Finite Group Databasebeta

← C67×#33223 #30712474

Subgroups — C67×#33223 #30712474

Order 17152.

714 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×#501×C67C211yesmaximal, characteristic, complemented
2C67×#3080C211yesmaximal, characteristic
3C67×#3079C211yesmaximal, characteristic
4C67×#2875C211yesmaximal, characteristic
5C67×#1998C211yesmaximal, characteristic, complemented
6C67×#1997C211yesmaximal, characteristic, complemented
7C67×#1952C212yesmaximal, complemented
8C67×#1951C212yesmaximal, complemented
9C67×#1849C211yesmaximal, characteristic, complemented
10C67×#1816C212yesmaximal
11C67×#1815C212yesmaximal
12C2×#131×C67C2212yes—
13C2×C4⋊Q8×C67C2211yescharacteristic
14C2×#125×C67C2211yescharacteristic
15C4×#41×C67C2211yescharacteristic
16#501×C67—21no—
17#501×C67—21no—
18#501×C67—22no—
19#479×C67—21no—
20#479×C67—21no—
21#478×C67—21no—
22#478×C67—21no—
23#477×C67—21no—
24#476×C67—21no—
25#475×C67—21no—
26#474×C67—21no—
27C2×#108×C67C2212yes—
28C2×#108×C67C2212yes—
29#428×C67C2212yes—
30#428×C67C2211yescharacteristic
31#428×C67C2211yescharacteristic
32C2×#107×C67C2212yes—
33C2×#106×C67C2212yes—
34#423×C67C2211yescharacteristic
35#423×C67C2211yescharacteristic
36#417×C67C2212yes—
37#417×C67C2211yescharacteristic
38#417×C67C2211yescharacteristic
39#405×C67—22no—
40#405×C67C2212yes—
41C4×M16×C67—21no—
42C4×M16×C67C2211yescharacteristic
43C2×#97×C67—21no—
44C2×#97×C67C2211yescharacteristic
45#343×C67C2212yes—
46#343×C67C2212yes—
47#343×C67C2212yes—
48#343×C67C2212yes—
49C2×#41×C67C2311yescharacteristic
50C22×Q8×C67—21no—
51#131×C67—24no—
52#131×C67Q818yescomplemented
53C2×M16×C67—22no—
54C2×M16×C67C2311yescharacteristic
55C2×M16×C67C2311yescharacteristic
56C2×M16×C67C2312yes—
57C2×M16×C67C2312yes—
58C22×C8×C67—22no—
59C22×C8×C67C2312yes—
60C4⋊Q8×C67—21no—
61C4⋊Q8×C67—21no—
62C4⋊Q8×C67—21no—
63C4⋊Q8×C67—21no—
64#125×C67—21no—
65#125×C67—21no—
66#125×C67—21no—
67#125×C67—21no—
68C4×Q8×C67D411yescharacteristic
69C4×Q8×C67D411yescharacteristic
70C4×D4×C67—21no—
71C4×D4×C67—21no—
72C4×D4×C67D411yescharacteristic
73C4×D4×C67D411yescharacteristic
74#117×C67—21no—
75#117×C67C2311yescharacteristic
76C2×C4⋊C4×C67—21no—
77C2×C4⋊C4×C67—21no—
78C2×C4⋊C4×C67—21no—
79C2×C4⋊C4×C67C2311yescharacteristic
80C2×C4⋊C4×C67C2311yescharacteristic
81C2×C4⋊C4×C67C2312yes—
82C2×C42×C67—21no—
83C2×C42×C67C2311yescharacteristic
84#108×C67—24no—
85#108×C67—24no—
86#107×C67—22no—
87#107×C67—22no—
88#107×C67—22no—
89#107×C67—21no—
90#107×C67—21no—
91#106×C67—22no—
92#106×C67—22no—
93#106×C67—22no—
94#106×C67—21no—
95#106×C67—21no—
96C4⋊C8×C67—21no—
97C4⋊C8×C67—21no—
98C4⋊C8×C67—21no—
99C4⋊C8×C67—21no—
100#103×C67—22no—
101#103×C67—21no—
102#103×C67—21no—
103#102×C67—22no—
