Finite Group Databasebeta

← C67×#9080 #30688331

Subgroups — C67×#9080 #30688331

Order 17152.

736 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#431×C67C211yesmaximal, characteristic, complemented
2C2×#388×C67C211yesmaximal, characteristic
3C67×#1817C211yesmaximal, characteristic, complemented
4C2×#323×C67C212yesmaximal
5C67×#1280C212yesmaximal, complemented
6C22×C4⋊C4×C67C411yescharacteristic
7C22×C22⋊C4×C67C411yescharacteristic
8C22×C22⋊C4×C67C411yescharacteristic
9C22×C22⋊C4×C67C2211yescharacteristic
10C22×C42×C67C2211yescharacteristic
11#431×C67—24no—
12C2×C4⋊C8×C67—21no—
13C2×C22⋊C8×C67C2212yes—
14C2×C22⋊C8×C67C2212yes—
15C2×C4×C8×C67—21no—
16#388×C67—21no—
17#388×C67—21no—
18#388×C67—21no—
19#388×C67—21no—
20C2×#95×C67C411yescharacteristic
21C2×#95×C67C2211yescharacteristic
22#336×C67—21no—
23#336×C67—21no—
24#323×C67—24no—
25C25×C67C2×C411yescharacteristic
26C23×C4×C67C2×C411yescharacteristic
27C23×C4×C67C2×C411yescharacteristic
28C23×C4×C67C2×C411yescharacteristic
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67C2×C411yescharacteristic
31C23×C4×C67C2311yescharacteristic
32C22×C8×C67—22no—
33C22×C8×C67—22no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—42no—
37C2×C22⋊C4×C67—21no—
38C2×C22⋊C4×C67—21no—
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67—21no—
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—22no—
46C2×C22⋊C4×C67—42no—
47C2×C22⋊C4×C67C814yescomplemented
48C2×C42×C67—22no—
49C2×C42×C67—22no—
50C2×C42×C67D411yescharacteristic
51C2×C42×C67D411yescharacteristic
52C2×C42×C67D411yescharacteristic
53C2×C42×C67D411yescharacteristic
54C4⋊C8×C67—24no—
55C22⋊C8×C67—22no—
56C22⋊C8×C67—22no—
57C22⋊C8×C67—22no—
58C22⋊C8×C67D414yes—
59C4×C8×C67—24no—
60#95×C67—21no—
61#95×C67—21no—
62#95×C67—21no—
63#95×C67—21no—
64C24×C67—21no—
65C24×C67—21no—
66C24×C67—22no—
67C24×C67—22no—
68C24×C67—41no—
69C24×C67—41no—
70C24×C67—41no—
71C24×C67—41no—
72C24×C67C2×C812yes—
73C24×C67C22×C411yescharacteristic
74C22×C4×C67—22no—
75C22×C4×C67—22no—
76C22×C4×C67—21no—
77C22×C4×C67—22no—
78C22×C4×C67—21no—
79C22×C4×C67—22no—
80C22×C4×C67—21no—
81C22×C4×C67—21no—
82C22×C4×C67—22no—
83C22×C4×C67—22no—
84C22×C4×C67—41no—
85C22×C4×C67—41no—
86C22×C4×C67—41no—
87C22×C4×C67—41no—
88C22×C4×C67—41no—
89C22×C4×C67—41no—
90C22×C4×C67—41no—
91C22×C4×C67—41no—
92C22×C4×C67—41no—
93C22×C4×C67C22⋊C411yescharacteristic
94C22×C4×C67C22⋊C411yescharacteristic
95C22×C4×C67C22⋊C411yescharacteristic
96C22×C4×C67C22⋊C411yescharacteristic
97C22×C4×C67C2×C812yes—
98C22×C4×C67C2×C812yes—
99C22×C4×C67M1612yes—
100C22×C4×C67C2×D411yescharacteristic
101C22×C4×C67C2×D411yescharacteristic
102C22×C4×C67C2×D412yes—
103C22×C4×C67#4112yes—
104C2×C8×C67—24no—
105C2×C8×C67—42no—
106C2×C8×C67—42no—
107C2×C8×C67—42no—
108C2×C8×C67—42no—
109C2×C8×C67—42no—
110C4⋊C4×C67—42no—
111C4⋊C4×C67—42no—
112C22⋊C4×C67—24no—
113C22⋊C4×C67—24no—
114C22⋊C4×C67—42no—
115C22⋊C4×C67—42no—
116C22⋊C4×C67—41no—
117C22⋊C4×C67—42no—
118C22⋊C4×C67—41no—
119C42×C67—21no—
120C42×C67—21no—
121C42×C67—22no—
122C42×C67—22no—
123C42×C67—21no—
124C42×C67—21no—
125C23×C67—21no—
126C23×C67—21no—
