Finite Group Databasebeta

← C67×#59322 #30738573

Subgroups — C67×#59322 #30738573

Order 17152.

1854 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2792C214yesmaximal, complemented
2C67×#2583C214yesmaximal
3C67×#2582C211yesmaximal, characteristic, complemented
4C67×#2567C214yesmaximal, complemented
5C67×#2551C211yesmaximal, characteristic, complemented
6C67×#2507C214yesmaximal, complemented
7C67×#2440C214yesmaximal
8C67×#2360C214yesmaximal
9Q8×C4⋊C4×C67C212yesmaximal, complemented
10C4×C22⋊Q8×C67C212yesmaximal
11C2×#384×C67C211yesmaximal, characteristic
12C2×C4⋊Q8×C67C2214yes—
13C2×C22⋊Q8×C67C2214yes—
14C2×C22⋊Q8×C67C2212yes—
15C2×C4×Q8×C67C2212yes—
16C2×C4×Q8×C67C2214yes—
17C22×C4⋊C4×C67C2212yes—
18C22×C4⋊C4×C67C2214yes—
19C22×C42×C67C2211yescharacteristic
20#398×C67C2214yes—
21#398×C67C2218yes—
22#398×C67C2214yes—
23#395×C67C2214yes—
24#395×C67C2218yes—
25#395×C67C2214yes—
26#391×C67C2212yes—
27#391×C67C2214yes—
28#389×C67C2212yes—
29#389×C67C2214yes—
30#389×C67C2218yes—
31#389×C67C2214yes—
32#389×C67C2214yes—
33#389×C67C2218yes—
34#389×C67C2218yes—
35#387×C67C2218yes—
36#387×C67C2214yes—
37#385×C67C2214yes—
38#385×C67C2218yes—
39#384×C67C2214yes—
40#384×C67C2212yes—
41#384×C67C2211yescharacteristic
42#384×C67C2211yescharacteristic
43#384×C67C2214yes—
44#380×C67C2212yes—
45C4×C4⋊C4×C67C2214yes—
46C4×C4⋊C4×C67C2214yes—
47C4×C4⋊C4×C67C2212yes—
48C4×C22⋊C4×C67C2214yes—
49C4×C22⋊C4×C67C2214yes—
50C22×Q8×C67C2312yes—
51C22×Q8×C67C2314yes—
52C23×C4×C67C2311yescharacteristic
53C23×C4×C67C2312yes—
54C23×C4×C67C2314yes—
55C4⋊Q8×C67—48no—
56C22⋊Q8×C67—48no—
57C22⋊Q8×C67Q8116yescomplemented
58C4×Q8×C67—28no—
59C4×Q8×C67—48no—
60C2×C4⋊C4×C67—48no—
61C2×C4⋊C4×C67—44no—
62C2×C4⋊C4×C67C2314yes—
63C2×C4⋊C4×C67C2314yes—
64C2×C4⋊C4×C67C2312yes—
65C2×C4⋊C4×C67C2314yes—
66C2×C4⋊C4×C67C2312yes—
67C2×C4⋊C4×C67C2314yes—
68C2×C4⋊C4×C67C2318yes—
69C2×C4⋊C4×C67C2312yes—
70C2×C4⋊C4×C67C2314yes—
71C2×C4⋊C4×C67C2312yes—
72C2×C4⋊C4×C67C2318yes—
73C2×C4⋊C4×C67C2318yes—
74C2×C4⋊C4×C67C2314yes—
75C2×C4⋊C4×C67C2318yes—
76C2×C4⋊C4×C67C2312yes—
77C2×C4⋊C4×C67C2314yes—
78C2×C22⋊C4×C67C2314yes—
79C2×C22⋊C4×C67C2314yes—
80C2×C42×C67—24no—
81C2×C42×C67C2312yes—
82C2×C42×C67C2311yescharacteristic
83C2×C42×C67C2314yes—
84C2×C42×C67C2314yes—
85C2×C42×C67C2318yes—
86C2×C42×C67C2311yescharacteristic
87#95×C67C2314yes—
88#95×C67C2314yes—
89#95×C67C2314yes—
90#95×C67C2314yes—
91#95×C67C2314yes—
92#95×C67C2314yes—
93#95×C67C2318yes—
94#95×C67C2314yes—
95#95×C67C2318yes—
96C24×C67C2411yescharacteristic
