Finite Group Databasebeta

← C67×#36698 #30715949

Subgroups — C67×#36698 #30715949

Order 17152.

1484 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2844C211yesmaximal, characteristic, complemented
2C67×#2838C214yesmaximal, complemented
3C67×#2836C214yesmaximal, complemented
4C67×#2833C212yesmaximal, complemented
5C67×#2830C212yesmaximal
6C67×#2822C211yesmaximal, characteristic, complemented
7C67×#2820C212yesmaximal
8C67×#2401C214yesmaximal, complemented
9C67×#2399C212yesmaximal, complemented
10C67×#2396C214yesmaximal, complemented
11C67×#2395C212yesmaximal
12C4×#125×C67C212yesmaximal, complemented
13C4×#124×C67C211yesmaximal, characteristic
14C2×#125×C67C2212yes—
15C2×#124×C67C2211yescharacteristic
16C2×C4×Q8×C67C2211yescharacteristic
17C2×C4×Q8×C67C2212yes—
18C2×C4×D4×C67C2211yescharacteristic
19C2×#117×C67C2212yes—
20#401×C67C2214yes—
21#400×C67C2214yes—
22#400×C67C2214yes—
23#400×C67C2214yes—
24#400×C67C2212yes—
25#400×C67C2214yes—
26#399×C67C2214yes—
27#399×C67C2212yes—
28#398×C67C2212yes—
29#398×C67C2212yes—
30#398×C67C2214yes—
31#398×C67C2214yes—
32#398×C67C2212yes—
33#398×C67C2214yes—
34#397×C67C2214yes—
35#397×C67C2214yes—
36#397×C67C2214yes—
37#397×C67C2212yes—
38#397×C67C2214yes—
39#396×C67C2214yes—
40#396×C67C2214yes—
41#396×C67C2214yes—
42#395×C67C2214yes—
43#395×C67C2212yes—
44#395×C67C2212yes—
45#395×C67C2214yes—
46#394×C67C2214yes—
47#393×C67C2214yes—
48#391×C67C2211yescharacteristic
49#390×C67C2211yescharacteristic
50#389×C67C2212yes—
51#389×C67C2212yes—
52#388×C67C2214yes—
53#387×C67C2214yes—
54#387×C67C2214yes—
55#382×C67C2211yescharacteristic
56#382×C67C2212yes—
57C4×C4⋊C4×C67C2214yes—
58C4×C4⋊C4×C67C2212yes—
59C4×C4⋊C4×C67C2214yes—
60C4×C4⋊C4×C67C2212yes—
61C4×C4⋊C4×C67C2214yes—
62C4×C22⋊C4×C67C2214yes—
63C4×C22⋊C4×C67C2214yes—
64C4×C22⋊C4×C67C2212yes—
65#376×C67C2212yes—
66C43×C67C2211yescharacteristic
67C22×Q8×C67C2311yescharacteristic
68C22×Q8×C67C2312yes—
69C22×D4×C67C2311yescharacteristic
70C23×C4×C67C2312yes—
71#125×C67—28no—
72#124×C67—24no—
73C4×Q8×C67—44no—
74C4×Q8×C67—42no—
75C4×D4×C67—42no—
76#117×C67—44no—
77C2×C4⋊C4×C67C2314yes—
78C2×C4⋊C4×C67C2314yes—
79C2×C4⋊C4×C67C2312yes—
80C2×C4⋊C4×C67C2312yes—
81C2×C4⋊C4×C67C2312yes—
82C2×C4⋊C4×C67C2312yes—
83C2×C4⋊C4×C67C2311yescharacteristic
84C2×C4⋊C4×C67C2311yescharacteristic
85C2×C4⋊C4×C67C2312yes—
86C2×C4⋊C4×C67C2314yes—
87C2×C4⋊C4×C67C2312yes—
88C2×C4⋊C4×C67C2312yes—
89C2×C4⋊C4×C67C2312yes—
90C2×C4⋊C4×C67C2314yes—
91C2×C4⋊C4×C67C2314yes—
92C2×C4⋊C4×C67C2314yes—
93C2×C4⋊C4×C67C2314yes—
94C2×C22⋊C4×C67C2314yes—
95C2×C22⋊C4×C67C2314yes—
96C2×C22⋊C4×C67C2312yes—
97C2×C22⋊C4×C67C2314yes—
98C2×C22⋊C4×C67C2314yes—
99C2×C22⋊C4×C67C2314yes—
100C2×C22⋊C4×C67C2314yes—
101C2×C42×C67C2314yes—
102C2×C42×C67C2311yescharacteristic
103C2×C42×C67C2312yes—
104C2×C42×C67C2312yes—
105C2×C42×C67C2314yes—
106C2×C42×C67C2311yescharacteristic
