Finite Group Databasebeta

← C67×#25923 #30705174

Subgroups — C67×#25923 #30705174

Order 17152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3503C211yesmaximal, characteristic
2C67×#3139C211yesmaximal, characteristic, complemented
3C67×#3039C211yesmaximal, characteristic
4C67×#2862C211yesmaximal, characteristic, complemented
5C67×#2028C214yesmaximal, complemented
6C67×#1997C211yesmaximal, characteristic, complemented
7C67×#1994C211yesmaximal, characteristic, complemented
8C67×#1882C214yesmaximal
9C4×#104×C67C211yesmaximal, characteristic
10C2×#137×C67C2212yes—
11Q82×C67—21no—
12#557×C67—21no—
13#554×C67—21no—
14#554×C67—22no—
15D4×Q8×C67—22no—
16D4×Q8×C67—21no—
17#544×C67—22no—
18#544×C67C2212yes—
19#533×C67—21no—
20#533×C67C2211yescharacteristic
21C2×C4⋊Q8×C67—21no—
22C2×C4⋊Q8×C67C2211yescharacteristic
23C2×C4⋊Q8×C67C2211yescharacteristic
24#520×C67C2212yes—
25C4×#41×C67C2211yescharacteristic
26#491×C67—22no—
27#487×C67—21no—
28#487×C67—21no—
29#462×C67—21no—
30#462×C67—21no—
31#461×C67—21no—
32#461×C67—21no—
33#423×C67C2211yescharacteristic
34C2×#104×C67C2212yes—
35#419×C67C2212yes—
36#419×C67C2214yes—
37C4×M16×C67C2211yescharacteristic
38#391×C67—22no—
39#391×C67C2214yes—
40#382×C67—22no—
41C43×C67C2211yescharacteristic
42#343×C67C2214yes—
43#339×C67C2214yes—
44#339×C67C2212yes—
45C2×#41×C67C2311yescharacteristic
46C22×Q8×C67—22no—
47C22×Q8×C67D412yes—
48C22×Q8×C67C2312yes—
49#137×C67—28no—
50C2×Q16×C67—22no—
51C2×SD16×C67—22no—
52C2×M16×C67C2312yes—
53C2×M16×C67C2312yes—
54C4⋊Q8×C67—22no—
55C4⋊Q8×C67—22no—
56C4⋊Q8×C67—24no—
57C4⋊Q8×C67—22no—
58C4⋊Q8×C67—22no—
59C4⋊Q8×C67—21no—
60C4⋊Q8×C67—21no—
61C4⋊Q8×C67—21no—
62C4⋊Q8×C67—21no—
63C4⋊Q8×C67—21no—
64C4⋊Q8×C67—21no—
65C4⋊Q8×C67—21no—
66C4⋊Q8×C67—21no—
67C4⋊Q8×C67D411yescharacteristic
68C4⋊Q8×C67D411yescharacteristic
69#125×C67—24no—
70#125×C67—22no—
71#125×C67—22no—
72#125×C67D412yes—
73C22⋊Q8×C67—22no—
74C22⋊Q8×C67—22no—
75C22⋊Q8×C67—24no—
76C22⋊Q8×C67—24no—
77C22⋊Q8×C67—21no—
78C22⋊Q8×C67—21no—
79C22⋊Q8×C67—24no—
80C4×Q8×C67—22no—
81C4×Q8×C67—22no—
82C4×Q8×C67—22no—
83C4×Q8×C67—21no—
84C4×Q8×C67—21no—
85C4×Q8×C67D411yescharacteristic
86C4×Q8×C67D411yescharacteristic
87C4×D4×C67—22no—
88C4×D4×C67—21no—
89C4×D4×C67—21no—
90C4×D4×C67—24no—
91C4×D4×C67D411yescharacteristic
92C4×D4×C67D411yescharacteristic
93#117×C67—22no—
94#117×C67—21no—
95#117×C67C2311yescharacteristic
96#117×C67C2312yes—
97C2×C4⋊C4×C67—22no—
98C2×C4⋊C4×C67—22no—
99C2×C4⋊C4×C67—21no—
100C2×C4⋊C4×C67—21no—
101C2×C4⋊C4×C67D412yes—
102C2×C4⋊C4×C67C2312yes—
103C2×C42×C67—21no—
104C2×C42×C67—21no—
105C2×C42×C67—21no—
106C2×C42×C67C2312yes—
107C2×C42×C67C2311yescharacteristic
108C4⋊C8×C67—21no—
109C4⋊C8×C67—21no—
110#104×C67—24no—
111#103×C67—22no—
