Finite Group Databasebeta

← C67×#59329 #30738580

Subgroups — C67×#59329 #30738580

Order 17152.

1694 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2794C214yesmaximal, complemented
2C67×#2582C211yesmaximal, characteristic, complemented
3C67×#2571C211yesmaximal, characteristic, complemented
4C67×#2570C214yesmaximal, complemented
5C67×#2511C214yesmaximal, complemented
6C67×#2503C214yesmaximal
7C67×#2484C214yesmaximal
8C67×#2355C212yesmaximal, complemented
9C67×#2344C214yesmaximal
10C4×C22⋊Q8×C67C212yesmaximal
11C2×#384×C67C211yesmaximal, characteristic
12C2×C22⋊Q8×C67C2212yes—
13C2×C4×Q8×C67C2212yes—
14C2×#117×C67C2214yes—
15C22×C4⋊C4×C67C2214yes—
16C22×C4⋊C4×C67C2212yes—
17C22×C42×C67C2211yescharacteristic
18#400×C67C2218yes—
19#400×C67C2218yes—
20#400×C67C2214yes—
21#400×C67C2214yes—
22#398×C67C2214yes—
23#398×C67C2214yes—
24#391×C67C2214yes—
25#391×C67C2214yes—
26#391×C67C2212yes—
27#389×C67C2214yes—
28#389×C67C2212yes—
29#389×C67C2214yes—
30#387×C67C2218yes—
31#387×C67C2218yes—
32#387×C67C2214yes—
33#387×C67C2218yes—
34#387×C67C2218yes—
35#385×C67C2214yes—
36#384×C67C2214yes—
37#384×C67C2212yes—
38#384×C67C2211yescharacteristic
39#384×C67C2211yescharacteristic
40#382×C67C2214yes—
41#382×C67C2214yes—
42#381×C67C2214yes—
43#381×C67C2218yes—
44#380×C67C2212yes—
45C4×C4⋊C4×C67C2214yes—
46C4×C4⋊C4×C67C2214yes—
47C4×C4⋊C4×C67C2212yes—
48C4×C4⋊C4×C67C2214yes—
49C4×C22⋊C4×C67C2214yes—
50C22×Q8×C67C2312yes—
51C23×C4×C67C2311yescharacteristic
52C23×C4×C67C2314yes—
53C23×C4×C67C2312yes—
54C22⋊Q8×C67—28no—
55C4×Q8×C67—28no—
56#117×C67—48no—
57C2×C4⋊C4×C67—216no—
58C2×C4⋊C4×C67—44no—
59C2×C4⋊C4×C67C2314yes—
60C2×C4⋊C4×C67C2314yes—
61C2×C4⋊C4×C67C2318yes—
62C2×C4⋊C4×C67C2312yes—
63C2×C4⋊C4×C67C2314yes—
64C2×C4⋊C4×C67C2312yes—
65C2×C4⋊C4×C67C2314yes—
66C2×C4⋊C4×C67C2312yes—
67C2×C4⋊C4×C67C2314yes—
68C2×C4⋊C4×C67C2312yes—
69C2×C4⋊C4×C67C2314yes—
70C2×C4⋊C4×C67C2312yes—
71C2×C4⋊C4×C67C2314yes—
72C2×C4⋊C4×C67C2314yes—
73C2×C22⋊C4×C67C2314yes—
74C2×C22⋊C4×C67C2314yes—
75C2×C42×C67—24no—
76C2×C42×C67C2312yes—
77C2×C42×C67C2311yescharacteristic
78C2×C42×C67C2314yes—
79C2×C42×C67C2314yes—
80C2×C42×C67C2318yes—
81C2×C42×C67C2311yescharacteristic
82#95×C67C2314yes—
83#95×C67C2314yes—
84#95×C67C2314yes—
85#95×C67C2318yes—
86#95×C67C2318yes—
87#95×C67C2314yes—
88#95×C67C2314yes—
89#95×C67C2314yes—
90#95×C67C2318yes—
91#95×C67C2318yes—
92#95×C67C2318yes—
93#95×C67C2314yes—
94C24×C67C2411yescharacteristic
95C2×Q8×C67—24no—
96C2×Q8×C67—24no—
97C2×Q8×C67—24no—
98C22×C4×C67—22no—
99C22×C4×C67—22no—
100C22×C4×C67—24no—
101C22×C4×C67—22no—
102C22×C4×C67—24no—
103C22×C4×C67—216no—
104C22×C4×C67—28no—
105C22×C4×C67#4118yes—
106C22×C4×C67C2414yes—
107C22×C4×C67C2414yes—
108C22×C4×C67C2414yes—
109C22×C4×C67C2414yes—
110C22×C4×C67C2414yes—
111C22×C4×C67C2414yes—
112C22×C4×C67C2411yescharacteristic
113C22×C4×C67C2412yes—
114C22×C4×C67C2411yescharacteristic
115C22×C4×C67C2412yes—
116C4⋊C4×C67—216no—
117C4⋊C4×C67—28no—
118C4⋊C4×C67—28no—
119C4⋊C4×C67—24no—
120C4⋊C4×C67—28no—
121C4⋊C4×C67—24no—
122C4⋊C4×C67—28no—
123C4⋊C4×C67—28no—
