Finite Group Databasebeta

← C67×#10236 #30689487

Subgroups — C67×#10236 #30689487

Order 17152.

628 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#477×C67C211yesmaximal, characteristic, complemented
2C2×#476×C67C211yesmaximal, characteristic
3C67×#2538C211yesmaximal, characteristic
4C67×#1956C211yesmaximal, characteristic, complemented
5C2×Q8⋊C8×C67C211yesmaximal, characteristic, complemented
6C2×D4⋊C8×C67C211yesmaximal, characteristic
7C67×#1296C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C67C2211yescharacteristic
9C2×C22⋊Q8×C67—21no—
10C2×C4×Q8×C67C2211yescharacteristic
11C2×C4×D4×C67C2211yescharacteristic
12#477×C67—22no—
13#477×C67—22no—
14#476×C67—22no—
15#476×C67—22no—
16C2×#107×C67—21no—
17C2×#106×C67—21no—
18C2×C4⋊C8×C67C2211yescharacteristic
19C2×C4⋊C8×C67C2211yescharacteristic
20C2×#103×C67—21no—
21C2×#103×C67—22no—
22C2×#102×C67—21no—
23C2×C4×C8×C67C2211yescharacteristic
24#396×C67—22no—
25#390×C67—21no—
26#389×C67—21no—
27#384×C67C2211yescharacteristic
28#380×C67—21no—
29Q8⋊C8×C67—22no—
30D4⋊C8×C67—22no—
31C22×Q8×C67—21no—
32C22×Q8×C67D411yescharacteristic
33C22×D4×C67D411yescharacteristic
34C23×C4×C67—21no—
35C22×C8×C67—21no—
36C22×C8×C67—21no—
37C22×C8×C67—21no—
38C4⋊Q8×C67—21no—
39C4⋊Q8×C67—21no—
40C4⋊Q8×C67—21no—
41C4⋊Q8×C67D412yes—
42C22⋊Q8×C67—42no—
43C22⋊Q8×C67—42no—
44C4×Q8×C67—21no—
45C4×Q8×C67—21no—
46C4×Q8×C67—21no—
47C4×Q8×C67—21no—
48C4×D4×C67—21no—
49C4×D4×C67—21no—
50C4×D4×C67—21no—
51C4×D4×C67—21no—
52C2×C4⋊C4×C67—21no—
53C2×C4⋊C4×C67—21no—
54C2×C4⋊C4×C67—22no—
55C2×C4⋊C4×C67—21no—
56C2×C4⋊C4×C67—21no—
57C2×C4⋊C4×C67—21no—
58C2×C4⋊C4×C67D411yescharacteristic
59C2×C4⋊C4×C67D411yescharacteristic
60C2×C4⋊C4×C67D412yes—
61C2×C22⋊C4×C67—22no—
62C2×C22⋊C4×C67—21no—
63C2×C22⋊C4×C67—22no—
64C2×C42×C67—21no—
65C2×C42×C67C2311yescharacteristic
66#107×C67—24no—
67#106×C67—24no—
68C4⋊C8×C67—21no—
69C4⋊C8×C67—21no—
70C4⋊C8×C67D412yes—
71C4⋊C8×C67D412yes—
72#103×C67—24no—
73#103×C67—44no—
74#102×C67—24no—
75C4×C8×C67—21no—
76C4×C8×C67—21no—
77#95×C67—22no—
78C24×C67—21no—
79C2×Q8×C67—21no—
80C2×Q8×C67—21no—
81C2×Q8×C67—21no—
82C2×Q8×C67—22no—
83C2×Q8×C67—21no—
84C2×Q8×C67—21no—
85C2×Q8×C67—42no—
86C2×Q8×C67—42no—
87C2×Q8×C67—41no—
88C2×Q8×C67SD1612yes—
89C2×D4×C67—21no—
90C2×D4×C67—21no—
91C2×D4×C67—21no—
92C2×D4×C67—21no—
93C2×D4×C67—21no—
94C2×D4×C67Q1612yes—
95C22×C4×C67—21no—
96C22×C4×C67—21no—
97C22×C4×C67—22no—
98C22×C4×C67—21no—
99C22×C4×C67—41no—
100C22×C4×C67—41no—
101C22×C4×C67—41no—
102C22×C4×C67—41no—
103C22×C4×C67—41no—
104C22×C4×C67—41no—
105C22×C4×C67—41no—
106C22×C4×C67C2×D411yescharacteristic
107C22×C4×C67C2×D411yescharacteristic
108C22×C4×C67C2×D411yescharacteristic
109C2×C8×C67—22no—
110C2×C8×C67—22no—
111C2×C8×C67—22no—
112C2×C8×C67—41no—
113C2×C8×C67—41no—
114C2×C8×C67—41no—
115C2×C8×C67—41no—
116C2×C8×C67—41no—
117C2×C8×C67—41no—
118C4⋊C4×C67—22no—
119C4⋊C4×C67—22no—
120C4⋊C4×C67—21no—
121C4⋊C4×C67—21no—
122C4⋊C4×C67—21no—
123C4⋊C4×C67—21no—
124C4⋊C4×C67—42no—
125C4⋊C4×C67—44no—
126C4⋊C4×C67—41no—
127C4⋊C4×C67—41no—
128C4⋊C4×C67—41no—
129C4⋊C4×C67—41no—
130C4⋊C4×C67—41no—
131C4⋊C4×C67—41no—
132C4⋊C4×C67—41no—
133C4⋊C4×C67—41no—
134C22⋊C4×C67—44no—
135C22⋊C4×C67—41no—
136C22⋊C4×C67—41no—
137C22⋊C4×C67—82no—
138C42×C67—41no—
139C42×C67—41no—
140C42×C67C2×D411yescharacteristic
141C42×C67C2×D411yescharacteristic
142C42×C67C2×D411yescharacteristic
143C42×C67#4111yescharacteristic
