteristic
155Q16×C73—24no—
156Q16×C73—24no—
157C2×C8×C73—21no—
158C2×C8×C73—21no—
159C2×C8×C73C22×C412yes—
160C2×C8×C73C2×D411yescharacteristic
161C2×C8×C73C2×D411yescharacteristic
162C2×C8×C73C2×D411yescharacteristic
163C2×C8×C73C2×D411yescharacteristic
164C2×C8×C73#4112yes—
165C4⋊C4×C73—22no—
166C4⋊C4×C73—22no—
167C4⋊C4×C73—42no—
168C4⋊C4×C73—42no—
169C42×C73—42no—
170C42×C73—42no—
171C42×C73C2×D412yes—
172C42×C73C2×D411yescharacteristic
173C42×C73C2×D411yescharacteristic
174C23×C73C2×#4111yescharacteristic
175C22×D73—21no—
176Q8×C73—24no—
177Q8×C73—24no—
178Q8×C73—42no—
179Q8×C73—42no—
180C2×C4×C73—24no—
181C2×C4×C73—42no—
182C2×C4×C73—42no—
183C2×C4×C73—42no—
184C2×C4×C73—42no—
185C2×C4×C73—82no—
186C2×C4×C73#12112yes—
187C2×C4×C73#12112yes—
188C2×C4×C73C23×C411yescharacteristic
189C2×C4×C73C22×D411yescharacteristic
190C2×C4×C73C22×D411yescharacteristic
191C2×C4×C73C2×#4111yescharacteristic
192C2×C4×C73C2×#4111yescharacteristic
193C2×C4×C73C2×#4111yescharacteristic
194C22⋊D73—22no—
195C22⋊D73—22no—
196C2×Dic73—21no—
197C2×Dic73—22no—
198D292—21no—
199D292C4×D412yes—
200C4×D73—22no—
201C4×D73—22no—
202C4×D73—82no—
203Dic146—21no—
204Dic146C4×D412yes—
205C8×C73—21no—
206C8×C73—21no—
207C8×C73C4×D412yes—
208C8×C73C4×D412yes—
209#10596831—22no—
210#10596831—24no—
211#10596831—22no—
212#10596831—42no—
213C22×C73—22no—
214C22×C73C4×#4111yescharacteristic
215C22×C73C2×#12111yescharacteristic
216C22×C73#52911yescharacteristic
217C2×D73—22no—
218C2×D73—42no—
219C4×C73—22no—
220C4×C73—22no—
221C4×C73—44no—
222C4×C73—82no—
223C4×C73C2×C4×D411yescharacteristic
224C4×C73C2×C4×D411yescharacteristic
225C4×C73#52511yescharacteristic
226C4×C73#53211yescharacteristic
227Dic73—22no—
228Dic73—42no—
229#21556—731nomaximal, Sylow
230C2×C73—21no—
231C2×C73—21no—
232C2×C73C4×#12111yescharacteristic
233C2×C73#306912yes—
234D73—42no—
235C2×#473—731no—
236C4×#137—732no—
237C2×#441D7311yescharacteristic, complemented
238#2875—731no—
239#1979—731no—
240#1951—731no—
241#1930—732no—
242#1908—732no—
243#1815—731no—
244#1778—731no—
245C4×#101—732no—
246C73#2155611yescharacteristic, complemented, Sylow
247C2×#137—732no—
248C22×Q16C2×D7311yescharacteristic
249C2×#131—731no—
250C2×#131—731no—
251C4×#41—731no—
252C2×C4×Q8C2×D7312yes—
253#473—738no—
254#441—21no—
255#441—21no—
256#441—21no—
257#441—21no—
258#441—1462no—
259#440—1462no—
260C4×Q16—1462no—
261C4×SD16—1462no—
262C2×#108—731no—
263C2×#108—731no—
264#428—732no—
265C2×#106C2×D7311yescharacteristic
266#419—732no—
267#417—732no—
268C2×#103C2×D7312yes—
269C2×#101—732no—
270C2×#101—732no—
271#405—731no—
272#405—731no—
273#405—1461no—
274#405—1461no—
275C4×M16—731no—
276C4×M16—732no—
277C4×M16—1461no—
278C2×#97—1461no—
279C2×#97C2×D7311yescharacteristic
280#343—732no—
281#343—732no—
282#341—732no—
283C2×#41—731no—
284C22×Q8C22×D7312yes—
285#137—738no—
286#137—1464no—
287C2×Q16—21no—
288C2×Q16—21no—
289C2×Q16—22no—
290C2×Q16—1462no—
291C2×Q16C4×D7314yescomplemented
292C2×SD16—1462no—
293#131—734no—
294#131—1462no—
295#131—1462no—
296#131—1462no—
297C2×M16—731no—
298C2×M16—731no—
299C2×M16—732no—
300C2×M16—732no—
301C2×M16—1461no—
302C2×M16—1461no—
303C22×C8—1461no—
304C22×C8—1461no—
305C22×C8C22×D7311yescharacteristic
306C22×C8C22×D7311yescharacteristic
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—732no—
312C4×D4—732no—
313C4×D4—1462no—
314#117—731no—
315#117—1461no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4C22×D7312yes—
318C2×C42—22no—
319C2×C42—1461no—
320C2×C42C22×D7311yescharacteristic
321#108—734no—
322#108—1462no—