124C4⋊C4×C67—24no—
125C4⋊C4×C67—42no—
126C4⋊C4×C67—44no—
127C4⋊C4×C67—42no—
128C4⋊C4×C67—44no—
129C4⋊C4×C67—44no—
130C22⋊C4×C67—28no—
131C22⋊C4×C67—28no—
132C42×C67—22no—
133C42×C67—24no—
134C42×C67—28no—
135C42×C67—28no—
136C42×C67—48no—
137C42×C67#4114yes—
138C23×C67—21no—
139C23×C67—22no—
140C23×C67—21no—
141C23×C67C2×#4114yes—
142C23×C67H(5,𝔽2)12yes—
143C23×C67C2511yescharacteristic
144Q8×C67—48no—
145C2×C4×C67—28no—
146C2×C4×C67—22no—
147C2×C4×C67—21no—
148C2×C4×C67—21no—
149C2×C4×C67—21no—
150C2×C4×C67—22no—
151C2×C4×C67—21no—
152C2×C4×C67—216no—
153C2×C4×C67—24no—
154C2×C4×C67—24no—
155C2×C4×C67—24no—
156C2×C4×C67—24no—
157C2×C4×C67—28no—
158C2×C4×C67—28no—
159C2×C4×C67—24no—
160C2×C4×C67—24no—
161C2×C4×C67—24no—
162C2×C4×C67—24no—
163C2×C4×C67—24no—
164C2×C4×C67—24no—
165C2×C4×C67—24no—
166C2×C4×C67—24no—
167C2×C4×C67—24no—
168C2×C4×C67—24no—
169C2×C4×C67—416no—
170C2×C4×C67C2×#4112yes—
171C2×C4×C67C2×#4114yes—
172C2×C4×C67C2×#4114yes—
173C2×C4×C67C2×#4114yes—
174C2×C4×C672−1+412yes—
175C2×C4×C672−1+414yes—
176C22×C67—24no—
177C22×C67—22no—
178C22×C67—24no—
179C22×C67—22no—
180C22×C67#55914yes—
181C22×C67C22×#4111yescharacteristic
182C22×C67C22×#4112yes—
183C22×C67C2×H(5,𝔽2)11yescharacteristic
184C22×C67C2×2−1+412yes—
185C22×C67C2×2−1+411yescharacteristic
186C4×C67—24no—
187C4×C67—22no—
188C4×C67—22no—
189C4×C67—44no—
190C4×C67—44no—
191C4×C67—44no—
192C4×C67—44no—
193C4×C67—44no—
194C4×C67—44no—
195C4×C67#55218yes—
196#59329C6711yesmaximal, characteristic, direct product factor, Sylow
197C2×C67—24no—
198C2×C67C2×#55212yes—
199C2×C67C2×#55911yescharacteristic
200C2×C67#349211yescharacteristic
201C2×C67#350912yes—
202C2×C67#354411yescharacteristic
203#2794C2×C6714yescomplemented
204#2582C2×C6711yescharacteristic, complemented
205#2571C2×C6711yescharacteristic, complemented
206#2570C2×C6714yescomplemented
207#2511C2×C6714yescomplemented
208#2503C2×C6714yes—
209#2484C2×C6714yes—
210#2355C2×C6712yescomplemented
211#2344C2×C6714yes—
212C4×C22⋊Q8C2×C6712yes—
213C2×#384C2×C6711yescharacteristic
214C67#5932911yescharacteristic, direct product factor, Sylow
215C2×C22⋊Q8C22×C6712yes—
216C2×C4×Q8C22×C6712yes—
217C2×#117C22×C6714yes—
218C22×C4⋊C4C22×C6714yes—
219C22×C4⋊C4C22×C6712yes—
220C22×C42C22×C6711yescharacteristic
221#400C22×C6718yes—
222#400C22×C6714yes—
223#400C22×C6714yes—
224#400C22×C6718yes—
225#398C22×C6714yes—
226#398C22×C6714yes—
227#391C22×C6714yes—
228#391C22×C6714yes—
229#391C22×C6712yes—
230#389C22×C6712yes—
231#389C22×C6714yes—
232#389C22×C6714yes—
233#387C22×C6718yes—
234#387C22×C6718yes—
235#387C22×C6718yes—
236#387C22×C6718yes—
237#387C22×C6714yes—
238#385C22×C6714yes—
239#384C22×C6714yes—
240#384C22×C6711yescharacteristic
241#384C22×C6711yescharacteristic
242#384C22×C6712yes—
243#382C22×C6714yes—
244#382C22×C6714yes—
245#381C22×C6718yes—
246#381C22×C6714yes—
247#380C22×C6712yes—
248C4×C4⋊C4C22×C6714yes—
249C4×C4⋊C4C22×C6714yes—
250C4×C4⋊C4C22×C6712yes—
251C4×C4⋊C4C22×C6714yes—
252C4×C22⋊C4C22×C6714yes—
253C22×Q8C23×C6712yes—
254C23×C4C23×C6714yes—
255C23×C4C23×C6711yescharacteristic
