Finite Group Databasebeta

← C67×#34976 #30714227

Subgroups — C67×#34976 #30714227

Order 17152.

2810 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#41×C67C212yesmaximal, complemented
2C67×#2609C214yesmaximal, complemented
3C67×#2582C211yesmaximal, characteristic, complemented
4C67×#2538C212yesmaximal, complemented
5C67×#2537C211yesmaximal, characteristic, complemented
6C67×#2485C214yesmaximal, complemented
7C67×#2407C214yesmaximal, complemented
8C67×#2392C214yesmaximal, complemented
9C4×C22⋊Q8×C67C214yesmaximal, complemented
10C4×#121×C67C214yesmaximal, complemented
11C67×#2285C211yesmaximal, characteristic, complemented
12C22×#41×C67C2212yes—
13C2×C22⋊Q8×C67C2214yes—
14C2×#121×C67C2214yes—
15C4×#41×C67—216no—
16C2×C4×Q8×C67C2212yes—
17C2×C4×Q8×C67C2212yes—
18C2×C4×D4×C67C2212yes—
19C2×C4×D4×C67C2212yes—
20C2×C4×D4×C67C2214yes—
21C2×C4×D4×C67C2214yes—
22C2×C4×D4×C67C2214yes—
23C2×#117×C67C2214yes—
24C2×#117×C67C2212yes—
25C2×#117×C67C2212yes—
26C22×C42×C67C2214yes—
27C22×C42×C67C2211yescharacteristic
28#398×C67C2214yes—
29#397×C67C2214yes—
30#396×C67C2218yes—
31#394×C67C2214yes—
32#394×C67C2218yes—
33#392×C67C2214yes—
34#391×C67C2214yes—
35#390×C67C2214yes—
36#389×C67C2214yes—
37#389×C67C2214yes—
38#388×C67C2218yes—
39#387×C67C2218yes—
40#386×C67C2218yes—
41#385×C67C2218yes—
42#384×C67C2211yescharacteristic
43#384×C67C2211yescharacteristic
44#382×C67C2212yes—
45#380×C67C2214yes—
46#380×C67C2214yes—
47#380×C67C2214yes—
48C4×C4⋊C4×C67C2214yes—
49C4×C4⋊C4×C67C2214yes—
50C4×C22⋊C4×C67C2218yes—
51C2×#41×C67—28no—
52C2×#41×C67—28no—
53C2×#41×C67—28no—
54C22×Q8×C67C2312yes—
55C22×D4×C67C2314yes—
56C22×D4×C67C2314yes—
57C22×D4×C67C2312yes—
58C23×C4×C67C2314yes—
59C23×C4×C67C2314yes—
60C23×C4×C67C2314yes—
61C23×C4×C67C2312yes—
62C23×C4×C67C2311yescharacteristic
63C22⋊Q8×C67—216no—
64#121×C67—216no—
65C4×Q8×C67—28no—
66C4×Q8×C67—28no—
67C4×D4×C67—216no—
68C4×D4×C67—216no—
69C4×D4×C67—28no—
70C4×D4×C67—28no—
71C4×D4×C67—48no—
72#117×C67—28no—
73#117×C67—216no—
74#117×C67—44no—
75C2×C4⋊C4×C67C2312yes—
76C2×C4⋊C4×C67C2312yes—
77C2×C4⋊C4×C67C2314yes—
78C2×C4⋊C4×C67C2314yes—
79C2×C4⋊C4×C67C2314yes—
80C2×C4⋊C4×C67C2314yes—
81C2×C4⋊C4×C67C2318yes—
82C2×C4⋊C4×C67C2312yes—
83C2×C4⋊C4×C67C2312yes—
84C2×C4⋊C4×C67C2314yes—
85C2×C22⋊C4×C67C2314yes—
86C2×C22⋊C4×C67C2314yes—
87C2×C22⋊C4×C67C2314yes—
88C2×C22⋊C4×C67C2318yes—
89C2×C22⋊C4×C67C2318yes—
90C2×C22⋊C4×C67C2314yes—
91C2×C22⋊C4×C67C2318yes—
92C2×C42×C67—216no—
93C2×C42×C67D418yescomplemented
94C2×C42×C67C2314yes—
95C2×C42×C67C2314yes—
96C2×C42×C67C2314yes—
97C2×C42×C67C2314yes—
98C2×C42×C67C2311yescharacteristic
99C2×C42×C67C2311yescharacteristic
100C2×C42×C67C2312yes—
101#95×C67C2318yes—
102#95×C67C2318yes—
103#95×C67C2314yes—
104#95×C67C2318yes—
105#95×C67C2314yes—
106C24×C67C2411yescharacteristic
107C24×C67C2414yes—
108#41×C67—264no—
109C2×Q8×C67—24no—
110C2×Q8×C67—24no—
111C2×Q8×C67#4118yescomplemented
112C2×D4×C67—28no—
