Finite Group Databasebeta

← C67×#36684 #30715935

Subgroups — C67×#36684 #30715935

Order 17152.

2138 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529×C67C212yesmaximal, complemented
2C67×#2823C214yesmaximal, complemented
3C67×#2822C211yesmaximal, characteristic, complemented
4C67×#2820C212yesmaximal, complemented
5C67×#2819C211yesmaximal, characteristic, complemented
6C67×#2814C214yesmaximal, complemented
7C67×#2813C214yesmaximal, complemented
8C67×#2396C214yesmaximal, complemented
9C67×#2392C214yesmaximal, complemented
10C67×#2392C214yesmaximal, complemented
11C67×#2285C211yesmaximal, characteristic, complemented
12#529×C67—216no—
13C2×#126×C67C2214yes—
14C2×#125×C67C2212yes—
15C2×#124×C67C2212yes—
16C2×#123×C67C2214yes—
17C2×C22⋊Q8×C67C2212yes—
18C2×#121×C67C2212yes—
19C2×C4×Q8×C67C2212yes—
20C2×C4×D4×C67C2214yes—
21C2×C4×D4×C67C2212yes—
22C2×C4×D4×C67C2214yes—
23C2×#117×C67C2214yes—
24C2×#117×C67C2212yes—
25C22×C42×C67C2211yescharacteristic
26#400×C67C2214yes—
27#399×C67C2214yes—
28#398×C67C2214yes—
29#398×C67C2218yes—
30#398×C67C2214yes—
31#397×C67C2214yes—
32#397×C67C2214yes—
33#396×C67C2218yes—
34#396×C67C2218yes—
35#396×C67C2218yes—
36#395×C67C2214yes—
37#394×C67C2218yes—
38#394×C67C2214yes—
39#394×C67C2218yes—
40#394×C67C2214yes—
41#393×C67C2214yes—
42#392×C67C2214yes—
43#392×C67C2214yes—
44#384×C67C2212yes—
45#382×C67C2211yescharacteristic
46#382×C67C2211yescharacteristic
47#380×C67C2214yes—
48C4×C4⋊C4×C67C2214yes—
49C4×C4⋊C4×C67C2214yes—
50C4×C22⋊C4×C67C2218yes—
51C22×Q8×C67C2312yes—
52C22×D4×C67C2314yes—
53C22×D4×C67C2312yes—
54C22×D4×C67C2314yes—
55C23×C4×C67C2314yes—
56C23×C4×C67C2311yescharacteristic
57C23×C4×C67C2312yes—
58#126×C67—216no—
59#125×C67—28no—
60#124×C67—28no—
61#123×C67—216no—
62C22⋊Q8×C67—28no—
63#121×C67—28no—
64C4×Q8×C67—28no—
65C4×D4×C67—216no—
66C4×D4×C67—28no—
67C4×D4×C67—48no—
68#117×C67—216no—
69#117×C67—44no—
70C2×C4⋊C4×C67C2314yes—
71C2×C4⋊C4×C67C2314yes—
72C2×C4⋊C4×C67C2312yes—
73C2×C4⋊C4×C67C2314yes—
74C2×C4⋊C4×C67C2312yes—
75C2×C4⋊C4×C67C2314yes—
76C2×C4⋊C4×C67C2312yes—
77C2×C4⋊C4×C67C2312yes—
78C2×C4⋊C4×C67C2314yes—
79C2×C4⋊C4×C67C2318yes—
80C2×C4⋊C4×C67C2318yes—
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×C67C2318yes—
88C2×C22⋊C4×C67C2318yes—
89C2×C22⋊C4×C67C2318yes—
90C2×C42×C67D418yescomplemented
91C2×C42×C67C2314yes—
92C2×C42×C67C2314yes—
93C2×C42×C67C2311yescharacteristic
94C2×C42×C67C2311yescharacteristic
95C2×C42×C67C2312yes—
96#95×C67C2318yes—
97#95×C67C2318yes—
98#95×C67C2314yes—
99#95×C67C2314yes—
100#95×C67C2314yes—
101#95×C67C2314yes—
102C24×C67C2414yes—
