Finite Group Databasebeta

← C67×#33021 #30712272

Subgroups — C67×#33021 #30712272

Order 17152.

1442 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3428C211yesmaximal, characteristic, complemented
2C67×#2482C212yesmaximal, complemented
3C67×#2404C211yesmaximal, characteristic
4C67×#1889C214yesmaximal, complemented
5C67×#1867C214yesmaximal, complemented
6C67×#1820C212yesmaximal, complemented
7C67×#1782C211yesmaximal, characteristic, complemented
8C22×#41×C67C2211yescharacteristic
9C2×#120×C67C2212yes—
10#520×C67—22no—
11#520×C67—22no—
12#519×C67—21no—
13#518×C67—21no—
14#518×C67C2214yes—
15C4×#41×C67—24no—
16C2×#117×C67—22no—
17C2×#117×C67C2212yes—
18C22×C22⋊C4×C67C2211yescharacteristic
19C2×#99×C67C2214yes—
20#399×C67—22no—
21#397×C67—22no—
22#393×C67—21no—
23#392×C67—21no—
24#388×C67—24no—
25#386×C67—22no—
26#386×C67C2212yes—
27#385×C67C2214yes—
28#381×C67C2212yes—
29#381×C67C2211yescharacteristic
30#379×C67C2214yes—
31#342×C67C2218yes—
32C25×C67C2311yescharacteristic
33C2×#41×C67—28no—
34C2×#41×C67—22no—
35C2×#41×C67—22no—
36C22×Q8×C67D411yescharacteristic
37C22×D4×C67—22no—
38C22×D4×C67D411yescharacteristic
39C22×D4×C67C2312yes—
40C23×C4×C67D412yes—
41C23×C4×C67C2311yescharacteristic
42C23×C4×C67C2311yescharacteristic
43#120×C67—28no—
44C4×Q8×C67—24no—
45C4×Q8×C67—24no—
46C4×D4×C67—24no—
47C4×D4×C67—24no—
48C4×D4×C67—28no—
49C4×D4×C67—28no—
50#117×C67—24no—
51#117×C67—24no—
52#117×C67—22no—
53#117×C67—24no—
54#117×C67—22no—
55#117×C67—22no—
56#117×C67—22no—
57C2×C4⋊C4×C67—22no—
58C2×C4⋊C4×C67—22no—
59C2×C4⋊C4×C67—24no—
60C2×C4⋊C4×C67D412yescomplemented
61C2×C4⋊C4×C67D412yescomplemented
62C2×C22⋊C4×C67—24no—
63C2×C22⋊C4×C67—21no—
64C2×C22⋊C4×C67—22no—
65C2×C22⋊C4×C67—21no—
66C2×C22⋊C4×C67—22no—
67C2×C22⋊C4×C67—22no—
68C2×C22⋊C4×C67—24no—
69C2×C22⋊C4×C67—22no—
70C2×C22⋊C4×C67—24no—
71C2×C22⋊C4×C67C2312yes—
72C2×C22⋊C4×C67C2314yes—
73C2×C22⋊C4×C67C2314yes—
74C2×C42×C67—24no—
75C2×C42×C67D414yescomplemented
76#99×C67—28no—
77#95×C67—21no—
78#95×C67—22no—
79#95×C67—21no—
80#95×C67—22no—
81#95×C67—24no—
82#95×C67—22no—
83C24×C67—22no—
84C24×C67—21no—
85C24×C67—24no—
86C24×C67—42no—
87C24×C67—81no—
88C24×C67C2×D412yes—
89C24×C67C2411yescharacteristic
90#41×C67—216no—
91#41×C67—216no—
92C2×Q8×C67—22no—
93C2×Q8×C67—24no—
94C2×D4×C67—24no—
95C2×D4×C67—24no—
96C2×D4×C67—24no—
97C2×D4×C67—22no—
98C2×D4×C67—44no—
99C2×D4×C67—44no—
100C2×D4×C67—82no—
101C2×D4×C67#4118yes—
102C22×C4×C67—24no—
103C22×C4×C67—24no—
104C22×C4×C67—22no—
105C22×C4×C67—22no—
106C22×C4×C67—24no—
107C22×C4×C67—24no—
108C22×C4×C67—24no—
109C22×C4×C67—21no—
110C22×C4×C67—24no—
111C22×C4×C67—22no—
112C22×C4×C67—22no—
113C22×C4×C67—81no—
114C22×C4×C67C2×D412yes—
115C22×C4×C67C2×D412yes—
116C22×C4×C67C2×D411yescharacteristic
117C22×C4×C67C2×D411yescharacteristic
118C22×C4×C67C2×D412yes—
119C22×C4×C67C2×D414yes—
120C22×C4×C67C2×D414yes—
121C4⋊C4×C67—24no—
122C4⋊C4×C67—24no—
123C4⋊C4×C67—22no—
124C4⋊C4×C67—22no—
125C4⋊C4×C67—22no—
126C4⋊C4×C67—22no—
127C4⋊C4×C67—84no—
128C22⋊C4×C67—28no—
129C22⋊C4×C67—24no—
130C22⋊C4×C67—24no—
131C22⋊C4×C67—24no—
132C22⋊C4×C67—28no—
133C22⋊C4×C67—48no—
