Finite Group Databasebeta

← C67×#9212 #30688463

Subgroups — C67×#9212 #30688463

Order 17152.

514 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×#3107C211yesmaximal, characteristic
2C4×#131×C67C211yesmaximal, characteristic
3C67×#1526C211yesmaximal, characteristic, complemented
4C67×#1493C211yesmaximal, characteristic, complemented
5C67×#1489C211yesmaximal, characteristic
6C67×#1299C211yesmaximal, characteristic, complemented
7C67×#1278C211yesmaximal, characteristic, complemented
8C2×#131×C67—21no—
9#533×C67C2211yescharacteristic
10C4×#41×C67C2211yescharacteristic
11#485×C67—21no—
12#475×C67C411yescharacteristic
13#474×C67C411yescharacteristic
14#462×C67C411yescharacteristic
15#461×C67C411yescharacteristic
16#452×C67—21no—
17C4×Q16×C67—21no—
18C4×SD16×C67—21no—
19#431×C67C2211yescharacteristic
20#423×C67C2211yescharacteristic
21#405×C67—22no—
22#405×C67—21no—
23C4×M16×C67—21no—
24C4×M16×C67C2211yescharacteristic
25C4×M16×C67C2211yescharacteristic
26C2×C4×C8×C67—21no—
27C2×C4×C8×C67C2211yescharacteristic
28#330×C67—22no—
29Q8⋊C8×C67—21no—
30Q8⋊C8×C67—21no—
31Q8⋊C8×C67—21no—
32D4⋊C8×C67—21no—
33C2×#41×C67D411yescharacteristic
34C2×Q16×C67—21no—
35C2×SD16×C67—21no—
36#131×C67—24no—
37#131×C67—24no—
38C2×M16×C67—22no—
39C2×M16×C67—21no—
40C2×M16×C67—21no—
41C2×M16×C67—21no—
42C22×C8×C67—22no—
43C22×C8×C67—21no—
44C4⋊Q8×C67C2×C411yescharacteristic
45C4⋊Q8×C67C2×C411yescharacteristic
46#125×C67—21no—
47C22⋊Q8×C67—21no—
48C22⋊Q8×C67—21no—
49C4×Q8×C67—21no—
50C4×Q8×C67—21no—
51C4×Q8×C67C2×C411yescharacteristic
52C4×Q8×C67D411yescharacteristic
53C4×D4×C67—21no—
54C4×D4×C67—21no—
55C4×D4×C67C2×C411yescharacteristic
56C4×D4×C67D411yescharacteristic
57#117×C67—21no—
58#117×C67—21no—
59#117×C67D411yescharacteristic
60C2×C42×C67—21no—
61C2×C42×C67C2311yescharacteristic
62#107×C67—21no—
63#106×C67—21no—
64C4⋊C8×C67—21no—
65C4⋊C8×C67—21no—
66C4⋊C8×C67C2×C411yescharacteristic
67C4⋊C8×C67C2×C411yescharacteristic
68#103×C67—21no—
69#103×C67—21no—
70#103×C67—21no—
71#102×C67—21no—
72C22⋊C8×C67—21no—
73#97×C67—22no—
74#97×C67—21no—
75#97×C67D412yes—
76C4×C8×C67—22no—
77C4×C8×C67—21no—
78C4×C8×C67—21no—
79C4×C8×C67—21no—
80C4×C8×C67—21no—
81C4×C8×C67D411yescharacteristic
82C4×C8×C67D411yescharacteristic
83#41×C67—22no—
84#41×C67—22no—
85C2×Q8×C67—21no—
86C2×Q8×C67—21no—
87C2×Q8×C67C22⋊C411yescharacteristic
88C2×D4×C67—21no—
89C2×D4×C67C22⋊C411yescharacteristic
90C22×C4×C67—21no—
91C22×C4×C67—21no—
92C22×C4×C67—21no—
93C22×C4×C67C2×D411yescharacteristic
94C22×C4×C67#4111yescharacteristic
95C22×C4×C67#4111yescharacteristic
96Q16×C67—42no—
97SD16×C67—42no—
98M16×C67—28no—
99M16×C67—22no—
100M16×C67—22no—
101M16×C67—22no—
102M16×C67—41no—
103M16×C67—81no—
104C2×C8×C67—28no—
105C2×C8×C67—22no—
106C2×C8×C67—22no—
107C2×C8×C67—21no—
108C2×C8×C67—22no—
109C2×C8×C67—21no—
110C2×C8×C67—21no—
111C2×C8×C67—21no—
112C2×C8×C67—41no—
113C2×C8×C67—41no—
114C4⋊C4×C67—21no—
115C4⋊C4×C67—21no—
116C4⋊C4×C67—21no—
117C4⋊C4×C67—21no—
118C4⋊C4×C67—21no—
119C4⋊C4×C67—21no—
120C4⋊C4×C67—41no—
121C4⋊C4×C67—41no—
122C4⋊C4×C67C22⋊C411yescharacteristic
