Finite Group Databasebeta

← C67×#34978 #30714229

Subgroups — C67×#34978 #30714229

Order 17152.

1834 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2612C214yesmaximal, complemented
2C67×#2571C211yesmaximal, characteristic, complemented
3C67×#2552C212yesmaximal, complemented
4C67×#2551C211yesmaximal, characteristic, complemented
5C67×#2441C214yesmaximal
6C67×#2439C214yesmaximal
7C67×#2396C214yesmaximal
8C4×C4⋊Q8×C67C214yesmaximal, complemented
9C4×#125×C67C214yesmaximal, complemented
10C67×#2285C211yesmaximal, characteristic
11C4×#117×C67C212yesmaximal
12C2×C4⋊Q8×C67C2214yes—
13C2×#125×C67C2214yes—
14C2×C4×Q8×C67C2214yes—
15C2×#117×C67C2212yes—
16C2×#117×C67C2212yes—
17C2×#117×C67C2212yes—
18C22×C42×C67C2211yescharacteristic
19#400×C67C2214yes—
20#399×C67C2214yes—
21#398×C67C2218yes—
22#396×C67C2218yes—
23#395×C67C2214yes—
24#393×C67C2214yes—
25#391×C67C2218yes—
26#389×C67C2214yes—
27#389×C67C2214yes—
28#389×C67C2214yes—
29#389×C67C2214yes—
30#389×C67C2218yes—
31#387×C67C2218yes—
32#387×C67C2218yes—
33#384×C67C2211yescharacteristic
34#384×C67C2211yescharacteristic
35#384×C67C2214yes—
36#382×C67C2212yes—
37#382×C67C2214yes—
38#380×C67C2214yes—
39C4×C4⋊C4×C67C2212yes—
40C4×C4⋊C4×C67C2212yes—
41C4×C4⋊C4×C67C2212yes—
42C4×C4⋊C4×C67C2212yes—
43C4×C4⋊C4×C67C2214yes—
44C4×C4⋊C4×C67C2214yes—
45C4×C4⋊C4×C67C2218yes—
46C4×C22⋊C4×C67C2214yes—
47C4×C22⋊C4×C67C2214yes—
48#376×C67C2214yes—
49C43×C67C2214yes—
50C22×Q8×C67C2314yes—
51C23×C4×C67C2312yes—
52C23×C4×C67C2311yescharacteristic
53C4⋊Q8×C67—216no—
54#125×C67—216no—
55C4×Q8×C67—48no—
56#117×C67—28no—
57#117×C67—28no—
58#117×C67—44no—
59C2×C4⋊C4×C67C2312yes—
60C2×C4⋊C4×C67C2312yes—
61C2×C4⋊C4×C67C2314yes—
62C2×C4⋊C4×C67C2318yes—
63C2×C4⋊C4×C67C2318yes—
64C2×C4⋊C4×C67C2314yes—
65C2×C4⋊C4×C67C2318yes—
66C2×C4⋊C4×C67C2312yes—
67C2×C4⋊C4×C67C2312yes—
68C2×C4⋊C4×C67C2318yes—
69C2×C4⋊C4×C67C2314yes—
70C2×C4⋊C4×C67C2312yes—
71C2×C4⋊C4×C67C2312yes—
72C2×C22⋊C4×C67C2314yes—
73C2×C22⋊C4×C67C2314yes—
74C2×C22⋊C4×C67C2314yes—
75C2×C42×C67Q818yescomplemented
76C2×C42×C67C2314yes—
77C2×C42×C67C2314yes—
78C2×C42×C67C2314yes—
79C2×C42×C67C2314yes—
80C2×C42×C67C2314yes—
81C2×C42×C67C2314yes—
82C2×C42×C67C2314yes—
83C2×C42×C67C2311yescharacteristic
84C2×C42×C67C2311yescharacteristic
85C2×C42×C67C2312yes—
86#95×C67C2314yes—
87#95×C67C2314yes—
88#95×C67C2314yes—
89#95×C67C2314yes—
90#95×C67C2318yes—
91#95×C67C2314yes—
92#95×C67C2318yes—
93#95×C67C2314yes—
94#95×C67C2318yes—
