Finite Group Databasebeta

← C67×#34311 #30713562

Subgroups — C67×#34311 #30713562

Order 17152.

2860 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Q8×C67C211yesmaximal, characteristic
2C2×#519×C67C212yesmaximal, complemented
3C67×#2365C213yesmaximal, complemented
4C67×#2362C216yesmaximal, complemented
5C67×#2354C213yesmaximal
6Q8×C4⋊C4×C67C211yesmaximal, characteristic, complemented
7Q8×C22⋊C4×C67C212yesmaximal, complemented
8C67×#2323C216yesmaximal, complemented
9C4×C4⋊Q8×C67C213yesmaximal, complemented
10C67×#2293C213yesmaximal
11C42×Q8×C67C211yesmaximal, characteristic, complemented
12C23×Q8×C67C2212yes—
13D4×Q8×C67C4116yescomplemented
14C2×C4⋊Q8×C67C2213yes—
15C2×C22⋊Q8×C67C2216yes—
16#519×C67—216no—
17C2×C4×Q8×C67C2211yescharacteristic
18C2×C4×Q8×C67C2216yes—
19C2×C4×Q8×C67C2212yes—
20C2×C4×Q8×C67C2216yes—
21C2×C4×Q8×C67C2211yescharacteristic
22C2×C4×Q8×C67C2211yescharacteristic
23C2×C4×D4×C67C2213yes—
24C2×#117×C67C2216yes—
25C2×#117×C67C2216yes—
26#391×C67C2216yes—
27#391×C67C2216yes—
28#390×C67C2211yescharacteristic
29#390×C67C2213yes—
30#389×C67C2213yes—
31#389×C67C2213yes—
32#389×C67C2216yes—
33#389×C67C2216yes—
34#388×C67C2216yes—
35#388×C67C2216yes—
36#388×C67C22112yes—
37#387×C67C22112yes—
38#384×C67C2213yes—
39#380×C67C2216yes—
40C4×C4⋊C4×C67C2213yes—
41C4×C4⋊C4×C67C2213yes—
42C4×C4⋊C4×C67C2213yes—
43C4×C4⋊C4×C67C2213yes—
44C4×C4⋊C4×C67C2216yes—
45C4×C4⋊C4×C67C2216yes—
46C4×C22⋊C4×C67C2216yes—
47C43×C67C2213yes—
48C22×Q8×C67—28no—
49C22×Q8×C67—28no—
50C22×Q8×C67C2×C4116yes—
51C22×Q8×C67C2316yes—
52C22×Q8×C67C2311yescharacteristic
53C22×D4×C67C2311yescharacteristic
54C23×C4×C67C2316yes—
55C4⋊Q8×C67C2×C4124yes—
56C22⋊Q8×C67C2×C4148yes—
57C4×Q8×C67—28no—
58C4×Q8×C67—24no—
59C4×Q8×C67—224no—
60C4×Q8×C67—224no—
61C4×Q8×C67C2×C418yes—
62C4×Q8×C67D418yescomplemented
63C4×D4×C67C2×C4124yes—
64#117×C67—224no—
65#117×C67—224no—
66C2×C4⋊C4×C67C2316yes—
67C2×C4⋊C4×C67C2316yes—
68C2×C4⋊C4×C67C2316yes—
69C2×C4⋊C4×C67C2313yes—
70C2×C4⋊C4×C67C2313yes—
71C2×C4⋊C4×C67C2311yescharacteristic
72C2×C4⋊C4×C67C2316yes—
73C2×C4⋊C4×C67C2313yes—
74C2×C4⋊C4×C67C2313yes—
75C2×C4⋊C4×C67C2316yes—
76C2×C4⋊C4×C67C2313yes—
77C2×C22⋊C4×C67C2316yes—
78C2×C22⋊C4×C67C2312yes—
79C2×C22⋊C4×C67C2312yes—
80C2×C22⋊C4×C67C2316yes—
81C2×C22⋊C4×C67C2316yes—
82C2×C42×C67C2316yes—
83C2×C42×C67C23112yes—
84C2×C42×C67C2313yes—
85C2×C42×C67C2311yescharacteristic
86C2×C42×C67C2313yes—
87C2×C42×C67C2313yes—
88C2×C42×C67C2313yes—
89C2×C42×C67C2313yes—
90C2×C42×C67C2313yes—
91#95×C67C2316yes—
92#95×C67C2316yes—
93#95×C67C2316yes—
94#95×C67C23112yes—
95#95×C67C2316yes—
96C24×C67C2412yes—
97C2×Q8×C67—264no—
98C2×Q8×C67—212no—
99C2×Q8×C67—212no—
100C2×Q8×C67C22×C4124yes—
101C2×Q8×C67C22×C414yes—
102C2×Q8×C67C2×D414yes—
103C2×Q8×C67#4114yes—
104C2×D4×C67—22no—
105C2×D4×C67—22no—
106C2×D4×C67C22×C414yes—
107C22×C4×C67—212no—
108C22×C4×C67—212no—
