Finite Group Databasebeta

← C67×#36695 #30715946

Subgroups — C67×#36695 #30715946

Order 17152.

1664 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2845C211yesmaximal, characteristic, complemented
2C67×#2837C214yesmaximal, complemented
3C67×#2831C214yesmaximal, complemented
4C67×#2829C214yesmaximal, complemented
5C67×#2820C212yesmaximal, complemented
6C67×#2819C211yesmaximal, characteristic
7C67×#2396C214yesmaximal, complemented
8C67×#2395C214yesmaximal, complemented
9C67×#2392C214yesmaximal, complemented
10C4×#125×C67C211yesmaximal, characteristic, complemented
11C4×#124×C67C212yesmaximal, complemented
12C2×#125×C67C2211yescharacteristic
13C2×#124×C67C2212yes—
14C2×C4×Q8×C67C2212yes—
15C2×C4×D4×C67C2212yes—
16C2×C4×D4×C67C2214yes—
17C2×#117×C67C2214yes—
18#400×C67C2214yes—
19#400×C67C2214yes—
20#399×C67C2214yes—
21#399×C67C2214yes—
22#398×C67C2214yes—
23#398×C67C2214yes—
24#398×C67C2214yes—
25#398×C67C2214yes—
26#397×C67C2214yes—
27#397×C67C2214yes—
28#397×C67C2214yes—
29#397×C67C2214yes—
30#396×C67C2214yes—
31#396×C67C2214yes—
32#396×C67C2214yes—
33#396×C67C2214yes—
34#395×C67C2214yes—
35#394×C67C2214yes—
36#394×C67C2214yes—
37#394×C67C2214yes—
38#394×C67C2214yes—
39#394×C67C2214yes—
40#393×C67C2214yes—
41#392×C67C2214yes—
42#391×C67C2212yes—
43#390×C67C2212yes—
44#389×C67C2212yes—
45#388×C67C2214yes—
46#388×C67C2214yes—
47#387×C67C2214yes—
48#382×C67C2212yes—
49#382×C67C2211yescharacteristic
50C4×C4⋊C4×C67C2214yes—
51C4×C4⋊C4×C67C2212yes—
52C4×C4⋊C4×C67C2214yes—
53C4×C22⋊C4×C67C2214yes—
54C4×C22⋊C4×C67C2214yes—
55C4×C22⋊C4×C67C2214yes—
56C43×C67C2211yescharacteristic
57C22×Q8×C67C2312yes—
58C22×D4×C67C2312yes—
59C22×D4×C67C2314yes—
60C23×C4×C67C2314yes—
61#125×C67—24no—
62#124×C67—28no—
63C4×Q8×C67—44no—
64C4×D4×C67—48no—
65C4×D4×C67—44no—
66#117×C67—48no—
67C2×C4⋊C4×C67C2314yes—
68C2×C4⋊C4×C67C2312yes—
69C2×C4⋊C4×C67C2314yes—
70C2×C4⋊C4×C67C2314yes—
71C2×C4⋊C4×C67C2312yes—
72C2×C4⋊C4×C67C2314yes—
73C2×C4⋊C4×C67C2312yes—
74C2×C4⋊C4×C67C2314yes—
75C2×C4⋊C4×C67C2312yes—
76C2×C4⋊C4×C67C2314yes—
77C2×C4⋊C4×C67C2314yes—
78C2×C4⋊C4×C67C2314yes—
79C2×C22⋊C4×C67C2314yes—
80C2×C22⋊C4×C67C2314yes—
81C2×C22⋊C4×C67C2314yes—
82C2×C22⋊C4×C67C2314yes—
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×C42×C67C2314yes—
91C2×C42×C67C2314yes—
92C2×C42×C67C2312yes—
93C2×C42×C67C2311yescharacteristic
94C2×C42×C67C2312yes—
95C2×C42×C67C2311yescharacteristic
96C2×C42×C67C2311yescharacteristic
97#95×C67C2314yes—
98#95×C67C2314yes—
99#95×C67C2314yes—
100#95×C67C2314yes—
101#95×C67C2314yes—
102#95×C67C2314yes—
103#95×C67C2314yes—
