Finite Group Databasebeta

← C2×C67×#3509 #30742055

Subgroups — C2×C67×#3509 #30742055

Order 17152.

7022 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3509C2132yesmaximal, direct product factor
2C2×#557×C67C214yesmaximal, complemented
3C2×#556×C67C214yesmaximal, complemented
4C2×#552×C67C218yesmaximal, complemented
5C2×#541×C67C214yesmaximal, complemented
6C2×#536×C67C212yesmaximal
7C2×#533×C67C211yesmaximal, characteristic, complemented
8C2×#533×C67C214yesmaximal, complemented
9C22×C4⋊Q8×C67C211yesmaximal, characteristic, complemented
10C2×#519×C67C212yesmaximal, complemented
11C2×C4×#41×C67C211yesmaximal, characteristic
12C22×#41×C67C2211yescharacteristic
13C23×Q8×C67C2212yes—
14#557×C67C22164yescomplemented
15#556×C67C22164yescomplemented
16#552×C67C221128yescomplemented
17#541×C67C22164yescomplemented
18#536×C67C22132yes—
19#533×C67C22116yescomplemented
20#533×C67C22164yescomplemented
21C2×C4⋊Q8×C67C2212yes—
22C2×C4⋊Q8×C67C2212yes—
23C2×C4⋊Q8×C67C2214yes—
24C2×C4⋊Q8×C67C2212yes—
25C2×C4⋊Q8×C67C2214yes—
26C2×C4⋊Q8×C67C22116yescomplemented
27C2×#126×C67C22116yes—
28C2×#125×C67C2212yes—
29C2×#125×C67C2218yes—
30C2×#125×C67C22116yes—
31C2×#124×C67C2214yes—
32C2×#123×C67C2218yes—
33C2×C22⋊Q8×C67C2218yes—
34C2×C22⋊Q8×C67C2214yes—
35C2×C22⋊Q8×C67C2218yes—
36#519×C67C22132yescomplemented
37C4×#41×C67C22116yes—
38C2×C4×Q8×C67C2212yes—
39C2×C4×Q8×C67C2218yes—
40C2×C4×Q8×C67C2218yes—
41C2×C4×Q8×C67C2218yes—
42C2×C4×Q8×C67C2214yes—
43C2×C4×D4×C67C2214yes—
44C2×C4×D4×C67C2212yes—
45C2×#117×C67C2211yescharacteristic
46C2×#117×C67C2212yes—
47C2×#117×C67C2212yes—
48C2×#117×C67C2218yes—
49C2×#117×C67C2218yes—
50C22×C4⋊C4×C67C2214yes—
51C22×C42×C67C2211yescharacteristic
52C22×C42×C67C2212yes—
53C2×#41×C67—28no—
54C2×#41×C67C2318yes—
55C22×Q8×C67—216no—
56C22×Q8×C67C2314yes—
57C22×Q8×C67C2314yes—
58C22×Q8×C67C2314yes—
59C22×Q8×C67C2311yescharacteristic
60C22×Q8×C67C23116yes—
61C22×D4×C67C2312yes—
62C22×D4×C67C2311yescharacteristic
63C23×C4×C67C2312yes—
64C23×C4×C67C2314yes—
65C23×C4×C67C2312yes—
66C23×C4×C67C2314yes—
67C23×C4×C67C2311yescharacteristic
68C4⋊Q8×C67C23116yes—
69C4⋊Q8×C67C23116yes—
70C4⋊Q8×C67C23132yes—
71C4⋊Q8×C67C23116yes—
72C4⋊Q8×C67C23132yes—
73#126×C67C231128yes—
74#125×C67C23116yes—
75#125×C67C23164yes—
76#125×C67C231128yes—
77#124×C67C23132yes—
78#123×C67C23164yes—
79C22⋊Q8×C67C23164yes—
80C22⋊Q8×C67C23132yes—
81C22⋊Q8×C67C23164yes—
82C4×Q8×C67C23116yes—
83C4×Q8×C67C23164yes—
84C4×Q8×C67C23164yes—
85C4×Q8×C67C23164yes—
86C4×Q8×C67C23132yes—