109C22×C4×C67C22×C4124yes—
110C22×C4×C67C2416yes—
111C22×C4×C67C2412yes—
112C22×C4×C67C2413yes—
113C22×C4×C67C2413yes—
114C22×C4×C67C2416yes—
115C22×C4×C67C2411yescharacteristic
116C22×C4×C67C2413yes—
117C22×C4×C67C2411yescharacteristic
118C22×C4×C67C2413yes—
119C22×C4×C67C2411yescharacteristic
120C4⋊C4×C67—212no—
121C4⋊C4×C67—212no—
122C4⋊C4×C67—212no—
123C4⋊C4×C67—26no—
124C4⋊C4×C67—22no—
125C4⋊C4×C67—26no—
126C4⋊C4×C67—46no—
127C4⋊C4×C67C22×C4112yes—
128C4⋊C4×C67C22×C4112yes—
129C4⋊C4×C67C22×C4124yes—
130C4⋊C4×C67C2×D4112yes—
131C22⋊C4×C67—212no—
132C22⋊C4×C67—24no—
133C22⋊C4×C67—24no—
134C22⋊C4×C67—46no—
135C22⋊C4×C67C22×C4124yes—
136C42×C67—224no—
137C42×C67—212no—
138C42×C67—26no—
139C42×C67—26no—
140C42×C67—22no—
141C42×C67—26no—
142C42×C67—43no—
143C42×C67C22×C4112yes—
144C42×C67C2×D4112yes—
145C23×C67—22no—
146C23×C67—22no—
147C23×C67—22no—
148C23×C67—22no—
149C23×C67—22no—
150C23×C67—22no—
151C23×C67C23×C414yes—
152C23×C67C2511yescharacteristic
153Q8×C67—248no—
154Q8×C67C4×D4116yes—
155D4×C67—28no—
156C2×C4×C67—26no—
157C2×C4×C67—21no—
158C2×C4×C67—26no—
159C2×C4×C67—23no—
160C2×C4×C67—21no—
161C2×C4×C67—23no—
162C2×C4×C67—23no—
163C2×C4×C67—21no—
164C2×C4×C67—21no—
165C2×C4×C67—23no—
166C2×C4×C67—248no—
167C2×C4×C67—26no—
168C2×C4×C67—22no—
169C2×C4×C67—23no—
170C2×C4×C67—23no—
171C2×C4×C67—22no—
172C2×C4×C67—26no—
173C2×C4×C67—22no—
174C2×C4×C67—23no—
175C2×C4×C67—26no—
176C2×C4×C67C23×C416yes—
177C2×C4×C67C23×C4112yes—
178C2×C4×C67C23×C416yes—
179C2×C4×C67C23×C412yes—
180C2×C4×C67C22×D416yes—
181C2×C4×C67C22×D412yes—
182C2×C4×C67C22×D416yes—
183C2×C4×C67C2×#4116yes—
184C2×C4×C672−1+412yes—
185C2×C4×C672−1+412yes—
186C2×C4×C672−1+412yes—
187C22×C67—24no—
188C22×C67—24no—
189C22×C67—24no—
190C22×C67—42no—
191C22×C67—42no—
192C22×C67—42no—
193C22×C67—42no—
194C22×C67C24×C411yescharacteristic
195C22×C67C23×D411yescharacteristic
196C22×C67C22×#4111yescharacteristic
197C22×C67C2×2−1+411yescharacteristic
198C22×C67C2×2−1+411yescharacteristic
199C22×C67C2×2−1+411yescharacteristic
200C22×C67#58511yescharacteristic
201C4×C67—212no—
202C4×C67—26no—
203C4×C67—22no—
204C4×C67—26no—
205C4×C67—22no—
206C4×C67—46no—
207C4×C67—42no—
208C4×C67—43no—
209C4×C67C2×C4×D4112yes—
210C4×C67#51914yescomplemented
211#34311C6711yesmaximal, characteristic, direct product factor, Sylow
212C2×C67—44no—
213C2×C67C22×C4×D411yescharacteristic
214C2×C67C2×#51911yescharacteristic
215C2×C67C4×2−1+411yescharacteristic
216C2×C67#349711yescharacteristic
217C2×C67#349911yescharacteristic
218C2×C67#351811yescharacteristic
219C2×C67#352311yescharacteristic
220C2×D4×Q8C2×C6711yescharacteristic
221C2×#519C2×C6712yescomplemented
222#2365C2×C6713yescomplemented
223#2362C2×C6716yescomplemented
224#2354C2×C6713yes—
225Q8×C4⋊C4C2×C6711yescharacteristic, complemented
226Q8×C22⋊C4C2×C6712yescomplemented
227#2323C2×C6716yescomplemented
