Finite Group Databasebeta

← C67×#34522 #30713773

Subgroups — C67×#34522 #30713773

Order 17152.

2216 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#562×C67C211yesmaximal, characteristic
2C67×#2382C211yesmaximal, characteristic, complemented
3C67×#2379C212yesmaximal, complemented
4C67×#2368C214yesmaximal, complemented
5C67×#2363C214yesmaximal, complemented
6C67×#2359C214yesmaximal, complemented
7C67×#2335C214yesmaximal, complemented
8C67×#2325C214yesmaximal, complemented
9C67×#2323C214yesmaximal, complemented
10C4×#125×C67C211yesmaximal, characteristic, complemented
11C4×#124×C67C212yesmaximal, complemented
12#562×C67C4116yescomplemented
13C2×#125×C67C2211yescharacteristic
14C2×#124×C67C2212yes—
15C2×#123×C67C2214yes—
16C2×C22⋊Q8×C67C2214yes—
17C2×#121×C67C2214yes—
18C2×C4×Q8×C67C2212yes—
19C2×C4×D4×C67C2212yes—
20C2×C4×D4×C67C2214yes—
21C2×#117×C67C2214yes—
22#391×C67C2212yes—
23#391×C67C2214yes—
24#390×C67C2212yes—
25#390×C67C2214yes—
26#389×C67C2212yes—
27#389×C67C2214yes—
28#389×C67C2214yes—
29#388×C67C2214yes—
30#388×C67C2214yes—
31#388×C67C2214yes—
32#388×C67C2214yes—
33#388×C67C2214yes—
34#388×C67C2214yes—
35#387×C67C2214yes—
36#387×C67C2214yes—
37#387×C67C2214yes—
38#387×C67C2214yes—
39#387×C67C2214yes—
40#386×C67C2214yes—
41#386×C67C2214yes—
42#386×C67C2214yes—
43#385×C67C2214yes—
44#385×C67C2214yes—
45#385×C67C2214yes—
46#382×C67C2212yes—
47#382×C67C2211yescharacteristic
48#380×C67C2214yes—
49C4×C4⋊C4×C67C2212yes—
50C4×C4⋊C4×C67C2214yes—
51C4×C4⋊C4×C67C2214yes—
52C4×C22⋊C4×C67C2214yes—
53C4×C22⋊C4×C67C2214yes—
54C4×C22⋊C4×C67C2214yes—
55C4×C22⋊C4×C67C2214yes—
56C2×#95×C67C2214yes—
57C43×C67C2211yescharacteristic
58C22×Q8×C67C2312yes—
59C22×D4×C67C2314yes—
60C22×D4×C67C2312yes—
61C23×C4×C67C2314yes—
62C23×C4×C67C2314yes—
63C23×C4×C67C2314yes—
64#125×C67C2×C418yescomplemented
65#124×C67C2×C4116yescomplemented
66#123×C67C2×C4132yescomplemented
67C22⋊Q8×C67C2×C4132yescomplemented
68#121×C67C2×C4132yescomplemented
69C4×Q8×C67—44no—
70C4×D4×C67—44no—
71C4×D4×C67—48no—
72#117×C67—48no—
73C2×C4⋊C4×C67C2314yes—
74C2×C4⋊C4×C67C2314yes—
75C2×C4⋊C4×C67C2312yes—
76C2×C4⋊C4×C67C2314yes—
77C2×C4⋊C4×C67C2312yes—
78C2×C4⋊C4×C67C2314yes—
79C2×C4⋊C4×C67C2312yes—
80C2×C4⋊C4×C67C2314yes—
81C2×C4⋊C4×C67C2314yes—
82C2×C4⋊C4×C67C2312yes—
83C2×C22⋊C4×C67C2314yes—
84C2×C22⋊C4×C67C2314yes—
85C2×C22⋊C4×C67C2314yes—
86C2×C22⋊C4×C67C2314yes—
87C2×C22⋊C4×C67C2314yes—
88C2×C22⋊C4×C67C2314yes—
89C2×C22⋊C4×C67C2314yes—
90C2×C22⋊C4×C67C2314yes—
91C2×C22⋊C4×C67C2314yes—
92C2×C42×C67C2314yes—
93C2×C42×C67C2312yes—
94C2×C42×C67C2311yescharacteristic
95C2×C42×C67C2314yes—
96C2×C42×C67C2314yes—
97C2×C42×C67C2311yescharacteristic
98C2×C42×C67C2312yes—
99C2×C42×C67C2314yes—
100C2×C42×C67C2311yescharacteristic
101#95×C67C2314yes—
102#95×C67C2314yes—
103#95×C67C2314yes—
104#95×C67C2314yes—
105#95×C67C2314yes—
106#95×C67C2314yes—
107#95×C67C2314yes—
108#95×C67C2314yes—
109#95×C67C2314yes—
