Finite Group Databasebeta

← #1952×D67 #30859373

Subgroups — #1952×D67 #30859373

Order 17152.

918 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#108×D67C211yesmaximal, characteristic
2#428×D67C212yesmaximal
3C2×#107×D67C211yesmaximal, characteristic
4#405×D67C211yesmaximal, characteristic
5#22560995C211yesmaximal, characteristic, complemented
6#22560994C211yesmaximal, characteristic, complemented
7#22560993C212yesmaximal, complemented
8#22560992C211yesmaximal, characteristic, complemented
9#22560991C212yesmaximal, complemented
10#343×D67C212yesmaximal
11C67×#1952C211yesmaximal, characteristic, complemented
12C2×M16×D67C2212yes—
13C2×M16×D67C2211yescharacteristic
14C22×C8×D67C2211yescharacteristic
15#117×D67C2211yescharacteristic
16C2×C4⋊C4×D67C2212yes—
17C2×#14523979C2211yescharacteristic
18C2×#14523978C2212yes—
19#108×D67C418yescomplemented
20#16863439C2212yes—
21#16863438C2212yes—
22#16863437—22no—
23#16863436C2212yes—
24#16863435—22no—
25C2×#14523977C2211yescharacteristic
26C2×#14523976C2212yes—
27#107×D67—21no—
28#107×D67—21no—
29#107×D67—21no—
30#107×D67—21no—
31#107×D67—22no—
32#106×D67—22no—
33#16863354C2211yescharacteristic
34#16863353—21no—
35#16863352C2211yescharacteristic
36#16863351C2211yescharacteristic
37#16863350—21no—
38#97×D67—21no—
39C4×C8×D67—21no—
40#16863165C2212yes—
41#16863164C2212yes—
42#16863164C2212yes—
43C2×#108×C67C2211yescharacteristic
44#428×C67C2212yes—
45C2×#107×C67C2211yescharacteristic
46#405×C67C2211yescharacteristic
47#343×C67C2212yes—
48C22×C4×D67—22no—
49C22×C4×D67C2311yescharacteristic
50C2×#11761612C2312yes—
51C2×#11761612C2311yescharacteristic
52C2×#11761611—21no—
53C2×#11761611—22no—
54M16×D67—24no—
55M16×D67—22no—
56M16×D67C2×C418yes—
57M16×D67Q814yes—
58C22×C67⋊C8C2311yescharacteristic
59C2×C8×D67—21no—
60C2×C8×D67—22no—
61C2×C8×D67C2×C414yes—
62C2×C8×D67D411yescharacteristic
63C2×C8×D67D411yescharacteristic
64C2×C8×D67D411yescharacteristic
65C2×C8×D67D411yescharacteristic
66C2×C8×D67Q814yes—
67#14524002C2311yescharacteristic
68#14524001D411yescharacteristic
69#14524001D411yescharacteristic
70#14524000—21no—
71#14523999—21no—
72C2×C4⋊Dic67C2312yes—
73C2×Dic67⋊C4—22no—
74C4⋊C4×D67—22no—
75C4⋊C4×D67—22no—
76C4⋊C4×D67—22no—
77C4⋊C4×D67—22no—
78C4⋊C4×D67D411yescharacteristic
79C4⋊C4×D67D411yescharacteristic
80C22⋊C4×D67—21no—
81C42×D67—21no—
82#14523979C2×C414yescomplemented
83#14523978C2×C418yescomplemented
84#14523977—22no—
85#14523977—21no—
86#14523977—21no—
87#14523976—22no—
88#14523976—22no—
89#14523976—22no—
90#14523976—22no—
91#14523975—22no—
92#14523974—22no—
93#14523974—22no—
94#14523954—21no—
95#14523953—21no—
96#14523953—21no—
97C4×C67⋊C8—21no—
98C8×Dic67—21no—
99C8×Dic67—21no—
100C2×M16×C67C2312yes—
101C2×M16×C67C2311yescharacteristic
102C22×C8×C67C2311yescharacteristic
103#117×C67C2311yescharacteristic
104C2×C4⋊C4×C67C2312yes—
105#108×C67C2×C414yescomplemented
106#107×C67—21no—
107#107×C67—21no—
108#107×C67—22no—
109#106×C67—22no—
110#97×C67—21no—
111C4×C8×C67—21no—
112C23×D67#4111yescharacteristic