228C4×C4⋊Q8C2×C6713yescomplemented
229#2293C2×C6713yes—
230C42×Q8C2×C6711yescharacteristic, complemented
231C67#3431111yescharacteristic, direct product factor, Sylow
232C23×Q8C22×C6712yes—
233D4×Q8C4×C67116yescomplemented
234C2×C4⋊Q8C22×C6713yes—
235C2×C22⋊Q8C22×C6716yes—
236#519—216no—
237C2×C4×Q8C22×C6716yes—
238C2×C4×Q8C22×C6711yescharacteristic
239C2×C4×Q8C22×C6712yes—
240C2×C4×Q8C22×C6716yes—
241C2×C4×Q8C22×C6711yescharacteristic
242C2×C4×Q8C22×C6711yescharacteristic
243C2×C4×D4C22×C6713yes—
244C2×#117C22×C6716yes—
245C2×#117C22×C6716yes—
246#391C22×C6716yes—
247#391C22×C6716yes—
248#390C22×C6713yes—
249#390C22×C6711yescharacteristic
250#389C22×C6713yes—
251#389C22×C6716yes—
252#389C22×C6716yes—
253#389C22×C6713yes—
254#388C22×C6716yes—
255#388C22×C67112yes—
256#388C22×C6716yes—
257#387C22×C67112yes—
258#384C22×C6713yes—
259#380C22×C6716yes—
260C4×C4⋊C4C22×C6716yes—
261C4×C4⋊C4C22×C6716yes—
262C4×C4⋊C4C22×C6713yes—
263C4×C4⋊C4C22×C6713yes—
264C4×C4⋊C4C22×C6713yes—
265C4×C4⋊C4C22×C6713yes—
266C4×C22⋊C4C22×C6716yes—
267C43C22×C6713yes—
268C22×Q8—28no—
269C22×Q8—28no—
270C22×Q8C2×C4×C67116yes—
271C22×Q8C23×C6716yes—
272C22×Q8C23×C6711yescharacteristic
273C22×D4C23×C6711yescharacteristic
274C23×C4C23×C6716yes—
275C4⋊Q8C2×C4×C67124yes—
276C22⋊Q8C2×C4×C67148yes—
277C4×Q8—28no—
278C4×Q8—24no—
279C4×Q8—224no—
280C4×Q8—224no—
281C4×Q8C2×C4×C6718yes—
282C4×Q8D4×C6718yescomplemented
283C4×D4C2×C4×C67124yes—
284#117—224no—
285#117—224no—
286C2×C4⋊C4C23×C6716yes—
287C2×C4⋊C4C23×C6716yes—
288C2×C4⋊C4C23×C6716yes—
289C2×C4⋊C4C23×C6713yes—
290C2×C4⋊C4C23×C6713yes—
291C2×C4⋊C4C23×C6711yescharacteristic
292C2×C4⋊C4C23×C6716yes—
293C2×C4⋊C4C23×C6713yes—
294C2×C4⋊C4C23×C6713yes—
295C2×C4⋊C4C23×C6716yes—
296C2×C4⋊C4C23×C6713yes—
297C2×C22⋊C4C23×C6712yes—
298C2×C22⋊C4C23×C6716yes—
299C2×C22⋊C4C23×C6712yes—
300C2×C22⋊C4C23×C6716yes—
301C2×C22⋊C4C23×C6716yes—
302C2×C42C23×C6716yes—
303C2×C42C23×C67112yes—
304C2×C42C23×C6713yes—
305C2×C42C23×C6711yescharacteristic
306C2×C42C23×C6713yes—
307C2×C42C23×C6713yes—
308C2×C42C23×C6713yes—
309C2×C42C23×C6713yes—
310C2×C42C23×C6713yes—
311#95C23×C6716yes—
312#95C23×C6716yes—
313#95C23×C6716yes—
314#95C23×C67112yes—
315#95C23×C6716yes—
316C24C24×C6712yes—
317C2×Q8—264no—
318C2×Q8—212no—
319C2×Q8—212no—
320C2×Q8C22×C4×C67124yes—
321C2×Q8C22×C4×C6714yes—
322C2×Q8C2×D4×C6714yes—
323C2×Q8#41×C6714yes—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4C22×C4×C6714yes—
327C22×C4—212no—
328C22×C4—212no—
329C22×C4C22×C4×C67124yes—
330C22×C4C24×C6713yes—
331C22×C4C24×C6712yes—
332C22×C4C24×C6716yes—
333C22×C4C24×C6713yes—
334C22×C4C24×C6716yes—
335C22×C4C24×C6713yes—
336C22×C4C24×C6711yescharacteristic
337C22×C4C24×C6711yescharacteristic
338C22×C4C24×C6713yes—
339C22×C4C24×C6711yescharacteristic
340C4⋊C4—212no—
341C4⋊C4—212no—
342C4⋊C4—212no—
343C4⋊C4—26no—
344C4⋊C4—22no—
345C4⋊C4—26no—
