Finite Group Databasebeta

← #34016010

Subgroups — #34016010

Order 18688.

1174 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626830C212yesmaximal, complemented
2#22625191C212yesmaximal, complemented
3#22625190C212yesmaximal, complemented
4#22625184C212yesmaximal, complemented
5#22625183C212yesmaximal, complemented
6#22625170C212yesmaximal, complemented
7#22625163C212yesmaximal, complemented
8C2×#17989066C211yesmaximal, characteristic
9#17989639—28no—
10#17989608C2214yes—
11#17989607C2214yes—
12#17989604C2214yes—
13#17989433C2214yes—
14#17989431C2214yes—
15#17989430C2214yes—
16#17989430C2214yes—
17C22×#11924027C411yescharacteristic
18#17989066—28no—
19C2×C4×Dic146C411yescharacteristic
20C2×#14556556C2211yescharacteristic
21C2×D4×Dic73C2211yescharacteristic
22C2×C4×C22⋊D73C2211yescharacteristic
23C2×C4×C22⋊D73C2211yescharacteristic
24C2×#14556551C411yescharacteristic
25C2×#14556551C411yescharacteristic
26C2×#14556548C2211yescharacteristic
27C2×#14556547C2211yescharacteristic
28C2×#14556547C2211yescharacteristic
29C22×#11924007C411yescharacteristic
30C22×#11924007C411yescharacteristic
31C2×C4⋊C4×D73C411yescharacteristic
32C2×C4×D4×C73C411yescharacteristic
33#14556658—24no—
34#14556658—24no—
35C4⋊#10596833—24no—
36C22⋊#10596833C2318yes—
37#14556646—24no—
38#14556646—24no—
39C4×#10596833—24no—
40C2×#11924027—24no—
41C2×#11924027—24no—
42C2×#11924027—24no—
43C22×Dic146C2×C411yescharacteristic
44C22×C22⋊D73C2×C411yescharacteristic
45C22×C22⋊D73C2×C411yescharacteristic
46C23×Dic73C2×C411yescharacteristic
47C23×Dic73C2×C411yescharacteristic
48C23×Dic73C2×C411yescharacteristic
49C23×Dic73C2×C411yescharacteristic
50C22×C4×D73C2×C411yescharacteristic
51C22×C4×D73C2×C411yescharacteristic
52C22×C4×D73C2×C411yescharacteristic
53C4×Dic146—24no—
54#14556556D418yes—
55D4×Dic73—24no—
56C4×C22⋊D73—24no—
57C4×C22⋊D73—24no—
58#14556551—24no—
59#14556551—24no—
60#14556548—24no—
61#14556547—24no—
62#14556547—24no—
63C2×C4⋊Dic73C2×C411yescharacteristic
64C2×#11924007—24no—
65C2×#11924007—24no—
66C2×#11924007C2×C411yescharacteristic
67C2×#11924007C2×C411yescharacteristic
68C2×#11924007C2×C411yescharacteristic
69C2×#11924007C2×C411yescharacteristic
70C2×#11924007C2×C411yescharacteristic
71C2×#11924007C2×C411yescharacteristic
72C2×#11924007C2×C411yescharacteristic
73C2×#11924007C2311yescharacteristic
74C4⋊C4×D73—24no—
75C2×C22⋊Dic73C2311yescharacteristic
76C2×C22⋊Dic73C2311yescharacteristic
77C2×#11924005C2×C411yescharacteristic
78C2×#11924005C2311yescharacteristic
79C2×C4×Dic73C2×C411yescharacteristic
80C2×C4×Dic73C2×C411yescharacteristic
81C2×C4×Dic73C2311yescharacteristic
82C2×C4×Dic73C2311yescharacteristic
83C2×C4×Dic73C2311yescharacteristic
84C22×D4×C73C2×C411yescharacteristic
85C23×C4×C73C2×C411yescharacteristic
86C23×C4×C73C2×C411yescharacteristic
87C4×D4×C73—24no—
88C2×C4⋊C4×C73C2×C411yescharacteristic
89C2×C22⋊C4×C73C2×C411yescharacteristic
90C2×C22⋊C4×C73C2×C411yescharacteristic
