Finite Group Databasebeta

← C67×#60343 #30739594

Subgroups — C67×#60343 #30739594

Order 17152.

1930 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×2−1+4×C67C211yesmaximal, characteristic
2C67×#2990C216yesmaximal, complemented
3C67×#2989C213yesmaximal, complemented
4C67×#2988C211yesmaximal, characteristic, complemented
5C67×#2984C212yesmaximal, complemented
6C67×#2983C213yesmaximal, complemented
7C67×#2970C216yesmaximal, complemented
8C67×#2965C213yesmaximal, complemented
9C8×#41×C67C211yesmaximal, characteristic, complemented
10Q8×M16×C67C212yesmaximal, complemented
11C2×#446×C67C213yesmaximal, complemented
12C2×2−1+4×C67C411yescharacteristic
13#520×C67C416yes—
14#520×C67C413yes—
15#520×C67C411yescharacteristic
16#519×C67C412yes—
17#519×C67C413yes—
18C4×#41×C67C2211yescharacteristic
19C4×#41×C67C2213yes—
20C4×#41×C67C2216yes—
21C2×C4×Q8×C67C2213yes—
22C2×C4×Q8×C67C2212yes—
23#446×C67C2213yes—
24#446×C67C2216yes—
25#446×C67C2216yes—
26#446×C67C2216yes—
27#446×C67C2216yes—
28#446×C67C2213yes—
29#446×C67C2216yes—
30C8×Q8×C67C2211yescharacteristic
31C8×Q8×C67C2212yes—
32C8×Q8×C67C2211yescharacteristic
33#436×C67C2211yescharacteristic
34#436×C67C2213yes—
35#435×C67C2216yes—
36#435×C67C2216yes—
37#435×C67C22112yes—
38C8×D4×C67C2213yes—
39C8×D4×C67C2213yes—
40C8×D4×C67C2216yes—
41#432×C67C22112yes—
42#432×C67C2216yes—
43#431×C67C2213yes—
44#431×C67C2213yes—
45#431×C67C2216yes—
46#423×C67C2216yes—
47C2×C4⋊C8×C67C2213yes—
48C2×C4⋊C8×C67C2216yes—
49C4×M16×C67C2216yes—
50C2×#97×C67C2216yes—
51C2×C4×C8×C67C2213yes—
522−1+4×C67—28no—
53C2×#41×C67C2×C416yes—
54C2×#41×C67C2×C413yes—
55C2×#41×C67C2×C411yescharacteristic
56C22×Q8×C67C2×C412yes—
57C22×Q8×C67C2×C413yes—
58C2×M16×C67C2312yes—
59C2×M16×C67C2316yes—
60C22×C8×C67C2313yes—
61C22×C8×C67C2313yes—
62C22×C8×C67C2316yes—
63C4×Q8×C67C2×C416yes—
64C4×Q8×C67C2×C416yes—
65C4×Q8×C67C2×C416yes—
66C4×Q8×C67C2×C412yes—
67C4×Q8×C67C2×C413yes—
68C4×Q8×C67C2×C416yes—
69C4×Q8×C67C2×C411yescharacteristic
70C4×Q8×C67C2311yescharacteristic
71C4×Q8×C67C2313yes—
72C4×Q8×C67C2316yes—
73C4×D4×C67C2×C416yes—
74C4×D4×C67C2×C413yes—
75C4×D4×C67C2×C4112yes—
76C4×D4×C67C2×C413yes—
77C4×D4×C67C2×C416yes—
78C4×D4×C67C2313yes—
79C4×D4×C67C2316yes—
80C4×D4×C67C2311yescharacteristic
81#117×C67C2×C416yes—
82#117×C67C2×C416yes—
83#117×C67C2×C4112yes—
84#117×C67C2×C413yes—
85#117×C67C2×C413yes—
86C2×C4⋊C4×C67C2×C416yes—
87C2×C4⋊C4×C67C2×C416yes—
88C2×C4⋊C4×C67C2×C413yes—
89C2×C42×C67C2313yes—
90C2×C42×C67C2316yes—
91C2×C42×C67C2316yes—
92C4⋊C8×C67C2316yes—
93C4⋊C8×C67C2313yes—
94C4⋊C8×C67C2316yes—
95C4⋊C8×C67C23112yes—
96C4⋊C8×C67C2313yes—
97C4⋊C8×C67C2311yescharacteristic
98C4⋊C8×C67C2313yes—
99C4⋊C8×C67C2316yes—
100C22⋊C8×C67C2312yes—
101C22⋊C8×C67C2316yes—
102C22⋊C8×C67C2316yes—
103C22⋊C8×C67C2316yes—
104#97×C67C2316yes—
105#97×C67C23112yes—
106#97×C67C2316yes—
107C4×C8×C67C2311yescharacteristic
108C4×C8×C67C2313yes—
109C4×C8×C67C2313yes—
