Finite Group Databasebeta

← C67×#60175 #30739426

Subgroups — C67×#60175 #30739426

Order 17152.

2650 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#578×C67C211yesmaximal, characteristic
2C4×2−1+4×C67C212yesmaximal, complemented
3C67×#2952C213yesmaximal, complemented
4C67×#2950C213yesmaximal, complemented
5C67×#2947C216yesmaximal, complemented
6C4×#137×C67C216yesmaximal, complemented
7C67×#2943C213yesmaximal, complemented
8C2×#441×C67C213yesmaximal, complemented
9C67×#2912C211yesmaximal, characteristic, complemented
10C67×#2898C212yesmaximal, complemented
11C67×#2875C211yesmaximal, characteristic, complemented
12C2×2−1+4×C67C2212yes—
13#578×C67C4116yescomplemented
14C2×#137×C67C2216yes—
15C2×#135×C67C2213yes—
16C22×Q16×C67C2213yes—
17C2×#131×C67C2211yescharacteristic
18C2×#131×C67C2211yescharacteristic
19#520×C67—26no—
20#520×C67C2216yes—
21#520×C67C2212yes—
22#519×C67—22no—
23#519×C67C2216yes—
24C4×#41×C67—26no—
25C4×#41×C67C2216yes—
26C4×#41×C67C2211yescharacteristic
27C2×C4×Q8×C67—22no—
28C2×C4×Q8×C67C2216yes—
29#442×C67C2211yescharacteristic
30#442×C67C2213yes—
31#441×C67C2213yes—
32#441×C67C2216yes—
33#441×C67C2216yes—
34#441×C67C2216yes—
35#441×C67C2213yes—
36#440×C67C2216yes—
37#440×C67C2216yes—
38C4×Q16×C67C22112yes—
39C4×SD16×C67C22112yes—
40#428×C67C2216yes—
41#427×C67C2213yes—
42C2×#106×C67C2213yes—
43#419×C67C2216yes—
44#417×C67C2216yes—
45#416×C67C2216yes—
46C2×#103×C67C2216yes—
47#405×C67C2213yes—
48#405×C67C2213yes—
49C4×M16×C67C2213yes—
50C2×#97×C67C2213yes—
512−1+4×C67—28no—
522−1+4×C67C2×C4116yescomplemented
53C2×#41×C67—26no—
54C2×#41×C67C2316yes—
55C2×#41×C67C2311yescharacteristic
56C22×Q8×C67—22no—
57C22×Q8×C67C2316yes—
58#137×C67C2×C4148yescomplemented
59#135×C67C2×C4124yescomplemented
60C2×Q16×C67C2×C4124yescomplemented
61C2×Q16×C67C2313yes—
62C2×Q16×C67C2316yes—
63C2×SD16×C67C2316yes—
64C2×D8×C67C2311yescharacteristic
65#131×C67—24no—
66#131×C67C2×C418yescomplemented
67C2×M16×C67C2313yes—
68C2×M16×C67C2313yes—
69C22×C8×C67C2313yes—
70C22×C8×C67C2313yes—
71C4×Q8×C67—22no—
72C4×Q8×C67—26no—
73C4×Q8×C67—26no—
74C4×Q8×C67—26no—
75C4×Q8×C67—26no—
76C4×Q8×C67D412yes—
77C4×Q8×C67C2316yes—
78C4×Q8×C67C23112yes—
79C4×Q8×C67C2316yes—
80C4×D4×C67—26no—
81C4×D4×C67—26no—
82C4×D4×C67—26no—
83C4×D4×C67—212no—
84C4×D4×C67D416yes—
85C4×D4×C67C2312yes—
86C4×D4×C67C2316yes—
87#117×C67—26no—
88#117×C67—212no—
89#117×C67—26no—
90#117×C67C2316yes—
91#117×C67C2313yes—
92C2×C4⋊C4×C67—26no—
93C2×C4⋊C4×C67—26no—
94C2×C4⋊C4×C67C2316yes—
95C2×C42×C67—26no—
96C2×C42×C67—26no—
97C2×C42×C67C2313yes—
98#107×C67C2316yes—
99#106×C67C2313yes—
100#106×C67C2316yes—
101#106×C67C2311yescharacteristic
102#103×C67C2316yes—
103#103×C67C23112yes—
104#103×C67C2316yes—
105#102×C67C2312yes—
106#102×C67C2316yes—