95C24×C67C2411yescharacteristic
96C2×Q8×C67—216no—
97C2×Q8×C67—44no—
98C22×C4×C67—24no—
99C22×C4×C67—24no—
100C22×C4×C67—22no—
101C22×C4×C67—22no—
102C22×C4×C67C2×Q818yes—
103C22×C4×C67C2×Q814yes—
104C22×C4×C67C2414yes—
105C22×C4×C67C2414yes—
106C22×C4×C67C2414yes—
107C22×C4×C67C2414yes—
108C22×C4×C67C2412yes—
109C22×C4×C67C2411yescharacteristic
110C22×C4×C67C2414yes—
111C22×C4×C67C2414yes—
112C22×C4×C67C2412yes—
113C22×C4×C67C2411yescharacteristic
114C4⋊C4×C67—24no—
115C4⋊C4×C67—24no—
116C4⋊C4×C67—216no—
117C4⋊C4×C67—216no—
118C4⋊C4×C67—28no—
119C4⋊C4×C67—216no—
120C4⋊C4×C67—28no—
121C4⋊C4×C67—42no—
122C4⋊C4×C67—42no—
123C4⋊C4×C67—48no—
124C4⋊C4×C67—44no—
125C4⋊C4×C67#4118yes—
126C4⋊C4×C67#4118yes—
127C22⋊C4×C67—28no—
128C22⋊C4×C67—28no—
129C22⋊C4×C67—44no—
130C42×C67—28no—
131C42×C67—28no—
132C42×C67—28no—
133C42×C67—28no—
134C42×C67—28no—
135C42×C67—28no—
136C42×C67—28no—
137C42×C67C2×Q814yes—
138C42×C67C2×Q814yes—
139C42×C67C2×Q818yes—
140C23×C67—22no—
141C23×C67—22no—
142C23×C67—21no—
143C23×C67—21no—
144C23×C67C22×Q812yes—
145C23×C67C2511yescharacteristic
146Q8×C67—416no—
147C2×C4×C67—22no—
148C2×C4×C67—22no—
149C2×C4×C67—24no—
150C2×C4×C67—24no—
151C2×C4×C67—24no—
152C2×C4×C67—21no—
153C2×C4×C67—24no—
154C2×C4×C67—21no—
155C2×C4×C67—216no—
156C2×C4×C67—28no—
157C2×C4×C67—24no—
158C2×C4×C67—24no—
159C2×C4×C67—24no—
160C2×C4×C67—24no—
161C2×C4×C67—28no—
162C2×C4×C67—28no—
163C2×C4×C67—24no—
164C2×C4×C67—24no—
165C2×C4×C67—24no—
166C2×C4×C67—24no—
167C2×C4×C67C22×Q814yes—
168C2×C4×C67C22×Q812yes—
169C2×C4×C67C22×Q814yes—
170C2×C4×C67C22×Q812yes—
171C2×C4×C67C2×#4118yes—
172C2×C4×C67C2×#4118yes—
173C2×C4×C67C2×#4114yes—
174C2×C4×C67C2×#4114yes—
175C2×C4×C67H(5,𝔽2)12yes—
176C2×C4×C672−1+412yes—
177C22×C67—24no—
178C22×C67—22no—
179C22×C67—41no—
180C22×C67—42no—
181C22×C67—41no—
182C22×C67C23×Q811yescharacteristic
183C22×C67C22×#4112yes—
184C22×C67C22×#4112yes—
185C22×C67C2×H(5,𝔽2)11yescharacteristic
186C22×C67C2×2−1+411yescharacteristic
187C4×C67—24no—
188C4×C67—22no—
189C4×C67—28no—
190C4×C67—28no—
191C4×C67—44no—
192C4×C67—44no—
193C4×C67—44no—
194C4×C67—44no—
195C4×C67#53318yes—
196C4×C67#54414yes—
197#34978C6711yesmaximal, characteristic, direct product factor, Sylow
198C2×C67—42no—
199C2×C67C2×#53312yes—
200C2×C67C2×#54411yescharacteristic
201C2×C67#348111yescharacteristic
202C2×C67#348212yes—
203C2×C67#348311yescharacteristic
204#2612C2×C6714yescomplemented
205#2571C2×C6711yescharacteristic, complemented