127C23×C67—21no—
128C23×C67—21no—
129C23×C67—22no—
130C23×C67—24no—
131C23×C67—42no—
132C23×C67—42no—
133C23×C67—42no—
134C23×C67—42no—
135C23×C67—41no—
136C23×C67—41no—
137C23×C67—41no—
138C23×C67—41no—
139C23×C67—41no—
140C23×C67—41no—
141C23×C67—42no—
142C23×C67—82no—
143C23×C67—82no—
144C23×C67—82no—
145C23×C67—82no—
146C23×C67C2×C22⋊C411yescharacteristic
147C23×C67C4×D412yes—
148C23×C67C22×C811yescharacteristic
149C23×C67C2×M1611yescharacteristic
150C23×C67#13112yes—
151C2×C4×C67—21no—
152C2×C4×C67—21no—
153C2×C4×C67—21no—
154C2×C4×C67—21no—
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—24no—
162C2×C4×C67—24no—
163C2×C4×C67—21no—
164C2×C4×C67—22no—
165C2×C4×C67—21no—
166C2×C4×C67—21no—
167C2×C4×C67—42no—
168C2×C4×C67—42no—
169C2×C4×C67—44no—
170C2×C4×C67—42no—
171C2×C4×C67—41no—
172C2×C4×C67—41no—
173C2×C4×C67—42no—
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×C67C22⋊C812yes—
183C2×C4×C67C22⋊C812yes—
184C2×C4×C67#10012yes—
185C2×C4×C67#12011yescharacteristic
186C2×C4×C67#12112yes—
187C2×C4×C67#12311yescharacteristic
188C8×C67—44no—
189C8×C67—82no—
190C22×C67—22no—
191C22×C67—22no—
192C22×C67—21no—
193C22×C67—21no—
194C22×C67—21no—
195C22×C67—21no—
196C22×C67—22no—
197C22×C67—22no—
198C22×C67—42no—
199C22×C67—42no—
200C22×C67—42no—
201C22×C67—84no—
202C22×C67—84no—
203C22×C67—81no—
204C22×C67—81no—
205C22×C67—81no—
206C22×C67—81no—
207C22×C67#38611yescharacteristic
208C22×C67C2×C22⋊C811yescharacteristic
209C22×C67#40811yescharacteristic
210C22×C67#41011yescharacteristic
211C22×C67#41011yescharacteristic
212C22×C67C2×#10011yescharacteristic
213C22×C67#41311yescharacteristic
214C22×C67C8×D412yes—
215C22×C67#43512yes—
216C4×C67—22no—
217C4×C67—22no—
218C4×C67—21no—
219C4×C67—22no—
220C4×C67—21no—
221C4×C67—22no—
222C4×C67—41no—
223C4×C67—42no—
224C4×C67—81no—
225C4×C67—81no—
226#9080C6711yesmaximal, characteristic, direct product factor, Sylow
227C2×C67—22no—
228C2×C67—22no—
229C2×C67—82no—
230C2×C67#145811yescharacteristic
231C2×C67#146011yescharacteristic
232C2×C67#146911yescharacteristic
233C2×C67#147311yescharacteristic
234C2×C67#186911yescharacteristic
235C2×C67#189911yescharacteristic
236C2×C67#190011yescharacteristic
237C2×#431C2×C6711yescharacteristic, complemented
238C2×#388C2×C6711yescharacteristic
239#1817C2×C6711yescharacteristic, complemented
240C2×#323C2×C6712yes—
241#1280C2×C6712yescomplemented
242C67#908011yescharacteristic, direct product factor, Sylow
243C22×C4⋊C4C4×C6711yescharacteristic
244C22×C22⋊C4C4×C6711yescharacteristic
245C22×C22⋊C4C4×C6711yescharacteristic
246C22×C22⋊C4C22×C6711yescharacteristic
247C22×C42C22×C6711yescharacteristic
248#431—24no—
249C2×C4⋊C8—21no—
250C2×C22⋊C8C22×C6712yes—
251C2×C22⋊C8C22×C6712yes—
252C2×C4×C8—21no—
253#388—21no—
254#388—21no—
255#388—21no—
256#388—21no—
257C2×#95C4×C6711yescharacteristic
258C2×#95C22×C6711yescharacteristic
259#336—21no—
260#336—21no—
261#323—24no—
262C25C2×C4×C6711yescharacteristic
263C23×C4C2×C4×C6711yescharacteristic
264C23×C4C2×C4×C6711yescharacteristic
265C23×C4C2×C4×C6711yescharacteristic
266C23×C4C2×C4×C6711yescharacteristic
