Finite Group Databasebeta

← C67×#34943 #30714194

Subgroups — C67×#34943 #30714194

Order 17152.

2036 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2572C214yesmaximal, complemented
2C67×#2571C211yesmaximal, characteristic, complemented
3C67×#2563C212yesmaximal, complemented
4C67×#2537C211yesmaximal, characteristic
5C67×#2495C214yesmaximal, complemented
6C67×#2420C214yesmaximal, complemented
7C67×#2398C214yesmaximal, complemented
8C4×C4⋊Q8×C67C211yesmaximal, characteristic, complemented
9C4×#121×C67C214yesmaximal, complemented
10C67×#2283C214yesmaximal, complemented
11C42×D4×C67C212yesmaximal, complemented
12C2×C4⋊Q8×C67C2211yescharacteristic
13C2×#121×C67C2214yes—
14C2×C4×Q8×C67C2212yes—
15C2×C4×D4×C67C2212yes—
16C2×C4×D4×C67C2212yes—
17C2×C4×D4×C67C2212yes—
18C2×C4×D4×C67C2214yes—
19C22×C42×C67C2214yes—
20#400×C67C2214yes—
21#400×C67C2214yes—
22#397×C67C2218yes—
23#394×C67C2214yes—
24#394×C67C2214yes—
25#393×C67C2218yes—
26#391×C67C2212yes—
27#390×C67C2214yes—
28#389×C67C2214yes—
29#389×C67C2214yes—
30#388×C67C2218yes—
31#386×C67C2218yes—
32#384×C67C2211yescharacteristic
33#383×C67C2218yes—
34#382×C67C2214yes—
35#382×C67C2214yes—
36#381×C67C2218yes—
37#380×C67C2214yes—
38#380×C67C2214yes—
39#380×C67C2214yes—
40C4×C4⋊C4×C67C2212yes—
41C4×C4⋊C4×C67C2212yes—
42C4×C4⋊C4×C67C2212yes—
43C4×C4⋊C4×C67C2214yes—
44C4×C22⋊C4×C67C2214yes—
45C4×C22⋊C4×C67C2214yes—
46C4×C22⋊C4×C67C2214yes—
47C4×C22⋊C4×C67C2218yes—
48#376×C67C2214yes—
49C43×C67C2211yescharacteristic
50C22×Q8×C67C2312yes—
51C22×D4×C67C2314yes—
52C22×D4×C67C2312yes—
53C23×C4×C67C2314yes—
54C23×C4×C67C2314yes—
55C23×C4×C67C2314yes—
56C4⋊Q8×C67—24no—
57#121×C67—216no—
58C4×Q8×C67—44no—
59C4×D4×C67—28no—
60C4×D4×C67—28no—
61C4×D4×C67—28no—
62C4×D4×C67—48no—
63C2×C4⋊C4×C67C2312yes—
64C2×C4⋊C4×C67C2312yes—
65C2×C4⋊C4×C67C2312yes—
66C2×C4⋊C4×C67C2314yes—
67C2×C4⋊C4×C67C2314yes—
68C2×C4⋊C4×C67C2314yes—
69C2×C4⋊C4×C67C2312yes—
70C2×C4⋊C4×C67C2314yes—
71C2×C22⋊C4×C67C2314yes—
72C2×C22⋊C4×C67C2314yes—
73C2×C22⋊C4×C67C2314yes—
74C2×C22⋊C4×C67C2318yes—
75C2×C22⋊C4×C67C2318yes—
76C2×C22⋊C4×C67C2318yes—
77C2×C42×C67—216no—
78C2×C42×C67C2314yes—
79C2×C42×C67C2314yes—
80C2×C42×C67C2312yes—
81C2×C42×C67C2314yes—
82C2×C42×C67C2314yes—
83C2×C42×C67C2311yescharacteristic
84C2×C42×C67C2311yescharacteristic
85C2×C42×C67C2311yescharacteristic
86C2×C42×C67C2312yes—
87#95×C67C2314yes—
88#95×C67C2318yes—
89#95×C67C2314yes—
90#95×C67C2314yes—
91#95×C67C2314yes—
92#95×C67C2314yes—
93#95×C67C2318yes—
94#95×C67C2318yes—
95#95×C67C2318yes—
96C24×C67C2414yes—
97C2×Q8×C67—24no—
