Finite Group Databasebeta

← C67×#59465 #30738716

Subgroups — C67×#59465 #30738716

Order 17152.

1980 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2784C214yesmaximal, complemented
2C67×#2571C211yesmaximal, characteristic, complemented
3C67×#2564C214yesmaximal, complemented
4C67×#2563C214yesmaximal, complemented
5C67×#2551C211yesmaximal, characteristic, complemented
6C67×#2454C214yesmaximal, complemented
7C67×#2452C214yesmaximal, complemented
8C67×#2366C212yesmaximal
9C67×#2349C214yesmaximal, complemented
10C4×#127×C67C211yesmaximal, characteristic
11C4×#125×C67C212yesmaximal, complemented
12C2×C4⋊Q8×C67C2214yes—
13C2×#127×C67C2211yescharacteristic
14C2×#125×C67C2212yes—
15C2×C22⋊Q8×C67C2214yes—
16C2×C4×D4×C67C2214yes—
17C2×C4×D4×C67C2212yes—
18C2×C4×D4×C67C2214yes—
19C2×C4×D4×C67C2214yes—
20#400×C67C2214yes—
21#399×C67C2214yes—
22#396×C67C2218yes—
23#395×C67C2214yes—
24#393×C67C2218yes—
25#393×C67C2214yes—
26#390×C67C2212yes—
27#390×C67C2214yes—
28#390×C67C2214yes—
29#390×C67C2214yes—
30#389×C67C2214yes—
31#389×C67C2212yes—
32#389×C67C2212yes—
33#389×C67C2214yes—
34#387×C67C2214yes—
35#387×C67C2214yes—
36#386×C67C2218yes—
37#385×C67C2218yes—
38#385×C67C2218yes—
39#384×C67C2211yescharacteristic
40#382×C67C2212yes—
41#382×C67C2214yes—
42#380×C67C2218yes—
43#380×C67C2218yes—
44C4×C4⋊C4×C67C2212yes—
45C4×C4⋊C4×C67C2214yes—
46C4×C4⋊C4×C67C2212yes—
47C4×C4⋊C4×C67C2214yes—
48C2×#95×C67C2214yes—
49C43×C67C2211yescharacteristic
50C22×Q8×C67C2314yes—
51C22×D4×C67C2314yes—
52C22×D4×C67C2312yes—
53C23×C4×C67C2314yes—
54C23×C4×C67C2318yes—
55C4⋊Q8×C67—48no—
56#127×C67Q818yescomplemented
57#125×C67—28no—
58C22⋊Q8×C67—48no—
59C4×D4×C67—216no—
60C4×D4×C67—48no—
61C4×D4×C67—48no—
62C4×D4×C67—44no—
63C2×C4⋊C4×C67C2312yes—
64C2×C4⋊C4×C67C2314yes—
65C2×C4⋊C4×C67C2314yes—
66C2×C4⋊C4×C67C2314yes—
67C2×C4⋊C4×C67C2314yes—
68C2×C4⋊C4×C67C2312yes—
69C2×C4⋊C4×C67C2312yes—
70C2×C4⋊C4×C67C2312yes—
71C2×C4⋊C4×C67C2314yes—
72C2×C4⋊C4×C67C2318yes—
73C2×C4⋊C4×C67C2314yes—
74C2×C4⋊C4×C67C2312yes—
75C2×C22⋊C4×C67C2318yes—
76C2×C22⋊C4×C67C2314yes—
77C2×C22⋊C4×C67C2318yes—
78C2×C22⋊C4×C67C2318yes—
79C2×C22⋊C4×C67C2318yes—
80C2×C42×C67C2312yes—
81C2×C42×C67C2311yescharacteristic
82C2×C42×C67C2314yes—
83C2×C42×C67C2314yes—
84C2×C42×C67C2312yes—
85C2×C42×C67C2311yescharacteristic
86C2×C42×C67C2311yescharacteristic
87#95×C67C2314yes—
88#95×C67C2314yes—
89#95×C67C2314yes—
90#95×C67C2314yes—
91#95×C67C2314yes—
92#95×C67C2314yes—
93#95×C67C2314yes—
94#95×C67C2314yes—
95#95×C67C2318yes—
96C24×C67C2414yes—
