Finite Group Databasebeta

← C67×#19401 #30698652

Subgroups — C67×#19401 #30698652

Order 17152.

632 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
1C67×#3207C211yesmaximal, characteristic
2C67×#3112C211yesmaximal, characteristic, complemented
3C67×#3103C211yesmaximal, characteristic, complemented
4C67×#2982C211yesmaximal, characteristic, complemented
5C67×#1684C212yesmaximal, complemented
6C67×#1678C212yesmaximal, complemented
7C67×#1668C212yesmaximal, complemented
8C67×#1666C212yesmaximal, complemented
9C67×#1572C211yesmaximal, characteristic, complemented
10C67×#1553C211yesmaximal, characteristic, complemented
11C67×#1543C211yesmaximal, characteristic, complemented
12#532×C67C2211yescharacteristic
13#532×C67C2211yescharacteristic
14C2×C4×Q8×C67C2211yescharacteristic
15#493×C67C2211yescharacteristic
16#492×C67C2211yescharacteristic
17#487×C67C2211yescharacteristic
18#486×C67C2211yescharacteristic
19#481×C67—21no—
20#480×C67—21no—
21#477×C67C2212yes—
22#475×C67C2212yes—
23#463×C67—22no—
24#460×C67C2212yes—
25#460×C67C2212yes—
26#449×C67—22no—
27#448×C67—21no—
28#448×C67—21no—
29C8×Q8×C67—22no—
30C8×Q8×C67C2212yes—
31C8×Q8×C67C2212yes—
32#440×C67—22no—
33C4×SD16×C67—22no—
34#432×C67C2211yescharacteristic
35#431×C67—22no—
36#431×C67C2211yescharacteristic
37#431×C67C2211yescharacteristic
38#431×C67C2211yescharacteristic
39#335×C67C2212yes—
40#334×C67C2212yes—
41#333×C67C2212yes—
42#332×C67C2212yes—
43#331×C67C2212yes—
44#331×C67C2212yes—
45C2×#41×C67—21no—
46C2×#41×C67—21no—
47C22×Q8×C67D411yescharacteristic
48C2×SD16×C67—22no—
49C2×SD16×C67—22no—
50C2×SD16×C67—22no—
51C2×D8×C67—22no—
52C4⋊Q8×C67C2311yescharacteristic
53C4⋊Q8×C67C2311yescharacteristic
54#127×C67C2311yescharacteristic
55#127×C67C2311yescharacteristic
56#124×C67—21no—
57#124×C67—21no—
58#121×C67—21no—
59#121×C67—21no—
60#121×C67—21no—
61#121×C67—21no—
62C4×Q8×C67—21no—
63C4×Q8×C67—21no—
64C4×Q8×C67D412yes—
65C4×Q8×C67C2312yes—
66C4×D4×C67—21no—
67C4×D4×C67—21no—
68C4×D4×C67—21no—
69C4×D4×C67—21no—
70C2×C4⋊C4×C67—21no—
71C2×C4⋊C4×C67D411yescharacteristic
72C2×C42×C67—21no—
73C2×C42×C67C2311yescharacteristic
74#107×C67—22no—
75#106×C67—22no—
76C4⋊C8×C67—22no—
77C4⋊C8×C67—22no—
78C4⋊C8×C67D412yes—
79C4⋊C8×C67C2312yes—
80C4⋊C8×C67C2312yes—
81C4⋊C8×C67C2312yes—
82#103×C67—22no—
83#103×C67—22no—
84#103×C67—22no—
85#102×C67—22no—
86#102×C67—22no—
87#102×C67—22no—
88C22⋊C8×C67—21no—
89C22⋊C8×C67—21no—
90C22⋊C8×C67—21no—
91C22⋊C8×C67D412yes—
92#97×C67—21no—
93C4×C8×C67—22no—
94C4×C8×C67—22no—
95C4×C8×C67—21no—
96C4×C8×C67—21no—
97C4×C8×C67C2311yescharacteristic
98C4×C8×C67C2311yescharacteristic
99#41×C67—82no—
100#41×C67—82no—
101C2×Q8×C67—24no—
102C2×Q8×C67—21no—
103C2×Q8×C67—21no—
104C2×Q8×C67—21no—
105C2×Q8×C67C2×D412yes—
106C2×D4×C67—21no—
107C2×D4×C67—21no—
108C2×D4×C67—21no—
109C2×D4×C67—21no—
110C2×D4×C67—21no—
111C2×D4×C67—21no—
112C2×D4×C67—41no—
113C2×D4×C67—41no—
114C22×C4×C67—21no—
115C22×C4×C67—21no—
116C22×C4×C67—41no—
117C22×C4×C67—41no—
118C22×C4×C67C2×D411yescharacteristic
119C22×C4×C67C2×D411yescharacteristic
120C22×C4×C67#4111yescharacteristic
121SD16×C67—44no—
122SD16×C67—44no—
123SD16×C67—82no—
124D8×C67—82no—
125C2×C8×C67—22no—
126C2×C8×C67—22no—
127C2×C8×C67—22no—
128C2×C8×C67C2×D412yes—
129C2×C8×C67C2×D412yes—
130C4⋊C4×C67—22no—
131C4⋊C4×C67—21no—