103C24×C67C2411yescharacteristic
104C2×Q8×C67—24no—
105C2×Q8×C67—44no—
106C2×D4×C67—28no—
107C2×D4×C67—24no—
108C2×D4×C67—44no—
109C2×D4×C67—44no—
110C2×D4×C67—48no—
111C2×D4×C67—48no—
112C22×C4×C67—22no—
113C22×C4×C67—22no—
114C22×C4×C67—28no—
115C22×C4×C67—28no—
116C22×C4×C67—28no—
117C22×C4×C67—44no—
118C22×C4×C67C2×D414yes—
119C22×C4×C67C2×D418yes—
120C22×C4×C67C2414yes—
121C22×C4×C67C2414yes—
122C22×C4×C67C2414yes—
123C22×C4×C67C2414yes—
124C22×C4×C67C2414yes—
125C22×C4×C67C2411yescharacteristic
126C22×C4×C67C2412yes—
127C22×C4×C67C2412yes—
128C22×C4×C67C2411yescharacteristic
129C4⋊C4×C67—28no—
130C4⋊C4×C67—28no—
131C4⋊C4×C67—24no—
132C4⋊C4×C67—28no—
133C4⋊C4×C67—24no—
134C4⋊C4×C67—28no—
135C4⋊C4×C67—48no—
136C4⋊C4×C67—48no—
137C4⋊C4×C67—42no—
138C4⋊C4×C67—42no—
139C4⋊C4×C67—44no—
140C22⋊C4×C67—28no—
141C22⋊C4×C67—28no—
142C22⋊C4×C67—28no—
143C22⋊C4×C67—28no—
144C22⋊C4×C67—28no—
145C22⋊C4×C67—48no—
146C22⋊C4×C67—48no—
147C22⋊C4×C67—44no—
148C22⋊C4×C67—48no—
149C42×C67—28no—
150C42×C67—28no—
151C42×C67C2×D418yes—
152C42×C67C2×D414yes—
153C42×C67C2×D414yes—
154C23×C67—21no—
155C23×C67—21no—
156C23×C67—24no—
157C23×C67—24no—
158C23×C67—24no—
159C23×C67—24no—
160C23×C67—28no—
161C23×C67—28no—
162C23×C67C22×D412yes—
163C23×C67C2511yescharacteristic
164Q8×C67—48no—
165D4×C67—48no—
166D4×C67—416no—
167D4×C67—416no—
168C2×C4×C67—21no—
169C2×C4×C67—21no—
170C2×C4×C67—24no—
171C2×C4×C67—24no—
172C2×C4×C67—28no—
173C2×C4×C67—24no—
174C2×C4×C67—24no—
175C2×C4×C67—24no—
176C2×C4×C67—24no—
177C2×C4×C67—24no—
178C2×C4×C67—24no—
179C2×C4×C67—24no—
180C2×C4×C67—28no—
181C2×C4×C67—28no—
182C2×C4×C67—28no—
183C2×C4×C67—28no—
184C2×C4×C67—48no—
185C2×C4×C67—416no—
186C2×C4×C67C22×D412yes—
187C2×C4×C67C22×D414yes—
188C2×C4×C67C22×D414yes—
189C2×C4×C67C22×D412yes—
190C2×C4×C67H(5,𝔽2)12yes—
191C2×C4×C672−1+412yes—
192C22×C67—22no—
193C22×C67—24no—
194C22×C67—42no—
195C22×C67—42no—
196C22×C67—44no—
197C22×C67—44no—
198C22×C67—44no—
199C22×C67—48no—
200C22×C67—48no—
201C22×C67C23×D411yescharacteristic
202C22×C67C2×H(5,𝔽2)11yescharacteristic
203C22×C67C2×2−1+411yescharacteristic
204C22×C67#58514yes—
205C4×C67—22no—
206C4×C67—24no—
207C4×C67—24no—
208C4×C67—44no—
209C4×C67—44no—
210C4×C67—44no—
211C4×C67—44no—
212C4×C67—44no—
213C4×C67#53714yes—
214#36684C6711yesmaximal, characteristic, direct product factor, Sylow
215C2×C67—42no—
216C2×C67—84no—
217C2×C67C2×#53711yescharacteristic
218C2×C67#344912yes—
219C2×C67#352912yes—
220C2×C67#353112yes—
221C2×#529C2×C6712yescomplemented
222#2823C2×C6714yescomplemented
223#2822C2×C6711yescharacteristic, complemented
