Finite Group Databasebeta

← #33055776

Subgroups — #33055776

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618892C211yesmaximal, characteristic
2C2×#17988718C211yesmaximal, characteristic, complemented
3C4×#14556622C211yesmaximal, characteristic
4#22608837C212yesmaximal, complemented
5#22608836C212yesmaximal, complemented
6#22608825C212yesmaximal, complemented
7#22608818C212yesmaximal, complemented
8#22608684C211yesmaximal, characteristic, complemented
9#22608319C211yesmaximal, characteristic
10#22608212C211yesmaximal, characteristic, complemented
11C73×#1479C211yesmaximal, characteristic, complemented
12C2×#14556622C411yescharacteristic
13C4×#11924028C2211yescharacteristic
14#17988718—21no—
15#17988718—21no—
16#17988718C2212yes—
17#17988718C2212yes—
18#17988714C2211yescharacteristic
19#17988714C2211yescharacteristic
20#17988692C412yes—
21#17988685C412yes—
22C4×#11924018C2212yes—
23C4×#11924015C2212yes—
24#17988664C2211yescharacteristic
25#17988664C2211yescharacteristic
26#17988655—21no—
27#17988655—21no—
28#17988638—21no—
29#17988627C411yescharacteristic
30C2×C4⋊#10596831—21no—
31C2×C4⋊#10596831C2211yescharacteristic
32#17988596C411yescharacteristic
33#17988586C411yescharacteristic
34C4×#11924012C2211yescharacteristic
35C4×#11924011—21no—
36C2×#14556502—21no—
37C2×C4×#10596831C2211yescharacteristic
38#17988339C2212yes—
39#17988337C2212yes—
40#17988319C2212yes—
41#17988316C2212yes—
42#17988309C2212yes—
43C8×#10596831C2212yes—
44C2×C4×Q8×C73C2211yescharacteristic
45#431×C73C2211yescharacteristic
46#431×C73C2211yescharacteristic
47Q8⋊C8×C73C2212yes—
48Q8⋊C8×C73C2212yes—
49C2×#11924028C2×C411yescharacteristic
50#14556622—24no—
51C2×#11924018C2×C412yes—
52C2×#11924015C2×C412yes—
53C2×#11924012C2×C411yescharacteristic
54C2×#11924012C2×C411yescharacteristic
55C2×#11924011—21no—
56C2×#11924011—21no—
57C22×#10596831—21no—
58C22×#10596831D411yescharacteristic
59C22×#10596831D411yescharacteristic
60C4×Dic146C2×C411yescharacteristic
61C4×Dic146C2311yescharacteristic
62C4×D292C2×C411yescharacteristic
63C4×D292C2311yescharacteristic
64C4×C22⋊D73—21no—
65C4×C22⋊D73—21no—
66#14556550C2×C411yescharacteristic
67#14556548—21no—
68C42×D73—21no—
69#14556524C2×C412yes—
70#14556522C2×C412yes—
71C4⋊#10596831—22no—
72C4⋊#10596831—22no—
73C4⋊#10596831—21no—
74C4⋊#10596831C2312yes—
75#14556520—21no—
76#14556520—21no—
77#14556515C2×C412yes—
78#14556515C2×C412yes—
79#14556514C2×C412yes—
80#14556511C2×C412yes—
81#14556503—21no—
82#14556503—21no—
83#14556502—22no—
84#14556502—22no—
85#14556502C2×C412yes—
86#14556501—21no—
87#14556501—21no—
88C4×#10596831D411yescharacteristic
89C4×#10596831D411yescharacteristic
90C4×#10596831C2312yes—
91C4×#10596831C2311yescharacteristic
92C4×#10596831C2311yescharacteristic
93C22×Q8×C73C2×C411yescharacteristic
94C4×Q8×C73—21no—
95C4×Q8×C73—21no—
96C4×Q8×C73C2×C412yes—
97C4×Q8×C73C2312yes—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73C2×C411yescharacteristic
100C2×C42×C73—21no—
101C2×C42×C73C2311yescharacteristic
102C4⋊C8×C73—21no—
103C4⋊C8×C73C2311yescharacteristic
104C4⋊C8×C73C2311yescharacteristic
105C22⋊C8×C73—21no—
106C22⋊C8×C73—21no—
107C4×C8×C73—21no—
108C4×C8×C73C2312yes—
109#11924028—22no—
110#11924028—22no—
111C2×Dic146C22×C411yescharacteristic
112C2×D292C22×C411yescharacteristic
113C2×C22⋊D73—21no—
114C2×C4×D73—21no—
115C2×C4×D73—21no—
116C2×C4×D73—21no—
117#11924018—24no—
118#11924015—24no—
119#11924012—22no—
120#11924012M1614yescomplemented
121#11924011—42no—
122#11924011—42no—
123C2×#10596831—22no—
124C2×#10596831—22no—
125C2×#10596831—41no—
126C2×#10596831—42no—
127C2×#10596831C22×C412yes—
128C2×#10596831C22×C412yes—
129C2×#10596831C2×D411yescharacteristic
130C2×#10596831C2×D411yescharacteristic
131C2×#10596831C2×D411yescharacteristic
