Finite Group Databasebeta

← C67×#33619 #30712870

Subgroups — C67×#33619 #30712870

Order 17152.

636 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#510×C67C211yesmaximal, characteristic, complemented
2C67×#3407C211yesmaximal, characteristic, complemented
3C67×#3067C211yesmaximal, characteristic
4C2×#473×C67C211yesmaximal, characteristic, complemented
5C67×#2223C214yesmaximal, complemented
6C67×#2222C214yesmaximal, complemented
7C67×#2161C211yesmaximal, characteristic, complemented
8C67×#2148C211yesmaximal, characteristic, complemented
9C2×#362×C67C211yesmaximal, characteristic, complemented
10#575×C67—21no—
11C2×#137×C67—21no—
12C2×#135×C67C2211yescharacteristic
13C22×Q16×C67C2211yescharacteristic
14C2×#131×C67C2211yescharacteristic
15#510×C67—22no—
16#510×C67—22no—
17C2×Q32×C67C2212yescomplemented
18C2×SD32×C67C2212yescomplemented
19#504×C67—22no—
20#504×C67C2214yes—
21C2×M32×C67—22no—
22C2×M32×C67C2211yescharacteristic
23C2×M32×C67C2211yescharacteristic
24#473×C67—22no—
25#473×C67C2214yes—
26#473×C67C2212yes—
27#472×C67C2212yes—
28#471×C67—22no—
29C2×#108×C67C2211yescharacteristic
30C2×#108×C67C2211yescharacteristic
31#413×C67—21no—
32C2×#101×C67—21no—
33#368×C67C2214yescomplemented
34#362×C67C2214yescomplemented
35#362×C67C2212yescomplemented
36#361×C67C2212yescomplemented
37C2×#41×C67—21no—
38C2×#41×C67D411yescharacteristic
39C22×Q8×C67—21no—
40#137×C67—22no—
41#137×C67—24no—
42#137×C67—42no—
43#137×C67—41no—
44AGL(1,ℤ/8)×C67—22no—
45AGL(1,ℤ/8)×C67—41no—
46#135×C67—21no—
47#135×C67—21no—
48#135×C67—21no—
49#135×C67—21no—
50#135×C67—42no—
51C2×Q16×C67—21no—
52C2×Q16×C67—21no—
53C2×Q16×C67—21no—
54C2×Q16×C67—21no—
55C2×Q16×C67—21no—
56C2×Q16×C67—41no—
57C2×Q16×C67C2312yes—
58C2×Q16×C67C2311yescharacteristic
59C2×SD16×C67—21no—
60C2×SD16×C67—41no—
61C2×D8×C67C2311yescharacteristic
62#131×C67—21no—
63#131×C67—21no—
64#131×C67D412yes—
65#131×C67C2312yes—
66C2×M16×C67—21no—
67C2×M16×C67—21no—
68C2×M16×C67—21no—
69C2×M16×C67D411yescharacteristic
70C22×C8×C67—21no—
71C22×C8×C67C2311yescharacteristic
72Q32×C67—24no—
73SD32×C67—24no—
74M32×C67—21no—
75M32×C67—24no—
76M32×C67D412yes—
77M32×C67D412yes—
78M32×C67C2312yes—
79C2×C16×C67—21no—
80C2×C16×C67—24no—
81C2×C16×C67C2312yes—
82#108×C67—21no—
83#108×C67—21no—
84#108×C67C2312yes—
85#108×C67C2312yes—
86#101×C67—22no—
87#101×C67—24no—
88#100×C67—22no—
89#41×C67—21no—
90#41×C67—21no—
91#41×C67—41no—
92#41×C67—41no—
93#41×C67—41no—
94#41×C67—41no—
95#41×C67D812yes—
96#41×C67C2×D412yes—
97C2×Q8×C67—21no—
98C2×Q8×C67—22no—
99C2×Q8×C67—41no—
100C2×Q8×C67—41no—
101C2×Q8×C67—41no—
102C2×Q8×C67—41no—
103C2×Q8×C67—41no—
104C2×Q8×C67D811yescharacteristic
105C2×D4×C67—21no—
106C2×D4×C67—21no—
107C2×D4×C67—41no—
108C2×D4×C67D811yescharacteristic
109C22×C4×C67—21no—
110C22×C4×C67—41no—
111C22×C4×C67—41no—
112C22×C4×C67C2×D411yescharacteristic
113Q16×C67—22no—
114Q16×C67—21no—
115Q16×C67—22no—
116Q16×C67—21no—
117Q16×C67—44no—
118Q16×C67—42no—
119Q16×C67—41no—
120Q16×C67—41no—
121SD16×C67—42no—
122SD16×C67—44no—
123SD16×C67—42no—
124SD16×C67—41no—
125SD16×C67—41no—
126D8×C67—21no—
127D8×C67—21no—
128D8×C67—42no—
129M16×C67—21no—
130M16×C67—22no—
131M16×C67—22no—
132M16×C67—22no—
133M16×C67—22no—
134M16×C67—41no—
135M16×C67—41no—
136M16×C67C2×D412yes—
137C2×C8×C67—21no—
138C2×C8×C67—22no—
139C2×C8×C67—22no—
140C2×C8×C67—41no—
141C2×C8×C67—41no—
142C2×C8×C67C2×D411yescharacteristic
143C2×C8×C67C2×D411yescharacteristic
144C2×C8×C67C2×D411yescharacteristic
145C2×C8×C67#4111yescharacteristic