97C2×Q8×C67—24no—
98C2×Q8×C67—24no—
99C2×Q8×C67—44no—
100C2×Q8×C67—48no—
101C2×Q8×C67C2×Q818yes—
102C22×C4×C67—22no—
103C22×C4×C67—22no—
104C22×C4×C67—24no—
105C22×C4×C67—24no—
106C22×C4×C67—22no—
107C22×C4×C67—28no—
108C22×C4×C67—48no—
109C22×C4×C67C2×Q818yes—
110C22×C4×C67C2414yes—
111C22×C4×C67C2414yes—
112C22×C4×C67C2414yes—
113C22×C4×C67C2414yes—
114C22×C4×C67C2414yes—
115C22×C4×C67C2414yes—
116C22×C4×C67C2412yes—
117C22×C4×C67C2411yescharacteristic
118C22×C4×C67C2411yescharacteristic
119C22×C4×C67C2412yes—
120C4⋊C4×C67—28no—
121C4⋊C4×C67—24no—
122C4⋊C4×C67—28no—
123C4⋊C4×C67—28no—
124C4⋊C4×C67—24no—
125C4⋊C4×C67—28no—
126C4⋊C4×C67—48no—
127C4⋊C4×C67—42no—
128C4⋊C4×C67—44no—
129C4⋊C4×C67—42no—
130C4⋊C4×C67—48no—
131C4⋊C4×C67—44no—
132C4⋊C4×C67—48no—
133C4⋊C4×C67—48no—
134C4⋊C4×C67C2×Q818yes—
135C4⋊C4×C67C2×Q8116yes—
136C22⋊C4×C67—28no—
137C22⋊C4×C67C2×Q8116yes—
138C42×C67—22no—
139C42×C67—24no—
140C42×C67—28no—
141C42×C67—28no—
142C42×C67—48no—
143C42×C67#4114yes—
144C23×C67—21no—
145C23×C67—22no—
146C23×C67—21no—
147C23×C67C22×Q814yes—
148C23×C67H(5,𝔽2)12yes—
149C23×C67C2511yescharacteristic
150Q8×C67—216no—
151Q8×C67—88no—
152C2×C4×C67—28no—
153C2×C4×C67—22no—
154C2×C4×C67—21no—
155C2×C4×C67—24no—
156C2×C4×C67—24no—
157C2×C4×C67—24no—
158C2×C4×C67—22no—
159C2×C4×C67—21no—
160C2×C4×C67—21no—
161C2×C4×C67—24no—
162C2×C4×C67—21no—
163C2×C4×C67—216no—
164C2×C4×C67—24no—
165C2×C4×C67—24no—
166C2×C4×C67—28no—
167C2×C4×C67—24no—
168C2×C4×C67—24no—
169C2×C4×C67—28no—
170C2×C4×C67—24no—
171C2×C4×C67—24no—
172C2×C4×C67—24no—
173C2×C4×C67—24no—
174C2×C4×C67—88no—
175C2×C4×C67C22×Q818yes—
176C2×C4×C67C22×Q818yes—
177C2×C4×C67C22×Q814yes—
178C2×C4×C67C22×Q814yes—
179C2×C4×C67C2×#4112yes—
180C2×C4×C67C2×#4114yes—
181C2×C4×C67H(5,𝔽2)14yes—
182C2×C4×C672−1+412yes—
183C22×C67—24no—
184C22×C67—22no—
185C22×C67—24no—
186C22×C67—22no—
187C22×C67Q8214yes—
188C22×C67C23×Q812yes—
189C22×C67C22×#4111yescharacteristic
190C22×C67C2×H(5,𝔽2)12yes—
191C22×C67C2×H(5,𝔽2)11yescharacteristic
192C22×C67C2×2−1+411yescharacteristic
193C4×C67—24no—
194C4×C67—28no—
195C4×C67—22no—
196C4×C67—22no—
197C4×C67—28no—
198C4×C67—44no—
199C4×C67—44no—
200C4×C67—44no—
201C4×C67—44no—
202C4×C67#55418yes—
203#59322C6711yesmaximal, characteristic, direct product factor, Sylow
204C2×C67—24no—
205C2×C67C2×#55412yes—
206C2×C67C2×Q8211yescharacteristic
207C2×C67#349211yescharacteristic
208C2×C67#350312yes—
209C2×C67#354311yescharacteristic
210#2792C2×C6714yescomplemented
211#2583C2×C6714yes—