97C2×Q8×C67—44no—
98C2×Q8×C67—48no—
99C2×D4×C67—28no—
100C2×D4×C67—28no—
101C2×D4×C67—42no—
102C2×D4×C67—42no—
103C2×D4×C67C2×Q8116yes—
104C2×D4×C67C2×Q818yes—
105C22×C4×C67—216no—
106C22×C4×C67—28no—
107C22×C4×C67—48no—
108C22×C4×C67—48no—
109C22×C4×C67—48no—
110C22×C4×C67C2411yescharacteristic
111C22×C4×C67C2412yes—
112C22×C4×C67C2414yes—
113C22×C4×C67C2414yes—
114C22×C4×C67C2414yes—
115C22×C4×C67C2414yes—
116C22×C4×C67C2411yescharacteristic
117C22×C4×C67C2414yes—
118C22×C4×C67C2411yescharacteristic
119C22×C4×C67C2412yes—
120C4⋊C4×C67—28no—
121C4⋊C4×C67—28no—
122C4⋊C4×C67—24no—
123C4⋊C4×C67—24no—
124C4⋊C4×C67—28no—
125C4⋊C4×C67—28no—
126C4⋊C4×C67—24no—
127C4⋊C4×C67—44no—
128C4⋊C4×C67—42no—
129C4⋊C4×C67—44no—
130C4⋊C4×C67—48no—
131C4⋊C4×C67#4118yes—
132C22⋊C4×C67—216no—
133C22⋊C4×C67—48no—
134C22⋊C4×C67—44no—
135C22⋊C4×C67—48no—
136C22⋊C4×C67—48no—
137C42×C67—28no—
138C42×C67—24no—
139C42×C67—22no—
140C42×C67—24no—
141C42×C67—28no—
142C42×C67—41no—
143C42×C67C2×Q814yes—
144C23×C67—28no—
145C23×C67—24no—
146C23×C67—28no—
147C23×C67—24no—
148C23×C67C22×Q818yes—
149C23×C67C2511yescharacteristic
150Q8×C67—88no—
151D4×C67—232no—
152D4×C67—48no—
153C2×C4×C67—24no—
154C2×C4×C67—21no—
155C2×C4×C67—22no—
156C2×C4×C67—21no—
157C2×C4×C67—24no—
158C2×C4×C67—24no—
159C2×C4×C67—22no—
160C2×C4×C67—22no—
161C2×C4×C67—24no—
162C2×C4×C67—24no—
163C2×C4×C67—21no—
164C2×C4×C67—22no—
165C2×C4×C67—24no—
166C2×C4×C67—24no—
167C2×C4×C67—24no—
168C2×C4×C67—24no—
169C2×C4×C67—28no—
170C2×C4×C67—24no—
171C2×C4×C67—24no—
172C2×C4×C67—22no—
173C2×C4×C67—24no—
174C2×C4×C67—816no—
175C2×C4×C67—88no—
176C2×C4×C67C22×Q812yes—
177C2×C4×C67C22×Q814yes—
178C2×C4×C67C2×#4118yes—
179C2×C4×C67C2×#4114yes—
180C2×C4×C67H(5,𝔽2)12yes—
181C2×C4×C67H(5,𝔽2)14yes—
182C22×C67—216no—
183C22×C67—28no—
184C22×C67—44no—
185C22×C67—48no—
186C22×C67—44no—
187C22×C67C23×Q811yescharacteristic
188C22×C67C22×#4112yes—
189C22×C67C2×H(5,𝔽2)12yes—
190C22×C67C2×H(5,𝔽2)11yescharacteristic
191C22×C67#58511yescharacteristic
192C4×C67—22no—
193C4×C67—28no—
194C4×C67—22no—
195C4×C67—44no—
196C4×C67—41no—
197C4×C67—42no—
198C4×C67—44no—
199C4×C67—44no—
200C4×C67—44no—
201C4×C67#55418yes—
202#59465C6711yesmaximal, characteristic, direct product factor, Sylow
203C2×C67—48no—
204C2×C67C2×#55412yes—
205C2×C67#348311yescharacteristic
206C2×C67#350111yescharacteristic
207C2×C67#351412yes—
208C2×C67#354311yescharacteristic
209#2784C2×C6714yescomplemented
210#2571C2×C6711yescharacteristic, complemented
211#2564C2×C6714yescomplemented