87C4×D4×C67C23132yes—
88C4×D4×C67C23116yes—
89#117×C67C2318yes—
90#117×C67C23116yes—
91#117×C67C23116yes—
92#117×C67C23164yes—
93#117×C67C23164yes—
94C2×C4⋊C4×C67C2318yes—
95C2×C4⋊C4×C67C2318yes—
96C2×C4⋊C4×C67C23132yes—
97C2×C4⋊C4×C67C2314yes—
98C2×C4⋊C4×C67C2312yes—
99C2×C4⋊C4×C67C2314yes—
100C2×C4⋊C4×C67C2314yes—
101C2×C4⋊C4×C67C23116yes—
102C2×C4⋊C4×C67C23116yes—
103C2×C4⋊C4×C67C2318yes—
104C2×C22⋊C4×C67C2312yes—
105C2×C22⋊C4×C67C2314yes—
106C2×C22⋊C4×C67C2314yes—
107C2×C22⋊C4×C67C23116yes—
108C2×C42×C67C2311yescharacteristic
109C2×C42×C67C2312yes—
110C2×C42×C67C2311yescharacteristic
111C2×C42×C67C2314yes—
112C2×C42×C67C2312yes—
113C2×C42×C67C2318yes—
114C2×C42×C67C23116yes—
115C2×C42×C67C23116yes—
116C2×C42×C67C2314yes—
117C24×C67C2412yes—
118C24×C67C2411yescharacteristic
119#41×C67—232no—
120C2×Q8×C67—216no—
121C2×Q8×C67—216no—
122C2×Q8×C67—264no—
123C2×Q8×C67—24no—
124C2×Q8×C67—216no—
125C2×Q8×C67C24116yes—
126C2×Q8×C67C24116yes—
127C2×Q8×C67C24116yes—
128C2×Q8×C67C2414yes—
129C2×D4×C67—28no—
130C2×D4×C67—24no—
131C2×D4×C67C2418yes—
132C2×D4×C67C2414yes—
133C22×C4×C67—216no—
134C22×C4×C67—28no—
135C22×C4×C67—28no—
136C22×C4×C67—216no—
137C22×C4×C67#4118yes—
138C22×C4×C67C2418yes—
139C22×C4×C67C24116yes—
140C22×C4×C67C2418yes—
141C22×C4×C67C24116yes—
142C22×C4×C67C2414yes—
143C22×C4×C67C2418yes—
144C22×C4×C67C2412yes—
145C22×C4×C67C2418yes—
146C22×C4×C67C2412yes—
147C22×C4×C67C2414yes—
148C22×C4×C67C2411yescharacteristic
149C22×C4×C67C2411yescharacteristic
150C22×C4×C67C2412yes—
151C4⋊C4×C67C24132yes—
152C4⋊C4×C67C24132yes—
153C4⋊C4×C67C24116yes—
154C4⋊C4×C67C2418yes—
155C4⋊C4×C67C24116yes—
156C4⋊C4×C67C24116yes—
157C4⋊C4×C67C24164yes—
158C4⋊C4×C67C24164yes—
159C4⋊C4×C67C24132yes—
160C22⋊C4×C67C2418yes—
161C22⋊C4×C67C24116yes—
162C22⋊C4×C67C24116yes—
163C22⋊C4×C67C24164yes—
164C42×C67C2414yes—
165C42×C67C2418yes—
166C42×C67C2414yes—
167C42×C67C24116yes—
168C42×C67C2418yes—
169C42×C67C24164yes—
170C42×C67C24116yes—
171C23×C67—24no—
172C23×C67—24no—
173C23×C67—28no—
174C23×C67C2×#4114yes—
175C23×C67C2514yes—
176C23×C67C2512yes—
177C23×C67C2511yescharacteristic
178Q8×C67—232no—
179Q8×C67—232no—
180Q8×C67—28no—
181Q8×C67—232no—
182D4×C67—216no—
183D4×C67—28no—
184C2×C4×C67—24no—
185C2×C4×C67—232no—
186C2×C4×C67—216no—
187C2×C4×C67—28no—
188C2×C4×C67—24no—
189C2×C4×C67—216no—
190C2×C4×C67—216no—
191C2×C4×C67—216no—
192C2×C4×C67—232no—
193C2×C4×C67C2×#4114yes—
194C2×C4×C67C2×#4114yes—
195C2×C4×C67C2×#41116yes—
196C2×C4×C672−1+418yes—
197C2×C4×C67C2518yes—