104#95×C67C2314yes—
105#95×C67C2314yes—
106#95×C67C2314yes—
107#95×C67C2314yes—
108C24×C67C2414yes—
109C2×Q8×C67—24no—
110C2×Q8×C67—42no—
111C2×Q8×C67—42no—
112C2×D4×C67—24no—
113C2×D4×C67—42no—
114C2×D4×C67—44no—
115C2×D4×C67—42no—
116C2×D4×C67—44no—
117C2×D4×C67—44no—
118C22×C4×C67—28no—
119C22×C4×C67—28no—
120C22×C4×C67—28no—
121C22×C4×C67C2412yes—
122C22×C4×C67C2411yescharacteristic
123C22×C4×C67C2414yes—
124C22×C4×C67C2414yes—
125C22×C4×C67C2414yes—
126C22×C4×C67C2414yes—
127C22×C4×C67C2414yes—
128C22×C4×C67C2411yescharacteristic
129C22×C4×C67C2412yes—
130C22×C4×C67C2411yescharacteristic
131C4⋊C4×C67—24no—
132C4⋊C4×C67—28no—
133C4⋊C4×C67—42no—
134C4⋊C4×C67—44no—
135C4⋊C4×C67—44no—
136C4⋊C4×C67—44no—
137C4⋊C4×C67—44no—
138C4⋊C4×C67—44no—
139C4⋊C4×C67—42no—
140C4⋊C4×C67—44no—
141C4⋊C4×C67—42no—
142C4⋊C4×C67—44no—
143C22⋊C4×C67—28no—
144C22⋊C4×C67—28no—
145C22⋊C4×C67—44no—
146C22⋊C4×C67—44no—
147C22⋊C4×C67—44no—
148C22⋊C4×C67—44no—
149C22⋊C4×C67—44no—
150C22⋊C4×C67—44no—
151C22⋊C4×C67—44no—
152C22⋊C4×C67—44no—
153C22⋊C4×C67—44no—
154C42×C67—28no—
155C42×C67—28no—
156C42×C67—24no—
157C42×C67—22no—
158C42×C67—41no—
159C42×C67—42no—
160C42×C67#4114yes—
161C23×C67—24no—
162C23×C67—24no—
163C23×C67—24no—
164C23×C67—24no—
165C23×C67—24no—
166C23×C67—24no—
167C23×C67—24no—
168C23×C67C2511yescharacteristic
169Q8×C67—48no—
170D4×C67—48no—
171D4×C67—416no—
172C2×C4×C67—22no—
173C2×C4×C67—21no—
174C2×C4×C67—21no—
175C2×C4×C67—22no—
176C2×C4×C67—24no—
177C2×C4×C67—24no—
178C2×C4×C67—24no—
179C2×C4×C67—22no—
180C2×C4×C67—24no—
181C2×C4×C67—24no—
182C2×C4×C67—24no—
183C2×C4×C67—24no—
184C2×C4×C67—24no—
185C2×C4×C67—21no—
186C2×C4×C67—24no—
187C2×C4×C67—22no—
188C2×C4×C67—22no—
189C2×C4×C67—22no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—24no—
193C2×C4×C67—24no—
194C2×C4×C67—24no—
195C2×C4×C67—24no—
196C2×C4×C67—416no—
197C2×C4×C67C2×#4112yes—
198C2×C4×C67C2×#4114yes—
199C2×C4×C67H(5,𝔽2)14yes—
200C2×C4×C672−1+412yes—
201C22×C67—44no—
202C22×C67—44no—
203C22×C67—44no—
204C22×C67—44no—
205C22×C67—44no—
206C22×C67—44no—
207C22×C67—44no—
208C22×C67C22×#4111yescharacteristic
209C22×C67C2×H(5,𝔽2)12yes—
210C22×C67C2×2−1+411yescharacteristic
211C22×C67#58512yes—
212C22×C67#58511yescharacteristic
213C4×C67—24no—
214C4×C67—22no—
215C4×C67—44no—
216C4×C67—44no—
217C4×C67—44no—
218C4×C67—42no—
219C4×C67—44no—
220C4×C67—44no—
221C4×C67—41no—
222C4×C67#56214yes—
223#36695C6711yesmaximal, characteristic, direct product factor, Sylow
224C2×C67—84no—
225C2×C67#346012yes—
226C2×C67#346111yescharacteristic
227C2×C67C2×#56211yescharacteristic
