Finite Group Databasebeta

← C67×#61292 #30740543

Subgroups — C67×#61292 #30740543

Order 17152.

1746 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3507C212yesmaximal
2C67×#3349C218yesmaximal, complemented
3C67×#3344C218yesmaximal, complemented
4C67×#3162C211yesmaximal, characteristic, complemented
5C67×#3097C214yesmaximal, complemented
6C67×#3078C212yesmaximal, complemented
7C67×#3059C212yesmaximal, complemented
8C2×#441×C67C211yesmaximal, characteristic, complemented
9C67×#2932C211yesmaximal, characteristic
10C67×#2912C212yesmaximal, complemented
11C22×Q16×C67C2211yescharacteristic
12C2×#131×C67C2212yes—
13#559×C67—22no—
14#557×C67—22no—
15#556×C67—24no—
16#556×C67C2214yes—
17#554×C67C2214yes—
18#552×C67C2218yes—
19#539×C67—24no—
20#536×C67C2214yes—
21#533×C67C2212yes—
22#532×C67C2211yescharacteristic
23#529×C67—24no—
24#520×C67C2214yes—
25C2×C4×Q8×C67C2212yes—
26#501×C67C2214yes—
27#501×C67C2214yes—
28#485×C67C2218yes—
29#485×C67C2218yes—
30#484×C67C2218yes—
31#483×C67C2218yes—
32#479×C67C2218yes—
33#477×C67C2214yes—
34#475×C67C2214yes—
35#470×C67C2214yes—
36#470×C67C2214yes—
37#467×C67C2218yes—
38#441×C67C2218yes—
39#435×C67C2218yes—
40#428×C67C2214yes—
41#428×C67C2218yes—
42#427×C67C2212yes—
43#427×C67C2214yes—
44#427×C67C2212yes—
45C2×#106×C67C2214yes—
46C2×#106×C67C2211yescharacteristic
47#423×C67C2211yescharacteristic
48#419×C67C2214yes—
49C2×#103×C67C2212yes—
50#408×C67C2212yes—
51C2×#97×C67C2211yescharacteristic
52C2×#41×C67C2312yes—
53C22×Q8×C67C2312yes—
54C2×Q16×C67—24no—
55C2×Q16×C67C2314yes—
56#131×C67—28no—
57C2×M16×C67C2314yes—
58C2×M16×C67C2312yes—
59C22×C8×C67C2314yes—
60C22×C8×C67C2311yescharacteristic
61C22×C8×C67C2311yescharacteristic
62C4⋊Q8×C67—24no—
63C4⋊Q8×C67D411yescharacteristic
64C4⋊Q8×C67C2312yes—
65C4⋊Q8×C67C2312yes—
66#127×C67D411yescharacteristic
67#126×C67—28no—
68#126×C67—28no—
69#125×C67—24no—
70#125×C67—28no—
71#125×C67C2314yes—
72#124×C67—24no—
73#124×C67—24no—
74#124×C67D412yes—
75#123×C67—28no—
76#123×C67—28no—
77C22⋊Q8×C67—22no—
78C22⋊Q8×C67—22no—
79C22⋊Q8×C67—28no—
80C22⋊Q8×C67C2318yes—
81C22⋊Q8×C67C2318yes—
82#121×C67D414yes—
83C4×Q8×C67—24no—
84C4×Q8×C67—24no—
85C4×Q8×C67—22no—
86C4×Q8×C67—22no—
87C4×Q8×C67—22no—
88C4×Q8×C67—22no—
89C4×Q8×C67C2314yes—
90C4×Q8×C67C2314yes—
91C4×D4×C67—28no—
92C4×D4×C67—24no—
93C4×D4×C67—24no—
94C4×D4×C67C2314yes—
95#117×C67—28no—
96#117×C67C2318yes—
97#117×C67C2314yes—
98C2×C4⋊C4×C67—22no—
99C2×C4⋊C4×C67C2318yes—
100C2×C4⋊C4×C67C2312yes—
101C2×C42×C67—22no—
102C2×C42×C67C2311yescharacteristic
103#107×C67C2314yes—
104#107×C67C2318yes—
