Finite Group Databasebeta

← C67×#9114 #30688365

Subgroups — C67×#9114 #30688365

Order 17152.

694 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#431×C67C211yesmaximal, characteristic, complemented
2C67×#2854C211yesmaximal, characteristic
3C2×#327×C67C212yesmaximal
4C67×#1434C211yesmaximal, characteristic, complemented
5C67×#1294C212yesmaximal, complemented
6C2×#123×C67—21no—
7C2×#121×C67C2211yescharacteristic
8C22×C42×C67C2211yescharacteristic
9#431×C67—24no—
10C2×C4⋊C8×C67—21no—
11C2×C22⋊C8×C67C2212yes—
12C2×C4×C8×C67—21no—
13#390×C67C411yescharacteristic, complemented
14#390×C67C411yescharacteristic, complemented
15#389×C67C411yescharacteristic, complemented
16#389×C67C411yescharacteristic, complemented
17#388×C67—21no—
18#388×C67—21no—
19#386×C67—21no—
20#380×C67C2211yescharacteristic
21C2×#95×C67—21no—
22C2×#95×C67—21no—
23C2×#95×C67C2212yes—
24#327×C67—24no—
25C22×D4×C67—21no—
26C22×D4×C67C2×C411yescharacteristic
27C23×C4×C67—22no—
28C23×C4×C67D411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C23×C4×C67C2311yescharacteristic
31C22×C8×C67—22no—
32#123×C67—42no—
33#123×C67—42no—
34#121×C67—22no—
35#121×C67—22no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67C2×C411yescharacteristic
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C22⋊C4×C67—21no—
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67—21no—
47C2×C42×C67—22no—
48C2×C42×C67—22no—
49C2×C42×C67C2×C411yescharacteristic
50C2×C42×C67C2×C411yescharacteristic
51C2×C42×C67D411yescharacteristic
52C2×C42×C67D411yescharacteristic
53C4⋊C8×C67—24no—
54C22⋊C8×C67—22no—
55C22⋊C8×C67D414yes—
56C4×C8×C67—24no—
57#95×C67—24no—
58#95×C67—21no—
59#95×C67—21no—
60#95×C67—21no—
61#95×C67—22no—
62#95×C67—22no—
63#95×C67—42no—
64C24×C67—21no—
65C24×C67C2×D411yescharacteristic
66C2×D4×C67—22no—
67C2×D4×C67—41no—
68C2×D4×C67—41no—
69C2×D4×C67—42no—
70C2×D4×C67—41no—
71C2×D4×C67—41no—
72C2×D4×C67—42no—
73C22×C4×C67—22no—
74C22×C4×C67—22no—
75C22×C4×C67—22no—
76C22×C4×C67—22no—
77C22×C4×C67—24no—
78C22×C4×C67—21no—
79C22×C4×C67—21no—
80C22×C4×C67—21no—
81C22×C4×C67—41no—
82C22×C4×C67—41no—
83C22×C4×C67—41no—
84C22×C4×C67—42no—
85C22×C4×C67—42no—
86C22×C4×C67—42no—
87C22×C4×C67—42no—
88C22×C4×C67—42no—
89C22×C4×C67—42no—
90C22×C4×C67C22⋊C411yescharacteristic
91C22×C4×C67C22⋊C411yescharacteristic
92C22×C4×C67C22⋊C411yescharacteristic
93C22×C4×C67C22⋊C411yescharacteristic
94C22×C4×C67D812yes—
95C22×C4×C67SD1612yes—
96C22×C4×C67C22×C411yescharacteristic
97C22×C4×C67C2×D411yescharacteristic
98C22×C4×C67C2×D412yes—
99C22×C4×C67#4112yes—
100C2×C8×C67—24no—
101C2×C8×C67—42no—
102C2×C8×C67—42no—
103C4⋊C4×C67—22no—
104C4⋊C4×C67—41no—
105C4⋊C4×C67—41no—
106C4⋊C4×C67—42no—
107C4⋊C4×C67—42no—
108C22⋊C4×C67—42no—
109C22⋊C4×C67—41no—
110C22⋊C4×C67—41no—
111C22⋊C4×C67—42no—
112C22⋊C4×C67—42no—
113C22⋊C4×C67—42no—
114C22⋊C4×C67—41no—
115C22⋊C4×C67—41no—
116C42×C67—22no—
117C42×C67—22no—
118C42×C67—21no—
119C42×C67—21no—
120C42×C67—21no—
121C42×C67—21no—
122C23×C67—21no—
123C23×C67—21no—
124C23×C67—21no—
125C23×C67—21no—
126C23×C67—41no—
127C23×C67—41no—