113C22×Dic67—22no—
114C22×Dic67#4111yescharacteristic
115C2×C4×D67—21no—
116C2×C4×D67—21no—
117C2×C4×D67—42no—
118C2×C4×D67—42no—
119C2×C4×D67—84no—
120C2×C4×D67C22×C412yes—
121C2×C4×D67C2×D411yescharacteristic
122C2×C4×D67C2×D411yescharacteristic
123C2×C4×D67C2×D411yescharacteristic
124C2×C4×D67C2×D411yescharacteristic
125C2×C4×D67C2×Q812yes—
126C2×C4×D67C2×Q812yes—
127C2×C4×D67#4112yes—
128#11761612—22no—
129#11761612—21no—
130#11761612C22×C414yes—
131#11761612C2×Q812yes—
132#11761611—28no—
133#11761611—24no—
134C2×C67⋊C8—21no—
135C2×C67⋊C8—22no—
136C2×C67⋊C8C22×C412yes—
137C2×C67⋊C8C2×D411yescharacteristic
138C2×C67⋊C8C2×D411yescharacteristic
139C2×C67⋊C8C2×Q812yes—
140C8×D67—28no—
141C8×D67—24no—
142C8×D67C4⋊C414yes—
143C8×D67C4⋊C414yes—
144C8×D67C4⋊C414yes—
145C8×D67C4⋊C414yes—
146C4⋊Dic67—22no—
147C4⋊Dic67—22no—
148C4⋊Dic67C2×D411yescharacteristic
149C4⋊Dic67C2×D411yescharacteristic
150Dic67⋊C4—21no—
151Dic67⋊C4—21no—
152Dic67⋊C4—42no—
153Dic67⋊C4—42no—
154C22⋊Dic67—21no—
155(C2×D67)⋊C4—21no—
156(C2×D67)⋊C4—21no—
157C4×Dic67—21no—
158C4×Dic67—21no—
159C4×Dic67—21no—
160C22×C4×C67—22no—
161C22×C4×C67C2411yescharacteristic
162M16×C67—22no—
163M16×C67—21no—
164M16×C67C22×C414yes—
165M16×C67C2×Q812yes—
166C2×C8×C67—22no—
167C2×C8×C67—21no—
168C2×C8×C67C22×C412yes—
169C2×C8×C67C2×D411yescharacteristic
170C2×C8×C67C2×D411yescharacteristic
171C2×C8×C67C2×Q812yes—
172C4⋊C4×C67—22no—
173C4⋊C4×C67—22no—
174C4⋊C4×C67C2×D411yescharacteristic
175C4⋊C4×C67C2×D411yescharacteristic
176C22⋊C4×C67—21no—
177C42×C67—21no—
178C23×C67C2×#4111yescharacteristic
179C22×D67—22no—
180C22×D67—22no—
181C22×D67C4×Q812yes—
182C22×D67#12111yescharacteristic
183C22×D67#12111yescharacteristic
184C22×D67#12512yes—
185C2×C4×C67—21no—
186C2×C4×C67—21no—
187C2×C4×C67—42no—
188C2×C4×C67—82no—
189C2×C4×C67C23×C411yescharacteristic
190C2×C4×C67C22×D411yescharacteristic
191C2×C4×C67C22×D411yescharacteristic
192C2×C4×C67C22×Q811yescharacteristic
193C2×C4×C67C22×Q811yescharacteristic
194C2×C4×C67C2×#4111yescharacteristic
195C2×Dic67—21no—
196C2×Dic67—21no—
197C2×Dic67—42no—
198C2×Dic67—82no—
199C2×Dic67C4×Q812yes—
200C2×Dic67#12111yescharacteristic
201C2×Dic67#12111yescharacteristic
202C2×Dic67#12512yes—
203C4×D67—42no—
204C4×D67—42no—
205C4×D67—84no—
206C4×D67—84no—
207C4×D67C2×C4⋊C412yes—
208C4×D67C2×C4⋊C412yes—
209C4×D67C4×D412yes—
210C4×D67C4×D412yes—
211C4×D67C22⋊Q812yes—
212C4×D67C22⋊Q812yes—
213C4×D67C4⋊Q812yes—
214C4×D67C4⋊Q812yes—
215C8×C67—24no—
216C8×C67—22no—
217C8×C67C2×C4⋊C412yes—
218C8×C67C2×C4⋊C412yes—
219C67⋊C8—24no—
220C67⋊C8—22no—
221C67⋊C8C2×C4⋊C412yes—
222C67⋊C8C2×C4⋊C412yes—
223C22×C67—21no—
224C22×C67—21no—
225C22×C67C2×C4×Q811yescharacteristic
226C22×C67C2×#12111yescharacteristic
227C22×C67C2×#12511yescharacteristic
228C2×D67—21no—
229C2×D67—21no—
230C2×D67—24no—
231C2×D67—24no—
232C2×D67#38912yes—
233C2×D67#38912yes—
234C2×D67#47212yes—
235C2×D67#47312yes—
236C4×C67—41no—
237C4×C67—41no—
238C4×C67—82no—
239C4×C67C22×C4⋊C411yescharacteristic