123C4⋊C4×C67C22⋊C411yescharacteristic
124C22⋊C4×C67—21no—
125C22⋊C4×C67—21no—
126C22⋊C4×C67—21no—
127C22⋊C4×C67—41no—
128C42×C67—21no—
129C42×C67—21no—
130C42×C67—21no—
131C42×C67—41no—
132C42×C67C22×C411yescharacteristic
133C42×C67C2×D411yescharacteristic
134C42×C67C2×D411yescharacteristic
135C42×C67C2×D411yescharacteristic
136C23×C67—21no—
137C23×C67#12311yescharacteristic
138Q8×C67—21no—
139Q8×C67—41no—
140Q8×C67—41no—
141Q8×C67—41no—
142Q8×C67—41no—
143Q8×C67#10412yes—
144D4×C67—24no—
145D4×C67—21no—
146D4×C67#10412yes—
147C2×C4×C67—24no—
148C2×C4×C67—21no—
149C2×C4×C67—21no—
150C2×C4×C67—21no—
151C2×C4×C67—21no—
152C2×C4×C67—21no—
153C2×C4×C67—21no—
154C2×C4×C67—21no—
155C2×C4×C67—21no—
156C2×C4×C67—21no—
157C2×C4×C67—41no—
158C2×C4×C67—41no—
159C2×C4×C67—41no—
160C2×C4×C67—41no—
161C2×C4×C67—41no—
162C2×C4×C67—41no—
163C2×C4×C67C2×C22⋊C411yescharacteristic
164C2×C4×C67C4×D411yescharacteristic
165C2×C4×C67C4×D411yescharacteristic
166C2×C4×C67#12011yescharacteristic
167C2×C4×C67#12111yescharacteristic
168C2×C4×C67#12111yescharacteristic
169C8×C67—22no—
170C8×C67—24no—
171C8×C67—22no—
172C8×C67—42no—
173C8×C67—41no—
174C8×C67—41no—
175C8×C67—81no—
176C22×C67—21no—
177C22×C67—21no—
178C22×C67—22no—
179C22×C67—21no—
180C22×C67—41no—
181C22×C67—41no—
182C22×C67#38611yescharacteristic
183C4×C67—21no—
184C4×C67—21no—
185C4×C67—22no—
186C4×C67—21no—
187C4×C67—41no—
188C4×C67—41no—
189C4×C67—81no—
190C4×C67—81no—
191C4×C67C2×#10411yescharacteristic
192C4×C67C2×#10411yescharacteristic
193C4×C67#44311yescharacteristic
194C4×C67#44411yescharacteristic
195C4×C67#47111yescharacteristic
196C4×C67#47311yescharacteristic
197#9212C6711yesmaximal, characteristic, direct product factor, Sylow
198C2×C67—41no—
199C2×C67—42no—
200C2×C67#187611yescharacteristic
201C2×C67#189611yescharacteristic
202C2×C67#190811yescharacteristic
203#3107C2×C6711yescharacteristic
204C4×#131C2×C6711yescharacteristic
205#1526C2×C6711yescharacteristic, complemented
206#1493C2×C6711yescharacteristic, complemented
207#1489C2×C6711yescharacteristic
208#1299C2×C6711yescharacteristic, complemented
209#1278C2×C6711yescharacteristic, complemented
210C67#921211yescharacteristic, direct product factor, Sylow
211C2×#131—21no—
212#533C22×C6711yescharacteristic
213C4×#41C22×C6711yescharacteristic
214#485—21no—
215#475C4×C6711yescharacteristic
216#474C4×C6711yescharacteristic
217#462C4×C6711yescharacteristic
218#461C4×C6711yescharacteristic
219#452—21no—
220C4×Q16—21no—
221C4×SD16—21no—
222#431C22×C6711yescharacteristic
223#423C22×C6711yescharacteristic
224#405—21no—
225#405—22no—
226C4×M16—21no—
227C4×M16C22×C6711yescharacteristic
228C4×M16C22×C6711yescharacteristic
229C2×C4×C8—21no—
230C2×C4×C8C22×C6711yescharacteristic
231#330—22no—
232Q8⋊C8—21no—
233Q8⋊C8—21no—
234Q8⋊C8—21no—
235D4⋊C8—21no—
236C2×#41D4×C6711yescharacteristic
237C2×Q16—21no—
238C2×SD16—21no—
239#131—24no—
240#131—24no—
241C2×M16—22no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C22×C8—22no—
246C22×C8—21no—
247C4⋊Q8C2×C4×C6711yescharacteristic
248C4⋊Q8C2×C4×C6711yescharacteristic