107C2×C42×C67C2312yes—
108C2×C42×C67C2311yescharacteristic
109#95×C67C2314yes—
110#95×C67C2312yes—
111#95×C67C2312yes—
112#95×C67C2314yes—
113#95×C67C2314yes—
114#95×C67C2314yes—
115#95×C67C2314yes—
116#95×C67C2312yes—
117#95×C67C2312yes—
118#95×C67C2314yes—
119#95×C67C2314yes—
120#95×C67C2314yes—
121#95×C67C2314yes—
122#95×C67C2314yes—
123#95×C67C2314yes—
124#95×C67C2314yes—
125#95×C67C2314yes—
126C24×C67C2412yes—
127C2×Q8×C67—22no—
128C2×Q8×C67—41no—
129C2×Q8×C67—42no—
130C2×Q8×C67—41no—
131C2×Q8×C67—44no—
132C2×D4×C67—22no—
133C2×D4×C67—41no—
134C2×D4×C67—41no—
135C22×C4×C67—24no—
136C22×C4×C67—28no—
137C22×C4×C67C2411yescharacteristic
138C22×C4×C67C2412yes—
139C22×C4×C67C2414yes—
140C22×C4×C67C2412yes—
141C22×C4×C67C2414yes—
142C22×C4×C67C2412yes—
143C22×C4×C67C2414yes—
144C22×C4×C67C2414yes—
145C22×C4×C67C2412yes—
146C22×C4×C67C2412yes—
147C22×C4×C67C2411yescharacteristic
148C22×C4×C67C2411yescharacteristic
149C4⋊C4×C67—28no—
150C4⋊C4×C67—24no—
151C4⋊C4×C67—24no—
152C4⋊C4×C67—28no—
153C4⋊C4×C67—42no—
154C4⋊C4×C67—42no—
155C4⋊C4×C67—44no—
156C4⋊C4×C67—44no—
157C4⋊C4×C67—44no—
158C4⋊C4×C67—44no—
159C4⋊C4×C67—42no—
160C4⋊C4×C67—42no—
161C4⋊C4×C67—41no—
162C4⋊C4×C67—41no—
163C4⋊C4×C67—42no—
164C4⋊C4×C67—42no—
165C4⋊C4×C67—44no—
166C22⋊C4×C67—28no—
167C22⋊C4×C67—44no—
168C22⋊C4×C67—44no—
169C22⋊C4×C67—44no—
170C22⋊C4×C67—44no—
171C22⋊C4×C67—44no—
172C22⋊C4×C67—42no—
173C42×C67—28no—
174C42×C67—22no—
175C42×C67—24no—
176C42×C67—24no—
177C42×C67—28no—
178C42×C67—42no—
179C42×C67—41no—
180C42×C67#4114yes—
181C23×C67—22no—
182C23×C67—22no—
183C23×C67—24no—
184C23×C67—24no—
185C23×C67—22no—
186C23×C67C2511yescharacteristic
187Q8×C67—44no—
188Q8×C67—48no—
189D4×C67—44no—
190C2×C4×C67—21no—
191C2×C4×C67—22no—
192C2×C4×C67—22no—
193C2×C4×C67—21no—
194C2×C4×C67—22no—
195C2×C4×C67—24no—
196C2×C4×C67—24no—
197C2×C4×C67—21no—
198C2×C4×C67—24no—
199C2×C4×C67—24no—
200C2×C4×C67—22no—
201C2×C4×C67—22no—
202C2×C4×C67—22no—
203C2×C4×C67—24no—
204C2×C4×C67—22no—
205C2×C4×C67—24no—
206C2×C4×C67—24no—
207C2×C4×C67—22no—
208C2×C4×C67—22no—
209C2×C4×C67—22no—
210C2×C4×C67—24no—
211C2×C4×C67—24no—
212C2×C4×C67—24no—
213C2×C4×C67—24no—
214C2×C4×C67—24no—
215C2×C4×C67—24no—
216C2×C4×C67—22no—
217C2×C4×C67—24no—
218C2×C4×C67—48no—
219C2×C4×C67C2×#4114yes—
220C2×C4×C67C2×#4112yes—
221C2×C4×C67H(5,𝔽2)12yes—
222C2×C4×C672−1+414yes—
223C22×C67—42no—
224C22×C67—44no—
225C22×C67—42no—
226C22×C67—42no—
227C22×C67—44no—
228C22×C67C22×#4111yescharacteristic
229C22×C67C2×H(5,𝔽2)11yescharacteristic
230C22×C67C2×2−1+412yes—
231C22×C67#58511yescharacteristic
232C22×C67#58512yes—
233C4×C67—22no—
234C4×C67—24no—
235C4×C67—44no—
236C4×C67—42no—
237C4×C67—44no—
238C4×C67—42no—
239C4×C67—41no—
240C4×C67—44no—
241C4×C67—42no—
242C4×C67—42no—
243C4×C67—44no—
244C4×C67#56314yes—
245#36698C6711yesmaximal, characteristic, direct product factor, Sylow
246C2×C67—82no—