212#2582C2×C6711yescharacteristic, complemented
213#2567C2×C6714yescomplemented
214#2551C2×C6711yescharacteristic, complemented
215#2507C2×C6714yescomplemented
216#2440C2×C6714yes—
217#2360C2×C6714yes—
218Q8×C4⋊C4C2×C6712yescomplemented
219C4×C22⋊Q8C2×C6712yes—
220C2×#384C2×C6711yescharacteristic
221C67#5932211yescharacteristic, direct product factor, Sylow
222C2×C4⋊Q8C22×C6714yes—
223C2×C22⋊Q8C22×C6714yes—
224C2×C22⋊Q8C22×C6712yes—
225C2×C4×Q8C22×C6712yes—
226C2×C4×Q8C22×C6714yes—
227C22×C4⋊C4C22×C6712yes—
228C22×C4⋊C4C22×C6714yes—
229C22×C42C22×C6711yescharacteristic
230#398C22×C6714yes—
231#398C22×C6714yes—
232#398C22×C6718yes—
233#395C22×C6714yes—
234#395C22×C6718yes—
235#395C22×C6714yes—
236#391C22×C6712yes—
237#391C22×C6714yes—
238#389C22×C6718yes—
239#389C22×C6718yes—
240#389C22×C6714yes—
241#389C22×C6718yes—
242#389C22×C6714yes—
243#389C22×C6714yes—
244#389C22×C6712yes—
245#387C22×C6718yes—
246#387C22×C6714yes—
247#385C22×C6714yes—
248#385C22×C6718yes—
249#384C22×C6714yes—
250#384C22×C6714yes—
251#384C22×C6711yescharacteristic
252#384C22×C6711yescharacteristic
253#384C22×C6712yes—
254#380C22×C6712yes—
255C4×C4⋊C4C22×C6714yes—
256C4×C4⋊C4C22×C6714yes—
257C4×C4⋊C4C22×C6712yes—
258C4×C22⋊C4C22×C6714yes—
259C4×C22⋊C4C22×C6714yes—
260C22×Q8C23×C6712yes—
261C22×Q8C23×C6714yes—
262C23×C4C23×C6711yescharacteristic
263C23×C4C23×C6712yes—
264C23×C4C23×C6714yes—
265C4⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8Q8×C67116yescomplemented
268C4×Q8—28no—
269C4×Q8—48no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C23×C6714yes—
273C2×C4⋊C4C23×C6714yes—
274C2×C4⋊C4C23×C6712yes—
275C2×C4⋊C4C23×C6714yes—
276C2×C4⋊C4C23×C6712yes—
277C2×C4⋊C4C23×C6714yes—
278C2×C4⋊C4C23×C6718yes—
279C2×C4⋊C4C23×C6712yes—
280C2×C4⋊C4C23×C6714yes—
281C2×C4⋊C4C23×C6712yes—
282C2×C4⋊C4C23×C6718yes—
283C2×C4⋊C4C23×C6718yes—
284C2×C4⋊C4C23×C6714yes—
285C2×C4⋊C4C23×C6718yes—
286C2×C4⋊C4C23×C6712yes—
287C2×C4⋊C4C23×C6714yes—
288C2×C22⋊C4C23×C6714yes—
289C2×C22⋊C4C23×C6714yes—
290C2×C42—24no—
291C2×C42C23×C6714yes—
292C2×C42C23×C6714yes—
293C2×C42C23×C6712yes—
294C2×C42C23×C6711yescharacteristic
295C2×C42C23×C6718yes—
296C2×C42C23×C6711yescharacteristic
297#95C23×C6714yes—
298#95C23×C6714yes—
299#95C23×C6714yes—
300#95C23×C6714yes—
301#95C23×C6714yes—
302#95C23×C6714yes—
303#95C23×C6718yes—
304#95C23×C6714yes—
305#95C23×C6718yes—
306C24C24×C6711yescharacteristic
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—44no—
310C2×Q8—48no—
311C2×Q8C2×Q8×C6718yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—28no—