134C22⋊C4×C67—41no—
135C22⋊C4×C67—42no—
136C22⋊C4×C67—41no—
137C22⋊C4×C67—44no—
138C22⋊C4×C67—84no—
139C22⋊C4×C67—82no—
140C42×C67—28no—
141C42×C67—24no—
142C42×C67—24no—
143C23×C67—22no—
144C23×C67—22no—
145C23×C67—24no—
146C23×C67—22no—
147C23×C67—41no—
148C23×C67—41no—
149C23×C67—42no—
150C23×C67—42no—
151C23×C67—44no—
152C23×C67—42no—
153C23×C67—42no—
154C23×C67—41no—
155C23×C67—41no—
156C23×C67—42no—
157C23×C67—44no—
158C23×C67—88no—
159C23×C67#12314yes—
160C23×C67C22×D411yescharacteristic
161C23×C67C22×D412yes—
162C23×C67C2×#4114yes—
163Q8×C67—28no—
164D4×C67—28no—
165D4×C67—216no—
166D4×C67—88no—
167C2×C4×C67—28no—
168C2×C4×C67—28no—
169C2×C4×C67—22no—
170C2×C4×C67—22no—
171C2×C4×C67—22no—
172C2×C4×C67—22no—
173C2×C4×C67—24no—
174C2×C4×C67—24no—
175C2×C4×C67—24no—
176C2×C4×C67—24no—
177C2×C4×C67—44no—
178C2×C4×C67—44no—
179C2×C4×C67—44no—
180C2×C4×C67—44no—
181C2×C4×C67—42no—
182C2×C4×C67—42no—
183C2×C4×C67—44no—
184C2×C4×C67—42no—
185C2×C4×C67—41no—
186C2×C4×C67—42no—
187C2×C4×C67—44no—
188C2×C4×C67—84no—
189C2×C4×C67—84no—
190C2×C4×C67#12011yescharacteristic
191C2×C4×C67#12012yes—
192C2×C4×C67#12011yescharacteristic
193C2×C4×C67#12314yes—
194C22×C67—22no—
195C22×C67—21no—
196C22×C67—22no—
197C22×C67—21no—
198C22×C67—24no—
199C22×C67—44no—
200C22×C67—44no—
201C22×C67—44no—
202C22×C67—48no—
203C22×C67—42no—
204C22×C67—42no—
205C22×C67—42no—
206C22×C67—82no—
207C22×C67—82no—
208C22×C67—84no—
209C22×C67—81no—
210C22×C67—81no—
211C22×C67C2×#12011yescharacteristic
212C22×C67C2×#12312yes—
213C22×C67#52514yes—
214C4×C67—21no—
215C4×C67—21no—
216C4×C67—22no—
217C4×C67—24no—
218C4×C67—44no—
219C4×C67—44no—
220C4×C67—84no—
221C4×C67—82no—
222C4×C67—81no—
223#33021C6711yesmaximal, characteristic, direct product factor, Sylow
224C2×C67—22no—
225C2×C67—24no—
226C2×C67—44no—
227C2×C67—82no—
228C2×C67#284911yescharacteristic
229C2×C67#302511yescharacteristic
230C2×C67#302711yescharacteristic
231#3428C2×C6711yescharacteristic, complemented
232#2482C2×C6712yescomplemented
233#2404C2×C6711yescharacteristic
234#1889C2×C6714yescomplemented
235#1867C2×C6714yescomplemented
236#1820C2×C6712yescomplemented
237#1782C2×C6711yescharacteristic, complemented
238C67#3302111yescharacteristic, direct product factor, Sylow
239C22×#41C22×C6711yescharacteristic
240C2×#120C22×C6712yes—
241#520—22no—
242#520—22no—
243#519—21no—
244#518—21no—
245#518C22×C6714yes—
246C4×#41—24no—
247C2×#117—22no—
248C2×#117C22×C6712yes—
249C22×C22⋊C4C22×C6711yescharacteristic
250C2×#99C22×C6714yes—
251#399—22no—
252#397—22no—
253#393—21no—
254#392—21no—
255#388—24no—
256#386—22no—
257#386C22×C6712yes—
258#385C22×C6714yes—
259#381C22×C6711yescharacteristic
260#381C22×C6712yes—
261#379C22×C6714yes—
262#342C22×C6718yes—
263C25C23×C6711yescharacteristic
264C2×#41—28no—
265C2×#41—22no—
266C2×#41—22no—
267C22×Q8D4×C6711yescharacteristic
268C22×D4—22no—
269C22×D4D4×C6711yescharacteristic
270C22×D4C23×C6712yes—
271C23×C4D4×C6712yes—
272C23×C4C23×C6711yescharacteristic
273C23×C4C23×C6711yescharacteristic
274#120—28no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—28no—
280C4×D4—28no—
281#117—24no—
