Finite Group Databasebeta

← C67×#9237 #30688488

Subgroups — C67×#9237 #30688488

Order 17152.

588 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×#3100C211yesmaximal, characteristic
2C67×#2293C211yesmaximal, characteristic, complemented
3C67×#1962C211yesmaximal, characteristic, complemented
4C67×#1502C211yesmaximal, characteristic, complemented
5C67×#1486C211yesmaximal, characteristic, complemented
6C67×#1303C211yesmaximal, characteristic, complemented
7C67×#1276C211yesmaximal, characteristic, complemented
8#532×C67C2211yescharacteristic
9C2×C4×D4×C67C2211yescharacteristic
10C2×#117×C67—21no—
11#480×C67—21no—
12#474×C67C412yescomplemented
13#459×C67C412yescomplemented
14#447×C67—21no—
15C4×D8×C67—22no—
16#431×C67C2211yescharacteristic
17#431×C67C2211yescharacteristic
18#423×C67C2211yescharacteristic
19C2×#104×C67—21no—
20C2×#104×C67—21no—
21#390×C67—21no—
22#389×C67—21no—
23#388×C67—22no—
24C4×C4⋊C4×C67C2211yescharacteristic
25C4×C22⋊C4×C67—21no—
26C43×C67C2211yescharacteristic
27#327×C67—22no—
28D4⋊C8×C67—22no—
29C2×#41×C67—21no—
30C22×D4×C67D411yescharacteristic
31C23×C4×C67—21no—
32C2×D8×C67—22no—
33C2×M16×C67—21no—
34C4⋊Q8×C67C2×C411yescharacteristic
35#127×C67C2×C411yescharacteristic
36#124×C67—21no—
37#121×C67—21no—
38#121×C67—21no—
39C4×D4×C67—21no—
40C4×D4×C67—21no—
41C4×D4×C67—21no—
42C4×D4×C67—21no—
43C4×D4×C67C2×C412yes—
44C4×D4×C67D412yes—
45#117×C67—42no—
46#117×C67—42no—
47C2×C4⋊C4×C67—22no—
48C2×C4⋊C4×C67—21no—
49C2×C4⋊C4×C67D411yescharacteristic
50C2×C22⋊C4×C67—21no—
51C2×C22⋊C4×C67—22no—
52C2×C22⋊C4×C67—22no—
53C2×C42×C67—21no—
54C2×C42×C67—21no—
55C2×C42×C67—21no—
56C2×C42×C67D411yescharacteristic
57C2×C42×C67D411yescharacteristic
58C2×C42×C67C2311yescharacteristic
59#107×C67—22no—
60C4⋊C8×C67—21no—
61C4⋊C8×C67—21no—
62C4⋊C8×C67—21no—
63C4⋊C8×C67C2×C412yes—
64#104×C67—42no—
65#104×C67—42no—
66#102×C67—22no—
67#102×C67—22no—
68C22⋊C8×C67—21no—
69C22⋊C8×C67—21no—
70C4×C8×C67—21no—
71C4×C8×C67—21no—
72#95×C67—22no—
73#95×C67D412yes—
74C24×C67—21no—
75#41×C67—41no—
76#41×C67—41no—
77#41×C67—41no—
78#41×C67—41no—
79C2×Q8×C67—21no—
80C2×D4×C67—21no—
81C2×D4×C67—21no—
82C2×D4×C67—21no—
83C2×D4×C67—21no—
84C2×D4×C67—42no—
85C2×D4×C67—41no—
86C2×D4×C67C22⋊C412yes—
87C22×C4×C67—21no—
88C22×C4×C67—21no—
89C22×C4×C67—21no—
90C22×C4×C67—22no—
91C22×C4×C67—22no—
92C22×C4×C67—22no—
93C22×C4×C67—41no—
94C22×C4×C67—41no—
95C22×C4×C67—41no—
96C22×C4×C67—41no—
97C22×C4×C67—41no—
98C22×C4×C67C2×D411yescharacteristic
99C22×C4×C67C2×D411yescharacteristic
100C22×C4×C67C2×D411yescharacteristic
101D8×C67—44no—
102M16×C67—41no—
103M16×C67—41no—
104C2×C8×C67—22no—
105C2×C8×C67—41no—
106C2×C8×C67—41no—
107C4⋊C4×C67—21no—
108C4⋊C4×C67—21no—
109C4⋊C4×C67—21no—
110C4⋊C4×C67—42no—
111C4⋊C4×C67—44no—
112C4⋊C4×C67C22⋊C412yes—
113C22⋊C4×C67—22no—
114C22⋊C4×C67—22no—
115C22⋊C4×C67—44no—
116C22⋊C4×C67—44no—
117C22⋊C4×C67—41no—
118C22⋊C4×C67—41no—
119C42×C67—21no—
120C42×C67—21no—
121C42×C67—21no—
122C42×C67—42no—
123C42×C67—42no—
124C42×C67—41no—
125C42×C67—41no—
126C42×C67D812yescomplemented
127C42×C67C22×C411yescharacteristic
128C42×C67C2×D411yescharacteristic