107#97×C67C2311yescharacteristic
108#97×C67C2313yes—
109#97×C67C2313yes—
110#97×C67C2313yes—
111C4×C8×C67C2316yes—
112#41×C67—248no—
113#41×C67—212no—
114#41×C67C22×C4124yes—
115#41×C67C22×C414yes—
116#41×C67#4114yes—
117C2×Q8×C67—216no—
118C2×Q8×C67—26no—
119C2×Q8×C67—212no—
120C2×Q8×C67C22×C4124yes—
121C2×Q8×C67C2×D411yescharacteristic
122C2×Q8×C67C2416yes—
123C2×D4×C67—26no—
124C2×D4×C67C2×D413yes—
125C2×D4×C67C2412yes—
126C22×C4×C67—26no—
127C22×C4×C67—26no—
128C22×C4×C67—26no—
129C22×C4×C67—22no—
130C22×C4×C67—26no—
131C22×C4×C67C2×D416yes—
132C22×C4×C67C2413yes—
133Q16×C67C22×C4112yes—
134Q16×C67C22×C4124yes—
135SD16×C67C22×C4124yes—
136D8×C67C22×C414yes—
137M16×C67—26no—
138M16×C67C22×C4112yes—
139C2×C8×C67—26no—
140C2×C8×C67C22×C4112yes—
141C2×C8×C67C2411yescharacteristic
142C2×C8×C67C2413yes—
143C2×C8×C67C2411yescharacteristic
144C2×C8×C67C2413yes—
145C4⋊C4×C67—26no—
146C4⋊C4×C67—212no—
147C4⋊C4×C67—26no—
148C4⋊C4×C67—26no—
149C4⋊C4×C67C2×D416yes—
150C4⋊C4×C67C2416yes—
151C4⋊C4×C67C2412yes—
152C22⋊C4×C67—26no—
153C22⋊C4×C67—22no—
154C22⋊C4×C67—26no—
155C22⋊C4×C67C2×D416yes—
156C42×C67—26no—
157C42×C67—26no—
158C42×C67—212no—
159C42×C67—26no—
160C42×C67C2×D416yes—
161C42×C67C2411yescharacteristic
162C42×C67C2413yes—
163C23×C67—22no—
164C23×C67C22×D413yes—
165Q8×C67—224no—
166Q8×C67—26no—
167Q8×C67C4×D414yes—
168Q8×C67C23×C4112yes—
169D4×C67—224no—
170D4×C67—22no—
171D4×C67C4×D4112yes—
172D4×C67C23×C414yes—
173C2×C4×C67—224no—
174C2×C4×C67—26no—
175C2×C4×C67—224no—
176C2×C4×C67—22no—
177C2×C4×C67—26no—
178C2×C4×C67—46no—
179C2×C4×C67—44no—
180C2×C4×C67—86no—
181C2×C4×C67—86no—
182C2×C4×C67C23×C416yes—
183C2×C4×C67C22×D412yes—
184C2×C4×C67C22×D416yes—
185C2×C4×C67C22×D413yes—
186C2×C4×C67C2×#4116yes—
187C2×C4×C67C2511yescharacteristic
188C8×C67—21no—
189C8×C67—23no—
190C8×C67C23×C412yes—
191C8×C67C23×C416yes—
192C22×C67—26no—
193C22×C67—24no—
194C22×C67—42no—
195C22×C67—42no—
196C22×C67C2×C4×D416yes—
197C22×C67C23×D411yescharacteristic
198C4×C67—22no—
199C4×C67—26no—
200C4×C67—212no—
201C4×C67—42no—
202C4×C67—46no—
203C4×C67C2×C4×D416yes—
204C4×C67C24×C411yescharacteristic
205C4×C67C22×#4111yescharacteristic
206#60175C6711yesmaximal, characteristic, direct product factor, Sylow
207C2×C67—26no—
208C2×C67—44no—
209C2×C67C22×C4×D411yescharacteristic
210C2×C67#358612yes—
211C2×#578C2×C6711yescharacteristic
212C4×2−1+4C2×C6712yescomplemented
213#2952C2×C6713yescomplemented
214#2950C2×C6713yescomplemented
215#2947C2×C6716yescomplemented
216C4×#137C2×C6716yescomplemented
217#2943C2×C6713yescomplemented
218C2×#441C2×C6713yescomplemented
219#2912C2×C6711yescharacteristic, complemented
220#2898C2×C6712yescomplemented
221#2875C2×C6711yescharacteristic, complemented
222C67#6017511yescharacteristic, direct product factor, Sylow