98C2×Q8×C67—42no—
99C2×Q8×C67—42no—
100C2×D4×C67—216no—
101C2×D4×C67—24no—
102C2×D4×C67—24no—
103C2×D4×C67—44no—
104C2×D4×C67#4118yes—
105C22×C4×C67—28no—
106C22×C4×C67—28no—
107C22×C4×C67—28no—
108C22×C4×C67—28no—
109C22×C4×C67—28no—
110C22×C4×C67—28no—
111C22×C4×C67—28no—
112C22×C4×C67—28no—
113C22×C4×C67—44no—
114C22×C4×C67C2411yescharacteristic
115C22×C4×C67C2414yes—
116C22×C4×C67C2414yes—
117C22×C4×C67C2414yes—
118C22×C4×C67C2414yes—
119C22×C4×C67C2414yes—
120C22×C4×C67C2412yes—
121C22×C4×C67C2411yescharacteristic
122C22×C4×C67C2412yes—
123C22×C4×C67C2411yescharacteristic
124C4⋊C4×C67—24no—
125C4⋊C4×C67—24no—
126C4⋊C4×C67—24no—
127C4⋊C4×C67—28no—
128C4⋊C4×C67—28no—
129C4⋊C4×C67—44no—
130C4⋊C4×C67—42no—
131C4⋊C4×C67—44no—
132C22⋊C4×C67—28no—
133C22⋊C4×C67—216no—
134C22⋊C4×C67—28no—
135C22⋊C4×C67—28no—
136C22⋊C4×C67—48no—
137C22⋊C4×C67—48no—
138C42×C67—28no—
139C42×C67—28no—
140C42×C67—28no—
141C42×C67—24no—
142C42×C67—28no—
143C42×C67—22no—
144C42×C67—41no—
145C42×C67#4118yes—
146C42×C67#4114yes—
147C23×C67—24no—
148C23×C67—24no—
149C23×C67—24no—
150C23×C67—24no—
151C23×C67—24no—
152C23×C67—24no—
153C23×C67C2×#4118yes—
154C23×C67C2511yescharacteristic
155Q8×C67—48no—
156D4×C67—216no—
157D4×C67—416no—
158C2×C4×C67—22no—
159C2×C4×C67—21no—
160C2×C4×C67—22no—
161C2×C4×C67—21no—
162C2×C4×C67—24no—
163C2×C4×C67—24no—
164C2×C4×C67—24no—
165C2×C4×C67—22no—
166C2×C4×C67—24no—
167C2×C4×C67—24no—
168C2×C4×C67—24no—
169C2×C4×C67—24no—
170C2×C4×C67—28no—
171C2×C4×C67—24no—
172C2×C4×C67—24no—
173C2×C4×C67—21no—
174C2×C4×C67—24no—
175C2×C4×C67—24no—
176C2×C4×C67—24no—
177C2×C4×C67—24no—
178C2×C4×C67—416no—
179C2×C4×C67—416no—
180C2×C4×C67—416no—
181C2×C4×C67C2×#4114yes—
182C2×C4×C67C2×#4112yes—
183C2×C4×C67C2×#4114yes—
184C2×C4×C67C2×#4114yes—
185C2×C4×C67C2×#4114yes—
186C2×C4×C67C2×#4114yes—
187C2×C4×C67H(5,𝔽2)12yes—
188C22×C67—28no—
189C22×C67—28no—
190C22×C67—28no—
191C22×C67—44no—
192C22×C67—44no—
193C22×C67—44no—
194C22×C67—44no—
195C22×C67C22×#4111yescharacteristic
196C22×C67C22×#4112yes—
197C22×C67C22×#4112yes—
198C22×C67C2×H(5,𝔽2)11yescharacteristic
199C22×C67#58511yescharacteristic
200C4×C67—22no—
201C4×C67—24no—
202C4×C67—44no—
203C4×C67—44no—
204C4×C67—44no—
205C4×C67—44no—
206C4×C67—41no—
207C4×C67—44no—
208C4×C67#52918yes—
209C4×C67#55514yes—
210#34943C6711yesmaximal, characteristic, direct product factor, Sylow
211C2×C67—48no—
212C2×C67C2×#52912yes—
213C2×C67#346011yescharacteristic
214C2×C67C2×#55511yescharacteristic
215C2×C67#347512yes—
216C2×C67#348111yescharacteristic
217#2572C2×C6714yescomplemented