112#103×C67—22no—
113#103×C67—24no—
114#103×C67—24no—
115#97×C67—21no—
116C4×C8×C67—21no—
117#95×C67—28no—
118#41×C67—22no—
119#41×C67#4114yes—
120C2×Q8×C67—24no—
121C2×Q8×C67—21no—
122C2×Q8×C67—22no—
123C2×Q8×C67—21no—
124C2×Q8×C67—22no—
125C2×Q8×C67—22no—
126C2×Q8×C67—22no—
127C2×Q8×C67—24no—
128C2×Q8×C67—24no—
129C2×Q8×C67—44no—
130C2×Q8×C67—84no—
131C2×Q8×C67C2×D411yescharacteristic
132C2×D4×C67—21no—
133C2×D4×C67C2×D411yescharacteristic
134C22×C4×C67—21no—
135C22×C4×C67—22no—
136C22×C4×C67—24no—
137C22×C4×C67—21no—
138C22×C4×C67—21no—
139C22×C4×C67—22no—
140C22×C4×C67C2×D411yescharacteristic
141C22×C4×C67C2×D411yescharacteristic
142C22×C4×C67C2×D414yes—
143C22×C4×C67C2411yescharacteristic
144Q16×C67—44no—
145SD16×C67—44no—
146M16×C67—22no—
147M16×C67—24no—
148M16×C67—22no—
149C2×C8×C67—22no—
150C2×C8×C67—22no—
151C4⋊C4×C67—24no—
152C4⋊C4×C67—24no—
153C4⋊C4×C67—24no—
154C4⋊C4×C67—24no—
155C4⋊C4×C67—21no—
156C4⋊C4×C67—22no—
157C4⋊C4×C67—22no—
158C4⋊C4×C67—21no—
159C4⋊C4×C67—21no—
160C4⋊C4×C67—21no—
161C4⋊C4×C67—22no—
162C4⋊C4×C67—22no—
163C4⋊C4×C67—22no—
164C4⋊C4×C67—22no—
165C4⋊C4×C67—41no—
166C4⋊C4×C67—41no—
167C4⋊C4×C67—41no—
168C4⋊C4×C67—41no—
169C4⋊C4×C67C2×D412yes—
170C4⋊C4×C67C2×D412yes—
171C4⋊C4×C67C2×D411yescharacteristic
172C4⋊C4×C67C2×D411yescharacteristic
173C22⋊C4×C67—24no—
174C22⋊C4×C67—22no—
175C22⋊C4×C67—21no—
176C22⋊C4×C67—21no—
177C22⋊C4×C67—21no—
178C42×C67—24no—
179C42×C67—24no—
180C42×C67—21no—
181C42×C67—21no—
182C42×C67—21no—
183C42×C67—22no—
184C42×C67—22no—
185C42×C67—22no—
186C42×C67—42no—
187C42×C67C2×D411yescharacteristic
188C42×C67C2×D411yescharacteristic
189C42×C67C2×D411yescharacteristic
190C42×C67C2×D411yescharacteristic
191C23×C67—21no—
192C23×C67C22×D411yescharacteristic
193Q8×C67—21no—
194Q8×C67—44no—
195Q8×C67—42no—
196Q8×C67—42no—
197Q8×C67—42no—
198Q8×C67—42no—
199Q8×C67—44no—
200Q8×C67—82no—
201Q8×C67—84no—
202Q8×C67#12412yes—
203D4×C67—24no—
204D4×C67—21no—
205D4×C67#12412yes—
206C2×C4×C67—21no—
207C2×C4×C67—24no—
208C2×C4×C67—22no—
209C2×C4×C67—22no—
210C2×C4×C67—24no—
211C2×C4×C67—22no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—24no—
215C2×C4×C67—22no—
216C2×C4×C67—42no—
217C2×C4×C67—42no—
218C2×C4×C67—42no—
219C2×C4×C67—42no—
220C2×C4×C67—44no—
221C2×C4×C67—42no—
222C2×C4×C67—44no—
223C2×C4×C67—84no—
224C2×C4×C67#12011yescharacteristic
225C2×C4×C67#12011yescharacteristic
226C2×C4×C67#12011yescharacteristic
227C2×C4×C67#12011yescharacteristic
228C2×C4×C67C22×D411yescharacteristic
229C2×C4×C67C22×D411yescharacteristic
230C2×C4×C67C2×#4112yes—
231C2×C4×C672−1+412yes—
232C8×C67—42no—
233C8×C67—42no—
234C22×C67—21no—
235C22×C67—21no—
236C22×C67—22no—
237C22×C67—41no—
238C22×C67—41no—
239C22×C67C2×#12011yescharacteristic