105#107×C67C2314yes—
106#106×C67C2312yes—
107#106×C67C2312yes—
108#106×C67C2318yes—
109#106×C67C2312yes—
110#106×C67C2312yes—
111#106×C67C2314yes—
112C4⋊C8×C67C2314yes—
113#103×C67C2318yes—
114#103×C67C2318yes—
115#103×C67C2318yes—
116C22⋊C8×C67C2318yes—
117#97×C67C2314yes—
118#41×C67—24no—
119#41×C67—24no—
120C2×Q8×C67—22no—
121C2×Q8×C67—22no—
122C2×Q8×C67—22no—
123C2×Q8×C67—24no—
124C2×Q8×C67—24no—
125C2×Q8×C67C2×D412yes—
126C2×Q8×C67C2414yes—
127C2×D4×C67C2×D414yes—
128C2×D4×C67C2×D412yes—
129C2×D4×C67C2×D414yes—
130C22×C4×C67—22no—
131C22×C4×C67—22no—
132C22×C4×C67—28no—
133C22×C4×C67C2×D412yes—
134C22×C4×C67C2414yes—
135C22×C4×C67C2411yescharacteristic
136Q16×C67—28no—
137M16×C67—24no—
138M16×C67—28no—
139C2×C8×C67—22no—
140C2×C8×C67—28no—
141C2×C8×C67#4114yes—
142C2×C8×C67C2414yes—
143C2×C8×C67C2412yes—
144C2×C8×C67C2412yes—
145C4⋊C4×C67—24no—
146C4⋊C4×C67—24no—
147C4⋊C4×C67—22no—
148C4⋊C4×C67—28no—
149C4⋊C4×C67—24no—
150C4⋊C4×C67—28no—
151C4⋊C4×C67—22no—
152C4⋊C4×C67—24no—
153C4⋊C4×C67—22no—
154C4⋊C4×C67—28no—
155C4⋊C4×C67C2×D414yes—
156C4⋊C4×C67C2414yes—
157C4⋊C4×C67C2414yes—
158C4⋊C4×C67C2414yes—
159C22⋊C4×C67—28no—
160C22⋊C4×C67—24no—
161C22⋊C4×C67—28no—
162C22⋊C4×C67—28no—
163C22⋊C4×C67C2×D418yes—
164C42×C67—24no—
165C42×C67—24no—
166C42×C67—28no—
167C42×C67—24no—
168C42×C67C2×D411yescharacteristic
169C42×C67C2×D411yescharacteristic
170C42×C67C2412yes—
171C23×C67C22×D414yes—
172C23×C67C22×D411yescharacteristic
173Q8×C67—24no—
174Q8×C67—24no—
175Q8×C67—24no—
176Q8×C67—44no—
177Q8×C67—44no—
178Q8×C67—82no—
179D4×C67—28no—
180D4×C67—24no—
181D4×C67—44no—
182C2×C4×C67—24no—
183C2×C4×C67—24no—
184C2×C4×C67—24no—
185C2×C4×C67—24no—
186C2×C4×C67—24no—
187C2×C4×C67—24no—
188C2×C4×C67—42no—
189C2×C4×C67—44no—
190C2×C4×C67—82no—
191C2×C4×C67—88no—
192C2×C4×C67C22×D412yes—
193C2×C4×C67C22×D412yes—
194C2×C4×C67C22×D414yes—
195C2×C4×C67C22×D411yescharacteristic
196C2×C4×C67C2×#4112yes—
197C2×C4×C672−1+412yes—
198C2×C4×C67C2511yescharacteristic
199C8×C67—24no—
200C8×C67—22no—
201C8×C67C2×#4114yes—
202C22×C67—22no—
203C22×C67—28no—
204C22×C67—24no—
205C22×C67#54712yes—
206C22×C67C23×D411yescharacteristic
207C4×C67—22no—
208C4×C67—22no—
209C4×C67—24no—
210C4×C67—44no—
211C4×C67—44no—
212C4×C67—44no—
213C4×C67—44no—
214C4×C67#54712yes—
215C4×C67C22×#4111yescharacteristic
216C4×C67C2×2−1+411yescharacteristic
217#61292C6711yesmaximal, characteristic, direct product factor, Sylow
218C2×C67—22no—
219C2×C67—44no—
220C2×C67C2×#54711yescharacteristic
221C2×C67#358612yes—
222#3507C2×C6712yes—
223#3349C2×C6718yescomplemented