223C2×2−1+4C22×C6712yes—
224#578C4×C67116yescomplemented
225C2×#137C22×C6716yes—
226C2×#135C22×C6713yes—
227C22×Q16C22×C6713yes—
228C2×#131C22×C6711yescharacteristic
229C2×#131C22×C6711yescharacteristic
230#520—26no—
231#520C22×C6716yes—
232#520C22×C6712yes—
233#519—22no—
234#519C22×C6716yes—
235C4×#41—26no—
236C4×#41C22×C6716yes—
237C4×#41C22×C6711yescharacteristic
238C2×C4×Q8—22no—
239C2×C4×Q8C22×C6716yes—
240#442C22×C6711yescharacteristic
241#442C22×C6713yes—
242#441C22×C6713yes—
243#441C22×C6716yes—
244#441C22×C6716yes—
245#441C22×C6716yes—
246#441C22×C6713yes—
247#440C22×C6716yes—
248#440C22×C6716yes—
249C4×Q16C22×C67112yes—
250C4×SD16C22×C67112yes—
251#428C22×C6716yes—
252#427C22×C6713yes—
253C2×#106C22×C6713yes—
254#419C22×C6716yes—
255#417C22×C6716yes—
256#416C22×C6716yes—
257C2×#103C22×C6716yes—
258#405C22×C6713yes—
259#405C22×C6713yes—
260C4×M16C22×C6713yes—
261C2×#97C22×C6713yes—
2622−1+4—28no—
2632−1+4C2×C4×C67116yescomplemented
264C2×#41—26no—
265C2×#41C23×C6716yes—
266C2×#41C23×C6711yescharacteristic
267C22×Q8—22no—
268C22×Q8C23×C6716yes—
269#137C2×C4×C67148yescomplemented
270#135C2×C4×C67124yescomplemented
271C2×Q16C2×C4×C67124yescomplemented
272C2×Q16C23×C6713yes—
273C2×Q16C23×C6716yes—
274C2×SD16C23×C6716yes—
275C2×D8C23×C6711yescharacteristic
276#131—24no—
277#131C2×C4×C6718yescomplemented
278C2×M16C23×C6713yes—
279C2×M16C23×C6713yes—
280C22×C8C23×C6713yes—
281C22×C8C23×C6713yes—
282C4×Q8—22no—
283C4×Q8—26no—
284C4×Q8—26no—
285C4×Q8—26no—
286C4×Q8—26no—
287C4×Q8D4×C6712yes—
288C4×Q8C23×C6716yes—
289C4×Q8C23×C67112yes—
290C4×Q8C23×C6716yes—
291C4×D4—26no—
292C4×D4—26no—
293C4×D4—26no—
294C4×D4—212no—
295C4×D4D4×C6716yes—
296C4×D4C23×C6712yes—
297C4×D4C23×C6716yes—
298#117—26no—
299#117—212no—
300#117—26no—
301#117C23×C6716yes—
302#117C23×C6713yes—
303C2×C4⋊C4—26no—
304C2×C4⋊C4—26no—
305C2×C4⋊C4C23×C6716yes—
306C2×C42—26no—
307C2×C42—26no—
308C2×C42C23×C6713yes—
309#107C23×C6716yes—
310#106C23×C6713yes—
311#106C23×C6716yes—
312#106C23×C6711yescharacteristic
313#103C23×C6716yes—
314#103C23×C67112yes—
315#103C23×C6716yes—
316#102C23×C6712yes—
317#102C23×C6716yes—
318#97C23×C6711yescharacteristic
319#97C23×C6713yes—
320#97C23×C6713yes—
321#97C23×C6713yes—
322C4×C8C23×C6716yes—
323#41—248no—
324#41—212no—
325#41C22×C4×C67124yes—
326#41C22×C4×C6714yes—
327#41#41×C6714yes—
328C2×Q8—216no—
329C2×Q8—26no—
330C2×Q8—212no—
331C2×Q8C22×C4×C67124yes—
332C2×Q8C2×D4×C6711yescharacteristic
333C2×Q8C24×C6716yes—
334C2×D4—26no—
335C2×D4C2×D4×C6713yes—
336C2×D4C24×C6712yes—
337C22×C4—26no—
338C22×C4—26no—
339C22×C4—26no—
340C22×C4—22no—
341C22×C4—26no—
342C22×C4C2×D4×C6716yes—
343C22×C4C24×C6713yes—
344Q16C22×C4×C67112yes—
345Q16C22×C4×C67124yes—
346SD16C22×C4×C67124yes—
347D8C22×C4×C6714yes—