346C4⋊C4—46no—
347C4⋊C4C22×C4×C67112yes—
348C4⋊C4C22×C4×C67112yes—
349C4⋊C4C22×C4×C67124yes—
350C4⋊C4C2×D4×C67112yes—
351C22⋊C4—212no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—46no—
355C22⋊C4C22×C4×C67124yes—
356C42—224no—
357C42—212no—
358C42—26no—
359C42—26no—
360C42—22no—
361C42—26no—
362C42—43no—
363C42C22×C4×C67112yes—
364C42C2×D4×C67112yes—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23C23×C4×C6714yes—
372C23C25×C6711yescharacteristic
373Q8—248no—
374Q8C4×D4×C67116yes—
375D4—28no—
376C2×C4—26no—
377C2×C4—21no—
378C2×C4—26no—
379C2×C4—21no—
380C2×C4—23no—
381C2×C4—21no—
382C2×C4—23no—
383C2×C4—23no—
384C2×C4—21no—
385C2×C4—23no—
386C2×C4—248no—
387C2×C4—26no—
388C2×C4—22no—
389C2×C4—23no—
390C2×C4—23no—
391C2×C4—22no—
392C2×C4—26no—
393C2×C4—22no—
394C2×C4—23no—
395C2×C4—26no—
396C2×C4C23×C4×C6716yes—
397C2×C4C23×C4×C67112yes—
398C2×C4C23×C4×C6716yes—
399C2×C4C23×C4×C6712yes—
400C2×C4C22×D4×C6716yes—
401C2×C4C22×D4×C6712yes—
402C2×C4C22×D4×C6716yes—
403C2×C4C2×#41×C6716yes—
404C2×C42−1+4×C6712yes—
405C2×C42−1+4×C6712yes—
406C2×C42−1+4×C6712yes—
407C22—24no—
408C22—24no—
409C22—24no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22C24×C4×C6711yescharacteristic
415C22C23×D4×C6711yescharacteristic
416C22C22×#41×C6711yescharacteristic
417C22C2×2−1+4×C6711yescharacteristic
418C22C2×2−1+4×C6711yescharacteristic
419C22C2×2−1+4×C6711yescharacteristic
420C22#585×C6711yescharacteristic
421C4—22no—
422C4—212no—
423C4—26no—
424C4—22no—
425C4—26no—
426C4—46no—
427C4—42no—
428C4—43no—
429C4C2×C4×D4×C67112yes—
430C4#519×C6714yescomplemented
431C2—44no—
432C2C22×C4×D4×C6711yescharacteristic
433C2C2×#519×C6711yescharacteristic
434C2C4×2−1+4×C6711yescharacteristic
435C2C67×#349711yescharacteristic
436C2C67×#349911yescharacteristic
437C2C67×#351811yescharacteristic
438C2C67×#352311yescharacteristic
subgroupG/H
center, socleC23×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC23C25×C67
prime radicalC22×D4×C67C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#519×C67C2semidirect
3. C67×#2365C2semidirect
4. C67×#2362C2semidirect
6. Q8×C4⋊C4×C67C2semidirect
7. Q8×C22⋊C4×C67C2semidirect
8. C67×#2323C2semidirect
9. C4×C4⋊Q8×C67C2semidirect
11. C42×Q8×C67C2semidirect
13. D4×Q8×C67C4semidirect
62. C4×Q8×C67D4semidirect
210. C4×C67#519semidirect
211. #34311C67direct (N × H)
221. C2×#519C2×C67semidirect
222. #2365C2×C67semidirect
223. #2362C2×C67semidirect
225. Q8×C4⋊C4C2×C67semidirect
226. Q8×C22⋊C4C2×C67semidirect
227. #2323C2×C67semidirect
228. C4×C4⋊Q8C2×C67semidirect
230. C42×Q8C2×C67semidirect
231. C67#34311direct (N × H)
233. D4×Q8C4×C67semidirect
282. C4×Q8D4×C67semidirect
430. C4#519×C67semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q8×C672
C2×#519×C672
C67×#23652
C67×#23622
C67×#23542
Q8×C4⋊C4×C672
Q8×C22⋊C4×C672
C67×#23232
C4×C4⋊Q8×C672
C67×#22932
C42×Q8×C672
#3431167