212#2563C2×C6714yescomplemented
213#2551C2×C6711yescharacteristic, complemented
214#2454C2×C6714yescomplemented
215#2452C2×C6714yescomplemented
216#2366C2×C6712yes—
217#2349C2×C6714yescomplemented
218C4×#127C2×C6711yescharacteristic
219C4×#125C2×C6712yescomplemented
220C67#5946511yescharacteristic, direct product factor, Sylow
221C2×C4⋊Q8C22×C6714yes—
222C2×#127C22×C6711yescharacteristic
223C2×#125C22×C6712yes—
224C2×C22⋊Q8C22×C6714yes—
225C2×C4×D4C22×C6714yes—
226C2×C4×D4C22×C6712yes—
227C2×C4×D4C22×C6714yes—
228C2×C4×D4C22×C6714yes—
229#400C22×C6714yes—
230#399C22×C6714yes—
231#396C22×C6718yes—
232#395C22×C6714yes—
233#393C22×C6714yes—
234#393C22×C6718yes—
235#390C22×C6714yes—
236#390C22×C6712yes—
237#390C22×C6714yes—
238#390C22×C6714yes—
239#389C22×C6712yes—
240#389C22×C6712yes—
241#389C22×C6714yes—
242#389C22×C6714yes—
243#387C22×C6714yes—
244#387C22×C6714yes—
245#386C22×C6718yes—
246#385C22×C6718yes—
247#385C22×C6718yes—
248#384C22×C6711yescharacteristic
249#382C22×C6712yes—
250#382C22×C6714yes—
251#380C22×C6718yes—
252#380C22×C6718yes—
253C4×C4⋊C4C22×C6714yes—
254C4×C4⋊C4C22×C6712yes—
255C4×C4⋊C4C22×C6714yes—
256C4×C4⋊C4C22×C6712yes—
257C2×#95C22×C6714yes—
258C43C22×C6711yescharacteristic
259C22×Q8C23×C6714yes—
260C22×D4C23×C6714yes—
261C22×D4C23×C6712yes—
262C23×C4C23×C6714yes—
263C23×C4C23×C6718yes—
264C4⋊Q8—48no—
265#127Q8×C6718yescomplemented
266#125—28no—
267C22⋊Q8—48no—
268C4×D4—216no—
269C4×D4—48no—
270C4×D4—48no—
271C4×D4—44no—
272C2×C4⋊C4C23×C6712yes—
273C2×C4⋊C4C23×C6714yes—
274C2×C4⋊C4C23×C6714yes—
275C2×C4⋊C4C23×C6714yes—
276C2×C4⋊C4C23×C6714yes—
277C2×C4⋊C4C23×C6712yes—
278C2×C4⋊C4C23×C6712yes—
279C2×C4⋊C4C23×C6712yes—
280C2×C4⋊C4C23×C6714yes—
281C2×C4⋊C4C23×C6718yes—
282C2×C4⋊C4C23×C6714yes—
283C2×C4⋊C4C23×C6712yes—
284C2×C22⋊C4C23×C6718yes—
285C2×C22⋊C4C23×C6714yes—
286C2×C22⋊C4C23×C6718yes—
287C2×C22⋊C4C23×C6718yes—
288C2×C22⋊C4C23×C6718yes—
289C2×C42C23×C6712yes—
290C2×C42C23×C6711yescharacteristic
291C2×C42C23×C6714yes—
292C2×C42C23×C6714yes—
293C2×C42C23×C6712yes—
294C2×C42C23×C6711yescharacteristic
295C2×C42C23×C6711yescharacteristic
296#95C23×C6714yes—
297#95C23×C6714yes—
298#95C23×C6714yes—
299#95C23×C6714yes—
300#95C23×C6714yes—
301#95C23×C6714yes—
302#95C23×C6714yes—
303#95C23×C6714yes—
304#95C23×C6718yes—
305C24C24×C6714yes—
306C2×Q8—44no—
307C2×Q8—48no—
308C2×D4—28no—
309C2×D4—28no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4C2×Q8×C67116yes—
313C2×D4C2×Q8×C6718yes—
314C22×C4—216no—