113C2×D4×C67—28no—
114C2×D4×C67—216no—
115C2×D4×C67—24no—
116C2×D4×C67—28no—
117C2×D4×C67—24no—
118C2×D4×C67—44no—
119C2×D4×C67#4118yescomplemented
120C22×C4×C67—28no—
121C22×C4×C67—28no—
122C22×C4×C67—28no—
123C22×C4×C67—24no—
124C22×C4×C67—24no—
125C22×C4×C67—22no—
126C22×C4×C67—22no—
127C22×C4×C67—28no—
128C22×C4×C67—28no—
129C22×C4×C67—28no—
130C22×C4×C67—28no—
131C22×C4×C67—28no—
132C22×C4×C67—28no—
133C22×C4×C67C2×D418yes—
134C22×C4×C67C2×D414yes—
135C22×C4×C67C2414yes—
136C22×C4×C67C2414yes—
137C22×C4×C67C2414yes—
138C22×C4×C67C2414yes—
139C22×C4×C67C2412yes—
140C22×C4×C67C2411yescharacteristic
141C22×C4×C67C2414yes—
142C22×C4×C67C2412yes—
143C22×C4×C67C2411yescharacteristic
144C4⋊C4×C67—28no—
145C4⋊C4×C67—28no—
146C4⋊C4×C67—24no—
147C4⋊C4×C67—24no—
148C4⋊C4×C67—28no—
149C4⋊C4×C67—28no—
150C4⋊C4×C67—216no—
151C4⋊C4×C67—42no—
152C4⋊C4×C67—42no—
153C4⋊C4×C67—44no—
154C22⋊C4×C67—28no—
155C22⋊C4×C67—216no—
156C22⋊C4×C67—28no—
157C22⋊C4×C67—28no—
158C22⋊C4×C67—216no—
159C22⋊C4×C67—44no—
160C22⋊C4×C67—48no—
161C42×C67—28no—
162C42×C67—28no—
163C42×C67—28no—
164C42×C67—28no—
165C42×C67C2×D418yes—
166C42×C67C2×D414yes—
167C42×C67C2×D414yes—
168C23×C67—24no—
169C23×C67—24no—
170C23×C67—22no—
171C23×C67—22no—
172C23×C67—24no—
173C23×C67—24no—
174C23×C67—21no—
175C23×C67—24no—
176C23×C67—21no—
177C23×C67—24no—
178C23×C67C22×D412yes—
179C23×C67C2×#4118yes—
180C23×C67C2511yescharacteristic
181Q8×C67—216no—
182D4×C67—232no—
183D4×C67—216no—
184D4×C67—416no—
185C2×C4×C67—232no—
186C2×C4×C67—24no—
187C2×C4×C67—22no—
188C2×C4×C67—22no—
189C2×C4×C67—24no—
190C2×C4×C67—21no—
191C2×C4×C67—21no—
192C2×C4×C67—216no—
193C2×C4×C67—28no—
194C2×C4×C67—24no—
195C2×C4×C67—24no—
196C2×C4×C67—24no—
197C2×C4×C67—28no—
198C2×C4×C67—28no—
199C2×C4×C67—24no—
200C2×C4×C67—24no—
201C2×C4×C67—24no—
202C2×C4×C67—24no—
203C2×C4×C67—24no—
204C2×C4×C67—416no—
205C2×C4×C67—416no—
206C2×C4×C67C22×D414yes—
207C2×C4×C67C22×D412yes—
208C2×C4×C67C22×D414yes—
209C2×C4×C67C22×D412yes—
210C2×C4×C67C2×#4118yes—
211C2×C4×C67C2×#4114yes—
212C2×C4×C67C2×#4114yes—
213C2×C4×C67H(5,𝔽2)12yes—
214C2×C4×C672−1+412yes—
215C22×C67—24no—
216C22×C67—28no—
217C22×C67—22no—
218C22×C67—28no—
219C22×C67—28no—
220C22×C67—44no—
221C22×C67—44no—
222C22×C67—41no—
223C22×C67—42no—
224C22×C67—44no—
225C22×C67—44no—
226C22×C67—41no—
227C22×C67C23×D411yescharacteristic
228C22×C67C22×#4112yes—
229C22×C67C22×#4112yes—
230C22×C67C2×H(5,𝔽2)11yescharacteristic
231C22×C67C2×2−1+411yescharacteristic
232C4×C67—24no—
233C4×C67—28no—
234C4×C67—22no—
235C4×C67—44no—
236C4×C67—44no—
237C4×C67—44no—
238C4×C67—44no—
239C4×C67#52518yes—
240C4×C67#53714yes—
241#34976C6711yesmaximal, characteristic, direct product factor, Sylow
242C2×C67—42no—
243C2×C67—48no—
244C2×C67C2×#52512yes—
245C2×C67C2×#53711yescharacteristic
246C2×C67#348111yescharacteristic
247C2×C67#348212yes—
248C2×C67#348311yescharacteristic
249C2×C4×#41C2×C6712yescomplemented