228C2×C67#352912yes—
229C2×C67#353111yescharacteristic
230#2845C2×C6711yescharacteristic, complemented
231#2837C2×C6714yescomplemented
232#2831C2×C6714yescomplemented
233#2829C2×C6714yescomplemented
234#2820C2×C6712yescomplemented
235#2819C2×C6711yescharacteristic
236#2396C2×C6714yescomplemented
237#2395C2×C6714yescomplemented
238#2392C2×C6714yescomplemented
239C4×#125C2×C6711yescharacteristic, complemented
240C4×#124C2×C6712yescomplemented
241C67#3669511yescharacteristic, direct product factor, Sylow
242C2×#125C22×C6711yescharacteristic
243C2×#124C22×C6712yes—
244C2×C4×Q8C22×C6712yes—
245C2×C4×D4C22×C6714yes—
246C2×C4×D4C22×C6712yes—
247C2×#117C22×C6714yes—
248#400C22×C6714yes—
249#400C22×C6714yes—
250#399C22×C6714yes—
251#399C22×C6714yes—
252#398C22×C6714yes—
253#398C22×C6714yes—
254#398C22×C6714yes—
255#398C22×C6714yes—
256#397C22×C6714yes—
257#397C22×C6714yes—
258#397C22×C6714yes—
259#397C22×C6714yes—
260#396C22×C6714yes—
261#396C22×C6714yes—
262#396C22×C6714yes—
263#396C22×C6714yes—
264#395C22×C6714yes—
265#394C22×C6714yes—
266#394C22×C6714yes—
267#394C22×C6714yes—
268#394C22×C6714yes—
269#394C22×C6714yes—
270#393C22×C6714yes—
271#392C22×C6714yes—
272#391C22×C6712yes—
273#390C22×C6712yes—
274#389C22×C6712yes—
275#388C22×C6714yes—
276#388C22×C6714yes—
277#387C22×C6714yes—
278#382C22×C6711yescharacteristic
279#382C22×C6712yes—
280C4×C4⋊C4C22×C6714yes—
281C4×C4⋊C4C22×C6712yes—
282C4×C4⋊C4C22×C6714yes—
283C4×C22⋊C4C22×C6714yes—
284C4×C22⋊C4C22×C6714yes—
285C4×C22⋊C4C22×C6714yes—
286C43C22×C6711yescharacteristic
287C22×Q8C23×C6712yes—
288C22×D4C23×C6712yes—
289C22×D4C23×C6714yes—
290C23×C4C23×C6714yes—
291#125—24no—
292#124—28no—
293C4×Q8—44no—
294C4×D4—48no—
295C4×D4—44no—
296#117—48no—
297C2×C4⋊C4C23×C6714yes—
298C2×C4⋊C4C23×C6712yes—
299C2×C4⋊C4C23×C6714yes—
300C2×C4⋊C4C23×C6714yes—
301C2×C4⋊C4C23×C6712yes—
302C2×C4⋊C4C23×C6714yes—
303C2×C4⋊C4C23×C6712yes—
304C2×C4⋊C4C23×C6714yes—
305C2×C4⋊C4C23×C6712yes—
306C2×C4⋊C4C23×C6714yes—
307C2×C4⋊C4C23×C6714yes—
308C2×C4⋊C4C23×C6714yes—
309C2×C22⋊C4C23×C6714yes—
310C2×C22⋊C4C23×C6714yes—
311C2×C22⋊C4C23×C6714yes—
312C2×C22⋊C4C23×C6714yes—
313C2×C22⋊C4C23×C6714yes—
314C2×C22⋊C4C23×C6714yes—
315C2×C22⋊C4C23×C6714yes—
316C2×C22⋊C4C23×C6714yes—
317C2×C22⋊C4C23×C6714yes—
318C2×C22⋊C4C23×C6714yes—
319C2×C22⋊C4C23×C6714yes—
320C2×C42C23×C6714yes—
321C2×C42C23×C6714yes—
322C2×C42C23×C6712yes—
323C2×C42C23×C6711yescharacteristic
324C2×C42C23×C6712yes—
325C2×C42C23×C6711yescharacteristic
326C2×C42C23×C6711yescharacteristic
327#95C23×C6714yes—
328#95C23×C6714yes—
329#95C23×C6714yes—
330#95C23×C6714yes—