240C22×C67#54812yes—
241C4×C67—22no—
242C4×C67—21no—
243C4×C67—21no—
244C4×C67—44no—
245C4×C67—41no—
246C4×C67—44no—
247C4×C67—42no—
248C4×C67—41no—
249C4×C67—42no—
250C4×C67—82no—
251C4×C67#45512yes—
252C4×C67#45612yes—
253C4×C67C2×#12411yescharacteristic
254C4×C67#53611yescharacteristic
255C4×C67#54812yes—
256#25923C6711yesmaximal, characteristic, direct product factor, Sylow
257C2×C67—22no—
258C2×C67—42no—
259C2×C67#243611yescharacteristic
260C2×C67C2×#45511yescharacteristic
261C2×C67C2×#45611yescharacteristic
262#3503C2×C6711yescharacteristic
263#3139C2×C6711yescharacteristic, complemented
264#3039C2×C6711yescharacteristic
265#2862C2×C6711yescharacteristic, complemented
266#2028C2×C6714yescomplemented
267#1997C2×C6711yescharacteristic, complemented
268#1994C2×C6711yescharacteristic, complemented
269#1882C2×C6714yes—
270C4×#104C2×C6711yescharacteristic
271C67#2592311yescharacteristic, direct product factor, Sylow
272C2×#137C22×C6712yes—
273Q82—21no—
274#557—21no—
275#554—21no—
276#554—22no—
277D4×Q8—22no—
278D4×Q8—21no—
279#544—22no—
280#544C22×C6712yes—
281#533—21no—
282#533C22×C6711yescharacteristic
283C2×C4⋊Q8—21no—
284C2×C4⋊Q8C22×C6711yescharacteristic
285C2×C4⋊Q8C22×C6711yescharacteristic
286#520C22×C6712yes—
287C4×#41C22×C6711yescharacteristic
288#491—22no—
289#487—21no—
290#487—21no—
291#462—21no—
292#462—21no—
293#461—21no—
294#461—21no—
295#423C22×C6711yescharacteristic
296C2×#104C22×C6712yes—
297#419C22×C6714yes—
298#419C22×C6712yes—
299C4×M16C22×C6711yescharacteristic
300#391—22no—
301#391C22×C6714yes—
302#382—22no—
303C43C22×C6711yescharacteristic
304#343C22×C6714yes—
305#339C22×C6714yes—
306#339C22×C6712yes—
307C2×#41C23×C6711yescharacteristic
308C22×Q8—22no—
309C22×Q8D4×C6712yes—
310C22×Q8C23×C6712yes—
311#137—28no—
312C2×Q16—22no—
313C2×SD16—22no—
314C2×M16C23×C6712yes—
315C2×M16C23×C6712yes—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
318C4⋊Q8—24no—
319C4⋊Q8—22no—
320C4⋊Q8—22no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8D4×C6711yescharacteristic
330C4⋊Q8D4×C6711yescharacteristic
331#125—24no—
332#125—22no—
333#125—22no—
334#125D4×C6712yes—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—24no—
338C22⋊Q8—24no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—24no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8D4×C6711yescharacteristic
348C4×Q8D4×C6711yescharacteristic
349C4×D4—22no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—24no—
353C4×D4D4×C6711yescharacteristic
354C4×D4D4×C6711yescharacteristic
355#117—22no—
356#117—21no—
357#117C23×C6711yescharacteristic
358#117C23×C6712yes—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4D4×C6712yes—
364C2×C4⋊C4C23×C6712yes—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42C23×C6712yes—