224#3344C2×C6718yescomplemented
225#3162C2×C6711yescharacteristic, complemented
226#3097C2×C6714yescomplemented
227#3078C2×C6712yescomplemented
228#3059C2×C6712yescomplemented
229C2×#441C2×C6711yescharacteristic, complemented
230#2932C2×C6711yescharacteristic
231#2912C2×C6712yescomplemented
232C67#6129211yescharacteristic, direct product factor, Sylow
233C22×Q16C22×C6711yescharacteristic
234C2×#131C22×C6712yes—
235#559—22no—
236#557—22no—
237#556—24no—
238#556C22×C6714yes—
239#554C22×C6714yes—
240#552C22×C6718yes—
241#539—24no—
242#536C22×C6714yes—
243#533C22×C6712yes—
244#532C22×C6711yescharacteristic
245#529—24no—
246#520C22×C6714yes—
247C2×C4×Q8C22×C6712yes—
248#501C22×C6714yes—
249#501C22×C6714yes—
250#485C22×C6718yes—
251#485C22×C6718yes—
252#484C22×C6718yes—
253#483C22×C6718yes—
254#479C22×C6718yes—
255#477C22×C6714yes—
256#475C22×C6714yes—
257#470C22×C6714yes—
258#470C22×C6714yes—
259#467C22×C6718yes—
260#441C22×C6718yes—
261#435C22×C6718yes—
262#428C22×C6714yes—
263#428C22×C6718yes—
264#427C22×C6714yes—
265#427C22×C6712yes—
266#427C22×C6712yes—
267C2×#106C22×C6714yes—
268C2×#106C22×C6711yescharacteristic
269#423C22×C6711yescharacteristic
270#419C22×C6714yes—
271C2×#103C22×C6712yes—
272#408C22×C6712yes—
273C2×#97C22×C6711yescharacteristic
274C2×#41C23×C6712yes—
275C22×Q8C23×C6712yes—
276C2×Q16—24no—
277C2×Q16C23×C6714yes—
278#131—28no—
279C2×M16C23×C6712yes—
280C2×M16C23×C6714yes—
281C22×C8C23×C6714yes—
282C22×C8C23×C6711yescharacteristic
283C22×C8C23×C6711yescharacteristic
284C4⋊Q8—24no—
285C4⋊Q8D4×C6711yescharacteristic
286C4⋊Q8C23×C6712yes—
287C4⋊Q8C23×C6712yes—
288#127D4×C6711yescharacteristic
289#126—28no—
290#126—28no—
291#125—24no—
292#125—28no—
293#125C23×C6714yes—
294#124—24no—
295#124—24no—
296#124D4×C6712yes—
297#123—28no—
298#123—28no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—28no—
302C22⋊Q8C23×C6718yes—
303C22⋊Q8C23×C6718yes—
304#121D4×C6714yes—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8C23×C6714yes—
312C4×Q8C23×C6714yes—
313C4×D4—28no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4C23×C6714yes—
317#117—28no—
318#117C23×C6714yes—
319#117C23×C6718yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4C23×C6712yes—
322C2×C4⋊C4C23×C6718yes—
323C2×C42—22no—
324C2×C42C23×C6711yescharacteristic
325#107C23×C6714yes—
326#107C23×C6718yes—
327#107C23×C6714yes—
328#106C23×C6714yes—
329#106C23×C6712yes—
330#106C23×C6718yes—
331#106C23×C6712yes—
332#106C23×C6712yes—
333#106C23×C6712yes—
334C4⋊C8C23×C6714yes—
335#103C23×C6718yes—
336#103C23×C6718yes—
337#103C23×C6718yes—