146C2×C8×C67#4111yescharacteristic
147C2×C8×C67C2411yescharacteristic
148C16×C67—22no—
149C16×C67C2×D414yes—
150C23×C67—21no—
151C23×C67—41no—
152C23×C67#13511yescharacteristic
153Q8×C67—21no—
154Q8×C67—41no—
155Q8×C67—42no—
156Q8×C67—42no—
157Q8×C67—41no—
158Q8×C67—81no—
159Q8×C67—81no—
160Q8×C67C2×D812yes—
161D4×C67—22no—
162D4×C67—22no—
163D4×C67—21no—
164D4×C67—41no—
165D4×C67—41no—
166D4×C67—81no—
167D4×C67—81no—
168D4×C67C2×D812yes—
169C2×C4×C67—21no—
170C2×C4×C67—22no—
171C2×C4×C67—22no—
172C2×C4×C67—41no—
173C2×C4×C67—42no—
174C2×C4×C67—81no—
175C2×C4×C67—81no—
176C2×C4×C67—81no—
177C2×C4×C67—81no—
178C2×C4×C67#12111yescharacteristic
179C2×C4×C67#12111yescharacteristic
180C2×C4×C67C2×D811yescharacteristic
181C2×C4×C67C2×D811yescharacteristic
182C2×C4×C67#13511yescharacteristic
183C2×C4×C67C22×D411yescharacteristic
184C8×C67—22no—
185C8×C67—42no—
186C8×C67—42no—
187C8×C67#12111yescharacteristic
188C8×C67#12111yescharacteristic
189C8×C67C22×D411yescharacteristic
190C8×C67C2×#4111yescharacteristic
191C22×C67—22no—
192C22×C67—41no—
193C22×C67—81no—
194C22×C67—81no—
195C22×C67—161no—
196C22×C67#46611yescharacteristic
197C22×C67#46611yescharacteristic
198C22×C67C2×#13511yescharacteristic
199C4×C67—22no—
200C4×C67—81no—
201C4×C67—82no—
202C4×C67#46611yescharacteristic
203C4×C67#46611yescharacteristic
204C4×C67C2×#12111yescharacteristic
205C4×C67C22×D811yescharacteristic
206#33619C6711yesmaximal, characteristic, direct product factor, Sylow
207C2×C67—21no—
208C2×C67—21no—
209C2×C67—21no—
210C2×C67—42no—
211C2×C67—161no—
212C2×C67C2×#46611yescharacteristic
213C2×#510C2×C6711yescharacteristic, complemented
214#3407C2×C6711yescharacteristic, complemented
215#3067C2×C6711yescharacteristic
216C2×#473C2×C6711yescharacteristic, complemented
217#2223C2×C6714yescomplemented
218#2222C2×C6714yescomplemented
219#2161C2×C6711yescharacteristic, complemented
220#2148C2×C6711yescharacteristic, complemented
221C2×#362C2×C6711yescharacteristic, complemented
222C67#3361911yescharacteristic, direct product factor, Sylow
223#575—21no—
224C2×#137—21no—
225C2×#135C22×C6711yescharacteristic
226C22×Q16C22×C6711yescharacteristic
227C2×#131C22×C6711yescharacteristic
228#510—22no—
229#510—22no—
230C2×Q32C22×C6712yescomplemented
231C2×SD32C22×C6712yescomplemented
232#504—22no—
233#504C22×C6714yes—
234C2×M32—22no—
235C2×M32C22×C6711yescharacteristic
236C2×M32C22×C6711yescharacteristic
237#473—22no—
238#473C22×C6712yes—
239#473C22×C6714yes—
240#472C22×C6712yes—
241#471—22no—
242C2×#108C22×C6711yescharacteristic
243C2×#108C22×C6711yescharacteristic
244#413—21no—
245C2×#101—21no—
246#368C22×C6714yescomplemented
247#362C22×C6714yescomplemented
248#362C22×C6712yescomplemented
249#361C22×C6712yescomplemented
250C2×#41—21no—
251C2×#41D4×C6711yescharacteristic
252C22×Q8—21no—
253#137—24no—
254#137—22no—
255#137—42no—
256#137—41no—
257AGL(1,ℤ/8)—22no—
258AGL(1,ℤ/8)—41no—
259#135—21no—
260#135—21no—
261#135—21no—
262#135—21no—
263#135—42no—
264C2×Q16—21no—
265C2×Q16—21no—
266C2×Q16—21no—
267C2×Q16—21no—
268C2×Q16—21no—
269C2×Q16—41no—
270C2×Q16C23×C6712yes—
271C2×Q16C23×C6711yescharacteristic
272C2×SD16—21no—
273C2×SD16—41no—
274C2×D8C23×C6711yescharacteristic
275#131—21no—
276#131—21no—
277#131D4×C6712yes—
278#131C23×C6712yes—
279C2×M16—21no—
280C2×M16—21no—
281C2×M16—21no—
282C2×M16D4×C6711yescharacteristic
283C22×C8—21no—
284C22×C8C23×C6711yescharacteristic
285Q32—24no—
286SD32—24no—
287M32—21no—
288M32—24no—
289M32D4×C6712yes—