110#95×C67C2314yes—
111#95×C67C2314yes—
112C24×C67C2414yes—
113C2×Q8×C67—42no—
114C2×Q8×C67—42no—
115C2×Q8×C67C22×C418yes—
116C2×D4×C67—42no—
117C2×D4×C67—44no—
118C2×D4×C67—44no—
119C2×D4×C67—42no—
120C2×D4×C67C22×C4116yes—
121C2×D4×C67C22×C418yes—
122C22×C4×C67—28no—
123C22×C4×C67—44no—
124C22×C4×C67—44no—
125C22×C4×C67—44no—
126C22×C4×C67—44no—
127C22×C4×C67—44no—
128C22×C4×C67—44no—
129C22×C4×C67—44no—
130C22×C4×C67C22×C4116yes—
131C22×C4×C67C2411yescharacteristic
132C22×C4×C67C2414yes—
133C22×C4×C67C2414yes—
134C22×C4×C67C2414yes—
135C22×C4×C67C2411yescharacteristic
136C22×C4×C67C2412yes—
137C22×C4×C67C2414yes—
138C22×C4×C67C2414yes—
139C22×C4×C67C2412yes—
140C22×C4×C67C2411yescharacteristic
141C4⋊C4×C67—28no—
142C4⋊C4×C67—42no—
143C4⋊C4×C67—44no—
144C4⋊C4×C67—44no—
145C4⋊C4×C67—42no—
146C4⋊C4×C67—44no—
147C4⋊C4×C67—42no—
148C4⋊C4×C67C22×C4116yes—
149C4⋊C4×C67C22×C4116yes—
150C4⋊C4×C67C22×C418yes—
151C22⋊C4×C67—28no—
152C22⋊C4×C67—28no—
153C22⋊C4×C67—44no—
154C22⋊C4×C67—44no—
155C22⋊C4×C67—44no—
156C22⋊C4×C67—44no—
157C22⋊C4×C67C22×C4116yes—
158C22⋊C4×C67C22×C4116yes—
159C22⋊C4×C67C22×C4116yes—
160C42×C67—28no—
161C42×C67—24no—
162C42×C67—22no—
163C42×C67—42no—
164C42×C67—44no—
165C42×C67—41no—
166C42×C67—44no—
167C42×C67—44no—
168C42×C67C22×C414yes—
169C23×C67—24no—
170C23×C67—24no—
171C23×C67—24no—
172C23×C67—24no—
173C23×C67—24no—
174C23×C67—24no—
175C23×C67C23×C418yes—
176C23×C67C2511yescharacteristic
177Q8×C67—48no—
178D4×C67—416no—
179D4×C67—48no—
180C2×C4×C67—24no—
181C2×C4×C67—22no—
182C2×C4×C67—24no—
183C2×C4×C67—21no—
184C2×C4×C67—22no—
185C2×C4×C67—21no—
186C2×C4×C67—22no—
187C2×C4×C67—22no—
188C2×C4×C67—24no—
189C2×C4×C67—22no—
190C2×C4×C67—24no—
191C2×C4×C67—22no—
192C2×C4×C67—24no—
193C2×C4×C67—24no—
194C2×C4×C67—24no—
195C2×C4×C67—24no—
196C2×C4×C67—21no—
197C2×C4×C67—24no—
198C2×C4×C67—24no—
199C2×C4×C67—24no—
200C2×C4×C67—24no—
201C2×C4×C67—24no—
202C2×C4×C67—416no—
203C2×C4×C67—88no—
204C2×C4×C67—88no—
205C2×C4×C67C23×C412yes—
206C2×C4×C67C23×C414yes—
207C2×C4×C67C23×C418yes—
208C2×C4×C67C23×C418yes—
209C2×C4×C67H(5,𝔽2)14yes—
210C2×C4×C672−1+412yes—
211C22×C67—28no—
212C22×C67—28no—
213C22×C67—28no—
214C22×C67—44no—
215C22×C67—44no—
216C22×C67—44no—
217C22×C67—44no—
218C22×C67C24×C411yescharacteristic
219C22×C67C2×H(5,𝔽2)12yes—
220C22×C67C2×2−1+411yescharacteristic
221C22×C67#58512yes—
222C22×C67#58511yescharacteristic
223C4×C67—28no—
224C4×C67—28no—
225C4×C67—24no—
226C4×C67—22no—
227C4×C67—24no—
228C4×C67—22no—
229C4×C67—44no—
230C4×C67—44no—
231C4×C67—41no—
232C4×C67—44no—
233#34522C6711yesmaximal, characteristic, direct product factor, Sylow
234C2×C67—48no—
235C2×C67C4×H(5,𝔽2)12yes—
236C2×C67C4×2−1+411yescharacteristic
237C2×C67#354512yes—
238C2×C67#354811yescharacteristic
239C2×C67#355311yescharacteristic
240C2×#562C2×C6711yescharacteristic
241#2382C2×C6711yescharacteristic, complemented
242#2379C2×C6712yescomplemented
243#2368C2×C6714yescomplemented