338C22⋊C8C23×C6718yes—
339#97C23×C6714yes—
340#41—24no—
341#41—24no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8C2×D4×C6712yes—
348C2×Q8C24×C6714yes—
349C2×D4C2×D4×C6714yes—
350C2×D4C2×D4×C6712yes—
351C2×D4C2×D4×C6714yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4C2×D4×C6712yes—
356C22×C4C24×C6714yes—
357C22×C4C24×C6711yescharacteristic
358Q16—28no—
359M16—24no—
360M16—28no—
361C2×C8—22no—
362C2×C8—28no—
363C2×C8#41×C6714yes—
364C2×C8C24×C6712yes—
365C2×C8C24×C6714yes—
366C2×C8C24×C6712yes—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—28no—
377C4⋊C4C2×D4×C6714yes—
378C4⋊C4C24×C6714yes—
379C4⋊C4C24×C6714yes—
380C4⋊C4C24×C6714yes—
381C22⋊C4—28no—
382C22⋊C4—24no—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4C2×D4×C6718yes—
386C42—24no—
387C42—24no—
388C42—28no—
389C42—24no—
390C42C2×D4×C6711yescharacteristic
391C42C2×D4×C6711yescharacteristic
392C42C24×C6712yes—
393C23C22×D4×C6714yes—
394C23C22×D4×C6711yescharacteristic
395Q8—24no—
396Q8—24no—
397Q8—24no—
398Q8—44no—
399Q8—44no—
400Q8—82no—
401D4—28no—
402D4—24no—
403D4—44no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—82no—
413C2×C4—88no—
414C2×C4C22×D4×C6712yes—
415C2×C4C22×D4×C6712yes—
416C2×C4C22×D4×C6714yes—
417C2×C4C22×D4×C6711yescharacteristic
418C2×C4C2×#41×C6712yes—
419C2×C42−1+4×C6712yes—
420C2×C4C25×C6711yescharacteristic
421C8—24no—
422C8—22no—
423C8C2×#41×C6714yes—
424C22—22no—
425C22—28no—
426C22—24no—
427C22#547×C6712yes—
428C22C23×D4×C6711yescharacteristic
429C4—22no—
430C4—22no—
431C4—24no—
432C4—44no—
433C4—44no—
434C4—44no—
435C4—44no—
436C4#547×C6712yes—
437C4C22×#41×C6711yescharacteristic
438C4C2×2−1+4×C6711yescharacteristic
439C2—22no—
440C2—44no—
441C2C2×#547×C6711yescharacteristic
442C2C67×#358612yes—
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
prime radical#532×C67C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ #532×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#3349C2semidirect
3. C67×#3344C2semidirect
4. C67×#3162C2semidirect
5. C67×#3097C2semidirect
6. C67×#3078C2semidirect
7. C67×#3059C2semidirect
8. C2×#441×C67C2semidirect
10. C67×#2912C2semidirect
217. #61292C67direct (N × H)
223. #3349C2×C67semidirect
224. #3344C2×C67semidirect
225. #3162C2×C67semidirect
226. #3097C2×C67semidirect
227. #3078C2×C67semidirect
228. #3059C2×C67semidirect
229. C2×#441C2×C67semidirect
231. #2912C2×C67semidirect
232. C67#61292direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#35072
C67×#33492
C67×#33442
C67×#31622
C67×#30972
C67×#30782
C67×#30592
C2×#441×C672
C67×#29322
C67×#29122
#6129267