198C2×C4×C67C25116yes—
199C2×C4×C67C2514yes—
200C2×C4×C67C25116yes—
201C2×C4×C67C2514yes—
202C2×C4×C67C2512yes—
203C2×C4×C67C2512yes—
204C2×C4×C67C2514yes—
205C22×C67—24no—
206C22×C67—28no—
207C22×C67—24no—
208C22×C67—28no—
209C22×C67—48no—
210C22×C67C22×#4114yes—
211C22×C67C22×#4112yes—
212C22×C67C2×2−1+414yes—
213C22×C67C2611yescharacteristic
214C4×C67—24no—
215C4×C67—216no—
216C4×C67—28no—
217C4×C67—24no—
218C4×C67—216no—
219C4×C67C22×#4114yes—
220C4×C67C22×#4114yes—
221C4×C67C2×2−1+418yes—
222C2×#3509C6711yesmaximal, characteristic, direct product factor, Sylow
223C2×C67—24no—
224C2×C67—44no—
225C2×C67#350914yesdirect product factor
226C2×C67C23×#4111yescharacteristic
227C2×C67C22×2−1+412yes—
228#3509C2×C67132yesdirect product factor
229C2×#557C2×C6714yescomplemented
230C2×#556C2×C6714yescomplemented
231C2×#552C2×C6718yescomplemented
232C2×#541C2×C6714yescomplemented
233C2×#536C2×C6712yes—
234C2×#533C2×C6711yescharacteristic, complemented
235C2×#533C2×C6714yescomplemented
236C22×C4⋊Q8C2×C6711yescharacteristic, complemented
237C2×#519C2×C6712yescomplemented
238C2×C4×#41C2×C6711yescharacteristic
239C67C2×#350911yescharacteristic, direct product factor, Sylow
240C22×#41C22×C6711yescharacteristic
241C23×Q8C22×C6712yes—
242#557C22×C67164yescomplemented
243#556C22×C67164yescomplemented
244#552C22×C671128yescomplemented
245#541C22×C67164yescomplemented
246#536C22×C67132yes—
247#533C22×C67116yescomplemented
248#533C22×C67164yescomplemented
249C2×C4⋊Q8C22×C6712yes—
250C2×C4⋊Q8C22×C6712yes—
251C2×C4⋊Q8C22×C6714yes—
252C2×C4⋊Q8C22×C6712yes—
253C2×C4⋊Q8C22×C6714yes—
254C2×C4⋊Q8C22×C67116yescomplemented
255C2×#126C22×C67116yes—
256C2×#125C22×C6712yes—
257C2×#125C22×C6718yes—
258C2×#125C22×C67116yes—
259C2×#124C22×C6714yes—
260C2×#123C22×C6718yes—
261C2×C22⋊Q8C22×C6718yes—
262C2×C22⋊Q8C22×C6714yes—
263C2×C22⋊Q8C22×C6718yes—
264#519C22×C67132yescomplemented
265C4×#41C22×C67116yes—
266C2×C4×Q8C22×C6712yes—
267C2×C4×Q8C22×C6718yes—
268C2×C4×Q8C22×C6718yes—
269C2×C4×Q8C22×C6718yes—
270C2×C4×Q8C22×C6714yes—
271C2×C4×D4C22×C6714yes—
272C2×C4×D4C22×C6712yes—
273C2×#117C22×C6711yescharacteristic
274C2×#117C22×C6712yes—
275C2×#117C22×C6712yes—
276C2×#117C22×C6718yes—
277C2×#117C22×C6718yes—
278C22×C4⋊C4C22×C6714yes—
279C22×C42C22×C6711yescharacteristic
280C22×C42C22×C6712yes—
281C2×#41—28no—
282C2×#41C23×C6718yes—
283C22×Q8—216no—
284C22×Q8C23×C6714yes—
285C22×Q8C23×C6714yes—
286C22×Q8C23×C6714yes—
287C22×Q8C23×C6711yescharacteristic
288C22×Q8C23×C67116yes—