331#95C23×C6714yes—
332#95C23×C6714yes—
333#95C23×C6714yes—
334#95C23×C6714yes—
335#95C23×C6714yes—
336#95C23×C6714yes—
337#95C23×C6714yes—
338C24C24×C6714yes—
339C2×Q8—24no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×D4—24no—
343C2×D4—42no—
344C2×D4—44no—
345C2×D4—42no—
346C2×D4—44no—
347C2×D4—44no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4C24×C6711yescharacteristic
352C22×C4C24×C6714yes—
353C22×C4C24×C6714yes—
354C22×C4C24×C6714yes—
355C22×C4C24×C6712yes—
356C22×C4C24×C6714yes—
357C22×C4C24×C6714yes—
358C22×C4C24×C6711yescharacteristic
359C22×C4C24×C6712yes—
360C22×C4C24×C6711yescharacteristic
361C4⋊C4—24no—
362C4⋊C4—28no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C22⋊C4—28no—
374C22⋊C4—28no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C42—28no—
385C42—28no—
386C42—24no—
387C42—22no—
388C42—41no—
389C42—42no—
390C42#41×C6714yes—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23C25×C6711yescharacteristic
399Q8—48no—
400D4—48no—
401D4—416no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—416no—
427C2×C4C2×#41×C6712yes—
428C2×C4C2×#41×C6714yes—
429C2×C4H(5,𝔽2)×C6714yes—
430C2×C42−1+4×C6712yes—
431C22—44no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22—44no—
437C22—44no—
438C22C22×#41×C6711yescharacteristic
439C22C2×H(5,𝔽2)×C6712yes—
440C22C2×2−1+4×C6711yescharacteristic
441C22#585×C6711yescharacteristic
442C22#585×C6712yes—
443C4—24no—
444C4—22no—
445C4—44no—
446C4—44no—
447C4—42no—
448C4—44no—
449C4—41no—
450C4—44no—
451C4—44no—
452C4#562×C6714yes—
453C2—84no—
454C2C67×#346012yes—
455C2C67×#346111yescharacteristic
456C2C2×#562×C6711yescharacteristic
457C2C67×#352912yes—
458C2C67×#353111yescharacteristic
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
prime radicalC67×#2819C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2845C2semidirect
2. C67×#2837C2semidirect
3. C67×#2831C2semidirect
4. C67×#2829C2semidirect
5. C67×#2820C2semidirect
7. C67×#2396C2semidirect
8. C67×#2395C2semidirect
9. C67×#2392C2semidirect
10. C4×#125×C67C2semidirect
11. C4×#124×C67C2semidirect
223. #36695C67direct (N × H)
230. #2845C2×C67semidirect
231. #2837C2×C67semidirect
232. #2831C2×C67semidirect
233. #2829C2×C67semidirect
234. #2820C2×C67semidirect
236. #2396C2×C67semidirect
237. #2395C2×C67semidirect
238. #2392C2×C67semidirect
239. C4×#125C2×C67semidirect
240. C4×#124C2×C67semidirect
241. C67#36695direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#28452
C67×#28372
C67×#28312
C67×#28292
C67×#28202
C67×#28192
C67×#23962
C67×#23952
C67×#23922
C4×#125×C672
C4×#124×C672
#3669567