244#2363C2×C6714yescomplemented
245#2359C2×C6714yescomplemented
246#2335C2×C6714yescomplemented
247#2325C2×C6714yescomplemented
248#2323C2×C6714yescomplemented
249C4×#125C2×C6711yescharacteristic, complemented
250C4×#124C2×C6712yescomplemented
251C67#3452211yescharacteristic, direct product factor, Sylow
252#562C4×C67116yescomplemented
253C2×#125C22×C6711yescharacteristic
254C2×#124C22×C6712yes—
255C2×#123C22×C6714yes—
256C2×C22⋊Q8C22×C6714yes—
257C2×#121C22×C6714yes—
258C2×C4×Q8C22×C6712yes—
259C2×C4×D4C22×C6712yes—
260C2×C4×D4C22×C6714yes—
261C2×#117C22×C6714yes—
262#391C22×C6714yes—
263#391C22×C6712yes—
264#390C22×C6714yes—
265#390C22×C6712yes—
266#389C22×C6712yes—
267#389C22×C6714yes—
268#389C22×C6714yes—
269#388C22×C6714yes—
270#388C22×C6714yes—
271#388C22×C6714yes—
272#388C22×C6714yes—
273#388C22×C6714yes—
274#388C22×C6714yes—
275#387C22×C6714yes—
276#387C22×C6714yes—
277#387C22×C6714yes—
278#387C22×C6714yes—
279#387C22×C6714yes—
280#386C22×C6714yes—
281#386C22×C6714yes—
282#386C22×C6714yes—
283#385C22×C6714yes—
284#385C22×C6714yes—
285#385C22×C6714yes—
286#382C22×C6711yescharacteristic
287#382C22×C6712yes—
288#380C22×C6714yes—
289C4×C4⋊C4C22×C6712yes—
290C4×C4⋊C4C22×C6714yes—
291C4×C4⋊C4C22×C6714yes—
292C4×C22⋊C4C22×C6714yes—
293C4×C22⋊C4C22×C6714yes—
294C4×C22⋊C4C22×C6714yes—
295C4×C22⋊C4C22×C6714yes—
296C2×#95C22×C6714yes—
297C43C22×C6711yescharacteristic
298C22×Q8C23×C6712yes—
299C22×D4C23×C6714yes—
300C22×D4C23×C6712yes—
301C23×C4C23×C6714yes—
302C23×C4C23×C6714yes—
303C23×C4C23×C6714yes—
304#125C2×C4×C6718yescomplemented
305#124C2×C4×C67116yescomplemented
306#123C2×C4×C67132yescomplemented
307C22⋊Q8C2×C4×C67132yescomplemented
308#121C2×C4×C67132yescomplemented
309C4×Q8—44no—
310C4×D4—44no—
311C4×D4—48no—
312#117—48no—
313C2×C4⋊C4C23×C6714yes—
314C2×C4⋊C4C23×C6714yes—
315C2×C4⋊C4C23×C6712yes—
316C2×C4⋊C4C23×C6714yes—
317C2×C4⋊C4C23×C6712yes—
318C2×C4⋊C4C23×C6714yes—
319C2×C4⋊C4C23×C6712yes—
320C2×C4⋊C4C23×C6714yes—
321C2×C4⋊C4C23×C6714yes—
322C2×C4⋊C4C23×C6712yes—
323C2×C22⋊C4C23×C6714yes—
324C2×C22⋊C4C23×C6714yes—
325C2×C22⋊C4C23×C6714yes—
326C2×C22⋊C4C23×C6714yes—
327C2×C22⋊C4C23×C6714yes—
328C2×C22⋊C4C23×C6714yes—
329C2×C22⋊C4C23×C6714yes—
330C2×C22⋊C4C23×C6714yes—
331C2×C22⋊C4C23×C6714yes—
332C2×C42C23×C6714yes—
333C2×C42C23×C6712yes—
334C2×C42C23×C6711yescharacteristic
335C2×C42C23×C6714yes—
336C2×C42C23×C6714yes—
337C2×C42C23×C6711yescharacteristic
338C2×C42C23×C6712yes—
339C2×C42C23×C6714yes—
340C2×C42C23×C6711yescharacteristic
341#95C23×C6714yes—
342#95C23×C6714yes—
343#95C23×C6714yes—
344#95C23×C6714yes—
345#95C23×C6714yes—
346#95C23×C6714yes—
347#95C23×C6714yes—
348#95C23×C6714yes—
349#95C23×C6714yes—
350#95C23×C6714yes—
351#95C23×C6714yes—
352C24C24×C6714yes—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8C22×C4×C6718yes—
356C2×D4—42no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4C22×C4×C67116yes—
361C2×D4C22×C4×C6718yes—