104#102×C67—21no—
105#102×C67—21no—
106#97×C67—22no—
107#97×C67—21no—
108#97×C67D411yescharacteristic
109#97×C67D411yescharacteristic
110#97×C67D412yes—
111C4×C8×C67—22no—
112C4×C8×C67—21no—
113#41×C67—22no—
114#41×C67C2×Q814yes—
115C2×Q8×C67—41no—
116C2×Q8×C67—41no—
117C2×Q8×C67—82no—
118C2×Q8×C67C2×D411yescharacteristic
119C2×D4×C67—21no—
120C2×D4×C67C2×D411yescharacteristic
121C22×C4×C67—21no—
122C22×C4×C67—21no—
123C22×C4×C67—21no—
124C22×C4×C67—21no—
125C22×C4×C67—21no—
126C22×C4×C67—22no—
127C22×C4×C67#4111yescharacteristic
128C22×C4×C67#4111yescharacteristic
129C22×C4×C67C2411yescharacteristic
130M16×C67—22no—
131M16×C67—28no—
132M16×C67—22no—
133M16×C67—22no—
134M16×C67—22no—
135M16×C67—22no—
136M16×C67C2×Q814yes—
137C2×C8×C67—22no—
138C2×C8×C67—28no—
139C2×C8×C67—22no—
140C2×C8×C67—21no—
141C2×C8×C67—21no—
142C2×C8×C67C2×D412yes—
143C2×C8×C67C2×D412yes—
144C2×C8×C67C2×Q814yes—
145C2×C8×C67#4114yes—
146C4⋊C4×C67—21no—
147C4⋊C4×C67—21no—
148C4⋊C4×C67—21no—
149C4⋊C4×C67—21no—
150C4⋊C4×C67—22no—
151C4⋊C4×C67—21no—
152C4⋊C4×C67—22no—
153C4⋊C4×C67—21no—
154C4⋊C4×C67—41no—
155C4⋊C4×C67—41no—
156C4⋊C4×C67—41no—
157C4⋊C4×C67—41no—
158C4⋊C4×C67—41no—
159C4⋊C4×C67—41no—
160C4⋊C4×C67C2×D411yescharacteristic
161C4⋊C4×C67C2×D411yescharacteristic
162C22⋊C4×C67—21no—
163C22⋊C4×C67—21no—
164C22⋊C4×C67—21no—
165C42×C67—21no—
166C42×C67—21no—
167C42×C67—21no—
168C42×C67C2×D411yescharacteristic
169C42×C67C2×D411yescharacteristic
170C42×C67C2×D411yescharacteristic
171C42×C67C2×D411yescharacteristic
172C23×C67—21no—
173C23×C67C2×#4111yescharacteristic
174Q8×C67—21no—
175Q8×C67—82no—
176Q8×C67—82no—
177Q8×C67C4⋊Q812yes—
178D4×C67—24no—
179D4×C67—21no—
180D4×C67C4⋊Q812yes—
181C2×C4×C67—21no—
182C2×C4×C67—24no—
183C2×C4×C67—21no—
184C2×C4×C67—21no—
185C2×C4×C67—41no—
186C2×C4×C67—41no—
187C2×C4×C67—41no—
188C2×C4×C67—41no—
189C2×C4×C67—42no—
190C2×C4×C67—81no—
191C2×C4×C67—81no—
192C2×C4×C67—81no—
193C2×C4×C67—81no—
194C2×C4×C67—82no—
195C2×C4×C67#12111yescharacteristic
196C2×C4×C67#12111yescharacteristic
197C2×C4×C67#12111yescharacteristic
198C2×C4×C67#12111yescharacteristic
199C2×C4×C67C22×D411yescharacteristic
200C2×C4×C67C22×D411yescharacteristic
201C2×C4×C67C22×Q812yes—
202C2×C4×C67C2×#4112yes—
203C8×C67—24no—
204C8×C67—42no—
205C8×C67—41no—
206C8×C67—41no—
207C8×C67C22⋊Q814yes—
208C8×C67C22⋊Q814yes—
209C22×C67—21no—
210C22×C67—21no—
211C22×C67—22no—
212C22×C67—41no—
213C22×C67—41no—
214C22×C67C2×#12111yescharacteristic
215C22×C67#53312yes—
216C4×C67—22no—
217C4×C67—21no—
218C4×C67—21no—
219C4×C67—21no—
220C4×C67—21no—
221C4×C67—41no—
222C4×C67—41no—
223C4×C67—81no—
224C4×C67—82no—
225C4×C67—81no—
226C4×C67C2×C22⋊Q812yes—
227C4×C67C2×C4⋊Q811yescharacteristic
