Finite Group Databasebeta

← C67×#10233 #30689484

Subgroups — C67×#10233 #30689484

Order 17152.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474×C67C211yesmaximal, characteristic
2C2×#462×C67C211yesmaximal, characteristic, complemented
3C67×#2538C211yesmaximal, characteristic
4C67×#1990C211yesmaximal, characteristic, complemented
5C2×Q8⋊C8×C67C211yesmaximal, characteristic, complemented
6C2×D4⋊C8×C67C211yesmaximal, characteristic
7C67×#1296C211yesmaximal, characteristic, complemented
8C22×Q16×C67—21no—
9C2×C4⋊Q8×C67C2211yescharacteristic
10C2×C22⋊Q8×C67—21no—
11C2×C4×Q8×C67C2211yescharacteristic
12C2×C4×D4×C67C2211yescharacteristic
13#474×C67—22no—
14#474×C67—22no—
15#462×C67—22no—
16#462×C67—22no—
17C2×#107×C67—21no—
18C2×C4⋊C8×C67C2211yescharacteristic
19C2×C4⋊C8×C67C2211yescharacteristic
20C2×#103×C67—21no—
21C2×#102×C67—21no—
22C2×C4×C8×C67C2211yescharacteristic
23#396×C67—22no—
24#390×C67—21no—
25#389×C67—21no—
26#384×C67C2211yescharacteristic
27#380×C67—21no—
28#340×C67—22no—
29Q8⋊C8×C67—22no—
30D4⋊C8×C67—22no—
31C22×Q8×C67—21no—
32C22×Q8×C67D411yescharacteristic
33C22×D4×C67D411yescharacteristic
34C23×C4×C67—21no—
35C2×Q16×C67—24no—
36C2×Q16×C67—42no—
37C2×Q16×C67—42no—
38C22×C8×C67—21no—
39C22×C8×C67—21no—
40C22×C8×C67—21no—
41C4⋊Q8×C67—21no—
42C4⋊Q8×C67—21no—
43C4⋊Q8×C67—21no—
44C4⋊Q8×C67D412yes—
45C22⋊Q8×C67—42no—
46C22⋊Q8×C67—42no—
47C4×Q8×C67—21no—
48C4×Q8×C67—21no—
49C4×Q8×C67—21no—
50C4×Q8×C67—21no—
51C4×D4×C67—21no—
52C4×D4×C67—21no—
53C4×D4×C67—21no—
54C4×D4×C67—21no—
55C2×C4⋊C4×C67—21no—
56C2×C4⋊C4×C67—21no—
57C2×C4⋊C4×C67—22no—
58C2×C4⋊C4×C67—21no—
59C2×C4⋊C4×C67—21no—
60C2×C4⋊C4×C67—21no—
61C2×C4⋊C4×C67D411yescharacteristic
62C2×C4⋊C4×C67D411yescharacteristic
63C2×C4⋊C4×C67D412yes—
64C2×C22⋊C4×C67—22no—
65C2×C22⋊C4×C67—21no—
66C2×C22⋊C4×C67—22no—
67C2×C42×C67—21no—
68C2×C42×C67C2311yescharacteristic
69#107×C67—24no—
70C4⋊C8×C67—21no—
71C4⋊C8×C67—21no—
72C4⋊C8×C67D412yes—
73C4⋊C8×C67D412yes—
74#103×C67—24no—
75#102×C67—24no—
76C4×C8×C67—21no—
77C4×C8×C67—21no—
78#95×C67—22no—
79C24×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—21no—
86C2×Q8×C67—41no—
87C2×Q8×C67—41no—
88C2×Q8×C67—41no—
89C2×Q8×C67—41no—
90C2×Q8×C67—41no—
91C2×Q8×C67D812yes—
92C2×D4×C67—21no—
93C2×D4×C67—21no—
94C2×D4×C67—21no—
95C2×D4×C67—21no—
96C2×D4×C67—21no—
97C2×D4×C67Q1612yescomplemented
98C22×C4×C67—21no—
99C22×C4×C67—21no—
100C22×C4×C67—22no—
101C22×C4×C67—21no—
102C22×C4×C67—41no—
103C22×C4×C67—41no—
104C22×C4×C67—41no—
105C22×C4×C67—41no—
106C22×C4×C67—41no—
107C22×C4×C67—41no—
108C22×C4×C67—41no—
109C22×C4×C67C2×D411yescharacteristic
110C22×C4×C67C2×D411yescharacteristic
111C22×C4×C67C2×D411yescharacteristic
112Q16×C67—48no—
113C2×C8×C67—22no—
114C2×C8×C67—22no—
115C2×C8×C67—22no—
116C2×C8×C67—41no—
117C2×C8×C67—41no—
118C2×C8×C67—41no—
119C2×C8×C67—41no—
120C2×C8×C67—41no—
121C2×C8×C67—41no—
122C4⋊C4×C67—22no—
123C4⋊C4×C67—21no—
124C4⋊C4×C67—21no—
125C4⋊C4×C67—22no—
126C4⋊C4×C67—21no—
127C4⋊C4×C67—21no—
128C4⋊C4×C67—42no—
129C4⋊C4×C67—44no—
130C4⋊C4×C67—41no—
131C4⋊C4×C67—41no—
132C4⋊C4×C67—41no—
133C4⋊C4×C67—41no—
134C4⋊C4×C67—41no—
135C4⋊C4×C67—41no—
136C4⋊C4×C67—41no—
137C4⋊C4×C67—41no—
138C22⋊C4×C67—44no—
139C22⋊C4×C67—41no—
140C22⋊C4×C67—41no—
141C22⋊C4×C67—82no—
142C42×C67—41no—
143C42×C67—41no—
144C42×C67C2×D411yescharacteristic
145C42×C67C2×D411yescharacteristic
146C42×C67C2×D411yescharacteristic