315C22×C4—28no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4C24×C6714yes—
320C22×C4C24×C6711yescharacteristic
321C22×C4C24×C6712yes—
322C22×C4C24×C6714yes—
323C22×C4C24×C6714yes—
324C22×C4C24×C6714yes—
325C22×C4C24×C6711yescharacteristic
326C22×C4C24×C6714yes—
327C22×C4C24×C6711yescharacteristic
328C22×C4C24×C6712yes—
329C4⋊C4—28no—
330C4⋊C4—28no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—24no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—48no—
339C4⋊C4—44no—
340C4⋊C4#41×C6718yes—
341C22⋊C4—216no—
342C22⋊C4—48no—
343C22⋊C4—44no—
344C22⋊C4—48no—
345C22⋊C4—48no—
346C42—28no—
347C42—24no—
348C42—22no—
349C42—24no—
350C42—28no—
351C42—41no—
352C42C2×Q8×C6714yes—
353C23—28no—
354C23—24no—
355C23—28no—
356C23—24no—
357C23C22×Q8×C6718yes—
358C23C25×C6711yescharacteristic
359Q8—88no—
360D4—232no—
361D4—48no—
362C2×C4—24no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—28no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—88no—
384C2×C4—816no—
385C2×C4C22×Q8×C6712yes—
386C2×C4C22×Q8×C6714yes—
387C2×C4C2×#41×C6718yes—
388C2×C4C2×#41×C6714yes—
389C2×C4H(5,𝔽2)×C6712yes—
390C2×C4H(5,𝔽2)×C6714yes—
391C22—216no—
392C22—28no—
393C22—44no—
394C22—44no—
395C22—48no—
396C22C23×Q8×C6711yescharacteristic
397C22C22×#41×C6712yes—
398C22C2×H(5,𝔽2)×C6712yes—
399C22C2×H(5,𝔽2)×C6711yescharacteristic
400C22#585×C6711yescharacteristic
401C4—22no—
402C4—22no—
403C4—28no—
404C4—44no—
405C4—44no—
406C4—44no—
407C4—41no—
408C4—44no—
409C4—42no—
410C4#554×C6718yes—
411C2—48no—
412C2C2×#554×C6712yes—
413C2C67×#348311yescharacteristic
414C2C67×#350111yescharacteristic
415C2C67×#351412yes—
416C2C67×#354311yescharacteristic
subgroupG/H
center, socleC23×C67C25
commutator, Frattini, cosocleC23C25×C67
prime radicalC2×#127×C67C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2784C2semidirect
2. C67×#2571C2semidirect
3. C67×#2564C2semidirect
4. C67×#2563C2semidirect
5. C67×#2551C2semidirect
6. C67×#2454C2semidirect
7. C67×#2452C2semidirect
9. C67×#2349C2semidirect
11. C4×#125×C67C2semidirect
56. #127×C67Q8semidirect
202. #59465C67direct (N × H)
209. #2784C2×C67semidirect
210. #2571C2×C67semidirect
211. #2564C2×C67semidirect
212. #2563C2×C67semidirect
213. #2551C2×C67semidirect
214. #2454C2×C67semidirect
215. #2452C2×C67semidirect
217. #2349C2×C67semidirect
219. C4×#125C2×C67semidirect
220. C67#59465direct (N × H)
265. #127Q8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#27842
C67×#25712
C67×#25642
C67×#25632
C67×#25512
C67×#24542
C67×#24522
C67×#23662
C67×#23492
C4×#127×C672
C4×#125×C672
#5946567