91C2×C42×C73C2×C411yescharacteristic
92C2×#10596833—28no—
93C23×D73C22×C411yescharacteristic
94#11924027—232no—
95C2×Dic146—22no—
96C2×Dic146—22no—
97C2×Dic146C22⋊C414yes—
98C2×C22⋊D73—22no—
99C2×C22⋊D73—22no—
100C2×C22⋊D73—22no—
101C2×C22⋊D73—22no—
102C2×C22⋊D73—22no—
103C2×C22⋊D73M1614yes—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—81no—
112C22×Dic73—81no—
113C22×Dic73M1614yes—
114C22×Dic73C22×C411yescharacteristic
115C22×Dic73C22×C411yescharacteristic
116C22×Dic73C22×C411yescharacteristic
117C22×Dic73C22×C411yescharacteristic
118C22×Dic73C22×C411yescharacteristic
119C22×Dic73C22×C411yescharacteristic
120C22×Dic73C2411yescharacteristic
121C2×C4×D73—22no—
122C2×C4×D73—22no—
123C2×C4×D73—22no—
124C2×C4×D73—22no—
125C2×C4×D73—22no—
126C2×C4×D73—81no—
127C2×C4×D73—81no—
128C4⋊Dic73—22no—
129#11924007—22no—
130#11924007—22no—
131#11924007—22no—
132#11924007—22no—
133#11924007—22no—
134#11924007—22no—
135#11924007C22⋊C414yes—
136#11924007C22⋊C414yes—
137C22⋊Dic73—22no—
138C22⋊Dic73—22no—
139#11924005—22no—
140#11924005—22no—
141C4×Dic73—22no—
142C4×Dic73—22no—
143C4×Dic73C2×D414yes—
144C4×Dic73C2×D414yes—
145C4×Dic73C2×D414yes—
146C24×C73C22×C411yescharacteristic
147C24×C73C22×C411yescharacteristic
148C2×D4×C73—22no—
149C2×D4×C73—22no—
150C2×D4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—81no—
154C22×C4×C73—81no—
155C22×C4×C73C22×C411yescharacteristic
156C22×C4×C73C22×C411yescharacteristic
157C22×C4×C73C22×C411yescharacteristic
158C22×C4×C73C22×C411yescharacteristic
159C22×C4×C73C22×C411yescharacteristic
160C4⋊C4×C73C22⋊C414yes—
161C22⋊C4×C73—22no—
162C22⋊C4×C73—22no—
163C42×C73—22no—
164C23×C73—21no—
165C23×C73—21no—
166C23×C73—21no—
167C23×C73—21no—
168C23×C73—41no—
169C23×C73—41no—
170C23×C73—41no—
171C23×C73—41no—
172C23×C73C2×M1612yes—
173C23×C73C2×M1612yes—
174C23×C73C23×C411yescharacteristic
175C22×D73—21no—
176C22×D73—21no—
177C22×D73—41no—
178C22×D73—41no—
179C22×D73C2×M1612yes—
180D4×C73—28no—
181C2×C4×C73—21no—
182C2×C4×C73—21no—
183C2×C4×C73—21no—
184C2×C4×C73—21no—
185C2×C4×C73—41no—
186C2×C4×C73—41no—
187C2×C4×C73—41no—
188C2×C4×C73—41no—
189C2×C4×C73—82no—
190C2×C4×C73—82no—
191C2×C4×C73C2×C22⋊C412yes—
192C2×C4×C73C2×C22⋊C412yes—
193C2×C4×C73C2×C22⋊C412yes—
194C22⋊D73—28no—
195C22⋊D73—28no—
196C2×Dic73—28no—
197C2×Dic73—21no—
198C2×Dic73—21no—
199C2×Dic73—21no—
200C2×Dic73—21no—
201C2×Dic73—28no—
202C2×Dic73—21no—
203C2×Dic73—21no—
204C2×Dic73—41no—
205C2×Dic73—41no—
206C2×Dic73—41no—
207C2×Dic73—41no—
208C2×Dic73—82no—
209C2×Dic73—82no—
210C2×Dic73C2×C22⋊C412yes—
211C2×Dic73C2×C22⋊C412yes—
212C2×Dic73C2×C22⋊C412yes—
213C2×Dic73C2×M1612yes—
214C2×Dic73C2×M1612yes—
215C2×Dic73C2×M1612yes—
216C2×Dic73C22×D412yes—
217C4×D73—28no—
218C4×D73—82no—
219C4×D73—82no—
220Dic146—28no—
221#10596833—48no—
222C22×C73—21no—
223C22×C73—21no—
224C22×C73—22no—
225C22×C73—22no—
226C22×C73—42no—
227C22×C73—42no—
228C22×C73—42no—