132C4⋊C4×C67—22no—
133C4⋊C4×C67—21no—
134C4⋊C4×C67—21no—
135C4⋊C4×C67—21no—
136C4⋊C4×C67—21no—
137C4⋊C4×C67C2×D412yes—
138C22⋊C4×C67—41no—
139C22⋊C4×C67—41no—
140C22⋊C4×C67—41no—
141C22⋊C4×C67—41no—
142C42×C67—22no—
143C42×C67—22no—
144C42×C67C2×D411yescharacteristic
145C42×C67C2×D411yescharacteristic
146C42×C67#4111yescharacteristic
147C42×C67C2411yescharacteristic
148C23×C67—41no—
149C23×C67—41no—
150C23×C67#12111yescharacteristic
151Q8×C67—24no—
152Q8×C67—22no—
153Q8×C67—42no—
154Q8×C67—81no—
155Q8×C67#13514yes—
156D4×C67—42no—
157D4×C67—42no—
158D4×C67—42no—
159D4×C67—42no—
160D4×C67—81no—
161D4×C67—81no—
162D4×C67—82no—
163D4×C67—82no—
164C2×C4×C67—24no—
165C2×C4×C67—22no—
166C2×C4×C67—22no—
167C2×C4×C67—22no—
168C2×C4×C67—22no—
169C2×C4×C67—41no—
170C2×C4×C67—41no—
171C2×C4×C67—41no—
172C2×C4×C67—82no—
173C2×C4×C67—82no—
174C2×C4×C67—81no—
175C2×C4×C67#12111yescharacteristic
176C2×C4×C67#12111yescharacteristic
177C2×C4×C67#12111yescharacteristic
178C2×C4×C67C22×D411yescharacteristic
179C2×C4×C67C22×D411yescharacteristic
180C2×C4×C67C2×#4111yescharacteristic
181C8×C67—22no—
182C8×C67—22no—
183C8×C67—42no—
184C8×C67—42no—
185C8×C67—42no—
186C22×C67—21no—
187C22×C67—21no—
188C22×C67—21no—
189C22×C67—81no—
190C22×C67—81no—
191C22×C67—81no—
192C22×C67—81no—
193C22×C67—81no—
194C22×C67—81no—
195C22×C67C2×#12111yescharacteristic
196C4×C67—21no—
197C4×C67—24no—
198C4×C67—21no—
199C4×C67—21no—
200C4×C67—21no—
201C4×C67—42no—
202C4×C67—81no—
203C4×C67—81no—
204C4×C67C2×#13512yes—
205C4×C67#57712yes—
206#19401C6711yesmaximal, characteristic, direct product factor, Sylow
207C2×C67—41no—
208C2×C67—161no—
209C2×C67—161no—
210C2×C67#304311yescharacteristic
211C2×C67#305011yescharacteristic
212C2×C67#306811yescharacteristic
213#3207C2×C6711yescharacteristic
214#3112C2×C6711yescharacteristic, complemented
215#3103C2×C6711yescharacteristic, complemented
216#2982C2×C6711yescharacteristic, complemented
217#1684C2×C6712yescomplemented
218#1678C2×C6712yescomplemented
219#1668C2×C6712yescomplemented
220#1666C2×C6712yescomplemented
221#1572C2×C6711yescharacteristic, complemented
222#1553C2×C6711yescharacteristic, complemented
223#1543C2×C6711yescharacteristic, complemented
224C67#1940111yescharacteristic, direct product factor, Sylow
225#532C22×C6711yescharacteristic
226#532C22×C6711yescharacteristic
227C2×C4×Q8C22×C6711yescharacteristic
228#493C22×C6711yescharacteristic
229#492C22×C6711yescharacteristic
230#487C22×C6711yescharacteristic
231#486C22×C6711yescharacteristic
232#481—21no—
233#480—21no—
234#477C22×C6712yes—
235#475C22×C6712yes—
236#463—22no—
237#460C22×C6712yes—
238#460C22×C6712yes—
239#449—22no—
240#448—21no—
241#448—21no—
242C8×Q8—22no—
243C8×Q8C22×C6712yes—
244C8×Q8C22×C6712yes—
245#440—22no—
246C4×SD16—22no—
247#432C22×C6711yescharacteristic
248#431—22no—
249#431C22×C6711yescharacteristic
250#431C22×C6711yescharacteristic
251#431C22×C6711yescharacteristic
252#335C22×C6712yes—
253#334C22×C6712yes—
254#333C22×C6712yes—
255#332C22×C6712yes—
256#331C22×C6712yes—
257#331C22×C6712yes—
258C2×#41—21no—
259C2×#41—21no—
260C22×Q8D4×C6711yescharacteristic
261C2×SD16—22no—
262C2×SD16—22no—
263C2×SD16—22no—
264C2×D8—22no—
265C4⋊Q8C23×C6711yescharacteristic
266C4⋊Q8C23×C6711yescharacteristic
267#127C23×C6711yescharacteristic