228C4×C67#53211yescharacteristic
229#33223C6711yesmaximal, characteristic, direct product factor, Sylow
230C2×C67—22no—
231C2×C67—42no—
232C2×C67#285711yescharacteristic
233C2×C67#306811yescharacteristic
234C2×C67#306911yescharacteristic
235C2×#501C2×C6711yescharacteristic, complemented
236#3080C2×C6711yescharacteristic
237#3079C2×C6711yescharacteristic
238#2875C2×C6711yescharacteristic
239#1998C2×C6711yescharacteristic, complemented
240#1997C2×C6711yescharacteristic, complemented
241#1952C2×C6712yescomplemented
242#1951C2×C6712yescomplemented
243#1849C2×C6711yescharacteristic, complemented
244#1816C2×C6712yes—
245#1815C2×C6712yes—
246C67#3322311yescharacteristic, direct product factor, Sylow
247C2×#131C22×C6712yes—
248C2×C4⋊Q8C22×C6711yescharacteristic
249C2×#125C22×C6711yescharacteristic
250C4×#41C22×C6711yescharacteristic
251#501—21no—
252#501—21no—
253#501—22no—
254#479—21no—
255#479—21no—
256#478—21no—
257#478—21no—
258#477—21no—
259#476—21no—
260#475—21no—
261#474—21no—
262C2×#108C22×C6712yes—
263C2×#108C22×C6712yes—
264#428C22×C6712yes—
265#428C22×C6711yescharacteristic
266#428C22×C6711yescharacteristic
267C2×#107C22×C6712yes—
268C2×#106C22×C6712yes—
269#423C22×C6711yescharacteristic
270#423C22×C6711yescharacteristic
271#417C22×C6712yes—
272#417C22×C6711yescharacteristic
273#417C22×C6711yescharacteristic
274#405—22no—
275#405C22×C6712yes—
276C4×M16—21no—
277C4×M16C22×C6711yescharacteristic
278C2×#97—21no—
279C2×#97C22×C6711yescharacteristic
280#343C22×C6712yes—
281#343C22×C6712yes—
282#343C22×C6712yes—
283#343C22×C6712yes—
284C2×#41C23×C6711yescharacteristic
285C22×Q8—21no—
286#131—24no—
287#131Q8×C6718yescomplemented
288C2×M16—22no—
289C2×M16C23×C6712yes—
290C2×M16C23×C6711yescharacteristic
291C2×M16C23×C6711yescharacteristic
292C2×M16C23×C6712yes—
293C22×C8—22no—
294C22×C8C23×C6712yes—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299#125—21no—
300#125—21no—
301#125—21no—
302#125—21no—
303C4×Q8D4×C6711yescharacteristic
304C4×Q8D4×C6711yescharacteristic
305C4×D4—21no—
306C4×D4—21no—
307C4×D4D4×C6711yescharacteristic
308C4×D4D4×C6711yescharacteristic
309#117—21no—
310#117C23×C6711yescharacteristic
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C23×C6711yescharacteristic
315C2×C4⋊C4C23×C6711yescharacteristic
316C2×C4⋊C4C23×C6712yes—
317C2×C42—21no—
318C2×C42C23×C6711yescharacteristic
319#108—24no—
320#108—24no—
321#107—22no—
322#107—22no—
323#107—22no—
324#107—21no—
325#107—21no—
326#106—22no—
327#106—22no—
328#106—22no—
329#106—21no—
330#106—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335#103—22no—
336#103—21no—
337#103—21no—
338#102—22no—
339#102—21no—
340#102—21no—
341#97—22no—
342#97—21no—
343#97D4×C6711yescharacteristic