247C2×C67#346011yescharacteristic
248C2×C67#346112yes—
249C2×C67C2×#56311yescharacteristic
250C2×C67#352911yescharacteristic
251C2×C67#353112yes—
252#2844C2×C6711yescharacteristic, complemented
253#2838C2×C6714yescomplemented
254#2836C2×C6714yescomplemented
255#2833C2×C6712yescomplemented
256#2830C2×C6712yes—
257#2822C2×C6711yescharacteristic, complemented
258#2820C2×C6712yes—
259#2401C2×C6714yescomplemented
260#2399C2×C6712yescomplemented
261#2396C2×C6714yescomplemented
262#2395C2×C6712yes—
263C4×#125C2×C6712yescomplemented
264C4×#124C2×C6711yescharacteristic
265C67#3669811yescharacteristic, direct product factor, Sylow
266C2×#125C22×C6712yes—
267C2×#124C22×C6711yescharacteristic
268C2×C4×Q8C22×C6712yes—
269C2×C4×Q8C22×C6711yescharacteristic
270C2×C4×D4C22×C6711yescharacteristic
271C2×#117C22×C6712yes—
272#401C22×C6714yes—
273#400C22×C6714yes—
274#400C22×C6714yes—
275#400C22×C6714yes—
276#400C22×C6712yes—
277#400C22×C6714yes—
278#399C22×C6714yes—
279#399C22×C6712yes—
280#398C22×C6712yes—
281#398C22×C6712yes—
282#398C22×C6714yes—
283#398C22×C6714yes—
284#398C22×C6712yes—
285#398C22×C6714yes—
286#397C22×C6714yes—
287#397C22×C6714yes—
288#397C22×C6714yes—
289#397C22×C6712yes—
290#397C22×C6714yes—
291#396C22×C6714yes—
292#396C22×C6714yes—
293#396C22×C6714yes—
294#395C22×C6714yes—
295#395C22×C6712yes—
296#395C22×C6712yes—
297#395C22×C6714yes—
298#394C22×C6714yes—
299#393C22×C6714yes—
300#391C22×C6711yescharacteristic
301#390C22×C6711yescharacteristic
302#389C22×C6712yes—
303#389C22×C6712yes—
304#388C22×C6714yes—
305#387C22×C6714yes—
306#387C22×C6714yes—
307#382C22×C6712yes—
308#382C22×C6711yescharacteristic
309C4×C4⋊C4C22×C6714yes—
310C4×C4⋊C4C22×C6712yes—
311C4×C4⋊C4C22×C6714yes—
312C4×C4⋊C4C22×C6712yes—
313C4×C4⋊C4C22×C6714yes—
314C4×C22⋊C4C22×C6714yes—
315C4×C22⋊C4C22×C6712yes—
316C4×C22⋊C4C22×C6714yes—
317#376C22×C6712yes—
318C43C22×C6711yescharacteristic
319C22×Q8C23×C6711yescharacteristic
320C22×Q8C23×C6712yes—
321C22×D4C23×C6711yescharacteristic
322C23×C4C23×C6712yes—
323#125—28no—
324#124—24no—
325C4×Q8—44no—
326C4×Q8—42no—
327C4×D4—42no—
328#117—44no—
329C2×C4⋊C4C23×C6714yes—
330C2×C4⋊C4C23×C6714yes—
331C2×C4⋊C4C23×C6712yes—
332C2×C4⋊C4C23×C6712yes—
333C2×C4⋊C4C23×C6712yes—
334C2×C4⋊C4C23×C6712yes—
335C2×C4⋊C4C23×C6711yescharacteristic
336C2×C4⋊C4C23×C6711yescharacteristic
337C2×C4⋊C4C23×C6712yes—
338C2×C4⋊C4C23×C6714yes—
339C2×C4⋊C4C23×C6712yes—
340C2×C4⋊C4C23×C6712yes—
341C2×C4⋊C4C23×C6712yes—
342C2×C4⋊C4C23×C6714yes—
343C2×C4⋊C4C23×C6714yes—
344C2×C4⋊C4C23×C6714yes—
345C2×C4⋊C4C23×C6714yes—
346C2×C22⋊C4C23×C6714yes—
347C2×C22⋊C4C23×C6714yes—
348C2×C22⋊C4C23×C6712yes—
349C2×C22⋊C4C23×C6714yes—
350C2×C22⋊C4C23×C6714yes—
351C2×C22⋊C4C23×C6714yes—
352C2×C22⋊C4C23×C6714yes—
353C2×C42C23×C6714yes—
354C2×C42C23×C6711yescharacteristic
355C2×C42C23×C6712yes—
356C2×C42C23×C6712yes—
357C2×C42C23×C6714yes—