206#2552C2×C6712yescomplemented
207#2551C2×C6711yescharacteristic, complemented
208#2441C2×C6714yes—
209#2439C2×C6714yes—
210#2396C2×C6714yes—
211C4×C4⋊Q8C2×C6714yescomplemented
212C4×#125C2×C6714yescomplemented
213#2285C2×C6711yescharacteristic
214C4×#117C2×C6712yes—
215C67#3497811yescharacteristic, direct product factor, Sylow
216C2×C4⋊Q8C22×C6714yes—
217C2×#125C22×C6714yes—
218C2×C4×Q8C22×C6714yes—
219C2×#117C22×C6712yes—
220C2×#117C22×C6712yes—
221C2×#117C22×C6712yes—
222C22×C42C22×C6711yescharacteristic
223#400C22×C6714yes—
224#399C22×C6714yes—
225#398C22×C6718yes—
226#396C22×C6718yes—
227#395C22×C6714yes—
228#393C22×C6714yes—
229#391C22×C6718yes—
230#389C22×C6714yes—
231#389C22×C6714yes—
232#389C22×C6714yes—
233#389C22×C6718yes—
234#389C22×C6714yes—
235#387C22×C6718yes—
236#387C22×C6718yes—
237#384C22×C6714yes—
238#384C22×C6711yescharacteristic
239#384C22×C6711yescharacteristic
240#382C22×C6714yes—
241#382C22×C6712yes—
242#380C22×C6714yes—
243C4×C4⋊C4C22×C6714yes—
244C4×C4⋊C4C22×C6712yes—
245C4×C4⋊C4C22×C6718yes—
246C4×C4⋊C4C22×C6714yes—
247C4×C4⋊C4C22×C6712yes—
248C4×C4⋊C4C22×C6712yes—
249C4×C4⋊C4C22×C6712yes—
250C4×C22⋊C4C22×C6714yes—
251C4×C22⋊C4C22×C6714yes—
252#376C22×C6714yes—
253C43C22×C6714yes—
254C22×Q8C23×C6714yes—
255C23×C4C23×C6711yescharacteristic
256C23×C4C23×C6712yes—
257C4⋊Q8—216no—
258#125—216no—
259C4×Q8—48no—
260#117—28no—
261#117—28no—
262#117—44no—
263C2×C4⋊C4C23×C6712yes—
264C2×C4⋊C4C23×C6712yes—
265C2×C4⋊C4C23×C6714yes—
266C2×C4⋊C4C23×C6718yes—
267C2×C4⋊C4C23×C6718yes—
268C2×C4⋊C4C23×C6714yes—
269C2×C4⋊C4C23×C6718yes—
270C2×C4⋊C4C23×C6712yes—
271C2×C4⋊C4C23×C6712yes—
272C2×C4⋊C4C23×C6718yes—
273C2×C4⋊C4C23×C6714yes—
274C2×C4⋊C4C23×C6712yes—
275C2×C4⋊C4C23×C6712yes—
276C2×C22⋊C4C23×C6714yes—
277C2×C22⋊C4C23×C6714yes—
278C2×C22⋊C4C23×C6714yes—
279C2×C42Q8×C6718yescomplemented
280C2×C42C23×C6714yes—
281C2×C42C23×C6714yes—
282C2×C42C23×C6714yes—
283C2×C42C23×C6714yes—
284C2×C42C23×C6714yes—
285C2×C42C23×C6714yes—
286C2×C42C23×C6714yes—
287C2×C42C23×C6712yes—
288C2×C42C23×C6711yescharacteristic
289C2×C42C23×C6711yescharacteristic
290#95C23×C6714yes—
291#95C23×C6714yes—
292#95C23×C6714yes—
293#95C23×C6714yes—
294#95C23×C6718yes—
295#95C23×C6714yes—
296#95C23×C6718yes—
297#95C23×C6714yes—
298#95C23×C6718yes—
299C24C24×C6711yescharacteristic
300C2×Q8—216no—
301C2×Q8—44no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4C2×Q8×C6718yes—
307C22×C4C2×Q8×C6714yes—