289C22×D4C23×C6712yes—
290C22×D4C23×C6711yescharacteristic
291C23×C4C23×C6714yes—
292C23×C4C23×C6712yes—
293C23×C4C23×C6714yes—
294C23×C4C23×C6712yes—
295C23×C4C23×C6711yescharacteristic
296C4⋊Q8C23×C67116yes—
297C4⋊Q8C23×C67116yes—
298C4⋊Q8C23×C67132yes—
299C4⋊Q8C23×C67116yes—
300C4⋊Q8C23×C67132yes—
301#126C23×C671128yes—
302#125C23×C67116yes—
303#125C23×C67164yes—
304#125C23×C671128yes—
305#124C23×C67132yes—
306#123C23×C67164yes—
307C22⋊Q8C23×C67164yes—
308C22⋊Q8C23×C67132yes—
309C22⋊Q8C23×C67164yes—
310C4×Q8C23×C67116yes—
311C4×Q8C23×C67164yes—
312C4×Q8C23×C67164yes—
313C4×Q8C23×C67164yes—
314C4×Q8C23×C67132yes—
315C4×D4C23×C67132yes—
316C4×D4C23×C67116yes—
317#117C23×C6718yes—
318#117C23×C67116yes—
319#117C23×C67116yes—
320#117C23×C67164yes—
321#117C23×C67164yes—
322C2×C4⋊C4C23×C6718yes—
323C2×C4⋊C4C23×C6718yes—
324C2×C4⋊C4C23×C67132yes—
325C2×C4⋊C4C23×C6714yes—
326C2×C4⋊C4C23×C6712yes—
327C2×C4⋊C4C23×C6714yes—
328C2×C4⋊C4C23×C6714yes—
329C2×C4⋊C4C23×C67116yes—
330C2×C4⋊C4C23×C67116yes—
331C2×C4⋊C4C23×C6718yes—
332C2×C22⋊C4C23×C6712yes—
333C2×C22⋊C4C23×C6714yes—
334C2×C22⋊C4C23×C6714yes—
335C2×C22⋊C4C23×C67116yes—
336C2×C42C23×C6712yes—
337C2×C42C23×C67116yes—
338C2×C42C23×C6711yescharacteristic
339C2×C42C23×C6714yes—
340C2×C42C23×C6714yes—
341C2×C42C23×C6712yes—
342C2×C42C23×C6718yes—
343C2×C42C23×C67116yes—
344C2×C42C23×C6711yescharacteristic
345C24C24×C6712yes—
346C24C24×C6711yescharacteristic
347#41—232no—
348C2×Q8—216no—
349C2×Q8—216no—
350C2×Q8—264no—
351C2×Q8—24no—
352C2×Q8—216no—
353C2×Q8C24×C67116yes—
354C2×Q8C24×C67116yes—
355C2×Q8C24×C67116yes—
356C2×Q8C24×C6714yes—
357C2×D4—28no—
358C2×D4—24no—
359C2×D4C24×C6718yes—
360C2×D4C24×C6714yes—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—216no—
365C22×C4#41×C6718yes—
366C22×C4C24×C6714yes—
367C22×C4C24×C6718yes—
368C22×C4C24×C6712yes—
369C22×C4C24×C6718yes—
370C22×C4C24×C6712yes—
371C22×C4C24×C6718yes—
372C22×C4C24×C67116yes—
373C22×C4C24×C6718yes—
374C22×C4C24×C67116yes—
375C22×C4C24×C6711yescharacteristic
376C22×C4C24×C6711yescharacteristic
377C22×C4C24×C6714yes—
378C22×C4C24×C6712yes—
379C4⋊C4C24×C67132yes—
380C4⋊C4C24×C67132yes—
381C4⋊C4C24×C67116yes—
382C4⋊C4C24×C6718yes—
383C4⋊C4C24×C67116yes—
384C4⋊C4C24×C67116yes—
385C4⋊C4C24×C67164yes—
386C4⋊C4C24×C67164yes—
387C4⋊C4C24×C67132yes—
388C22⋊C4C24×C6718yes—
389C22⋊C4C24×C67116yes—
390C22⋊C4C24×C67116yes—
391C22⋊C4C24×C67164yes—
392C42C24×C6714yes—
393C42C24×C6718yes—
394C42C24×C6714yes—
395C42C24×C67116yes—
396C42C24×C6718yes—
397C42C24×C67164yes—