147C42×C67#4111yescharacteristic
148C23×C67—22no—
149C23×C67—41no—
150C23×C67—41no—
151C23×C67—41no—
152C23×C67—41no—
153C23×C67—41no—
154C23×C67—41no—
155C23×C67#12011yescharacteristic
156Q8×C67—22no—
157Q8×C67—22no—
158Q8×C67—41no—
159Q8×C67—41no—
160Q8×C67—42no—
161Q8×C67—42no—
162Q8×C67—81no—
163Q8×C67—81no—
164D4×C67—22no—
165D4×C67—22no—
166D4×C67—41no—
167D4×C67—41no—
168C2×C4×C67—21no—
169C2×C4×C67—21no—
170C2×C4×C67—22no—
171C2×C4×C67—41no—
172C2×C4×C67—41no—
173C2×C4×C67—41no—
174C2×C4×C67—41no—
175C2×C4×C67—42no—
176C2×C4×C67—42no—
177C2×C4×C67—41no—
178C2×C4×C67—41no—
179C2×C4×C67—42no—
180C2×C4×C67—41no—
181C2×C4×C67—42no—
182C2×C4×C67—41no—
183C2×C4×C67—81no—
184C2×C4×C67—81no—
185C2×C4×C67—81no—
186C2×C4×C67—81no—
187C2×C4×C67—81no—
188C2×C4×C67—81no—
189C2×C4×C67#12011yescharacteristic
190C2×C4×C67#12011yescharacteristic
191C2×C4×C67#12111yescharacteristic
192C2×C4×C67#12111yescharacteristic
193C2×C4×C67#12111yescharacteristic
194C2×C4×C67#12711yescharacteristic
195C2×C4×C67C2×D811yescharacteristic
196C2×C4×C67C2×Q1611yescharacteristic
197C2×C4×C67#13511yescharacteristic
198C2×C4×C67#13511yescharacteristic
199C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
200C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
201C2×C4×C67#13711yescharacteristic
202C2×C4×C67#13711yescharacteristic
203C8×C67—42no—
204C8×C67—42no—
205C8×C67—42no—
206C22×C67—42no—
207C22×C67—42no—
208C22×C67—42no—
209C22×C67—81no—
210C22×C67—81no—
211C22×C67—81no—
212C22×C67—81no—
213C22×C67#39211yescharacteristic
214C22×C67#44711yescharacteristic
215C22×C67#44911yescharacteristic
216C22×C67#45111yescharacteristic
217C22×C67#45211yescharacteristic
218C22×C67#45511yescharacteristic
219C22×C67#45611yescharacteristic
220C4×C67—21no—
221C4×C67—21no—
222C4×C67—42no—
223C4×C67—82no—
224C4×C67—81no—
225C4×C67—82no—
226C4×C67—81no—
227C4×C67#44711yescharacteristic
228C4×C67#44911yescharacteristic
229C4×C67#45111yescharacteristic
230C4×C67#45211yescharacteristic
231C4×C67#46611yescharacteristic
232C4×C67#46711yescharacteristic
233C4×C67#46911yescharacteristic
234C4×C67#47011yescharacteristic
235#10233C6711yesmaximal, characteristic, direct product factor, Sylow
236C2×C67—82no—
237C2×C67#164111yescharacteristic
238C2×C67#164411yescharacteristic
239C2×C67#164911yescharacteristic
240C2×C67#165211yescharacteristic
241C2×C67#199911yescharacteristic
242C2×C67#200111yescharacteristic
243C2×C67#200711yescharacteristic
244C2×#474C2×C6711yescharacteristic
245C2×#462C2×C6711yescharacteristic, complemented
246#2538C2×C6711yescharacteristic
247#1990C2×C6711yescharacteristic, complemented
248C2×Q8⋊C8C2×C6711yescharacteristic, complemented
249C2×D4⋊C8C2×C6711yescharacteristic
250#1296C2×C6711yescharacteristic, complemented
251C67#1023311yescharacteristic, direct product factor, Sylow
252C22×Q16—21no—
253C2×C4⋊Q8C22×C6711yescharacteristic
254C2×C22⋊Q8—21no—
255C2×C4×Q8C22×C6711yescharacteristic
256C2×C4×D4C22×C6711yescharacteristic
257#474—22no—
258#474—22no—
259#462—22no—
260#462—22no—
261C2×#107—21no—
262C2×C4⋊C8C22×C6711yescharacteristic
263C2×C4⋊C8C22×C6711yescharacteristic
264C2×#103—21no—
265C2×#102—21no—
266C2×C4×C8C22×C6711yescharacteristic
267#396—22no—
268#390—21no—
269#389—21no—
270#384C22×C6711yescharacteristic
271#380—21no—
272#340—22no—
273Q8⋊C8—22no—
274D4⋊C8—22no—
275C22×Q8—21no—
276C22×Q8D4×C6711yescharacteristic
277C22×D4D4×C6711yescharacteristic