308C22×C4C24×C6714yes—
309C22×C4C24×C6714yes—
310C22×C4C24×C6714yes—
311C22×C4C24×C6714yes—
312C22×C4C24×C6712yes—
313C22×C4C24×C6711yescharacteristic
314C22×C4C24×C6714yes—
315C22×C4C24×C6714yes—
316C22×C4C24×C6712yes—
317C22×C4C24×C6711yescharacteristic
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—216no—
321C4⋊C4—216no—
322C4⋊C4—28no—
323C4⋊C4—216no—
324C4⋊C4—28no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—48no—
328C4⋊C4—44no—
329C4⋊C4#41×C6718yes—
330C4⋊C4#41×C6718yes—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C42—28no—
335C42—28no—
336C42—28no—
337C42—28no—
338C42—28no—
339C42—28no—
340C42—28no—
341C42C2×Q8×C6714yes—
342C42C2×Q8×C6714yes—
343C42C2×Q8×C6718yes—
344C23—22no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23C22×Q8×C6712yes—
349C23C25×C6711yescharacteristic
350Q8—416no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—21no—
359C2×C4—216no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4C22×Q8×C6714yes—
372C2×C4C22×Q8×C6712yes—
373C2×C4C22×Q8×C6714yes—
374C2×C4C22×Q8×C6712yes—
375C2×C4C2×#41×C6718yes—
376C2×C4C2×#41×C6718yes—
377C2×C4C2×#41×C6714yes—
378C2×C4C2×#41×C6714yes—
379C2×C4H(5,𝔽2)×C6712yes—
380C2×C42−1+4×C6712yes—
381C22—24no—
382C22—22no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22C23×Q8×C6711yescharacteristic
387C22C22×#41×C6712yes—
388C22C22×#41×C6712yes—
389C22C2×H(5,𝔽2)×C6711yescharacteristic
390C22C2×2−1+4×C6711yescharacteristic
391C4—24no—
392C4—22no—
393C4—28no—
394C4—28no—
395C4—44no—
396C4—44no—
397C4—44no—
398C4—44no—
399C4#533×C6718yes—
400C4#544×C6714yes—
401C2—42no—
402C2C2×#533×C6712yes—
403C2C2×#544×C6711yescharacteristic
404C2C67×#348111yescharacteristic
405C2C67×#348212yes—
406C2C67×#348311yescharacteristic
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
prime radicalC24×C67C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2612C2semidirect
2. C67×#2571C2semidirect
3. C67×#2552C2semidirect
4. C67×#2551C2semidirect
8. C4×C4⋊Q8×C67C2semidirect
9. C4×#125×C67C2semidirect
75. C2×C42×C67Q8semidirect
197. #34978C67direct (N × H)
204. #2612C2×C67semidirect
205. #2571C2×C67semidirect
206. #2552C2×C67semidirect
207. #2551C2×C67semidirect
211. C4×C4⋊Q8C2×C67semidirect
212. C4×#125C2×C67semidirect
215. C67#34978direct (N × H)
279. C2×C42Q8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#26122
C67×#25712
C67×#25522
C67×#25512
C67×#24412
C67×#24392
C67×#23962
C4×C4⋊Q8×C672
C4×#125×C672
C67×#22852
C4×#117×C672
#3497867