110C4×C8×C67C2313yes—
111C4×C8×C67C2316yes—
112#41×C67—224no—
113#41×C67—212no—
114#41×C67M1618yescomplemented
115C2×Q8×C67—28no—
116C2×Q8×C67—212no—
117C2×Q8×C67C22×C416yes—
118C2×Q8×C67C22×C413yes—
119C2×Q8×C67C22×C411yescharacteristic
120C2×D4×C67C22×C416yes—
121C2×D4×C67C22×C411yescharacteristic
122C2×D4×C67C22×C413yes—
123C22×C4×C67C22×C416yes—
124C22×C4×C67C22×C416yes—
125C22×C4×C67C22×C416yes—
126C22×C4×C67C22×C413yes—
127C22×C4×C67C22×C416yes—
128C22×C4×C67C22×C413yes—
129C22×C4×C67C2413yes—
130C22×C4×C67C2412yes—
131M16×C67—24no—
132M16×C67—46no—
133C2×C8×C67—26no—
134C2×C8×C67—43no—
135C2×C8×C67—46no—
136C2×C8×C67C2411yescharacteristic
137C2×C8×C67C2416yes—
138C2×C8×C67C2413yes—
139C2×C8×C67C2413yes—
140C2×C8×C67C2412yes—
141C2×C8×C67C2411yescharacteristic
142C4⋊C4×C67C22×C416yes—
143C4⋊C4×C67C22×C4112yes—
144C4⋊C4×C67C22×C413yes—
145C4⋊C4×C67C22×C416yes—
146C4⋊C4×C67C22×C413yes—
147C4⋊C4×C67C22×C416yes—
148C4⋊C4×C67C22×C413yes—
149C4⋊C4×C67C22×C411yescharacteristic
150C22⋊C4×C67C22×C416yes—
151C22⋊C4×C67C22×C412yes—
152C22⋊C4×C67C22×C416yes—
153C22⋊C4×C67C22×C416yes—
154C42×C67C22×C4112yes—
155C42×C67C22×C416yes—
156C42×C67C22×C413yes—
157C42×C67C22×C413yes—
158C42×C67C22×C416yes—
159C42×C67C2411yescharacteristic
160C42×C67C2413yes—
161C42×C67C2416yes—
162C23×C67C23×C412yes—
163C23×C67C23×C413yes—
164Q8×C67—212no—
165Q8×C67—26no—
166Q8×C67C2×M1614yes—
167D4×C67—212no—
168D4×C67—22no—
169D4×C67C2×M16112yes—
170C2×C4×C67—212no—
171C2×C4×C67—212no—
172C2×C4×C67—23no—
173C2×C4×C67—22no—
174C2×C4×C67—23no—
175C2×C4×C67—22no—
176C2×C4×C67—43no—
177C2×C4×C67—46no—
178C2×C4×C67—46no—
179C2×C4×C67C2×M16112yes—
180C2×C4×C67C23×C413yes—
181C2×C4×C67C23×C412yes—
182C2×C4×C67C23×C413yes—
183C2×C4×C67C23×C411yescharacteristic
184C2×C4×C67C23×C416yes—
185C2×C4×C67C23×C416yes—
186C2×C4×C67C23×C413yes—
187C2×C4×C67C23×C411yescharacteristic
188C2×C4×C67C2511yescharacteristic
189C8×C67—23no—
190C8×C67—22no—
191C8×C67—21no—
192C8×C67—26no—
193C8×C67—23no—
194C8×C672−1+412yescomplemented
195C22×C67—23no—
196C22×C67—23no—
197C22×C67—22no—
198C22×C67—22no—
199C22×C67—22no—
200C22×C67C22×M1616yes—
201C22×C67C24×C411yescharacteristic
202C4×C67—23no—
203C4×C67—22no—
204C4×C67—26no—
205C4×C67—21no—
206C4×C67—26no—
207C4×C67—23no—
208C4×C67C22×M1616yes—
209C4×C67C22×M1612yes—
210C4×C67C2×2−1+411yescharacteristic
211C4×C67#58511yescharacteristic
212#60343C6711yesmaximal, characteristic, direct product factor, Sylow
213C2×C67—26no—
214C2×C67—42no—
215C2×C67C4×2−1+411yescharacteristic
216C2×C67C23×M1611yescharacteristic
217C2×C67#357311yescharacteristic
218C4×2−1+4C2×C6711yescharacteristic
219#2990C2×C6716yescomplemented
220#2989C2×C6713yescomplemented
221#2988C2×C6711yescharacteristic, complemented
222#2984C2×C6712yescomplemented
223#2983C2×C6713yescomplemented
224#2970C2×C6716yescomplemented
225#2965C2×C6713yescomplemented
226C8×#41C2×C6711yescharacteristic, complemented
227Q8×M16C2×C6712yescomplemented