348M16—26no—
349M16C22×C4×C67112yes—
350C2×C8—26no—
351C2×C8C22×C4×C67112yes—
352C2×C8C24×C6713yes—
353C2×C8C24×C6711yescharacteristic
354C2×C8C24×C6713yes—
355C2×C8C24×C6711yescharacteristic
356C4⋊C4—26no—
357C4⋊C4—212no—
358C4⋊C4—26no—
359C4⋊C4—26no—
360C4⋊C4C2×D4×C6716yes—
361C4⋊C4C24×C6716yes—
362C4⋊C4C24×C6712yes—
363C22⋊C4—26no—
364C22⋊C4—22no—
365C22⋊C4—26no—
366C22⋊C4C2×D4×C6716yes—
367C42—26no—
368C42—26no—
369C42—212no—
370C42—26no—
371C42C2×D4×C6716yes—
372C42C24×C6713yes—
373C42C24×C6711yescharacteristic
374C23—22no—
375C23C22×D4×C6713yes—
376Q8—224no—
377Q8—26no—
378Q8C4×D4×C6714yes—
379Q8C23×C4×C67112yes—
380D4—224no—
381D4—22no—
382D4C4×D4×C67112yes—
383D4C23×C4×C6714yes—
384C2×C4—224no—
385C2×C4—26no—
386C2×C4—224no—
387C2×C4—22no—
388C2×C4—26no—
389C2×C4—46no—
390C2×C4—44no—
391C2×C4—86no—
392C2×C4—86no—
393C2×C4C23×C4×C6716yes—
394C2×C4C22×D4×C6712yes—
395C2×C4C22×D4×C6716yes—
396C2×C4C22×D4×C6713yes—
397C2×C4C2×#41×C6716yes—
398C2×C4C25×C6711yescharacteristic
399C8—21no—
400C8—23no—
401C8C23×C4×C6712yes—
402C8C23×C4×C6716yes—
403C22—26no—
404C22—24no—
405C22—42no—
406C22—42no—
407C22C2×C4×D4×C6716yes—
408C22C23×D4×C6711yescharacteristic
409C4—22no—
410C4—26no—
411C4—212no—
412C4—42no—
413C4—46no—
414C4C2×C4×D4×C6716yes—
415C4C24×C4×C6711yescharacteristic
416C4C22×#41×C6711yescharacteristic
417C2—26no—
418C2—44no—
419C2C22×C4×D4×C6711yescharacteristic
420C2C67×#358612yes—
subgroupG/H
center, socleC22×C67C23×D4
commutatorC4C24×C4×C67
Frattini, cosocleC2×C4C25×C67
prime radicalC2×#578×C67C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C4×#41×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×2−1+4×C67C2semidirect
3. C67×#2952C2semidirect
4. C67×#2950C2semidirect
5. C67×#2947C2semidirect
6. C4×#137×C67C2semidirect
7. C67×#2943C2semidirect
8. C2×#441×C67C2semidirect
9. C67×#2912C2semidirect
10. C67×#2898C2semidirect
11. C67×#2875C2semidirect
13. #578×C67C4semidirect
52. 2−1+4×C67C2×C4semidirect
58. #137×C67C2×C4semidirect
59. #135×C67C2×C4semidirect
60. C2×Q16×C67C2×C4semidirect
66. #131×C67C2×C4semidirect
206. #60175C67direct (N × H)
212. C4×2−1+4C2×C67semidirect
213. #2952C2×C67semidirect
214. #2950C2×C67semidirect
215. #2947C2×C67semidirect
216. C4×#137C2×C67semidirect
217. #2943C2×C67semidirect
218. C2×#441C2×C67semidirect
219. #2912C2×C67semidirect
220. #2898C2×C67semidirect
221. #2875C2×C67semidirect
222. C67#60175direct (N × H)
224. #578C4×C67semidirect
263. 2−1+4C2×C4×C67semidirect
269. #137C2×C4×C67semidirect
270. #135C2×C4×C67semidirect
271. C2×Q16C2×C4×C67semidirect
277. #131C2×C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#578×C672
C4×2−1+4×C672
C67×#29522
C67×#29502
C67×#29472
C4×#137×C672
C67×#29432
C2×#441×C672
C67×#29122
C67×#28982
C67×#28752
#6017567