144C23×C67—22no—
145C23×C67—41no—
146C23×C67—41no—
147C23×C67—41no—
148C23×C67—41no—
149C23×C67—41no—
150C23×C67—41no—
151C23×C67#12011yescharacteristic
152Q8×C67—22no—
153Q8×C67—22no—
154Q8×C67—44no—
155Q8×C67—41no—
156Q8×C67—41no—
157Q8×C67—82no—
158D4×C67—22no—
159D4×C67—22no—
160D4×C67—41no—
161D4×C67—41no—
162C2×C4×C67—21no—
163C2×C4×C67—21no—
164C2×C4×C67—22no—
165C2×C4×C67—41no—
166C2×C4×C67—41no—
167C2×C4×C67—41no—
168C2×C4×C67—41no—
169C2×C4×C67—42no—
170C2×C4×C67—42no—
171C2×C4×C67—41no—
172C2×C4×C67—42no—
173C2×C4×C67—41no—
174C2×C4×C67—41no—
175C2×C4×C67—42no—
176C2×C4×C67—41no—
177C2×C4×C67—81no—
178C2×C4×C67—81no—
179C2×C4×C67—81no—
180C2×C4×C67—81no—
181C2×C4×C67—81no—
182C2×C4×C67—81no—
183C2×C4×C67#12011yescharacteristic
184C2×C4×C67#12011yescharacteristic
185C2×C4×C67#12111yescharacteristic
186C2×C4×C67#12111yescharacteristic
187C2×C4×C67#12111yescharacteristic
188C2×C4×C67#12711yescharacteristic
189C2×C4×C67C2×SD1611yescharacteristic
190C2×C4×C67C2×Q1611yescharacteristic
191C2×C4×C67#13511yescharacteristic
192C2×C4×C67#13511yescharacteristic
193C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
194C2×C4×C67#13711yescharacteristic
195C2×C4×C67#13711yescharacteristic
196C2×C4×C67#13711yescharacteristic
197C8×C67—42no—
198C8×C67—42no—
199C8×C67—42no—
200C22×C67—42no—
201C22×C67—42no—
202C22×C67—42no—
203C22×C67—81no—
204C22×C67—81no—
205C22×C67—81no—
206C22×C67—81no—
207C22×C67#39211yescharacteristic
208C22×C67#45011yescharacteristic
209C22×C67#45111yescharacteristic
210C22×C67#45211yescharacteristic
211C22×C67#45211yescharacteristic
212C22×C67#45311yescharacteristic
213C22×C67#45411yescharacteristic
214C4×C67—21no—
215C4×C67—21no—
216C4×C67—42no—
217C4×C67—82no—
218C4×C67—81no—
219C4×C67—81no—
220C4×C67—82no—
221C4×C67#44811yescharacteristic
222C4×C67#44911yescharacteristic
223C4×C67#45111yescharacteristic
224C4×C67#45211yescharacteristic
225C4×C67#46511yescharacteristic
226C4×C67#46711yescharacteristic
227C4×C67#46811yescharacteristic
228C4×C67#47011yescharacteristic
229#10236C6711yesmaximal, characteristic, direct product factor, Sylow
230C2×C67—82no—
231C2×C67#163111yescharacteristic
232C2×C67#163211yescharacteristic
233C2×C67#163711yescharacteristic
234C2×C67#163811yescharacteristic
235C2×C67#200011yescharacteristic
236C2×C67#200111yescharacteristic
237C2×C67#200611yescharacteristic
238C2×#477C2×C6711yescharacteristic, complemented
239C2×#476C2×C6711yescharacteristic
240#2538C2×C6711yescharacteristic
241#1956C2×C6711yescharacteristic, complemented
242C2×Q8⋊C8C2×C6711yescharacteristic, complemented
243C2×D4⋊C8C2×C6711yescharacteristic
244#1296C2×C6711yescharacteristic, complemented
245C67#1023611yescharacteristic, direct product factor, Sylow
246C2×C4⋊Q8C22×C6711yescharacteristic
247C2×C22⋊Q8—21no—
248C2×C4×Q8C22×C6711yescharacteristic
249C2×C4×D4C22×C6711yescharacteristic
250#477—22no—
251#477—22no—
252#476—22no—
253#476—22no—
254C2×#107—21no—
255C2×#106—21no—
256C2×C4⋊C8C22×C6711yescharacteristic
257C2×C4⋊C8C22×C6711yescharacteristic
258C2×#103—21no—
259C2×#103—22no—
260C2×#102—21no—
261C2×C4×C8C22×C6711yescharacteristic
262#396—22no—
263#390—21no—
264#389—21no—
265#384C22×C6711yescharacteristic
266#380—21no—
267Q8⋊C8—22no—
268D4⋊C8—22no—
269C22×Q8—21no—
270C22×Q8D4×C6711yescharacteristic
271C22×D4D4×C6711yescharacteristic
272C23×C4—21no—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8D4×C6712yes—