318C22×C4—48no—
319C22×C4C2×Q8×C6718yes—
320C22×C4C24×C6714yes—
321C22×C4C24×C6714yes—
322C22×C4C24×C6714yes—
323C22×C4C24×C6714yes—
324C22×C4C24×C6714yes—
325C22×C4C24×C6714yes—
326C22×C4C24×C6712yes—
327C22×C4C24×C6711yescharacteristic
328C22×C4C24×C6711yescharacteristic
329C22×C4C24×C6712yes—
330C4⋊C4—28no—
331C4⋊C4—24no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—24no—
335C4⋊C4—28no—
336C4⋊C4—48no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—48no—
344C4⋊C4C2×Q8×C6718yes—
345C4⋊C4C2×Q8×C67116yes—
346C22⋊C4—28no—
347C22⋊C4C2×Q8×C67116yes—
348C42—22no—
349C42—24no—
350C42—28no—
351C42—28no—
352C42—48no—
353C42#41×C6714yes—
354C23—21no—
355C23—22no—
356C23—21no—
357C23C22×Q8×C6714yes—
358C23H(5,𝔽2)×C6712yes—
359C23C25×C6711yescharacteristic
360Q8—216no—
361Q8—88no—
362C2×C4—28no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—21no—
373C2×C4—216no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—28no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—88no—
385C2×C4C22×Q8×C6718yes—
386C2×C4C22×Q8×C6718yes—
387C2×C4C22×Q8×C6714yes—
388C2×C4C22×Q8×C6714yes—
389C2×C4C2×#41×C6712yes—
390C2×C4C2×#41×C6714yes—
391C2×C4H(5,𝔽2)×C6714yes—
392C2×C42−1+4×C6712yes—
393C22—22no—
394C22—24no—
395C22—22no—
396C22—24no—
397C22Q82×C6714yes—
398C22C23×Q8×C6712yes—
399C22C22×#41×C6711yescharacteristic
400C22C2×H(5,𝔽2)×C6712yes—
401C22C2×H(5,𝔽2)×C6711yescharacteristic
402C22C2×2−1+4×C6711yescharacteristic
403C4—24no—
404C4—28no—
405C4—22no—
406C4—22no—
407C4—28no—
408C4—44no—
409C4—44no—
410C4—44no—
411C4—44no—
412C4#554×C6718yes—
413C2—24no—
414C2C2×#554×C6712yes—
415C2C2×Q82×C6711yescharacteristic
416C2C67×#349211yescharacteristic
417C2C67×#350312yes—
418C2C67×#354311yescharacteristic
subgroupG/H
center, socleC23×C67C25
commutator, Frattini, cosocleC23C25×C67
prime radicalC24×C67C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2792C2semidirect
3. C67×#2582C2semidirect
4. C67×#2567C2semidirect
5. C67×#2551C2semidirect
6. C67×#2507C2semidirect
9. Q8×C4⋊C4×C67C2semidirect
57. C22⋊Q8×C67Q8semidirect
203. #59322C67direct (N × H)
210. #2792C2×C67semidirect
212. #2582C2×C67semidirect
213. #2567C2×C67semidirect
214. #2551C2×C67semidirect
215. #2507C2×C67semidirect
218. Q8×C4⋊C4C2×C67semidirect
221. C67#59322direct (N × H)
267. C22⋊Q8Q8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#27922
C67×#25832
C67×#25822
C67×#25672
C67×#25512
C67×#25072
C67×#24402
C67×#23602
Q8×C4⋊C4×C672
C4×C22⋊Q8×C672
C2×#384×C672
#5932267