132C2×#10596831C2×D411yescharacteristic
133C4⋊Dic73C22×C411yescharacteristic
134C4⋊Dic73C22×C411yescharacteristic
135#11924007—21no—
136#11924007—21no—
137C22⋊Dic73—21no—
138#11924005—21no—
139#11924005—21no—
140C4×Dic73—21no—
141C4×Dic73—21no—
142C4×Dic73—21no—
143C2×Q8×C73—21no—
144C2×Q8×C73—22no—
145C2×Q8×C73—22no—
146C2×Q8×C73C22×C412yes—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73C22×C411yescharacteristic
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73#4111yescharacteristic
152C2×C8×C73—21no—
153C2×C8×C73—21no—
154C2×C8×C73—22no—
155C4⋊C4×C73—22no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—22no—
158C4⋊C4×C73C22×C412yes—
159C42×C73—22no—
160C42×C73—22no—
161C42×C73C22×C411yescharacteristic
162C42×C73C2×D411yescharacteristic
163C42×C73#4111yescharacteristic
164C42×C73C2411yescharacteristic
165C23×C73C4×D411yescharacteristic
166C22×D73—21no—
167Q8×C73—22no—
168Q8×C73—22no—
169Q8×C73—42no—
170C2×C4×C73—21no—
171C2×C4×C73—21no—
172C2×C4×C73—22no—
173C2×C4×C73—22no—
174C2×C4×C73—41no—
175C2×C4×C73—42no—
176C2×C4×C73—42no—
177C2×C4×C73C4×D411yescharacteristic
178C2×C4×C73C4×D411yescharacteristic
179C2×C4×C73C4×D411yescharacteristic
180C2×C4×C73C2×M1612yes—
181C2×C4×C73#13112yes—
182C2×C4×C73C23×C411yescharacteristic
183C2×C4×C73C22×D411yescharacteristic
184C2×C4×C73C2×#4111yescharacteristic
185C22⋊D73—42no—
186C2×Dic73—21no—
187C2×Dic73—21no—
188C2×Dic73—21no—
189D292—21no—
190D292—21no—
191C4×D73—42no—
192C4×D73—41no—
193C4×D73—41no—
194C4×D73—42no—
195Dic146—21no—
196Dic146—21no—
197C8×C73—42no—
198C8×C73—41no—
199C8×C73—41no—
200#10596831—22no—
201#10596831—21no—
202#10596831—21no—
203#10596831—21no—
204#10596831—42no—
205#10596831C2×M1614yes—
206#10596831#13712yescomplemented
207C22×C73—21no—
208C22×C73—21no—
209C22×C73#43612yes—
210C22×C73C2×C4×D411yescharacteristic
211C2×D73—41no—
212C2×D73—41no—
213C2×D73—41no—
214C4×C73—21no—
215C4×C73—21no—
216C4×C73—21no—
217C4×C73—42no—
218C4×C73—42no—
219C4×C73#43612yes—
220C4×C73C22×M1611yescharacteristic
221C4×C73C2×#13111yescharacteristic
222C4×C73C2×#13711yescharacteristic
223C4×C73#57511yescharacteristic
224Dic73—41no—
225Dic73—41no—
226Dic73—41no—
227#17892—731nomaximal, Sylow
228C2×C73—22no—
229C2×C73C2×#43611yescharacteristic
230C2×C73C4×#13711yescharacteristic
231C2×C73#295511yescharacteristic
232D73—81no—
233#2965—731no—
234C2×#446—731no—
235C4×#137—731no—
236#1593—732no—
237#1592—732no—
238#1591—732no—
239#1590—732no—
240#1565—731no—
241#1479D7311yescharacteristic, complemented
242#1477—731no—
243C8×M16—731no—
244C73#1789211yescharacteristic, complemented, Sylow
245C2×#137—731no—
246C4×#41—731no—
247C2×C4×Q8C2×D7311yescharacteristic
248#446—732no—
249#446—732no—
250#446—731no—
251#446—731no—
252#446—1461no—
253#446—1461no—
254#441—732no—
255#440—732no—
256C4×Q16—732no—
257C4×SD16—732no—
258#436—731no—
259#436—731no—
260#435—1461no—
261#435—1461no—
262#431—1461no—
263#431C2×D7311yescharacteristic
264#431C2×D7311yescharacteristic
265#428—731no—
266C2×C4⋊C8—731no—
267C2×C4⋊C8—1461no—
268#419—731no—
269#417—731no—
270C4×M16—731no—
271C4×M16—1461no—
272C2×#97—1461no—
273C2×C4×C8—731no—
274#335—732no—
275#334—732no—
276Q8⋊C8—732no—
277Q8⋊C8C2×D7312yes—
278Q8⋊C8C2×D7312yes—
279D4⋊C8—732no—
280#322—732no—
281C82—732no—
282C2×#41—731no—
283C22×Q8C4×D7311yescharacteristic
284#137—1464no—
285C2×Q16—732no—
286C2×SD16—732no—
287C2×M16—731no—
288C2×M16—731no—
289C2×M16—1461no—
290C2×M16—1461no—
291C22×C8—731no—
292C22×C8—731no—
293C22×C8—1461no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—731no—
297C4×Q8—731no—
298C4×Q8C4×D7312yes—
299C4×Q8C22×D7312yes—
300C4×D4—731no—
301C4×D4—731no—
302C4×D4—1461no—
303C4×D4—1461no—
304#117—731no—