344#97D4×C6711yescharacteristic
345#97D4×C6712yes—
346C4×C8—22no—
347C4×C8—21no—
348#41—22no—
349#41C2×Q8×C6714yes—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—82no—
353C2×Q8C2×D4×C6711yescharacteristic
354C2×D4—21no—
355C2×D4C2×D4×C6711yescharacteristic
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4#41×C6711yescharacteristic
363C22×C4#41×C6711yescharacteristic
364C22×C4C24×C6711yescharacteristic
365M16—22no—
366M16—28no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—22no—
371M16C2×Q8×C6714yes—
372C2×C8—22no—
373C2×C8—28no—
374C2×C8—22no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8C2×D4×C6712yes—
378C2×C8C2×D4×C6712yes—
379C2×C8C2×Q8×C6714yes—
380C2×C8#41×C6714yes—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4C2×D4×C6711yescharacteristic
396C4⋊C4C2×D4×C6711yescharacteristic
397C22⋊C4—21no—
398C22⋊C4—21no—
399C22⋊C4—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42C2×D4×C6711yescharacteristic
404C42C2×D4×C6711yescharacteristic
405C42C2×D4×C6711yescharacteristic
406C42C2×D4×C6711yescharacteristic
407C23—21no—
408C23C2×#41×C6711yescharacteristic
409Q8—21no—
410Q8—82no—
411Q8—82no—
412Q8C4⋊Q8×C6712yes—
413D4—24no—
414D4—21no—
415D4C4⋊Q8×C6712yes—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4#121×C6711yescharacteristic
431C2×C4#121×C6711yescharacteristic
432C2×C4#121×C6711yescharacteristic
433C2×C4#121×C6711yescharacteristic
434C2×C4C22×D4×C6711yescharacteristic
435C2×C4C22×D4×C6711yescharacteristic
436C2×C4C22×Q8×C6712yes—
437C2×C4C2×#41×C6712yes—
438C8—24no—
439C8—42no—
440C8—41no—
441C8—41no—
442C8C22⋊Q8×C6714yes—
443C8C22⋊Q8×C6714yes—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22C2×#121×C6711yescharacteristic
450C22#533×C6712yes—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—82no—
459C4—81no—
460C4—81no—
461C4C2×C22⋊Q8×C6712yes—
462C4C2×C4⋊Q8×C6711yescharacteristic
463C4#532×C6711yescharacteristic
464C2—22no—
465C2—42no—
466C2C67×#285711yescharacteristic
467C2C67×#306811yescharacteristic
468C2C67×#306911yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC2×#41×C67C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#501×C67C2semidirect
5. C67×#1998C2semidirect
6. C67×#1997C2semidirect
7. C67×#1952C2semidirect
8. C67×#1951C2semidirect
9. C67×#1849C2semidirect
52. #131×C67Q8semidirect
229. #33223C67direct (N × H)
235. C2×#501C2×C67semidirect
239. #1998C2×C67semidirect
240. #1997C2×C67semidirect
241. #1952C2×C67semidirect
242. #1951C2×C67semidirect
243. #1849C2×C67semidirect
246. C67#33223direct (N × H)
287. #131Q8×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#501×C672
C67×#30802
C67×#30792
C67×#28752
C67×#19982
C67×#19972
C67×#19522
C67×#19512
C67×#18492
C67×#18162
C67×#18152
#3322367