362C22×C4—28no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4C22×C4×C67116yes—
371C22×C4C24×C6714yes—
372C22×C4C24×C6714yes—
373C22×C4C24×C6711yescharacteristic
374C22×C4C24×C6714yes—
375C22×C4C24×C6711yescharacteristic
376C22×C4C24×C6712yes—
377C22×C4C24×C6714yes—
378C22×C4C24×C6714yes—
379C22×C4C24×C6712yes—
380C22×C4C24×C6711yescharacteristic
381C4⋊C4—28no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4C22×C4×C67116yes—
389C4⋊C4C22×C4×C67116yes—
390C4⋊C4C22×C4×C6718yes—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4C22×C4×C67116yes—
398C22⋊C4C22×C4×C67116yes—
399C22⋊C4C22×C4×C67116yes—
400C42—28no—
401C42—24no—
402C42—22no—
403C42—44no—
404C42—44no—
405C42—42no—
406C42—44no—
407C42—41no—
408C42C22×C4×C6714yes—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23C23×C4×C6718yes—
416C23C25×C6711yescharacteristic
417Q8—48no—
418D4—416no—
419D4—48no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—416no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4C23×C4×C6712yes—
446C2×C4C23×C4×C6714yes—
447C2×C4C23×C4×C6718yes—
448C2×C4C23×C4×C6718yes—
449C2×C4H(5,𝔽2)×C6714yes—
450C2×C42−1+4×C6712yes—
451C22—28no—
452C22—28no—
453C22—28no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22C24×C4×C6711yescharacteristic
459C22C2×H(5,𝔽2)×C6712yes—
460C22C2×2−1+4×C6711yescharacteristic
461C22#585×C6711yescharacteristic
462C22#585×C6712yes—
463C4—24no—
464C4—22no—
465C4—28no—
466C4—28no—
467C4—24no—
468C4—22no—
469C4—44no—
470C4—44no—
471C4—41no—
472C4—44no—
473C2—48no—
474C2C4×H(5,𝔽2)×C6712yes—
475C2C4×2−1+4×C6711yescharacteristic
476C2C67×#354512yes—
477C2C67×#354811yescharacteristic
478C2C67×#355311yescharacteristic
subgroupG/H
center, socleC23×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC23C25×C67
prime radicalC2×#562×C67C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2382C2semidirect
3. C67×#2379C2semidirect
4. C67×#2368C2semidirect
5. C67×#2363C2semidirect
6. C67×#2359C2semidirect
7. C67×#2335C2semidirect
8. C67×#2325C2semidirect
9. C67×#2323C2semidirect
10. C4×#125×C67C2semidirect
11. C4×#124×C67C2semidirect
12. #562×C67C4semidirect
64. #125×C67C2×C4semidirect
65. #124×C67C2×C4semidirect
66. #123×C67C2×C4semidirect
67. C22⋊Q8×C67C2×C4semidirect
68. #121×C67C2×C4semidirect
233. #34522C67direct (N × H)
241. #2382C2×C67semidirect
242. #2379C2×C67semidirect
243. #2368C2×C67semidirect
244. #2363C2×C67semidirect
245. #2359C2×C67semidirect
246. #2335C2×C67semidirect
247. #2325C2×C67semidirect
248. #2323C2×C67semidirect
249. C4×#125C2×C67semidirect
250. C4×#124C2×C67semidirect
251. C67#34522direct (N × H)
252. #562C4×C67semidirect
304. #125C2×C4×C67semidirect
305. #124C2×C4×C67semidirect
306. #123C2×C4×C67semidirect
307. C22⋊Q8C2×C4×C67semidirect
308. #121C2×C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#562×C672
C67×#23822
C67×#23792
C67×#23682
C67×#23632
C67×#23592
C67×#23352
C67×#23252
C67×#23232
C4×#125×C672
C4×#124×C672
#3452267