228C2×#446C2×C6713yescomplemented
229C67#6034311yescharacteristic, direct product factor, Sylow
230C2×2−1+4C4×C6711yescharacteristic
231#520C4×C6716yes—
232#520C4×C6713yes—
233#520C4×C6711yescharacteristic
234#519C4×C6712yes—
235#519C4×C6713yes—
236C4×#41C22×C6711yescharacteristic
237C4×#41C22×C6713yes—
238C4×#41C22×C6716yes—
239C2×C4×Q8C22×C6713yes—
240C2×C4×Q8C22×C6712yes—
241#446C22×C6713yes—
242#446C22×C6716yes—
243#446C22×C6716yes—
244#446C22×C6716yes—
245#446C22×C6716yes—
246#446C22×C6713yes—
247#446C22×C6716yes—
248C8×Q8C22×C6711yescharacteristic
249C8×Q8C22×C6712yes—
250C8×Q8C22×C6711yescharacteristic
251#436C22×C6711yescharacteristic
252#436C22×C6713yes—
253#435C22×C6716yes—
254#435C22×C6716yes—
255#435C22×C67112yes—
256C8×D4C22×C6713yes—
257C8×D4C22×C6713yes—
258C8×D4C22×C6716yes—
259#432C22×C67112yes—
260#432C22×C6716yes—
261#431C22×C6713yes—
262#431C22×C6713yes—
263#431C22×C6716yes—
264#423C22×C6716yes—
265C2×C4⋊C8C22×C6713yes—
266C2×C4⋊C8C22×C6716yes—
267C4×M16C22×C6716yes—
268C2×#97C22×C6716yes—
269C2×C4×C8C22×C6713yes—
2702−1+4—28no—
271C2×#41C2×C4×C6716yes—
272C2×#41C2×C4×C6713yes—
273C2×#41C2×C4×C6711yescharacteristic
274C22×Q8C2×C4×C6712yes—
275C22×Q8C2×C4×C6713yes—
276C2×M16C23×C6712yes—
277C2×M16C23×C6716yes—
278C22×C8C23×C6713yes—
279C22×C8C23×C6713yes—
280C22×C8C23×C6716yes—
281C4×Q8C2×C4×C6716yes—
282C4×Q8C2×C4×C6716yes—
283C4×Q8C2×C4×C6716yes—
284C4×Q8C2×C4×C6712yes—
285C4×Q8C2×C4×C6713yes—
286C4×Q8C2×C4×C6716yes—
287C4×Q8C2×C4×C6711yescharacteristic
288C4×Q8C23×C6711yescharacteristic
289C4×Q8C23×C6713yes—
290C4×Q8C23×C6716yes—
291C4×D4C2×C4×C6716yes—
292C4×D4C2×C4×C6713yes—
293C4×D4C2×C4×C67112yes—
294C4×D4C2×C4×C6713yes—
295C4×D4C2×C4×C6716yes—
296C4×D4C23×C6713yes—
297C4×D4C23×C6716yes—
298C4×D4C23×C6711yescharacteristic
299#117C2×C4×C6716yes—
300#117C2×C4×C6716yes—
301#117C2×C4×C67112yes—
302#117C2×C4×C6713yes—
303#117C2×C4×C6713yes—
304C2×C4⋊C4C2×C4×C6716yes—
305C2×C4⋊C4C2×C4×C6716yes—
306C2×C4⋊C4C2×C4×C6713yes—
307C2×C42C23×C6713yes—
308C2×C42C23×C6716yes—
309C2×C42C23×C6716yes—
310C4⋊C8C23×C6716yes—
311C4⋊C8C23×C6713yes—
312C4⋊C8C23×C6716yes—
313C4⋊C8C23×C67112yes—
314C4⋊C8C23×C6713yes—
315C4⋊C8C23×C6711yescharacteristic
316C4⋊C8C23×C6713yes—
317C4⋊C8C23×C6716yes—
318C22⋊C8C23×C6712yes—
319C22⋊C8C23×C6716yes—
320C22⋊C8C23×C6716yes—
321C22⋊C8C23×C6716yes—
322#97C23×C6716yes—
323#97C23×C67112yes—
324#97C23×C6716yes—
325C4×C8C23×C6711yescharacteristic
326C4×C8C23×C6713yes—
327C4×C8C23×C6713yes—
328C4×C8C23×C6713yes—
329C4×C8C23×C6716yes—
330#41—224no—
331#41—212no—
332#41M16×C6718yescomplemented
333C2×Q8—28no—
334C2×Q8—212no—
335C2×Q8C22×C4×C6716yes—
336C2×Q8C22×C4×C6713yes—
337C2×Q8C22×C4×C6711yescharacteristic
338C2×D4C22×C4×C6716yes—
339C2×D4C22×C4×C6711yescharacteristic
340C2×D4C22×C4×C6713yes—
341C22×C4C22×C4×C6716yes—
342C22×C4C22×C4×C6716yes—
343C22×C4C22×C4×C6716yes—
344C22×C4C22×C4×C6713yes—
345C22×C4C22×C4×C6716yes—