229C22×C73—42no—
230C22×C73—41no—
231C22×C73—41no—
232C22×C73—41no—
233C22×C73—41no—
234C22×C73C22×C22⋊C411yescharacteristic
235C22×C73C22×M1611yescharacteristic
236C22×C73C22×M1611yescharacteristic
237C2×D73—22no—
238C2×D73—42no—
239C2×D73—41no—
240C2×D73—41no—
241C2×D73—42no—
242C4×C73—22no—
243C4×C73—22no—
244C4×C73—22no—
245C4×C73—41no—
246C4×C73—41no—
247Dic73—22no—
248Dic73—22no—
249Dic73—22no—
250Dic73—22no—
251Dic73—41no—
252Dic73—41no—
253Dic73#40714yes—
254#17372—731nomaximal, Sylow
255C2×C73—21no—
256C2×C73—21no—
257C2×C73—42no—
258C2×C73—42no—
259C2×C73C2×#40711yescharacteristic
260C2×C73#288411yescharacteristic
261C2×C73#288811yescharacteristic
262D73—42no—
263C2×#520—731no—
264#2983—732no—
265#1466—732no—
266#1466—732no—
267#1464—732no—
268#1464—732no—
269#1460—732no—
270#1458—732no—
271C73#1737211yescharacteristic, complemented, Sylow
272C22×#41—731no—
273#520—1468no—
274C2×C4×Q8—731no—
275C2×C4×Q8—731no—
276C2×C4×D4—731no—
277C2×C4×D4—731no—
278C2×C4×D4—731no—
279C2×C4×D4—731no—
280C2×C4×D4—731no—
281C2×C4×D4#6440811yescharacteristic
282C2×#117—731no—
283C2×#117—731no—
284C2×#117—731no—
285C22×C4⋊C4—731no—
286C22×C4⋊C4—731no—
287C22×C4⋊C4—731no—
288#446—1468no—
289#432—734no—
290#432—734no—
291#431—734no—
292#324—734no—
293#323—734no—
294#323—734no—
295#323—734no—
296C2×#41—1464no—
297C2×#41—1464no—
298C2×#41—1464no—
299C22×Q8—731no—
300C22×D4—731no—
301C22×D4—731no—
302C22×D4C2×#6440811yescharacteristic
303C23×C4—731no—
304C23×C4—731no—
305C23×C4—731no—
306C23×C4—731no—
307C23×C4—731no—
308C23×C4—731no—
309C23×C4—731no—
310C23×C4C2×#6440811yescharacteristic
311C23×C4C2×#6440811yescharacteristic
312C4×Q8—738no—
313C4×Q8—1464no—
314C4×D4—24no—
315C4×D4—1464no—
316C4×D4—1464no—
317C4×D4—1464no—
318C4×D4—1464no—
319C4×D4—1464no—
320#117—1464no—
321#117—1464no—
322#117—1464no—
323C2×C4⋊C4—731no—
324C2×C4⋊C4—731no—
325C2×C4⋊C4—731no—
326C2×C4⋊C4—731no—
327C2×C4⋊C4—731no—
328C2×C4⋊C4—731no—
329C2×C4⋊C4—731no—
330C2×C4⋊C4—731no—
331C2×C4⋊C4—731no—
332C2×C4⋊C4—1464no—
333C2×C4⋊C4—1464no—
334C2×C4⋊C4—1464no—
335C2×C4⋊C4C2×#6440811yescharacteristic
336C2×C22⋊C4—731no—
337C2×C22⋊C4—731no—
338C2×C22⋊C4—731no—
339C2×C22⋊C4—731no—
340C2×C22⋊C4C2×#6440811yescharacteristic
341C2×C22⋊C4C2×#6440811yescharacteristic
342C2×C42—731no—
343C2×C42—731no—
344C2×C42—731no—
345C2×C42—731no—
346C2×C42—731no—
347C2×C42C2×#6440811yescharacteristic
348C4⋊C8—1464no—
349C4⋊C8—1464no—
350C4⋊C8—1464no—
351C22⋊C8—738no—
352#97—1464no—
353#97—1464no—
354C4×C8—1464no—
355C24—731no—
356C24C22×#6440811yescharacteristic
357C24C22×#6440811yescharacteristic
358#41—14632no—
359C2×Q8—734no—
360C2×Q8—1462no—
361C2×Q8—1462no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—734no—
366C2×D4—1462no—
367C2×D4—1462no—
368C2×D4—1462no—
369C2×D4—1462no—
370C2×D4—1462no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—5841no—
376C22×C4—5841no—
377C22×C4—5841no—
378C22×C4—5841no—