218#2571C2×C6711yescharacteristic, complemented
219#2563C2×C6712yescomplemented
220#2537C2×C6711yescharacteristic
221#2495C2×C6714yescomplemented
222#2420C2×C6714yescomplemented
223#2398C2×C6714yescomplemented
224C4×C4⋊Q8C2×C6711yescharacteristic, complemented
225C4×#121C2×C6714yescomplemented
226#2283C2×C6714yescomplemented
227C42×D4C2×C6712yescomplemented
228C67#3494311yescharacteristic, direct product factor, Sylow
229C2×C4⋊Q8C22×C6711yescharacteristic
230C2×#121C22×C6714yes—
231C2×C4×Q8C22×C6712yes—
232C2×C4×D4C22×C6712yes—
233C2×C4×D4C22×C6712yes—
234C2×C4×D4C22×C6712yes—
235C2×C4×D4C22×C6714yes—
236C22×C42C22×C6714yes—
237#400C22×C6714yes—
238#400C22×C6714yes—
239#397C22×C6718yes—
240#394C22×C6714yes—
241#394C22×C6714yes—
242#393C22×C6718yes—
243#391C22×C6712yes—
244#390C22×C6714yes—
245#389C22×C6714yes—
246#389C22×C6714yes—
247#388C22×C6718yes—
248#386C22×C6718yes—
249#384C22×C6711yescharacteristic
250#383C22×C6718yes—
251#382C22×C6714yes—
252#382C22×C6714yes—
253#381C22×C6718yes—
254#380C22×C6714yes—
255#380C22×C6714yes—
256#380C22×C6714yes—
257C4×C4⋊C4C22×C6714yes—
258C4×C4⋊C4C22×C6712yes—
259C4×C4⋊C4C22×C6712yes—
260C4×C4⋊C4C22×C6712yes—
261C4×C22⋊C4C22×C6714yes—
262C4×C22⋊C4C22×C6714yes—
263C4×C22⋊C4C22×C6718yes—
264C4×C22⋊C4C22×C6714yes—
265#376C22×C6714yes—
266C43C22×C6711yescharacteristic
267C22×Q8C23×C6712yes—
268C22×D4C23×C6714yes—
269C22×D4C23×C6712yes—
270C23×C4C23×C6714yes—
271C23×C4C23×C6714yes—
272C23×C4C23×C6714yes—
273C4⋊Q8—24no—
274#121—216no—
275C4×Q8—44no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—48no—
280C2×C4⋊C4C23×C6712yes—
281C2×C4⋊C4C23×C6712yes—
282C2×C4⋊C4C23×C6712yes—
283C2×C4⋊C4C23×C6714yes—
284C2×C4⋊C4C23×C6714yes—
285C2×C4⋊C4C23×C6714yes—
286C2×C4⋊C4C23×C6712yes—
287C2×C4⋊C4C23×C6714yes—
288C2×C22⋊C4C23×C6714yes—
289C2×C22⋊C4C23×C6714yes—
290C2×C22⋊C4C23×C6714yes—
291C2×C22⋊C4C23×C6718yes—
292C2×C22⋊C4C23×C6718yes—
293C2×C22⋊C4C23×C6718yes—
294C2×C42—216no—
295C2×C42C23×C6714yes—
296C2×C42C23×C6712yes—
297C2×C42C23×C6711yescharacteristic
298C2×C42C23×C6714yes—
299C2×C42C23×C6714yes—
300C2×C42C23×C6714yes—
301C2×C42C23×C6711yescharacteristic
302C2×C42C23×C6711yescharacteristic
303C2×C42C23×C6712yes—
304#95C23×C6714yes—
305#95C23×C6718yes—
306#95C23×C6714yes—
307#95C23×C6714yes—
308#95C23×C6714yes—
309#95C23×C6714yes—
310#95C23×C6718yes—
311#95C23×C6718yes—
312#95C23×C6718yes—
313C24C24×C6714yes—
314C2×Q8—24no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×D4—216no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—44no—