305#117—1461no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C4×D7311yescharacteristic
308C2×C42—21no—
309C2×C42—1461no—
310C2×C42C22×D7311yescharacteristic
311#107—732no—
312#106—732no—
313C4⋊C8—21no—
314C4⋊C8—732no—
315C4⋊C8—1462no—
316C4⋊C8—1462no—
317C4⋊C8—1461no—
318C4⋊C8—1461no—
319C4⋊C8—1461no—
320C4⋊C8C22×D7311yescharacteristic
321C4⋊C8C22×D7311yescharacteristic
322#103—732no—
323#103—732no—
324#103—732no—
325#102—732no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—1461no—
329C22⋊C8—1461no—
330#97—732no—
331#97—1462no—
332#97—1462no—
333#97—1461no—
334#97—1461no—
335C4×C8—21no—
336C4×C8—731no—
337C4×C8—731no—
338C4×C8—731no—
339C4×C8—732no—
340C4×C8—731no—
341C4×C8C22×D7312yes—
342#41—1462no—
343#41—1462no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—731no—
348C2×Q8C2×C4×D7312yes—
349C2×D4—731no—
350C2×D4—1461no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—1461no—
354C22×C4—1461no—
355C22×C4—1461no—
356C22×C4C2×C4×D7311yescharacteristic
357C22×C4D4×D7311yescharacteristic
358C22×C4#1192402711yescharacteristic
359Q16—1464no—
360SD16—1464no—
361M16—2922no—
362M16—2922no—
363M16—734no—
364M16—1462no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—2922no—
369C2×C8—2921no—
370C2×C8—732no—
371C2×C8—731no—
372C2×C8—732no—
373C2×C8—731no—
374C2×C8—731no—
375C2×C8—731no—
376C2×C8—1462no—
377C2×C8—1462no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—731no—
382C4⋊C4—731no—
383C4⋊C4—1461no—
384C4⋊C4—1461no—
385C4⋊C4C2×C4×D7312yes—
386C22⋊C4—1461no—
387C22⋊C4—1461no—
388C22⋊C4—1461no—
389C42—22no—
390C42—22no—
391C42—1461no—
392C42—1461no—
393C42—1461no—
394C42C2×C4×D7311yescharacteristic
395C42D4×D7311yescharacteristic
396C42#1192402711yescharacteristic
397C42C23×D7311yescharacteristic
398C23—1461no—
399C23#1455655111yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—42no—
403Q8—1461no—
404Q8—1461no—
405D4—2922no—
406D4—1461no—
407D4—1461no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—2922no—
416C2×C4—2921no—
417C2×C4—2921no—
418C2×C4—2922no—
419C2×C4—1461no—
420C2×C4—1461no—
421C2×C4—1461no—
422C2×C4#1455655111yescharacteristic
423C2×C4#1455655111yescharacteristic
424C2×C4#1455655111yescharacteristic
425C2×C4M16×D7312yes—
426C2×C4#1455659712yes—
427C2×C4C22×C4×D7311yescharacteristic
428C2×C4C2×D4×D7311yescharacteristic
429C2×C4C2×#1192402711yescharacteristic
430C8—42no—
431C8—41no—
432C8—41no—
433C8—2922no—
434C8—732no—
435C8—734no—
436C8—1461no—
437C8—1461no—
438C8—1461no—
439C8—1462no—
440C22—21no—
441C22—21no—
442C22—2921no—
443C22—2921no—
444C22—2921no—
445C22#1798866212yes—
446C22C2×#1455655111yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—42no—
451C4—42no—
452C4—2921no—
453C4—2921no—
454C4—2921no—
455C4#1798866212yes—
456C4C2×M16×D7311yescharacteristic
457C4C2×#1455659711yescharacteristic
458C4#137×D7311yescharacteristic
459C4#1798938611yescharacteristic
460C2—22no—
461C2—5841no—
462C2C2×#1798866211yescharacteristic
463C2#2261874611yescharacteristic
464C2#2261883011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1479C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C2×C4
cosocleC42×C73C24
hypercenter#1479D73
hypocenterC73#17892
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. C2×#17988718C2semidirect
4. #22608837C2semidirect
5. #22608836C2semidirect
6. #22608825C2semidirect
7. #22608818C2semidirect
8. #22608684C2semidirect
10. #22608212C2semidirect
11. C73×#1479C2semidirect
120. #11924012M16semidirect
206. #10596831#137semidirect
241. #1479D73semidirect
244. C73#17892semidirect

Maximal subgroups

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

subgroupindex
#226188922
C2×#179887182
C4×#145566222
#226088372
#226088362
#226088252
#226088182
#226086842
#226083192
#226082122
C73×#14792
#1789273