268#127C23×C6711yescharacteristic
269#124—21no—
270#124—21no—
271#121—21no—
272#121—21no—
273#121—21no—
274#121—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8D4×C6712yes—
278C4×Q8C23×C6712yes—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4D4×C6711yescharacteristic
285C2×C42—21no—
286C2×C42C23×C6711yescharacteristic
287#107—22no—
288#106—22no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8D4×C6712yes—
292C4⋊C8C23×C6712yes—
293C4⋊C8C23×C6712yes—
294C4⋊C8C23×C6712yes—
295#103—22no—
296#103—22no—
297#103—22no—
298#102—22no—
299#102—22no—
300#102—22no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8D4×C6712yes—
305#97—21no—
306C4×C8—22no—
307C4×C8—22no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8C23×C6711yescharacteristic
311C4×C8C23×C6711yescharacteristic
312#41—82no—
313#41—82no—
314C2×Q8—24no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8C2×D4×C6712yes—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—41no—
326C2×D4—41no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×D4×C6711yescharacteristic
332C22×C4C2×D4×C6711yescharacteristic
333C22×C4#41×C6711yescharacteristic
334SD16—44no—
335SD16—44no—
336SD16—82no—
337D8—82no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8C2×D4×C6712yes—
342C2×C8C2×D4×C6712yes—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4C2×D4×C6712yes—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—22no—
356C42—22no—
357C42C2×D4×C6711yescharacteristic
358C42C2×D4×C6711yescharacteristic
359C42#41×C6711yescharacteristic
360C42C24×C6711yescharacteristic
361C23—41no—
362C23—41no—
363C23#121×C6711yescharacteristic
364Q8—24no—
365Q8—22no—
366Q8—42no—
367Q8—81no—
368Q8#135×C6714yes—
369D4—42no—
370D4—42no—
371D4—42no—
372D4—42no—
373D4—81no—
374D4—81no—
375D4—82no—
376D4—82no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4#121×C6711yescharacteristic
389C2×C4#121×C6711yescharacteristic
390C2×C4#121×C6711yescharacteristic
391C2×C4C22×D4×C6711yescharacteristic
392C2×C4C22×D4×C6711yescharacteristic
393C2×C4C2×#41×C6711yescharacteristic
394C8—22no—
395C8—22no—
396C8—42no—
397C8—42no—
398C8—42no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22C2×#121×C6711yescharacteristic
409C4—21no—
410C4—24no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—81no—
416C4—81no—
417C4C2×#135×C6712yes—
418C4#577×C6712yes—
419C2—41no—
420C2—161no—
421C2—161no—
422C2C67×#304311yescharacteristic
423C2C67×#305011yescharacteristic
424C2C67×#306811yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3207C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#3112C2semidirect
3. C67×#3103C2semidirect
4. C67×#2982C2semidirect
5. C67×#1684C2semidirect
6. C67×#1678C2semidirect
7. C67×#1668C2semidirect
8. C67×#1666C2semidirect
9. C67×#1572C2semidirect
10. C67×#1553C2semidirect
11. C67×#1543C2semidirect
206. #19401C67direct (N × H)
214. #3112C2×C67semidirect
215. #3103C2×C67semidirect
216. #2982C2×C67semidirect
217. #1684C2×C67semidirect
218. #1678C2×C67semidirect
219. #1668C2×C67semidirect
220. #1666C2×C67semidirect
221. #1572C2×C67semidirect
222. #1553C2×C67semidirect
223. #1543C2×C67semidirect
224. C67#19401direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#32072
C67×#31122
C67×#31032
C67×#29822
C67×#16842
C67×#16782
C67×#16682
C67×#16662
C67×#15722
C67×#15532
C67×#15432
#1940167