256C23×C4C23×C6712yes—
257C22⋊Q8—28no—
258C4×Q8—28no—
259#117—48no—
260C2×C4⋊C4—216no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C23×C6714yes—
263C2×C4⋊C4C23×C6714yes—
264C2×C4⋊C4C23×C6718yes—
265C2×C4⋊C4C23×C6712yes—
266C2×C4⋊C4C23×C6714yes—
267C2×C4⋊C4C23×C6712yes—
268C2×C4⋊C4C23×C6714yes—
269C2×C4⋊C4C23×C6712yes—
270C2×C4⋊C4C23×C6714yes—
271C2×C4⋊C4C23×C6712yes—
272C2×C4⋊C4C23×C6714yes—
273C2×C4⋊C4C23×C6712yes—
274C2×C4⋊C4C23×C6714yes—
275C2×C4⋊C4C23×C6714yes—
276C2×C22⋊C4C23×C6714yes—
277C2×C22⋊C4C23×C6714yes—
278C2×C42—24no—
279C2×C42C23×C6714yes—
280C2×C42C23×C6714yes—
281C2×C42C23×C6712yes—
282C2×C42C23×C6711yescharacteristic
283C2×C42C23×C6718yes—
284C2×C42C23×C6711yescharacteristic
285#95C23×C6714yes—
286#95C23×C6714yes—
287#95C23×C6714yes—
288#95C23×C6718yes—
289#95C23×C6718yes—
290#95C23×C6714yes—
291#95C23×C6714yes—
292#95C23×C6714yes—
293#95C23×C6718yes—
294#95C23×C6718yes—
295#95C23×C6718yes—
296#95C23×C6714yes—
297C24C24×C6711yescharacteristic
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—216no—
307C22×C4—28no—
308C22×C4#41×C6718yes—
309C22×C4C24×C6714yes—
310C22×C4C24×C6714yes—
311C22×C4C24×C6714yes—
312C22×C4C24×C6714yes—
313C22×C4C24×C6714yes—
314C22×C4C24×C6714yes—
315C22×C4C24×C6712yes—
316C22×C4C24×C6711yescharacteristic
317C22×C4C24×C6711yescharacteristic
318C22×C4C24×C6712yes—
319C4⋊C4—216no—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C22⋊C4—28no—
334C22⋊C4—28no—
335C42—22no—
336C42—24no—
337C42—28no—
338C42—28no—
339C42—48no—
340C42#41×C6714yes—
341C23—21no—
342C23—22no—
343C23—21no—
344C23C2×#41×C6714yes—
345C23H(5,𝔽2)×C6712yes—
346C23C25×C6711yescharacteristic
347Q8—48no—
348C2×C4—28no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—216no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—416no—
373C2×C4C2×#41×C6712yes—
374C2×C4C2×#41×C6714yes—
375C2×C4C2×#41×C6714yes—
376C2×C4C2×#41×C6714yes—
377C2×C42−1+4×C6712yes—
378C2×C42−1+4×C6714yes—
379C22—22no—
380C22—24no—
381C22—22no—
382C22—24no—
383C22#559×C6714yes—
384C22C22×#41×C6711yescharacteristic
385C22C22×#41×C6712yes—
386C22C2×H(5,𝔽2)×C6711yescharacteristic
387C22C2×2−1+4×C6712yes—
388C22C2×2−1+4×C6711yescharacteristic
389C4—24no—
390C4—22no—
391C4—22no—
392C4—44no—
393C4—44no—
394C4—44no—
395C4—44no—
396C4—44no—
397C4—44no—
398C4#552×C6718yes—
399C2—24no—
400C2C2×#552×C6712yes—
401C2C2×#559×C6711yescharacteristic
402C2C67×#349211yescharacteristic
403C2C67×#350912yes—
404C2C67×#354411yescharacteristic
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×#2794C2semidirect
2. C67×#2582C2semidirect
3. C67×#2571C2semidirect
4. C67×#2570C2semidirect
5. C67×#2511C2semidirect
8. C67×#2355C2semidirect
196. #59329C67direct (N × H)
203. #2794C2×C67semidirect
204. #2582C2×C67semidirect
205. #2571C2×C67semidirect
206. #2570C2×C67semidirect
207. #2511C2×C67semidirect
210. #2355C2×C67semidirect
214. C67#59329direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#27942
C67×#25822
C67×#25712
C67×#25702
C67×#25112
C67×#25032
C67×#24842
C67×#23552
C67×#23442
C4×C22⋊Q8×C672
C2×#384×C672
#5932967