Finite Group Databasebeta

← C67×#9443 #30688694

Subgroups — C67×#9443 #30688694

Order 17152.

712 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2879C211yesmaximal, characteristic
2C67×#2432C211yesmaximal, characteristic
3C67×#1782C211yesmaximal, characteristic
4C67×#1511C212yesmaximal, complemented
5C67×#1284C212yesmaximal, complemented
6C22×#41×C67C2211yescharacteristic
7C22×M16×C67—21no—
8C2×C4⋊Q8×C67C411yescharacteristic, complemented
9C2×#124×C67C411yescharacteristic, complemented
10C2×C22⋊Q8×C67C2211yescharacteristic
11C2×#99×C67—21no—
12C2×#99×C67—21no—
13#408×C67—24no—
14#407×C67—21no—
15#407×C67C2212yes—
16#394×C67—21no—
17#394×C67—21no—
18#391×C67C411yescharacteristic, complemented
19#390×C67C411yescharacteristic, complemented
20#385×C67—21no—
21#385×C67—21no—
22#381×C67C2212yes—
23C4×C22⋊C4×C67C2211yescharacteristic
24#328×C67—24no—
25C2×#41×C67—22no—
26C2×#41×C67—22no—
27C2×#41×C67—42no—
28C2×#41×C67—42no—
29C22×Q8×C67C2×C411yescharacteristic
30C22×Q8×C67C2×C411yescharacteristic
31C22×D4×C67C2×C411yescharacteristic
32C22×D4×C67D411yescharacteristic
33C22×D4×C67D411yescharacteristic
34C23×C4×C67D411yescharacteristic
35C23×C4×C67D411yescharacteristic
36C23×C4×C67C2311yescharacteristic
37C2×M16×C67—22no—
38C2×M16×C67—22no—
39C2×M16×C67—42no—
40C2×M16×C67—42no—
41C22×C8×C67—22no—
42C4⋊Q8×C67—42no—
43#124×C67—42no—
44C22⋊Q8×C67—22no—
45C22⋊Q8×C67—22no—
46C2×C4⋊C4×C67—21no—
47C2×C4⋊C4×C67—21no—
48C2×C4⋊C4×C67—21no—
49C2×C4⋊C4×C67C2×C411yescharacteristic
50C2×C22⋊C4×C67—21no—
51C2×C22⋊C4×C67—21no—
52C2×C22⋊C4×C67—21no—
53C2×C22⋊C4×C67—21no—
54C2×C22⋊C4×C67—21no—
55C2×C22⋊C4×C67—21no—
56C2×C22⋊C4×C67—22no—
57C2×C42×C67C2×C411yescharacteristic
58C2×C42×C67C2×C411yescharacteristic
59#99×C67—42no—
60#99×C67—42no—
61C22⋊C8×C67—24no—
62C22⋊C8×C67—22no—
63C22⋊C8×C67D414yes—
64#95×C67—21no—
65#95×C67—21no—
66#95×C67—21no—
67#95×C67—22no—
68#95×C67—22no—
69C24×C67C22⋊C411yescharacteristic
70C24×C67C22⋊C411yescharacteristic
71C24×C67C2×D411yescharacteristic
72#41×C67—44no—
73#41×C67—44no—
74#41×C67—48no—
75C2×Q8×C67—22no—
76C2×Q8×C67—22no—
77C2×Q8×C67—22no—
78C2×Q8×C67—41no—
79C2×Q8×C67—41no—
80C2×Q8×C67—41no—
81C2×D4×C67—22no—
82C2×D4×C67—22no—
83C2×D4×C67—22no—
84C2×D4×C67—22no—
85C2×D4×C67—22no—
86C2×D4×C67—22no—
87C2×D4×C67—41no—
88C2×D4×C67—41no—
89C2×D4×C67—41no—
90C22×C4×C67—22no—
91C22×C4×C67—22no—
92C22×C4×C67—21no—
93C22×C4×C67—21no—
94C22×C4×C67—21no—
95C22×C4×C67—21no—
96C22×C4×C67—21no—
97C22×C4×C67—21no—
98C22×C4×C67—21no—
99C22×C4×C67—21no—
100C22×C4×C67—41no—
101C22×C4×C67—41no—
102C22×C4×C67—41no—
103C22×C4×C67—41no—
104C22×C4×C67—42no—
105C22×C4×C67C22⋊C411yescharacteristic
106C22×C4×C67C22⋊C411yescharacteristic
107C22×C4×C67C22×C411yescharacteristic
108C22×C4×C67C2×D411yescharacteristic
109C22×C4×C67C2×D412yes—
110C22×C4×C67#4112yes—
111M16×C67—44no—
112M16×C67—42no—
113M16×C67—42no—