250#2609C2×C6714yescomplemented
251#2582C2×C6711yescharacteristic, complemented
252#2538C2×C6712yescomplemented
253#2537C2×C6711yescharacteristic, complemented
254#2485C2×C6714yescomplemented
255#2407C2×C6714yescomplemented
256#2392C2×C6714yescomplemented
257C4×C22⋊Q8C2×C6714yescomplemented
258C4×#121C2×C6714yescomplemented
259#2285C2×C6711yescharacteristic, complemented
260C67#3497611yescharacteristic, direct product factor, Sylow
261C22×#41C22×C6712yes—
262C2×C22⋊Q8C22×C6714yes—
263C2×#121C22×C6714yes—
264C4×#41—216no—
265C2×C4×Q8C22×C6712yes—
266C2×C4×Q8C22×C6712yes—
267C2×C4×D4C22×C6714yes—
268C2×C4×D4C22×C6714yes—
269C2×C4×D4C22×C6712yes—
270C2×C4×D4C22×C6712yes—
271C2×C4×D4C22×C6714yes—
272C2×#117C22×C6714yes—
273C2×#117C22×C6712yes—
274C2×#117C22×C6712yes—
275C22×C42C22×C6714yes—
276C22×C42C22×C6711yescharacteristic
277#398C22×C6714yes—
278#397C22×C6714yes—
279#396C22×C6718yes—
280#394C22×C6714yes—
281#394C22×C6718yes—
282#392C22×C6714yes—
283#391C22×C6714yes—
284#390C22×C6714yes—
285#389C22×C6714yes—
286#389C22×C6714yes—
287#388C22×C6718yes—
288#387C22×C6718yes—
289#386C22×C6718yes—
290#385C22×C6718yes—
291#384C22×C6711yescharacteristic
292#384C22×C6711yescharacteristic
293#382C22×C6712yes—
294#380C22×C6714yes—
295#380C22×C6714yes—
296#380C22×C6714yes—
297C4×C4⋊C4C22×C6714yes—
298C4×C4⋊C4C22×C6714yes—
299C4×C22⋊C4C22×C6718yes—
300C2×#41—28no—
301C2×#41—28no—
302C2×#41—28no—
303C22×Q8C23×C6712yes—
304C22×D4C23×C6714yes—
305C22×D4C23×C6714yes—
306C22×D4C23×C6712yes—
307C23×C4C23×C6714yes—
308C23×C4C23×C6714yes—
309C23×C4C23×C6714yes—
310C23×C4C23×C6711yescharacteristic
311C23×C4C23×C6712yes—
312C22⋊Q8—216no—
313#121—216no—
314C4×Q8—28no—
315C4×Q8—28no—
316C4×D4—216no—
317C4×D4—216no—
318C4×D4—28no—
319C4×D4—28no—
320C4×D4—48no—
321#117—28no—
322#117—216no—
323#117—44no—
324C2×C4⋊C4C23×C6712yes—
325C2×C4⋊C4C23×C6712yes—
326C2×C4⋊C4C23×C6714yes—
327C2×C4⋊C4C23×C6714yes—
328C2×C4⋊C4C23×C6714yes—
329C2×C4⋊C4C23×C6714yes—
330C2×C4⋊C4C23×C6718yes—
331C2×C4⋊C4C23×C6712yes—
332C2×C4⋊C4C23×C6712yes—
333C2×C4⋊C4C23×C6714yes—
334C2×C22⋊C4C23×C6714yes—
335C2×C22⋊C4C23×C6714yes—
336C2×C22⋊C4C23×C6714yes—
337C2×C22⋊C4C23×C6718yes—
338C2×C22⋊C4C23×C6718yes—
339C2×C22⋊C4C23×C6714yes—
340C2×C22⋊C4C23×C6718yes—
341C2×C42—216no—
342C2×C42D4×C6718yescomplemented
343C2×C42C23×C6714yes—
344C2×C42C23×C6714yes—
345C2×C42C23×C6714yes—
346C2×C42C23×C6714yes—
347C2×C42C23×C6711yescharacteristic
348C2×C42C23×C6711yescharacteristic
349C2×C42C23×C6712yes—
350#95C23×C6718yes—
351#95C23×C6718yes—
352#95C23×C6714yes—
353#95C23×C6718yes—
354#95C23×C6714yes—
355C24C24×C6711yescharacteristic
356C24C24×C6714yes—
357#41—264no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8#41×C6718yescomplemented
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—24no—
365C2×D4—28no—
366C2×D4—24no—
367C2×D4—44no—
368C2×D4#41×C6718yescomplemented
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4C2×D4×C6718yes—