369C2×C42C23×C6711yescharacteristic
370C4⋊C8—21no—
371C4⋊C8—21no—
372#104—24no—
373#103—22no—
374#103—22no—
375#103—24no—
376#103—24no—
377#97—21no—
378C4×C8—21no—
379#95—28no—
380#41—22no—
381#41#41×C6714yes—
382C2×Q8—24no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—44no—
392C2×Q8—84no—
393C2×Q8C2×D4×C6711yescharacteristic
394C2×D4—21no—
395C2×D4C2×D4×C6711yescharacteristic
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4C2×D4×C6711yescharacteristic
403C22×C4C2×D4×C6711yescharacteristic
404C22×C4C2×D4×C6714yes—
405C22×C4C24×C6711yescharacteristic
406Q16—44no—
407SD16—44no—
408M16—22no—
409M16—24no—
410M16—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4C2×D4×C6712yes—
432C4⋊C4C2×D4×C6712yes—
433C4⋊C4C2×D4×C6711yescharacteristic
434C4⋊C4C2×D4×C6711yescharacteristic
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C42—24no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—42no—
449C42C2×D4×C6711yescharacteristic
450C42C2×D4×C6711yescharacteristic
451C42C2×D4×C6711yescharacteristic
452C42C2×D4×C6711yescharacteristic
453C23—21no—
454C23C22×D4×C6711yescharacteristic
455Q8—21no—
456Q8—44no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460Q8—42no—
461Q8—44no—
462Q8—82no—
463Q8—84no—
464Q8#124×C6712yes—
465D4—24no—
466D4—21no—
467D4#124×C6712yes—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—84no—
486C2×C4#120×C6711yescharacteristic
487C2×C4#120×C6711yescharacteristic
488C2×C4#120×C6711yescharacteristic
489C2×C4#120×C6711yescharacteristic
490C2×C4C22×D4×C6711yescharacteristic
491C2×C4C22×D4×C6711yescharacteristic
492C2×C4C2×#41×C6712yes—
493C2×C42−1+4×C6712yes—
494C8—42no—
495C8—42no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22C2×#120×C6711yescharacteristic
502C22#548×C6712yes—
503C4—21no—
504C4—21no—
505C4—22no—
506C4—44no—
507C4—41no—
508C4—44no—
509C4—42no—
510C4—41no—
511C4—42no—
512C4—82no—
513C4#455×C6712yes—
514C4#456×C6712yes—
515C4C2×#124×C6711yescharacteristic
516C4#536×C6711yescharacteristic
517C4#548×C6712yes—
518C2—22no—
519C2—42no—
520C2C67×#243611yescharacteristic
521C2C2×#455×C6711yescharacteristic
522C2C2×#456×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
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
2. C67×#3139C2semidirect
4. C67×#2862C2semidirect
5. C67×#2028C2semidirect
6. C67×#1997C2semidirect
7. C67×#1994C2semidirect
256. #25923C67direct (N × H)
263. #3139C2×C67semidirect
265. #2862C2×C67semidirect
266. #2028C2×C67semidirect
267. #1997C2×C67semidirect
268. #1994C2×C67semidirect
271. C67#25923direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#35032
C67×#31392
C67×#30392
C67×#28622
C67×#20282
C67×#19972
C67×#19942
C67×#18822
C4×#104×C672
#2592367