398C42C24×C67116yes—
399C23—24no—
400C23—24no—
401C23—28no—
402C23C2×#41×C6714yes—
403C23C25×C6714yes—
404C23C25×C6712yes—
405C23C25×C6711yescharacteristic
406Q8—232no—
407Q8—232no—
408Q8—28no—
409Q8—232no—
410D4—216no—
411D4—28no—
412C2×C4—24no—
413C2×C4—232no—
414C2×C4—216no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—216no—
418C2×C4—216no—
419C2×C4—216no—
420C2×C4—232no—
421C2×C4C2×#41×C6714yes—
422C2×C4C2×#41×C6714yes—
423C2×C4C2×#41×C67116yes—
424C2×C42−1+4×C6718yes—
425C2×C4C25×C6718yes—
426C2×C4C25×C67116yes—
427C2×C4C25×C6714yes—
428C2×C4C25×C67116yes—
429C2×C4C25×C6714yes—
430C2×C4C25×C6712yes—
431C2×C4C25×C6712yes—
432C2×C4C25×C6714yes—
433C22—24no—
434C22—28no—
435C22—24no—
436C22—28no—
437C22—48no—
438C22C22×#41×C6714yes—
439C22C22×#41×C6712yes—
440C22C2×2−1+4×C6714yes—
441C22C26×C6711yescharacteristic
442C4—24no—
443C4—216no—
444C4—28no—
445C4—24no—
446C4—216no—
447C4C22×#41×C6714yes—
448C4C22×#41×C6714yes—
449C4C2×2−1+4×C6718yes—
450C2—24no—
451C2—44no—
452C2C67×#350914yesdirect product factor
453C2C23×#41×C6711yescharacteristic
454C2C22×2−1+4×C6712yes—
subgroupG/H
center, socleC23×C67C25
commutator, Frattini, cosocleC22C26×C67
prime radicalC22×#41×C67C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3509C2direct (N × H)
2. C2×#557×C67C2semidirect
3. C2×#556×C67C2semidirect
4. C2×#552×C67C2semidirect
5. C2×#541×C67C2semidirect
7. C2×#533×C67C2semidirect
8. C2×#533×C67C2semidirect
9. C22×C4⋊Q8×C67C2semidirect
10. C2×#519×C67C2semidirect
14. #557×C67C22semidirect
15. #556×C67C22semidirect
16. #552×C67C22semidirect
17. #541×C67C22semidirect
19. #533×C67C22semidirect
20. #533×C67C22semidirect
26. C2×C4⋊Q8×C67C22semidirect
36. #519×C67C22semidirect
222. C2×#3509C67direct (N × H)
225. C2×C67#3509direct (N × H)
228. #3509C2×C67direct (N × H)
229. C2×#557C2×C67semidirect
230. C2×#556C2×C67semidirect
231. C2×#552C2×C67semidirect
232. C2×#541C2×C67semidirect
234. C2×#533C2×C67semidirect
235. C2×#533C2×C67semidirect
236. C22×C4⋊Q8C2×C67semidirect
237. C2×#519C2×C67semidirect
239. C67C2×#3509direct (N × H)
242. #557C22×C67semidirect
243. #556C22×C67semidirect
244. #552C22×C67semidirect
245. #541C22×C67semidirect
247. #533C22×C67semidirect
248. #533C22×C67semidirect
254. C2×C4⋊Q8C22×C67semidirect
264. #519C22×C67semidirect
452. C2C67×#3509direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#35092
C2×#557×C672
C2×#556×C672
C2×#552×C672
C2×#541×C672
C2×#536×C672
C2×#533×C672
C22×C4⋊Q8×C672
C2×#519×C672
C2×C4×#41×C672
C2×#350967