290M32D4×C6712yes—
291M32C23×C6712yes—
292C2×C16—21no—
293C2×C16—24no—
294C2×C16C23×C6712yes—
295#108—21no—
296#108—21no—
297#108C23×C6712yes—
298#108C23×C6712yes—
299#101—24no—
300#101—22no—
301#100—22no—
302#41—21no—
303#41—21no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41D8×C6712yes—
309#41C2×D4×C6712yes—
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8D8×C6711yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4D8×C6711yescharacteristic
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4C2×D4×C6711yescharacteristic
326Q16—22no—
327Q16—21no—
328Q16—22no—
329Q16—21no—
330Q16—44no—
331Q16—42no—
332Q16—41no—
333Q16—41no—
334SD16—42no—
335SD16—44no—
336SD16—42no—
337SD16—41no—
338SD16—41no—
339D8—21no—
340D8—21no—
341D8—42no—
342M16—21no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—41no—
348M16—41no—
349M16C2×D4×C6712yes—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8C2×D4×C6711yescharacteristic
356C2×C8C2×D4×C6711yescharacteristic
357C2×C8C2×D4×C6711yescharacteristic
358C2×C8#41×C6711yescharacteristic
359C2×C8#41×C6711yescharacteristic
360C2×C8C24×C6711yescharacteristic
361C16—22no—
362C16C2×D4×C6714yes—
363C23—21no—
364C23—41no—
365C23#135×C6711yescharacteristic
366Q8—21no—
367Q8—41no—
368Q8—42no—
369Q8—42no—
370Q8—41no—
371Q8—81no—
372Q8—81no—
373Q8C2×D8×C6712yes—
374D4—22no—
375D4—22no—
376D4—21no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381D4C2×D8×C6712yes—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4#121×C6711yescharacteristic
392C2×C4#121×C6711yescharacteristic
393C2×C4C2×D8×C6711yescharacteristic
394C2×C4C2×D8×C6711yescharacteristic
395C2×C4#135×C6711yescharacteristic
396C2×C4C22×D4×C6711yescharacteristic
397C8—22no—
398C8—42no—
399C8—42no—
400C8#121×C6711yescharacteristic
401C8#121×C6711yescharacteristic
402C8C22×D4×C6711yescharacteristic
403C8C2×#41×C6711yescharacteristic
404C22—22no—
405C22—41no—
406C22—81no—
407C22—81no—
408C22—161no—
409C22#466×C6711yescharacteristic
410C22#466×C6711yescharacteristic
411C22C2×#135×C6711yescharacteristic
412C4—22no—
413C4—81no—
414C4—82no—
415C4#466×C6711yescharacteristic
416C4#466×C6711yescharacteristic
417C4C2×#121×C6711yescharacteristic
418C4C22×D8×C6711yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—161no—
424C2C2×#466×C6711yescharacteristic
subgroupG/H
center, socleC2×C67C2×#466
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC67×#3067C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C67 ◁ C22×C4×C67 ◁ C2×#131×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#510×C67C2semidirect
2. C67×#3407C2semidirect
4. C2×#473×C67C2semidirect
5. C67×#2223C2semidirect
6. C67×#2222C2semidirect
7. C67×#2161C2semidirect
8. C67×#2148C2semidirect
9. C2×#362×C67C2semidirect
17. C2×Q32×C67C22semidirect
18. C2×SD32×C67C22semidirect
33. #368×C67C22semidirect
34. #362×C67C22semidirect
35. #362×C67C22semidirect
36. #361×C67C22semidirect
206. #33619C67direct (N × H)
213. C2×#510C2×C67semidirect
214. #3407C2×C67semidirect
216. C2×#473C2×C67semidirect
217. #2223C2×C67semidirect
218. #2222C2×C67semidirect
219. #2161C2×C67semidirect
220. #2148C2×C67semidirect
221. C2×#362C2×C67semidirect
222. C67#33619direct (N × H)
230. C2×Q32C22×C67semidirect
231. C2×SD32C22×C67semidirect
246. #368C22×C67semidirect
247. #362C22×C67semidirect
248. #362C22×C67semidirect
249. #361C22×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#510×C672
C67×#34072
C67×#30672
C2×#473×C672
C67×#22232
C67×#22222
C67×#21612
C67×#21482
C2×#362×C672
#3361967