129C42×C67#4111yescharacteristic
130C42×C67#4111yescharacteristic
131C23×C67—22no—
132C23×C67—42no—
133C23×C67—42no—
134C23×C67—41no—
135C23×C67—41no—
136C23×C67—41no—
137C23×C67#12011yescharacteristic
138Q8×C67—41no—
139Q8×C67—41no—
140D4×C67—22no—
141D4×C67—22no—
142D4×C67—42no—
143D4×C67—42no—
144D4×C67—41no—
145D4×C67—41no—
146D4×C67—81no—
147D4×C67—81no—
148D4×C67—81no—
149C2×C4×C67—22no—
150C2×C4×C67—22no—
151C2×C4×C67—22no—
152C2×C4×C67—22no—
153C2×C4×C67—41no—
154C2×C4×C67—41no—
155C2×C4×C67—41no—
156C2×C4×C67—41no—
157C2×C4×C67—41no—
158C2×C4×C67—41no—
159C2×C4×C67—41no—
160C2×C4×C67—42no—
161C2×C4×C67—42no—
162C2×C4×C67—42no—
163C2×C4×C67—42no—
164C2×C4×C67—42no—
165C2×C4×C67—42no—
166C2×C4×C67—41no—
167C2×C4×C67—42no—
168C2×C4×C67—81no—
169C2×C4×C67—81no—
170C2×C4×C67C2×C22⋊C411yescharacteristic
171C2×C4×C67C4×D411yescharacteristic
172C2×C4×C67C4×D411yescharacteristic
173C2×C4×C67#12111yescharacteristic
174C2×C4×C67#12111yescharacteristic
175C2×C4×C67#12311yescharacteristic
176C2×C4×C67C2×D811yescharacteristic
177C2×C4×C67#13511yescharacteristic
178C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
180C8×C67—42no—
181C8×C67—81no—
182C8×C67—81no—
183C22×C67—21no—
184C22×C67—21no—
185C22×C67—42no—
186C22×C67—42no—
187C22×C67—42no—
188C22×C67—82no—
189C22×C67—82no—
190C22×C67—81no—
191C22×C67—81no—
192C22×C67—81no—
193C22×C67#38611yescharacteristic
194C22×C67#44711yescharacteristic
195C22×C67#44911yescharacteristic
196C4×C67—21no—
197C4×C67—21no—
198C4×C67—21no—
199C4×C67—42no—
200C4×C67—42no—
201C4×C67—42no—
202C4×C67—82no—
203C4×C67—81no—
204C4×C67#42112yes—
205C4×C67C4×D811yescharacteristic
206C4×C67#44211yescharacteristic
207C4×C67#45911yescharacteristic
208C4×C67#46311yescharacteristic
209#9237C6711yesmaximal, characteristic, direct product factor, Sylow
210C2×C67—21no—
211C2×C67—21no—
212C2×C67—82no—
213C2×C67—161no—
214C2×C67#187911yescharacteristic
215C2×C67#189711yescharacteristic
216C2×C67#190611yescharacteristic
217#3100C2×C6711yescharacteristic
218#2293C2×C6711yescharacteristic, complemented
219#1962C2×C6711yescharacteristic, complemented
220#1502C2×C6711yescharacteristic, complemented
221#1486C2×C6711yescharacteristic, complemented
222#1303C2×C6711yescharacteristic, complemented
223#1276C2×C6711yescharacteristic, complemented
224C67#923711yescharacteristic, direct product factor, Sylow
225#532C22×C6711yescharacteristic
226C2×C4×D4C22×C6711yescharacteristic
227C2×#117—21no—
228#480—21no—
229#474C4×C6712yescomplemented
230#459C4×C6712yescomplemented
231#447—21no—
232C4×D8—22no—
233#431C22×C6711yescharacteristic
234#431C22×C6711yescharacteristic
235#423C22×C6711yescharacteristic
236C2×#104—21no—
237C2×#104—21no—
238#390—21no—
239#389—21no—
240#388—22no—
241C4×C4⋊C4C22×C6711yescharacteristic
242C4×C22⋊C4—21no—
243C43C22×C6711yescharacteristic
244#327—22no—
245D4⋊C8—22no—
246C2×#41—21no—
247C22×D4D4×C6711yescharacteristic
248C23×C4—21no—
249C2×D8—22no—
250C2×M16—21no—
251C4⋊Q8C2×C4×C6711yescharacteristic
252#127C2×C4×C6711yescharacteristic
253#124—21no—
254#121—21no—