358C2×C42C23×C6711yescharacteristic
359C2×C42C23×C6712yes—
360C2×C42C23×C6711yescharacteristic
361#95C23×C6714yes—
362#95C23×C6712yes—
363#95C23×C6712yes—
364#95C23×C6714yes—
365#95C23×C6714yes—
366#95C23×C6714yes—
367#95C23×C6714yes—
368#95C23×C6712yes—
369#95C23×C6712yes—
370#95C23×C6714yes—
371#95C23×C6714yes—
372#95C23×C6714yes—
373#95C23×C6714yes—
374#95C23×C6714yes—
375#95C23×C6714yes—
376#95C23×C6714yes—
377#95C23×C6714yes—
378C24C24×C6712yes—
379C2×Q8—22no—
380C2×Q8—41no—
381C2×Q8—42no—
382C2×Q8—41no—
383C2×Q8—44no—
384C2×D4—22no—
385C2×D4—41no—
386C2×D4—41no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4C24×C6712yes—
390C22×C4C24×C6712yes—
391C22×C4C24×C6714yes—
392C22×C4C24×C6714yes—
393C22×C4C24×C6712yes—
394C22×C4C24×C6714yes—
395C22×C4C24×C6712yes—
396C22×C4C24×C6711yescharacteristic
397C22×C4C24×C6714yes—
398C22×C4C24×C6712yes—
399C22×C4C24×C6711yescharacteristic
400C22×C4C24×C6711yescharacteristic
401C4⋊C4—28no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—28no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C22⋊C4—28no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C42—28no—
426C42—22no—
427C42—24no—
428C42—24no—
429C42—28no—
430C42—42no—
431C42—41no—
432C42#41×C6714yes—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23C25×C6711yescharacteristic
439Q8—44no—
440Q8—48no—
441D4—44no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—48no—
471C2×C4C2×#41×C6714yes—
472C2×C4C2×#41×C6712yes—
473C2×C4H(5,𝔽2)×C6712yes—
474C2×C42−1+4×C6714yes—
475C22—42no—
476C22—42no—
477C22—44no—
478C22—42no—
479C22—44no—
480C22C22×#41×C6711yescharacteristic
481C22C2×H(5,𝔽2)×C6711yescharacteristic
482C22C2×2−1+4×C6712yes—
483C22#585×C6712yes—
484C22#585×C6711yescharacteristic
485C4—22no—
486C4—24no—
487C4—42no—
488C4—44no—
489C4—42no—
490C4—41no—
491C4—42no—
492C4—44no—
493C4—42no—
494C4—44no—
495C4—44no—
496C4#563×C6714yes—
497C2—82no—
498C2C67×#346011yescharacteristic
499C2C67×#346112yes—
500C2C2×#563×C6711yescharacteristic
501C2C67×#352911yescharacteristic
502C2C67×#353112yes—
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
prime radicalC22×D4×C67C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2844C2semidirect
2. C67×#2838C2semidirect
3. C67×#2836C2semidirect
4. C67×#2833C2semidirect
6. C67×#2822C2semidirect
8. C67×#2401C2semidirect
9. C67×#2399C2semidirect
10. C67×#2396C2semidirect
12. C4×#125×C67C2semidirect
245. #36698C67direct (N × H)
252. #2844C2×C67semidirect
253. #2838C2×C67semidirect
254. #2836C2×C67semidirect
255. #2833C2×C67semidirect
257. #2822C2×C67semidirect
259. #2401C2×C67semidirect
260. #2399C2×C67semidirect
261. #2396C2×C67semidirect
263. C4×#125C2×C67semidirect
265. C67#36698direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#28442
C67×#28382
C67×#28362
C67×#28332
C67×#28302
C67×#28222
C67×#28202
C67×#24012
C67×#23992
C67×#23962
C67×#23952
C4×#125×C672
C4×#124×C672
#3669867