224#2820C2×C6712yescomplemented
225#2819C2×C6711yescharacteristic, complemented
226#2814C2×C6714yescomplemented
227#2813C2×C6714yescomplemented
228#2396C2×C6714yescomplemented
229#2392C2×C6714yescomplemented
230#2392C2×C6714yescomplemented
231#2285C2×C6711yescharacteristic, complemented
232C67#3668411yescharacteristic, direct product factor, Sylow
233#529—216no—
234C2×#126C22×C6714yes—
235C2×#125C22×C6712yes—
236C2×#124C22×C6712yes—
237C2×#123C22×C6714yes—
238C2×C22⋊Q8C22×C6712yes—
239C2×#121C22×C6712yes—
240C2×C4×Q8C22×C6712yes—
241C2×C4×D4C22×C6714yes—
242C2×C4×D4C22×C6712yes—
243C2×C4×D4C22×C6714yes—
244C2×#117C22×C6714yes—
245C2×#117C22×C6712yes—
246C22×C42C22×C6711yescharacteristic
247#400C22×C6714yes—
248#399C22×C6714yes—
249#398C22×C6714yes—
250#398C22×C6714yes—
251#398C22×C6718yes—
252#397C22×C6714yes—
253#397C22×C6714yes—
254#396C22×C6718yes—
255#396C22×C6718yes—
256#396C22×C6718yes—
257#395C22×C6714yes—
258#394C22×C6714yes—
259#394C22×C6714yes—
260#394C22×C6718yes—
261#394C22×C6718yes—
262#393C22×C6714yes—
263#392C22×C6714yes—
264#392C22×C6714yes—
265#384C22×C6712yes—
266#382C22×C6711yescharacteristic
267#382C22×C6711yescharacteristic
268#380C22×C6714yes—
269C4×C4⋊C4C22×C6714yes—
270C4×C4⋊C4C22×C6714yes—
271C4×C22⋊C4C22×C6718yes—
272C22×Q8C23×C6712yes—
273C22×D4C23×C6714yes—
274C22×D4C23×C6712yes—
275C22×D4C23×C6714yes—
276C23×C4C23×C6714yes—
277C23×C4C23×C6711yescharacteristic
278C23×C4C23×C6712yes—
279#126—216no—
280#125—28no—
281#124—28no—
282#123—216no—
283C22⋊Q8—28no—
284#121—28no—
285C4×Q8—28no—
286C4×D4—216no—
287C4×D4—28no—
288C4×D4—48no—
289#117—216no—
290#117—44no—
291C2×C4⋊C4C23×C6714yes—
292C2×C4⋊C4C23×C6714yes—
293C2×C4⋊C4C23×C6712yes—
294C2×C4⋊C4C23×C6714yes—
295C2×C4⋊C4C23×C6712yes—
296C2×C4⋊C4C23×C6714yes—
297C2×C4⋊C4C23×C6712yes—
298C2×C4⋊C4C23×C6712yes—
299C2×C4⋊C4C23×C6714yes—
300C2×C4⋊C4C23×C6718yes—
301C2×C4⋊C4C23×C6718yes—
302C2×C22⋊C4C23×C6714yes—
303C2×C22⋊C4C23×C6714yes—
304C2×C22⋊C4C23×C6714yes—
305C2×C22⋊C4C23×C6714yes—
306C2×C22⋊C4C23×C6714yes—
307C2×C22⋊C4C23×C6714yes—
308C2×C22⋊C4C23×C6718yes—
309C2×C22⋊C4C23×C6718yes—
310C2×C22⋊C4C23×C6718yes—
311C2×C42D4×C6718yescomplemented
312C2×C42C23×C6714yes—
313C2×C42C23×C6714yes—
314C2×C42C23×C6712yes—
315C2×C42C23×C6711yescharacteristic
316C2×C42C23×C6711yescharacteristic
317#95C23×C6718yes—
318#95C23×C6718yes—
319#95C23×C6714yes—
320#95C23×C6714yes—
321#95C23×C6714yes—
322#95C23×C6714yes—
323C24C24×C6714yes—
324C24C24×C6711yescharacteristic
325C2×Q8—24no—
326C2×Q8—44no—
327C2×D4—28no—