255#121—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C2×C4×C6712yes—
261C4×D4D4×C6712yes—
262#117—42no—
263#117—42no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4D4×C6711yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42D4×C6711yescharacteristic
274C2×C42D4×C6711yescharacteristic
275C2×C42C23×C6711yescharacteristic
276#107—22no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8C2×C4×C6712yes—
281#104—42no—
282#104—42no—
283#102—22no—
284#102—22no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289#95—22no—
290#95D4×C6712yes—
291C24—21no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4C22⋊C4×C6712yes—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4C2×D4×C6711yescharacteristic
316C22×C4C2×D4×C6711yescharacteristic
317C22×C4C2×D4×C6711yescharacteristic
318D8—44no—
319M16—41no—
320M16—41no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4C22⋊C4×C6712yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—42no—
340C42—42no—
341C42—41no—
342C42—41no—
343C42D8×C6712yescomplemented
344C42C22×C4×C6711yescharacteristic
345C42C2×D4×C6711yescharacteristic
346C42#41×C6711yescharacteristic
347C42#41×C6711yescharacteristic
348C23—22no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23#120×C6711yescharacteristic
355Q8—41no—
356Q8—41no—
357D4—22no—
358D4—22no—
359D4—42no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4C2×C22⋊C4×C6711yescharacteristic
388C2×C4C4×D4×C6711yescharacteristic
389C2×C4C4×D4×C6711yescharacteristic
390C2×C4#121×C6711yescharacteristic
391C2×C4#121×C6711yescharacteristic
392C2×C4#123×C6711yescharacteristic
393C2×C4C2×D8×C6711yescharacteristic
394C2×C4#135×C6711yescharacteristic
395C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
396C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
397C8—42no—
398C8—81no—
399C8—81no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#386×C6711yescharacteristic
411C22#447×C6711yescharacteristic
412C22#449×C6711yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—82no—
420C4—81no—
421C4#421×C6712yes—
422C4C4×D8×C6711yescharacteristic
423C4#442×C6711yescharacteristic
424C4#459×C6711yescharacteristic
425C4#463×C6711yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2C67×#187911yescharacteristic
431C2C67×#189711yescharacteristic
432C2C67×#190611yescharacteristic
subgroupG/H
centerC2×C4×C67#121
commutatorC42C22×C4×C67
Frattini, cosocleC2×C42C23×C67
socleC22×C67#386
prime radicalC67×#3100C2
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
2. C67×#2293C2semidirect
3. C67×#1962C2semidirect
4. C67×#1502C2semidirect
5. C67×#1486C2semidirect
6. C67×#1303C2semidirect
7. C67×#1276C2semidirect
12. #474×C67C4semidirect
13. #459×C67C4semidirect
126. C42×C67D8semidirect
209. #9237C67direct (N × H)
218. #2293C2×C67semidirect
219. #1962C2×C67semidirect
220. #1502C2×C67semidirect
221. #1486C2×C67semidirect
222. #1303C2×C67semidirect
223. #1276C2×C67semidirect
224. C67#9237direct (N × H)
229. #474C4×C67semidirect
230. #459C4×C67semidirect
343. C42D8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#31002
C67×#22932
C67×#19622
C67×#15022
C67×#14862
C67×#13032
C67×#12762
#923767