240C4×C67C2×C4×D411yescharacteristic
241C4×C67C2×C22⋊Q811yescharacteristic
242C4×C67C2×C4⋊Q811yescharacteristic
243Dic67—41no—
244Dic67—41no—
245Dic67—82no—
246Dic67#38912yes—
247Dic67#38912yes—
248C2×#1952—671nomaximal, Sylow
249C2×C67—21no—
250C2×C67—21no—
251C2×C67C2×#38911yescharacteristic
252C2×C67C2×#47211yescharacteristic
253C2×C67C2×#47311yescharacteristic
254D67—22no—
255D67#195214yesdirect product factor
256C22×#108—671no—
257C2×#428—672no—
258C22×#107—671no—
259C2×#405—671no—
260#1952—671no—
261#1952—672no—
262#1952—671no—
263#1952—671no—
264#1952—672no—
265#1952D6711yescharacteristic, direct product factor
266C2×#343—672no—
267C67C2×#195211yescharacteristic, complemented, Sylow
268C22×M16—671no—
269C22×M16—672no—
270C23×C8—671no—
271C2×#117—671no—
272C22×C4⋊C4—672no—
273C2×#108—672no—
274C2×#108—671no—
275C2×#108—678no—
276C2×#108C2×D6711yescharacteristic
277#428—672no—
278#428—672no—
279#428—672no—
280#428—1342no—
281#428—1342no—
282#428C2×D6712yes—
283C2×#107—671no—
284C2×#107—672no—
285C2×#107—1342no—
286C2×#107—1341no—
287C2×#107—1341no—
288C2×#107—1341no—
289C2×#107—1341no—
290C2×#107C2×D6711yescharacteristic
291C2×#106—1342no—
292#405—671no—
293#405—671no—
294#405—671no—
295#405—1341no—
296#405—1341no—
297#405C2×D6711yescharacteristic
298C2×#97—1341no—
299C2×C4×C8—1341no—
300#343—672no—
301#343—672no—
302#343—672no—
303#343C2×D6712yes—
304C23×C4—671no—
305C23×C4—1342no—
306C2×M16—671no—
307C2×M16—672no—
308C2×M16—678no—
309C2×M16—674no—
310C2×M16—1341no—
311C2×M16—1342no—
312C2×M16—1342no—
313C2×M16—1344no—
314C2×M16C22×D6712yes—
315C2×M16C22×D6711yescharacteristic
316C22×C8—671no—
317C22×C8—674no—
318C22×C8—671no—
319C22×C8—671no—
320C22×C8—671no—
321C22×C8—671no—
322C22×C8—674no—
323C22×C8—1341no—
324C22×C8—1342no—
325C22×C8C22×D6711yescharacteristic
326#117—671no—
327#117—671no—
328#117—671no—
329#117—1341no—
330#117—1341no—
331#117C22×D6711yescharacteristic
332C2×C4⋊C4—671no—
333C2×C4⋊C4—671no—
334C2×C4⋊C4—672no—
335C2×C4⋊C4—1342no—
336C2×C4⋊C4—1342no—
337C2×C4⋊C4—1342no—
338C2×C4⋊C4—1342no—
339C2×C4⋊C4—1342no—
340C2×C4⋊C4C22×D6712yes—
341C2×C22⋊C4—1341no—
342C2×C42—1341no—
343#108—678no—
344#108—674no—
345#108C4×D6714yescomplemented
346#107—21no—
347#107—21no—
348#107—22no—
349#107—1342no—
350#107—1342no—
351#107—1342no—
352#107—1342no—
353#107—1341no—
354#107—1341no—
355#107—1342no—
356#106—22no—
357#106—1342no—
358#106—1342no—
359#106—1342no—
360#97—21no—
361#97—1341no—
362#97—1341no—
363#97—1341no—
364C4×C8—21no—
365C4×C8—1341no—
366C4×C8—1341no—
367C4×C8—1341no—
368C24—671no—
369C22×C4—22no—
370C22×C4—2682no—
371C22×C4—2682no—
372C22×C4—5364no—
373C22×C4—671no—
374C22×C4—672no—
375C22×C4—671no—
376C22×C4—671no—
377C22×C4—671no—
378C22×C4—671no—
379C22×C4—672no—
380C22×C4—672no—
381C22×C4—672no—
382C22×C4—1341no—
383C22×C4—1341no—
384C22×C4—1342no—
385C22×C4C23×D6711yescharacteristic
386M16—22no—
387M16—21no—
388M16—674no—
389M16—672no—
390M16—1348no—
391M16—1344no—
392M16—1342no—
393M16—1341no—
394M16C2×C4×D6714yes—