114C2×C8×C67—24no—
115C2×C8×C67—44no—
116C4⋊C4×C67—22no—
117C4⋊C4×C67—42no—
118C4⋊C4×C67—42no—
119C4⋊C4×C67—41no—
120C4⋊C4×C67—41no—
121C22⋊C4×C67—42no—
122C22⋊C4×C67—42no—
123C22⋊C4×C67—42no—
124C22⋊C4×C67—42no—
125C22⋊C4×C67—42no—
126C22⋊C4×C67—42no—
127C22⋊C4×C67—42no—
128C22⋊C4×C67—42no—
129C42×C67—41no—
130C42×C67—41no—
131C23×C67—21no—
132C23×C67—21no—
133C23×C67—21no—
134C23×C67—21no—
135C23×C67—21no—
136C23×C67—21no—
137C23×C67—21no—
138C23×C67—41no—
139C23×C67—41no—
140C23×C67—41no—
141C23×C67—41no—
142C23×C67—41no—
143C23×C67—41no—
144C23×C67#9912yes—
145C23×C67C2×C22⋊C411yescharacteristic
146C23×C67#12112yes—
147Q8×C67—44no—
148Q8×C67—82no—
149D4×C67—44no—
150D4×C67—44no—
151D4×C67—44no—
152C2×C4×C67—22no—
153C2×C4×C67—22no—
154C2×C4×C67—24no—
155C2×C4×C67—21no—
156C2×C4×C67—21no—
157C2×C4×C67—21no—
158C2×C4×C67—21no—
159C2×C4×C67—21no—
160C2×C4×C67—21no—
161C2×C4×C67—22no—
162C2×C4×C67—21no—
163C2×C4×C67—21no—
164C2×C4×C67—21no—
165C2×C4×C67—42no—
166C2×C4×C67—42no—
167C2×C4×C67—42no—
168C2×C4×C67—42no—
169C2×C4×C67—41no—
170C2×C4×C67—41no—
171C2×C4×C67—41no—
172C2×C4×C67—41no—
173C2×C4×C67—41no—
174C2×C4×C67—41no—
175C2×C4×C67—41no—
176C2×C4×C67—41no—
177C2×C4×C67—41no—
178C2×C4×C67—41no—
179C2×C4×C67—41no—
180C2×C4×C67—82no—
181C2×C4×C67—82no—
182C2×C4×C67—82no—
183C2×C4×C67#9912yes—
184C2×C4×C67C4×D412yes—
185C2×C4×C67#12011yescharacteristic
186C2×C4×C67#12311yescharacteristic
187C8×C67—44no—
188C22×C67—21no—
189C22×C67—21no—
190C22×C67—22no—
191C22×C67—22no—
192C22×C67—22no—
193C22×C67—22no—
194C22×C67—42no—
195C22×C67—42no—
196C22×C67—41no—
197C22×C67—41no—
198C22×C67—41no—
199C22×C67—41no—
200C22×C67—41no—
201C22×C67—81no—
202C22×C67—81no—
203C22×C67—81no—
204C22×C67—81no—
205C22×C67#38611yescharacteristic
206C22×C67C2×#9911yescharacteristic
207C22×C67C2×#9911yescharacteristic
208C4×C67—21no—
209C4×C67—21no—
210C4×C67—22no—
211C4×C67—22no—
212C4×C67—42no—
213C4×C67—41no—
214C4×C67—81no—
215C4×C67—81no—
216C4×C67—81no—
217C4×C67—162no—
218#9443C6711yesmaximal, characteristic, direct product factor, Sylow
219C2×C67—21no—
220C2×C67—21no—
221C2×C67—42no—
222C2×C67—42no—
223C2×C67—81no—
224C2×C67#188411yescharacteristic
225C2×C67#189811yescharacteristic
226C2×C67#190911yescharacteristic
227#2879C2×C6711yescharacteristic
228#2432C2×C6711yescharacteristic
229#1782C2×C6711yescharacteristic
230#1511C2×C6712yescomplemented
231#1284C2×C6712yescomplemented
232C67#944311yescharacteristic, direct product factor, Sylow
233C22×#41C22×C6711yescharacteristic
234C22×M16—21no—
235C2×C4⋊Q8C4×C6711yescharacteristic, complemented
236C2×#124C4×C6711yescharacteristic, complemented
237C2×C22⋊Q8C22×C6711yescharacteristic
238C2×#99—21no—
239C2×#99—21no—
240#408—24no—
241#407—21no—
242#407C22×C6712yes—
243#394—21no—
244#394—21no—
245#391C4×C6711yescharacteristic, complemented
246#390C4×C6711yescharacteristic, complemented