346C22×C4C22×C4×C6713yes—
347C22×C4C24×C6713yes—
348C22×C4C24×C6712yes—
349M16—24no—
350M16—46no—
351C2×C8—26no—
352C2×C8—43no—
353C2×C8—46no—
354C2×C8C24×C6711yescharacteristic
355C2×C8C24×C6712yes—
356C2×C8C24×C6713yes—
357C2×C8C24×C6713yes—
358C2×C8C24×C6716yes—
359C2×C8C24×C6711yescharacteristic
360C4⋊C4C22×C4×C6716yes—
361C4⋊C4C22×C4×C67112yes—
362C4⋊C4C22×C4×C6713yes—
363C4⋊C4C22×C4×C6716yes—
364C4⋊C4C22×C4×C6713yes—
365C4⋊C4C22×C4×C6716yes—
366C4⋊C4C22×C4×C6713yes—
367C4⋊C4C22×C4×C6711yescharacteristic
368C22⋊C4C22×C4×C6716yes—
369C22⋊C4C22×C4×C6712yes—
370C22⋊C4C22×C4×C6716yes—
371C22⋊C4C22×C4×C6716yes—
372C42C22×C4×C67112yes—
373C42C22×C4×C6716yes—
374C42C22×C4×C6713yes—
375C42C22×C4×C6713yes—
376C42C22×C4×C6716yes—
377C42C24×C6711yescharacteristic
378C42C24×C6713yes—
379C42C24×C6716yes—
380C23C23×C4×C6712yes—
381C23C23×C4×C6713yes—
382Q8—212no—
383Q8—26no—
384Q8C2×M16×C6714yes—
385D4—212no—
386D4—22no—
387D4C2×M16×C67112yes—
388C2×C4—212no—
389C2×C4—212no—
390C2×C4—23no—
391C2×C4—22no—
392C2×C4—23no—
393C2×C4—22no—
394C2×C4—43no—
395C2×C4—46no—
396C2×C4—46no—
397C2×C4C2×M16×C67112yes—
398C2×C4C23×C4×C6713yes—
399C2×C4C23×C4×C6712yes—
400C2×C4C23×C4×C6713yes—
401C2×C4C23×C4×C6711yescharacteristic
402C2×C4C23×C4×C6716yes—
403C2×C4C23×C4×C6716yes—
404C2×C4C23×C4×C6713yes—
405C2×C4C23×C4×C6711yescharacteristic
406C2×C4C25×C6711yescharacteristic
407C8—23no—
408C8—22no—
409C8—21no—
410C8—26no—
411C8—23no—
412C82−1+4×C6712yescomplemented
413C22—23no—
414C22—23no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22C22×M16×C6716yes—
419C22C24×C4×C6711yescharacteristic
420C4—26no—
421C4—21no—
422C4—26no—
423C4—23no—
424C4—23no—
425C4—22no—
426C4C22×M16×C6716yes—
427C4C22×M16×C6712yes—
428C4C2×2−1+4×C6711yescharacteristic
429C4#585×C6711yescharacteristic
430C2—26no—
431C2—42no—
432C2C4×2−1+4×C6711yescharacteristic
433C2C23×M16×C6711yescharacteristic
434C2C67×#357311yescharacteristic
subgroupG/H
centerC2×C4×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC2×2−1+4×C67C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2990C2semidirect
3. C67×#2989C2semidirect
4. C67×#2988C2semidirect
5. C67×#2984C2semidirect
6. C67×#2983C2semidirect
7. C67×#2970C2semidirect
8. C67×#2965C2semidirect
9. C8×#41×C67C2semidirect
10. Q8×M16×C67C2semidirect
11. C2×#446×C67C2semidirect
114. #41×C67M16semidirect
194. C8×C672−1+4semidirect
212. #60343C67direct (N × H)
219. #2990C2×C67semidirect
220. #2989C2×C67semidirect
221. #2988C2×C67semidirect
222. #2984C2×C67semidirect
223. #2983C2×C67semidirect
224. #2970C2×C67semidirect
225. #2965C2×C67semidirect
226. C8×#41C2×C67semidirect
227. Q8×M16C2×C67semidirect
228. C2×#446C2×C67semidirect
229. C67#60343direct (N × H)
332. #41M16×C67semidirect
412. C82−1+4×C67semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+4×C672
C67×#29902
C67×#29892
C67×#29882
C67×#29842
C67×#29832
C67×#29702
C67×#29652
C8×#41×C672
Q8×M16×C672
C2×#446×C672
#6034367