328C2×D4—24no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—48no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—44no—
339C22×C4C2×D4×C6714yes—
340C22×C4C2×D4×C6718yes—
341C22×C4C24×C6714yes—
342C22×C4C24×C6714yes—
343C22×C4C24×C6714yes—
344C22×C4C24×C6714yes—
345C22×C4C24×C6714yes—
346C22×C4C24×C6712yes—
347C22×C4C24×C6711yescharacteristic
348C22×C4C24×C6712yes—
349C22×C4C24×C6711yescharacteristic
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—28no—
354C4⋊C4—24no—
355C4⋊C4—28no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C22⋊C4—28no—
362C22⋊C4—28no—
363C22⋊C4—28no—
364C22⋊C4—28no—
365C22⋊C4—28no—
366C22⋊C4—44no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C42—28no—
371C42—28no—
372C42C2×D4×C6718yes—
373C42C2×D4×C6714yes—
374C42C2×D4×C6714yes—
375C23—21no—
376C23—21no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—28no—
382C23—28no—
383C23C22×D4×C6712yes—
384C23C25×C6711yescharacteristic
385Q8—48no—
386D4—48no—
387D4—416no—
388D4—416no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—48no—
406C2×C4—416no—
407C2×C4C22×D4×C6712yes—
408C2×C4C22×D4×C6714yes—
409C2×C4C22×D4×C6714yes—
410C2×C4C22×D4×C6712yes—
411C2×C4H(5,𝔽2)×C6712yes—
412C2×C42−1+4×C6712yes—
413C22—22no—
414C22—24no—
415C22—42no—
416C22—42no—
417C22—48no—
418C22—48no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22C23×D4×C6711yescharacteristic
423C22C2×H(5,𝔽2)×C6711yescharacteristic
424C22C2×2−1+4×C6711yescharacteristic
425C22#585×C6714yes—
426C4—22no—
427C4—24no—
428C4—24no—
429C4—44no—
430C4—44no—
431C4—44no—
432C4—44no—
433C4—44no—
434C4#537×C6714yes—
435C2—42no—
436C2—84no—
437C2C2×#537×C6711yescharacteristic
438C2C67×#344912yes—
439C2C67×#352912yes—
440C2C67×#353112yes—
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×#529×C67C2semidirect
2. C67×#2823C2semidirect
3. C67×#2822C2semidirect
4. C67×#2820C2semidirect
5. C67×#2819C2semidirect
6. C67×#2814C2semidirect
7. C67×#2813C2semidirect
8. C67×#2396C2semidirect
9. C67×#2392C2semidirect
10. C67×#2392C2semidirect
11. C67×#2285C2semidirect
90. C2×C42×C67D4semidirect
214. #36684C67direct (N × H)
221. C2×#529C2×C67semidirect
222. #2823C2×C67semidirect
223. #2822C2×C67semidirect
224. #2820C2×C67semidirect
225. #2819C2×C67semidirect
226. #2814C2×C67semidirect
227. #2813C2×C67semidirect
228. #2396C2×C67semidirect
229. #2392C2×C67semidirect
230. #2392C2×C67semidirect
231. #2285C2×C67semidirect
232. C67#36684direct (N × H)
311. C2×C42D4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#529×C672
C67×#28232
C67×#28222
C67×#28202
C67×#28192
C67×#28142
C67×#28132
C67×#23962
C67×#23922
C67×#22852
#3668467