247#385—21no—
248#385—21no—
249#381C22×C6712yes—
250C4×C22⋊C4C22×C6711yescharacteristic
251#328—24no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—42no—
255C2×#41—42no—
256C22×Q8C2×C4×C6711yescharacteristic
257C22×Q8C2×C4×C6711yescharacteristic
258C22×D4C2×C4×C6711yescharacteristic
259C22×D4D4×C6711yescharacteristic
260C22×D4D4×C6711yescharacteristic
261C23×C4D4×C6711yescharacteristic
262C23×C4D4×C6711yescharacteristic
263C23×C4C23×C6711yescharacteristic
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—42no—
267C2×M16—42no—
268C22×C8—22no—
269C4⋊Q8—42no—
270#124—42no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C2×C4×C6711yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C42C2×C4×C6711yescharacteristic
285C2×C42C2×C4×C6711yescharacteristic
286#99—42no—
287#99—42no—
288C22⋊C8—24no—
289C22⋊C8—22no—
290C22⋊C8D4×C6714yes—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—22no—
295#95—22no—
296C24C22⋊C4×C6711yescharacteristic
297C24C22⋊C4×C6711yescharacteristic
298C24C2×D4×C6711yescharacteristic
299#41—44no—
300#41—44no—
301#41—48no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4C22⋊C4×C6711yescharacteristic
333C22×C4C22⋊C4×C6711yescharacteristic
334C22×C4C22×C4×C6711yescharacteristic
335C22×C4C2×D4×C6711yescharacteristic
336C22×C4C2×D4×C6712yes—
337C22×C4#41×C6712yes—
338M16—44no—
339M16—42no—
340M16—42no—
341C2×C8—24no—
342C2×C8—44no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C42—41no—
357C42—41no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23#99×C6712yes—
372C23C2×C22⋊C4×C6711yescharacteristic
373C23#121×C6712yes—
374Q8—44no—
375Q8—82no—
376D4—44no—
377D4—44no—
378D4—44no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4#99×C6712yes—
411C2×C4C4×D4×C6712yes—
412C2×C4#120×C6711yescharacteristic
413C2×C4#123×C6711yescharacteristic
414C8—44no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22#386×C6711yescharacteristic
433C22C2×#99×C6711yescharacteristic
434C22C2×#99×C6711yescharacteristic
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—162no—
445C2—21no—
446C2—21no—
447C2—42no—
448C2—42no—
449C2—81no—
450C2C67×#188411yescharacteristic
451C2C67×#189811yescharacteristic
452C2C67×#190911yescharacteristic
subgroupG/H
center, socleC22×C67#386
commutatorC22×C4C22×C4×C67
Frattini, cosocleC23×C4C23×C67
prime radicalC22×#41×C67C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C67×#1511C2semidirect
5. C67×#1284C2semidirect
8. C2×C4⋊Q8×C67C4semidirect
9. C2×#124×C67C4semidirect
18. #391×C67C4semidirect
19. #390×C67C4semidirect
218. #9443C67direct (N × H)
230. #1511C2×C67semidirect
231. #1284C2×C67semidirect
232. C67#9443direct (N × H)
235. C2×C4⋊Q8C4×C67semidirect
236. C2×#124C4×C67semidirect
245. #391C4×C67semidirect
246. #390C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#28792
C67×#24322
C67×#17822
C67×#15112
C67×#12842
#944367