128C23×C67—41no—
129C23×C67—41no—
130C23×C67—41no—
131C23×C67—41no—
132C23×C67—41no—
133C23×C67C2×C22⋊C411yescharacteristic
134C23×C67#12112yes—
135C23×C67C2×D811yescharacteristic
136C23×C67C2×SD1611yescharacteristic
137C23×C67#13512yes—
138D4×C67—82no—
139D4×C67—82no—
140D4×C67—82no—
141C2×C4×C67—21no—
142C2×C4×C67—21no—
143C2×C4×C67—21no—
144C2×C4×C67—21no—
145C2×C4×C67—21no—
146C2×C4×C67—21no—
147C2×C4×C67—21no—
148C2×C4×C67—21no—
149C2×C4×C67—21no—
150C2×C4×C67—21no—
151C2×C4×C67—24no—
152C2×C4×C67—24no—
153C2×C4×C67—21no—
154C2×C4×C67—21no—
155C2×C4×C67—21no—
156C2×C4×C67—42no—
157C2×C4×C67—42no—
158C2×C4×C67—44no—
159C2×C4×C67—44no—
160C2×C4×C67—44no—
161C2×C4×C67—44no—
162C2×C4×C67—41no—
163C2×C4×C67—41no—
164C2×C4×C67—41no—
165C2×C4×C67—41no—
166C2×C4×C67—41no—
167C2×C4×C67—41no—
168C2×C4×C67—41no—
169C2×C4×C67—41no—
170C2×C4×C67—41no—
171C2×C4×C67—82no—
172C2×C4×C67—82no—
173C2×C4×C67—82no—
174C2×C4×C67—82no—
175C2×C4×C67#10212yes—
176C2×C4×C67#10212yes—
177C2×C4×C67#10412yes—
178C2×C4×C67C4×D412yes—
179C2×C4×C67#12011yescharacteristic
180C2×C4×C67#12311yescharacteristic
181C8×C67—44no—
182C22×C67—22no—
183C22×C67—22no—
184C22×C67—22no—
185C22×C67—21no—
186C22×C67—21no—
187C22×C67—21no—
188C22×C67—21no—
189C22×C67—22no—
190C22×C67—81no—
191C22×C67—81no—
192C22×C67—81no—
193C22×C67—81no—
194C22×C67—81no—
195C22×C67—81no—
196C22×C67—81no—
197C22×C67#38611yescharacteristic
198C22×C67#41011yescharacteristic
199C22×C67#41011yescharacteristic
200C22×C67C2×#10211yescharacteristic
201C22×C67#41611yescharacteristic
202C22×C67C2×#10411yescharacteristic
203C22×C67#42111yescharacteristic
204C22×C67#46512yes—
205C22×C67#46612yes—
206C4×C67—22no—
207C4×C67—22no—
208C4×C67—22no—
209C4×C67—21no—
210C4×C67—22no—
211C4×C67—21no—
212C4×C67—41no—
213C4×C67—84no—
214C4×C67—81no—
215C4×C67—81no—
216C4×C67—81no—
217#9114C6711yesmaximal, characteristic, direct product factor, Sylow
218C2×C67—22no—
219C2×C67—22no—
220C2×C67—161no—
221C2×C67#150011yescharacteristic
222C2×C67#150211yescharacteristic
223C2×C67#151711yescharacteristic
224C2×C67#152111yescharacteristic
225C2×C67#187111yescharacteristic
226C2×C67#189311yescharacteristic
227C2×C67#189911yescharacteristic
228C2×#431C2×C6711yescharacteristic, complemented
229#2854C2×C6711yescharacteristic
230C2×#327C2×C6712yes—
231#1434C2×C6711yescharacteristic, complemented
232#1294C2×C6712yescomplemented
233C67#911411yescharacteristic, direct product factor, Sylow
234C2×#123—21no—
235C2×#121C22×C6711yescharacteristic
236C22×C42C22×C6711yescharacteristic
237#431—24no—
238C2×C4⋊C8—21no—
239C2×C22⋊C8C22×C6712yes—
240C2×C4×C8—21no—
241#390C4×C6711yescharacteristic, complemented
242#390C4×C6711yescharacteristic, complemented
243#389C4×C6711yescharacteristic, complemented
244#389C4×C6711yescharacteristic, complemented
245#388—21no—
246#388—21no—
247#386—21no—
248#380C22×C6711yescharacteristic
249C2×#95—21no—
250C2×#95—21no—
251C2×#95C22×C6712yes—
252#327—24no—
253C22×D4—21no—
254C22×D4C2×C4×C6711yescharacteristic
255C23×C4—22no—
256C23×C4D4×C6711yescharacteristic
257C23×C4D4×C6711yescharacteristic