383C22×C4C2×D4×C6714yes—
384C22×C4C24×C6714yes—
385C22×C4C24×C6714yes—
386C22×C4C24×C6714yes—
387C22×C4C24×C6714yes—
388C22×C4C24×C6712yes—
389C22×C4C24×C6711yescharacteristic
390C22×C4C24×C6714yes—
391C22×C4C24×C6712yes—
392C22×C4C24×C6711yescharacteristic
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—216no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C22⋊C4—28no—
404C22⋊C4—216no—
405C22⋊C4—28no—
406C22⋊C4—28no—
407C22⋊C4—216no—
408C22⋊C4—44no—
409C22⋊C4—48no—
410C42—28no—
411C42—28no—
412C42—28no—
413C42—28no—
414C42C2×D4×C6718yes—
415C42C2×D4×C6714yes—
416C42C2×D4×C6714yes—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—24no—
423C23—21no—
424C23—24no—
425C23—21no—
426C23—24no—
427C23C22×D4×C6712yes—
428C23C2×#41×C6718yes—
429C23C25×C6711yescharacteristic
430Q8—216no—
431D4—232no—
432D4—216no—
433D4—416no—
434C2×C4—232no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—216no—
442C2×C4—28no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—416no—
454C2×C4—416no—
455C2×C4C22×D4×C6714yes—
456C2×C4C22×D4×C6712yes—
457C2×C4C22×D4×C6714yes—
458C2×C4C22×D4×C6712yes—
459C2×C4C2×#41×C6718yes—
460C2×C4C2×#41×C6714yes—
461C2×C4C2×#41×C6714yes—
462C2×C4H(5,𝔽2)×C6712yes—
463C2×C42−1+4×C6712yes—
464C22—24no—
465C22—28no—
466C22—22no—
467C22—28no—
468C22—28no—
469C22—44no—
470C22—44no—
471C22—41no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—41no—
476C22C23×D4×C6711yescharacteristic
477C22C22×#41×C6712yes—
478C22C22×#41×C6712yes—
479C22C2×H(5,𝔽2)×C6711yescharacteristic
480C22C2×2−1+4×C6711yescharacteristic
481C4—24no—
482C4—28no—
483C4—22no—
484C4—44no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4#525×C6718yes—
489C4#537×C6714yes—
490C2—42no—
491C2—48no—
492C2C2×#525×C6712yes—
493C2C2×#537×C6711yescharacteristic
494C2C67×#348111yescharacteristic
495C2C67×#348212yes—
496C2C67×#348311yescharacteristic
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4×#41×C67C2semidirect
2. C67×#2609C2semidirect
3. C67×#2582C2semidirect
4. C67×#2538C2semidirect
5. C67×#2537C2semidirect
6. C67×#2485C2semidirect
7. C67×#2407C2semidirect
8. C67×#2392C2semidirect
9. C4×C22⋊Q8×C67C2semidirect
10. C4×#121×C67C2semidirect
11. C67×#2285C2semidirect
93. C2×C42×C67D4semidirect
111. C2×Q8×C67#41semidirect
119. C2×D4×C67#41semidirect
241. #34976C67direct (N × H)
249. C2×C4×#41C2×C67semidirect
250. #2609C2×C67semidirect
251. #2582C2×C67semidirect
252. #2538C2×C67semidirect
253. #2537C2×C67semidirect
254. #2485C2×C67semidirect
255. #2407C2×C67semidirect
256. #2392C2×C67semidirect
257. C4×C22⋊Q8C2×C67semidirect
258. C4×#121C2×C67semidirect
259. #2285C2×C67semidirect
260. C67#34976direct (N × H)
342. C2×C42D4×C67semidirect
360. C2×Q8#41×C67semidirect
368. C2×D4#41×C67semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#41×C672
C67×#26092
C67×#25822
C67×#25382
C67×#25372
C67×#24852
C67×#24072
C67×#23922
C4×C22⋊Q8×C672
C4×#121×C672
C67×#22852
#3497667