379C22×C4—731no—
380C22×C4—731no—
381C22×C4—731no—
382C22×C4—731no—
383C22×C4—731no—
384C22×C4—734no—
385C22×C4—731no—
386C22×C4—731no—
387C22×C4—1462no—
388C22×C4—1462no—
389C22×C4—1462no—
390C22×C4—1462no—
391C22×C4—1462no—
392C22×C4—1462no—
393C22×C4—1462no—
394C22×C4—1462no—
395C22×C4—1462no—
396C22×C4—1462no—
397C22×C4—1462no—
398C22×C4—1462no—
399C22×C4C22×#6440811yescharacteristic
400C22×C4C22×#6440811yescharacteristic
401C22×C4C22×#6440811yescharacteristic
402C22×C4C22×#6440811yescharacteristic
403C22×C4C22×#6440811yescharacteristic
404C2×C8—1468no—
405C4⋊C4—734no—
406C4⋊C4—734no—
407C4⋊C4—1462no—
408C4⋊C4—1462no—
409C4⋊C4—1462no—
410C4⋊C4—1462no—
411C4⋊C4—1462no—
412C4⋊C4—1462no—
413C4⋊C4—1462no—
414C4⋊C4C22⋊#6440814yes—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—1462no—
418C22⋊C4—1462no—
419C22⋊C4—1462no—
420C22⋊C4—1462no—
421C42—22no—
422C42—734no—
423C42—734no—
424C42—734no—
425C42—1462no—
426C42—1462no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—2921no—
436C23—2921no—
437C23—732no—
438C23—1461no—
439C23—1461no—
440C23D73⋊M1612yes—
441C23D73⋊M1612yes—
442C23C23×#6440811yescharacteristic
443Q8—1468no—
444D4—28no—
445D4—1468no—
446D4—1468no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—2921no—
458C2×C4—2921no—
459C2×C4—2921no—
460C2×C4—2921no—
461C2×C4—5842no—
462C2×C4—5842no—
463C2×C4—5842no—
464C2×C4—5842no—
465C2×C4—732no—
466C2×C4—732no—
467C2×C4—732no—
468C2×C4—732no—
469C2×C4—732no—
470C2×C4—732no—
471C2×C4—732no—
472C2×C4—1468no—
473C2×C4—1461no—
474C2×C4—1461no—
475C2×C4—1468no—
476C2×C4—1468no—
477C2×C4—1461no—
478C2×C4—1461no—
479C2×C4—1461no—
480C2×C4—1461no—
481C2×C4C2×C22⋊#6440812yes—
482C2×C4C2×C22⋊#6440812yes—
483C2×C4C2×C22⋊#6440812yes—
484C8—2928no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—2922no—
498C22—2922no—
499C22—2921no—
500C22—2921no—
501C22—1462no—
502C22C22×C22⋊#6440811yescharacteristic
503C22C2×D73⋊M1611yescharacteristic
504C22C2×D73⋊M1611yescharacteristic
505C4—22no—
506C4—22no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—2921no—
511C4—2921no—
512C4—734no—
513C4—1462no—
514C4—1462no—
515C4—1462no—
516C4—1462no—
517C2—21no—
518C2—21no—
519C2—42no—
520C2—42no—
521C2—2922no—
522C2D73⋊#40711yescharacteristic
523C2#2262653311yescharacteristic
524C2#2262654111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C4×D4×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC22×#11924027C4
cosocleC22×Dic73C24
hypercenterC2×C4×D4#64408
hypocenterC73#17372
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22626830C2semidirect
2. #22625191C2semidirect
3. #22625190C2semidirect
4. #22625184C2semidirect
5. #22625183C2semidirect
6. #22625170C2semidirect
7. #22625163C2semidirect
271. C73#17372semidirect

Maximal subgroups

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

subgroupindex
#226268302
#226251912
#226251902
#226251842
#226251832
#226251702
#226251632
C2×#179890662
#1737273