395M16Q8×D6712yes—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—672no—
399C2×C8—671no—
400C2×C8—671no—
401C2×C8—672no—
402C2×C8—674no—
403C2×C8—674no—
404C2×C8—674no—
405C2×C8—674no—
406C2×C8—1348no—
407C2×C8—1344no—
408C2×C8—1341no—
409C2×C8—1342no—
410C2×C8C2×C4×D6712yes—
411C2×C8D4×D6711yescharacteristic
412C2×C8D4×D6711yescharacteristic
413C2×C8Q8×D6712yes—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—2682no—
417C4⋊C4—2682no—
418C4⋊C4—671no—
419C4⋊C4—671no—
420C4⋊C4—1341no—
421C4⋊C4—1341no—
422C4⋊C4—1342no—
423C4⋊C4—1342no—
424C4⋊C4D4×D6711yescharacteristic
425C4⋊C4D4×D6711yescharacteristic
426C22⋊C4—21no—
427C22⋊C4—1341no—
428C22⋊C4—1341no—
429C22⋊C4—1341no—
430C42—21no—
431C42—1341no—
432C42—1341no—
433C42—1341no—
434C23—671no—
435C23—672no—
436C23—671no—
437C23—672no—
438C23—1342no—
439C23—1342no—
440C23#41×D6711yescharacteristic
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—2682no—
446C2×C4—2682no—
447C2×C4—2682no—
448C2×C4—5364no—
449C2×C4—5364no—
450C2×C4—5362no—
451C2×C4—672no—
452C2×C4—671no—
453C2×C4—671no—
454C2×C4—672no—
455C2×C4—672no—
456C2×C4—672no—
457C2×C4—672no—
458C2×C4—672no—
459C2×C4—672no—
460C2×C4—672no—
461C2×C4—672no—
462C2×C4—672no—
463C2×C4—1341no—
464C2×C4—1341no—
465C2×C4C22×C4×D6711yescharacteristic
466C2×C4C2×D4×D6711yescharacteristic
467C2×C4C2×D4×D6711yescharacteristic
468C2×C4C2×Q8×D6711yescharacteristic
469C2×C4C2×Q8×D6711yescharacteristic
470C2×C4#41×D6711yescharacteristic
471C8—24no—
472C8—22no—
473C8—672no—
474C8—672no—
475C8—1344no—
476C8—1342no—
477C8C4⋊C4×D6712yes—
478C8C4⋊C4×D6712yes—
479C22—21no—
480C22—21no—
481C22—672no—
482C22—672no—
483C22—672no—
484C22—672no—
485C22—1344no—
486C22—1344no—
487C22—1341no—
488C22—1341no—
489C22C4×Q8×D6711yescharacteristic
490C22#121×D6711yescharacteristic
491C22#125×D6711yescharacteristic
492C4—41no—
493C4—41no—
494C4—82no—
495C4—2681no—
496C4—2681no—
497C4—5362no—
498C4—672no—
499C4—672no—
500C4C2×C4⋊C4×D6711yescharacteristic
501C4C4×D4×D6711yescharacteristic
502C4C22⋊Q8×D6711yescharacteristic
503C4C4⋊Q8×D6711yescharacteristic
504C2—21no—
505C2—21no—
506C2—674no—
507C2—1342no—
508C2#389×D6711yescharacteristic
509C2#472×D6711yescharacteristic
510C2#473×D6711yescharacteristic
subgroupG/H
centerC22#121×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#1952C2
socleC22×C67C2×#121
prime radicalC23×D67#41
cosocleC22×C4×C67C24
hypercenter#1952D67
hypocenterC67C2×#1952
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
5. #22560995C2semidirect
6. #22560994C2semidirect
7. #22560993C2semidirect
8. #22560992C2semidirect
9. #22560991C2semidirect
11. C67×#1952C2semidirect
19. #108×D67C4semidirect
82. #14523979C2×C4semidirect
83. #14523978C2×C4semidirect
105. #108×C67C2×C4semidirect
255. D67#1952direct (N × H)
265. #1952D67direct (N × H)
267. C67C2×#1952semidirect
345. #108C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#108×D672
#428×D672
C2×#107×D672
#405×D672
#225609952
#225609942
#225609932
#225609922
#225609912
#343×D672
C67×#19522
C2×#195267