280C22⋊Q8—42no—
281C22⋊Q8—42no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4D4×C6711yescharacteristic
297C2×C4⋊C4D4×C6711yescharacteristic
298C2×C4⋊C4D4×C6712yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C42—21no—
303C2×C42C23×C6711yescharacteristic
304#107—24no—
305#106—24no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8D4×C6712yes—
309C4⋊C8D4×C6712yes—
310#103—24no—
311#103—44no—
312#102—24no—
313C4×C8—21no—
314C4×C8—21no—
315#95—22no—
316C24—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—22no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×Q8SD16×C6712yes—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4Q16×C6712yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4C2×D4×C6711yescharacteristic
345C22×C4C2×D4×C6711yescharacteristic
346C22×C4C2×D4×C6711yescharacteristic
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C22⋊C4—44no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—82no—
376C42—41no—
377C42—41no—
378C42C2×D4×C6711yescharacteristic
379C42C2×D4×C6711yescharacteristic
380C42C2×D4×C6711yescharacteristic
381C42#41×C6711yescharacteristic
382C23—22no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23#120×C6711yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—44no—
393Q8—41no—
394Q8—41no—
395Q8—82no—
396D4—22no—
397D4—22no—
398D4—41no—
399D4—41no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#120×C6711yescharacteristic
422C2×C4#120×C6711yescharacteristic
423C2×C4#121×C6711yescharacteristic
424C2×C4#121×C6711yescharacteristic
425C2×C4#121×C6711yescharacteristic
426C2×C4#127×C6711yescharacteristic
427C2×C4C2×SD16×C6711yescharacteristic
428C2×C4C2×Q16×C6711yescharacteristic
429C2×C4#135×C6711yescharacteristic
430C2×C4#135×C6711yescharacteristic
431C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
432C2×C4#137×C6711yescharacteristic
433C2×C4#137×C6711yescharacteristic
434C2×C4#137×C6711yescharacteristic
435C8—42no—
436C8—42no—
437C8—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22#392×C6711yescharacteristic
446C22#450×C6711yescharacteristic
447C22#451×C6711yescharacteristic
448C22#452×C6711yescharacteristic
449C22#452×C6711yescharacteristic
450C22#453×C6711yescharacteristic
451C22#454×C6711yescharacteristic
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—82no—
458C4—82no—
459C4#448×C6711yescharacteristic
460C4#449×C6711yescharacteristic
461C4#451×C6711yescharacteristic
462C4#452×C6711yescharacteristic
463C4#465×C6711yescharacteristic
464C4#467×C6711yescharacteristic
465C4#468×C6711yescharacteristic
466C4#470×C6711yescharacteristic
467C2—82no—
468C2C67×#163111yescharacteristic
469C2C67×#163211yescharacteristic
470C2C67×#163711yescharacteristic
471C2C67×#163811yescharacteristic
472C2C67×#200011yescharacteristic
473C2C67×#200111yescharacteristic
474C2C67×#200611yescharacteristic
subgroupG/H
center, socleC23×C67#120
commutator, Frattini, cosocleC2×C42C23×C67
prime radicalC22×D4×C67D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#477×C67C2semidirect
4. C67×#1956C2semidirect
5. C2×Q8⋊C8×C67C2semidirect
7. C67×#1296C2semidirect
229. #10236C67direct (N × H)
238. C2×#477C2×C67semidirect
241. #1956C2×C67semidirect
242. C2×Q8⋊C8C2×C67semidirect
244. #1296C2×C67semidirect
245. C67#10236direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#477×C672
C2×#476×C672
C67×#25382
C67×#19562
C2×Q8⋊C8×C672
C2×D4⋊C8×C672
C67×#12962
#1023667