323#107—1462no—
324#106—21no—
325#106—21no—
326#106—1462no—
327#103—22no—
328#103—22no—
329#103—1462no—
330#103—1462no—
331#103—1462no—
332#102—1462no—
333#101—738no—
334#101—1464no—
335#97—1461no—
336#97—1461no—
337#97—1462no—
338#97—1461no—
339#97C22⋊D7311yescharacteristic
340#97C22⋊D7311yescharacteristic
341#97C22⋊D7311yescharacteristic
342#97C22⋊D7311yescharacteristic
343C4×C8—1461no—
344C4×C8—1461no—
345C4×C8—1462no—
346C4×C8—1461no—
347#41—732no—
348#41—732no—
349#41—1461no—
350#41—1461no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—731no—
357C2×Q8C2×C4×D7314yes—
358C2×D4—731no—
359C2×D4—1461no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—1462no—
363C22×C4—1461no—
364C22×C4#1192402712yes—
365C22×C4C23×D7311yescharacteristic
366Q16—24no—
367Q16—24no—
368Q16—1468no—
369SD16—1468no—
370M16—732no—
371M16—734no—
372M16—732no—
373M16—1464no—
374M16—1462no—
375M16—1462no—
376M16—1461no—
377M16—1462no—
378M16—1461no—
379M16—1464no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—1464no—
383C2×C8—1462no—
384C2×C8—1462no—
385C2×C8—1461no—
386C2×C8—1461no—
387C2×C8—1464no—
388C2×C8C2×C4×D7312yes—
389C2×C8C2×C22⋊D7311yescharacteristic
390C2×C8C2×C22⋊D7311yescharacteristic
391C2×C8C2×C22⋊D7311yescharacteristic
392C2×C8C2×C22⋊D7311yescharacteristic
393C2×C8#1192402812yes—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—732no—
399C4⋊C4—1462no—
400C22⋊C4—1461no—
401C22⋊C4—1462no—
402C42—42no—
403C42—42no—
404C42—1462no—
405C42—1461no—
406C42D4×D7312yes—
407C42C2×C22⋊D7311yescharacteristic
408C42C2×C22⋊D7311yescharacteristic
409C23—1461no—
410C23C2×#1192402711yescharacteristic
411Q8—24no—
412Q8—24no—
413Q8—42no—
414Q8—42no—
415Q8—732no—
416Q8—1461no—
417D4—732no—
418D4—1462no—
419D4—1462no—
420D4—1461no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—5842no—
428C2×C4—1462no—
429C2×C4—1461no—
430C2×C4—1462no—
431C2×C4—1462no—
432C2×C4#1455656212yes—
433C2×C4#1455656212yes—
434C2×C4C22×C4×D7311yescharacteristic
435C2×C4C2×D4×D7311yescharacteristic
436C2×C4C22×C22⋊D7311yescharacteristic
437C2×C4#41×D7311yescharacteristic
438C2×C4#41×D7311yescharacteristic
439C2×C4C2×#1192402811yescharacteristic
440C8—21no—
441C8—21no—
442C8—2922no—
443C8—1464no—
444C8—1462no—
445C8—1462no—
446C8C4×C22⋊D7312yes—
447C8C4×C22⋊D7312yes—
448C22—22no—
449C22—2922no—
450C22—1462no—
451C22C4×#1192402711yescharacteristic
452C22C2×#1455656211yescharacteristic
453C22#1798911911yescharacteristic
454C4—22no—
455C4—22no—
456C4—44no—
457C4—82no—
458C4—2922no—
459C4—1462no—
460C4C4×D4×D7311yescharacteristic
461C4C2×C4×C22⋊D7311yescharacteristic
462C4#1798909411yescharacteristic
463C4#1798913411yescharacteristic
464C2—21no—
465C2—21no—
466C2—2922no—
467C2C4×#1455656211yescharacteristic
468C2#2261966412yes—
subgroupG/H
centerC22C2×#14556562
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#441×C73C2
socleC22×C73C2×#121
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenterC2×#441D73
hypocenterC73#21556
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
4. #22610847C2semidirect
5. #22610709C2semidirect
6. #22610583C2semidirect
9. #22609837C2semidirect
10. C4×#14556510C2semidirect
11. C2×#441×C73C2semidirect
16. #17988859C4semidirect
72. #14556527C2×C4semidirect
80. #14556510C2×C4semidirect
93. C2×Q16×C73C2×C4semidirect
237. C2×#441D73semidirect
246. C73#21556semidirect
291. C2×Q16C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888592
C4×#145566222
#226183562
#226108472
#226107092
#226105832
#226104492
#226099912
#226098372
C4×#145565102
C2×#441×C732
#2155673