282#117—24no—
283#117—22no—
284#117—24no—
285#117—22no—
286#117—22no—
287#117—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4D4×C6712yescomplemented
292C2×C4⋊C4D4×C6712yescomplemented
293C2×C22⋊C4—24no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4C23×C6712yes—
303C2×C22⋊C4C23×C6714yes—
304C2×C22⋊C4C23×C6714yes—
305C2×C42—24no—
306C2×C42D4×C6714yescomplemented
307#99—28no—
308#95—21no—
309#95—22no—
310#95—21no—
311#95—22no—
312#95—24no—
313#95—22no—
314C24—22no—
315C24—21no—
316C24—24no—
317C24—42no—
318C24—81no—
319C24C2×D4×C6712yes—
320C24C24×C6711yescharacteristic
321#41—216no—
322#41—216no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—82no—
332C2×D4#41×C6718yes—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—81no—
345C22×C4C2×D4×C6712yes—
346C22×C4C2×D4×C6712yes—
347C22×C4C2×D4×C6711yescharacteristic
348C22×C4C2×D4×C6711yescharacteristic
349C22×C4C2×D4×C6712yes—
350C22×C4C2×D4×C6714yes—
351C22×C4C2×D4×C6714yes—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—84no—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—28no—
363C22⋊C4—24no—
364C22⋊C4—48no—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—41no—
368C22⋊C4—44no—
369C22⋊C4—84no—
370C22⋊C4—82no—
371C42—28no—
372C42—24no—
373C42—24no—
374C23—22no—
375C23—22no—
376C23—24no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—44no—
383C23—42no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—44no—
389C23—88no—
390C23#123×C6714yes—
391C23C22×D4×C6711yescharacteristic
392C23C22×D4×C6712yes—
393C23C2×#41×C6714yes—
394Q8—28no—
395D4—28no—
396D4—216no—
397D4—88no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4#120×C6711yescharacteristic
422C2×C4#120×C6712yes—
423C2×C4#120×C6711yescharacteristic
424C2×C4#123×C6714yes—
425C22—22no—
426C22—21no—
427C22—22no—
428C22—21no—
429C22—24no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22—48no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—82no—
438C22—82no—
439C22—84no—
440C22—81no—
441C22—81no—
442C22C2×#120×C6711yescharacteristic
443C22C2×#123×C6712yes—
444C22#525×C6714yes—
445C4—21no—
446C4—21no—
447C4—22no—
448C4—24no—
449C4—44no—
450C4—44no—
451C4—82no—
452C4—84no—
453C4—81no—
454C2—22no—
455C2—24no—
456C2—44no—
457C2—82no—
458C2C67×#284911yescharacteristic
459C2C67×#302511yescharacteristic
460C2C67×#302711yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC24C24×C67
prime radicalC67×#2404C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3428C2semidirect
2. C67×#2482C2semidirect
4. C67×#1889C2semidirect
5. C67×#1867C2semidirect
6. C67×#1820C2semidirect
7. C67×#1782C2semidirect
60. C2×C4⋊C4×C67D4semidirect
61. C2×C4⋊C4×C67D4semidirect
75. C2×C42×C67D4semidirect
223. #33021C67direct (N × H)
231. #3428C2×C67semidirect
232. #2482C2×C67semidirect
234. #1889C2×C67semidirect
235. #1867C2×C67semidirect
236. #1820C2×C67semidirect
237. #1782C2×C67semidirect
238. C67#33021direct (N × H)
291. C2×C4⋊C4D4×C67semidirect
292. C2×C4⋊C4D4×C67semidirect
306. C2×C42D4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34282
C67×#24822
C67×#24042
C67×#18892
C67×#18672
C67×#18202
C67×#17822
#3302167