267C23×C4C2×C4×C6711yescharacteristic
268C23×C4C23×C6711yescharacteristic
269C22×C8—22no—
270C22×C8—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—42no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4C8×C6714yescomplemented
285C2×C42—22no—
286C2×C42—22no—
287C2×C42D4×C6711yescharacteristic
288C2×C42D4×C6711yescharacteristic
289C2×C42D4×C6711yescharacteristic
290C2×C42D4×C6711yescharacteristic
291C4⋊C8—24no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8D4×C6714yes—
296C4×C8—24no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301C24—21no—
302C24—21no—
303C24—22no—
304C24—22no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24C2×C8×C6712yes—
310C24C22×C4×C6711yescharacteristic
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4C22⋊C4×C6711yescharacteristic
331C22×C4C22⋊C4×C6711yescharacteristic
332C22×C4C22⋊C4×C6711yescharacteristic
333C22×C4C22⋊C4×C6711yescharacteristic
334C22×C4C2×C8×C6712yes—
335C22×C4C2×C8×C6712yes—
336C22×C4M16×C6712yes—
337C22×C4C2×D4×C6711yescharacteristic
338C22×C4C2×D4×C6711yescharacteristic
339C22×C4C2×D4×C6712yes—
340C22×C4#41×C6712yes—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—41no—
354C22⋊C4—42no—
355C22⋊C4—41no—
356C42—21no—
357C42—21no—
358C42—22no—
359C42—22no—
360C42—21no—
361C42—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—24no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23C2×C22⋊C4×C6711yescharacteristic
384C23C4×D4×C6712yes—
385C23C22×C8×C6711yescharacteristic
386C23C2×M16×C6711yescharacteristic
387C23#131×C6712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4C22⋊C8×C6712yes—
420C2×C4C22⋊C8×C6712yes—
421C2×C4#100×C6712yes—
422C2×C4#120×C6711yescharacteristic
423C2×C4#121×C6712yes—
424C2×C4#123×C6711yescharacteristic
425C8—44no—
426C8—82no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—84no—
439C22—84no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22#386×C6711yescharacteristic
445C22C2×C22⋊C8×C6711yescharacteristic
446C22#408×C6711yescharacteristic
447C22#410×C6711yescharacteristic
448C22#410×C6711yescharacteristic
449C22C2×#100×C6711yescharacteristic
450C22#413×C6711yescharacteristic
451C22C8×D4×C6712yes—
452C22#435×C6712yes—
453C4—22no—
454C4—22no—
455C4—21no—
456C4—22no—
457C4—21no—
458C4—22no—
459C4—41no—
460C4—42no—
461C4—81no—
462C4—81no—
463C2—22no—
464C2—22no—
465C2—82no—
466C2C67×#145811yescharacteristic
467C2C67×#146011yescharacteristic
468C2C67×#146911yescharacteristic
469C2C67×#147311yescharacteristic
470C2C67×#186911yescharacteristic
471C2C67×#189911yescharacteristic
472C2C67×#190011yescharacteristic
subgroupG/H
center, socleC23×C67C2×C22⋊C4
commutatorC23C22×C8×C67
Frattini, cosocleC23×C4C23×C67
prime radicalC25×C67C2×C4
LCSG ▷ C23 ▷ C2 ▷ 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
3. C67×#1817C2semidirect
5. C67×#1280C2semidirect
47. C2×C22⋊C4×C67C8semidirect
226. #9080C67direct (N × H)
237. C2×#431C2×C67semidirect
239. #1817C2×C67semidirect
241. #1280C2×C67semidirect
242. C67#9080direct (N × H)
284. C2×C22⋊C4C8×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#431×C672
C2×#388×C672
C67×#18172
C2×#323×C672
C67×#12802
#908067