278C23×C4—21no—
279C2×Q16—24no—
280C2×Q16—42no—
281C2×Q16—42no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8D4×C6712yes—
289C22⋊Q8—42no—
290C22⋊Q8—42no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4D4×C6711yescharacteristic
306C2×C4⋊C4D4×C6711yescharacteristic
307C2×C4⋊C4D4×C6712yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C42—21no—
312C2×C42C23×C6711yescharacteristic
313#107—24no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8D4×C6712yes—
317C4⋊C8D4×C6712yes—
318#103—24no—
319#102—24no—
320C4×C8—21no—
321C4×C8—21no—
322#95—22no—
323C24—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8D8×C6712yes—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4Q16×C6712yescomplemented
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C2×D4×C6711yescharacteristic
354C22×C4C2×D4×C6711yescharacteristic
355C22×C4C2×D4×C6711yescharacteristic
356Q16—48no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C22⋊C4—44no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—82no—
386C42—41no—
387C42—41no—
388C42C2×D4×C6711yescharacteristic
389C42C2×D4×C6711yescharacteristic
390C42C2×D4×C6711yescharacteristic
391C42#41×C6711yescharacteristic
392C23—22no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23#120×C6711yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—42no—
405Q8—42no—
406Q8—81no—
407Q8—81no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4#120×C6711yescharacteristic
434C2×C4#120×C6711yescharacteristic
435C2×C4#121×C6711yescharacteristic
436C2×C4#121×C6711yescharacteristic
437C2×C4#121×C6711yescharacteristic
438C2×C4#127×C6711yescharacteristic
439C2×C4C2×D8×C6711yescharacteristic
440C2×C4C2×Q16×C6711yescharacteristic
441C2×C4#135×C6711yescharacteristic
442C2×C4#135×C6711yescharacteristic
443C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
444C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
445C2×C4#137×C6711yescharacteristic
446C2×C4#137×C6711yescharacteristic
447C8—42no—
448C8—42no—
449C8—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22#392×C6711yescharacteristic
458C22#447×C6711yescharacteristic
459C22#449×C6711yescharacteristic
460C22#451×C6711yescharacteristic
461C22#452×C6711yescharacteristic
462C22#455×C6711yescharacteristic
463C22#456×C6711yescharacteristic
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—82no—
469C4—81no—
470C4—82no—
471C4#447×C6711yescharacteristic
472C4#449×C6711yescharacteristic
473C4#451×C6711yescharacteristic
474C4#452×C6711yescharacteristic
475C4#466×C6711yescharacteristic
476C4#467×C6711yescharacteristic
477C4#469×C6711yescharacteristic
478C4#470×C6711yescharacteristic
479C2—82no—
480C2C67×#164111yescharacteristic
481C2C67×#164411yescharacteristic
482C2C67×#164911yescharacteristic
483C2C67×#165211yescharacteristic
484C2C67×#199911yescharacteristic
485C2C67×#200111yescharacteristic
486C2C67×#200711yescharacteristic
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
2. C2×#462×C67C2semidirect
4. C67×#1990C2semidirect
5. C2×Q8⋊C8×C67C2semidirect
7. C67×#1296C2semidirect
97. C2×D4×C67Q16semidirect
235. #10233C67direct (N × H)
245. C2×#462C2×C67semidirect
247. #1990C2×C67semidirect
248. C2×Q8⋊C8C2×C67semidirect
250. #1296C2×C67semidirect
251. C67#10233direct (N × H)
341. C2×D4Q16×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#474×C672
C2×#462×C672
C67×#25382
C67×#19902
C2×Q8⋊C8×C672
C2×D4⋊C8×C672
C67×#12962
#1023367