249#125—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C2×C4×C6711yescharacteristic
255C4×Q8D4×C6711yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C2×C4×C6711yescharacteristic
259C4×D4D4×C6711yescharacteristic
260#117—21no—
261#117—21no—
262#117D4×C6711yescharacteristic
263C2×C42—21no—
264C2×C42C23×C6711yescharacteristic
265#107—21no—
266#106—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8C2×C4×C6711yescharacteristic
270C4⋊C8C2×C4×C6711yescharacteristic
271#103—21no—
272#103—21no—
273#103—21no—
274#102—21no—
275C22⋊C8—21no—
276#97—22no—
277#97—21no—
278#97D4×C6712yes—
279C4×C8—22no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8D4×C6711yescharacteristic
285C4×C8D4×C6711yescharacteristic
286#41—22no—
287#41—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8C22⋊C4×C6711yescharacteristic
291C2×D4—21no—
292C2×D4C22⋊C4×C6711yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4C2×D4×C6711yescharacteristic
297C22×C4#41×C6711yescharacteristic
298C22×C4#41×C6711yescharacteristic
299Q16—42no—
300SD16—42no—
301M16—28no—
302M16—22no—
303M16—22no—
304M16—22no—
305M16—41no—
306M16—81no—
307C2×C8—28no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4C22⋊C4×C6711yescharacteristic
326C4⋊C4C22⋊C4×C6711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—41no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—41no—
335C42C22×C4×C6711yescharacteristic
336C42C2×D4×C6711yescharacteristic
337C42C2×D4×C6711yescharacteristic
338C42C2×D4×C6711yescharacteristic
339C23—21no—
340C23#123×C6711yescharacteristic
341Q8—21no—
342Q8—41no—
343Q8—41no—
344Q8—41no—
345Q8—41no—
346Q8#104×C6712yes—
347D4—24no—
348D4—21no—
349D4#104×C6712yes—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4C2×C22⋊C4×C6711yescharacteristic
367C2×C4C4×D4×C6711yescharacteristic
368C2×C4C4×D4×C6711yescharacteristic
369C2×C4#120×C6711yescharacteristic
370C2×C4#121×C6711yescharacteristic
371C2×C4#121×C6711yescharacteristic
372C8—22no—
373C8—24no—
374C8—22no—
375C8—42no—
376C8—41no—
377C8—41no—
378C8—81no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22#386×C6711yescharacteristic
386C4—21no—
387C4—21no—
388C4—22no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—81no—
393C4—81no—
394C4C2×#104×C6711yescharacteristic
395C4C2×#104×C6711yescharacteristic
396C4#443×C6711yescharacteristic
397C4#444×C6711yescharacteristic
398C4#471×C6711yescharacteristic
399C4#473×C6711yescharacteristic
400C2—41no—
401C2—42no—
402C2C67×#187611yescharacteristic
403C2C67×#189611yescharacteristic
404C2C67×#190811yescharacteristic
subgroupG/H
centerC2×C4×C67#121
commutatorC42C22×C4×C67
Frattini, cosocleC2×C42C23×C67
socleC22×C67#386
prime radicalC2×#41×C67D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C67×#1526C2semidirect
4. C67×#1493C2semidirect
6. C67×#1299C2semidirect
7. C67×#1278C2semidirect
197. #9212C67direct (N × H)
205. #1526C2×C67semidirect
206. #1493C2×C67semidirect
208. #1299C2×C67semidirect
209. #1278C2×C67semidirect
210. C67#9212direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#31072
C4×#131×C672
C67×#15262
C67×#14932
C67×#14892
C67×#12992
C67×#12782
#921267