258C23×C4C23×C6711yescharacteristic
259C22×C8—22no—
260#123—42no—
261#123—42no—
262#121—22no—
263#121—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×C4×C6711yescharacteristic
267C2×C4⋊C4C2×C4×C6711yescharacteristic
268C2×C4⋊C4C2×C4×C6711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C2×C4×C6711yescharacteristic
278C2×C42C2×C4×C6711yescharacteristic
279C2×C42D4×C6711yescharacteristic
280C2×C42D4×C6711yescharacteristic
281C4⋊C8—24no—
282C22⋊C8—22no—
283C22⋊C8D4×C6714yes—
284C4×C8—24no—
285#95—24no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—22no—
290#95—22no—
291#95—42no—
292C24—21no—
293C24C2×D4×C6711yescharacteristic
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4C22⋊C4×C6711yescharacteristic
319C22×C4C22⋊C4×C6711yescharacteristic
320C22×C4C22⋊C4×C6711yescharacteristic
321C22×C4C22⋊C4×C6711yescharacteristic
322C22×C4D8×C6712yes—
323C22×C4SD16×C6712yes—
324C22×C4C22×C4×C6711yescharacteristic
325C22×C4C2×D4×C6711yescharacteristic
326C22×C4C2×D4×C6712yes—
327C22×C4#41×C6712yes—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C42—22no—
345C42—22no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C2×C22⋊C4×C6711yescharacteristic
362C23#121×C6712yes—
363C23C2×D8×C6711yescharacteristic
364C23C2×SD16×C6711yescharacteristic
365C23#135×C6712yes—
366D4—82no—
367D4—82no—
368D4—82no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4#102×C6712yes—
404C2×C4#102×C6712yes—
405C2×C4#104×C6712yes—
406C2×C4C4×D4×C6712yes—
407C2×C4#120×C6711yescharacteristic
408C2×C4#123×C6711yescharacteristic
409C8—44no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22#386×C6711yescharacteristic
426C22#410×C6711yescharacteristic
427C22#410×C6711yescharacteristic
428C22C2×#102×C6711yescharacteristic
429C22#416×C6711yescharacteristic
430C22C2×#104×C6711yescharacteristic
431C22#421×C6711yescharacteristic
432C22#465×C6712yes—
433C22#466×C6712yes—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—21no—
438C4—22no—
439C4—21no—
440C4—41no—
441C4—84no—
442C4—81no—
443C4—81no—
444C4—81no—
445C2—22no—
446C2—22no—
447C2—161no—
448C2C67×#150011yescharacteristic
449C2C67×#150211yescharacteristic
450C2C67×#151711yescharacteristic
451C2C67×#152111yescharacteristic
452C2C67×#187111yescharacteristic
453C2C67×#189311yescharacteristic
454C2C67×#189911yescharacteristic
subgroupG/H
center, socleC23×C67C2×C22⋊C4
commutatorC22×C4C22×C4×C67
Frattini, cosocleC23×C4C23×C67
prime radicalC2×#121×C67C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C22×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#431×C67C2semidirect
4. C67×#1434C2semidirect
5. C67×#1294C2semidirect
13. #390×C67C4semidirect
14. #390×C67C4semidirect
15. #389×C67C4semidirect
16. #389×C67C4semidirect
217. #9114C67direct (N × H)
228. C2×#431C2×C67semidirect
231. #1434C2×C67semidirect
232. #1294C2×C67semidirect
233. C67#9114direct (N × H)
241. #390C4×C67semidirect
242. #390C4×C67semidirect
243. #389C4×C67semidirect
244. #389C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#431×C672
C67×#28542
C2×#327×C672
C67×#14342
C67×#12942
#911467