321C2×D4#41×C6718yes—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—44no—
331C22×C4C24×C6714yes—
332C22×C4C24×C6714yes—
333C22×C4C24×C6714yes—
334C22×C4C24×C6714yes—
335C22×C4C24×C6714yes—
336C22×C4C24×C6711yescharacteristic
337C22×C4C24×C6712yes—
338C22×C4C24×C6711yescharacteristic
339C22×C4C24×C6712yes—
340C22×C4C24×C6711yescharacteristic
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—28no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C22⋊C4—28no—
350C22⋊C4—216no—
351C22⋊C4—28no—
352C22⋊C4—28no—
353C22⋊C4—48no—
354C22⋊C4—48no—
355C42—28no—
356C42—28no—
357C42—28no—
358C42—24no—
359C42—28no—
360C42—22no—
361C42—41no—
362C42#41×C6718yes—
363C42#41×C6714yes—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23C2×#41×C6718yes—
371C23C25×C6711yescharacteristic
372Q8—48no—
373D4—216no—
374D4—416no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—28no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—416no—
396C2×C4—416no—
397C2×C4—416no—
398C2×C4C2×#41×C6714yes—
399C2×C4C2×#41×C6712yes—
400C2×C4C2×#41×C6714yes—
401C2×C4C2×#41×C6714yes—
402C2×C4C2×#41×C6714yes—
403C2×C4C2×#41×C6714yes—
404C2×C4H(5,𝔽2)×C6712yes—
405C22—28no—
406C22—28no—
407C22—28no—
408C22—44no—
409C22—44no—
410C22—44no—
411C22—44no—
412C22C22×#41×C6711yescharacteristic
413C22C22×#41×C6712yes—
414C22C22×#41×C6712yes—
415C22C2×H(5,𝔽2)×C6711yescharacteristic
416C22#585×C6711yescharacteristic
417C4—22no—
418C4—24no—
419C4—44no—
420C4—44no—
421C4—44no—
422C4—44no—
423C4—44no—
424C4—41no—
425C4#529×C6718yes—
426C4#555×C6714yes—
427C2—48no—
428C2C2×#529×C6712yes—
429C2C67×#346011yescharacteristic
430C2C2×#555×C6711yescharacteristic
431C2C67×#347512yes—
432C2C67×#348111yescharacteristic
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
prime radicalC67×#2537C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2572C2semidirect
2. C67×#2571C2semidirect
3. C67×#2563C2semidirect
5. C67×#2495C2semidirect
6. C67×#2420C2semidirect
7. C67×#2398C2semidirect
8. C4×C4⋊Q8×C67C2semidirect
9. C4×#121×C67C2semidirect
10. C67×#2283C2semidirect
11. C42×D4×C67C2semidirect
210. #34943C67direct (N × H)
217. #2572C2×C67semidirect
218. #2571C2×C67semidirect
219. #2563C2×C67semidirect
221. #2495C2×C67semidirect
222. #2420C2×C67semidirect
223. #2398C2×C67semidirect
224. C4×C4⋊Q8C2×C67semidirect
225. C4×#121C2×C67semidirect
226. #2283C2×C67semidirect
227. C42×D4C2×C67semidirect
228. C67#34943direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#25722
C67×#25712
C67×#25632
C67×#25372
C67×#24952
C67×#24202
C67×#23982
C4×C4⋊Q8×C672
C4×#121×C672
C67×#22832
C42×D4×C672
#3494367