Finite Group Databasebeta

← C67×#9305 #30688556

Subgroups — C67×#9305 #30688556

Order 17152.

542 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3036C211yesmaximal, characteristic
2C67×#2972C211yesmaximal, characteristic, complemented
3C67×#1804C211yesmaximal, characteristic, complemented
4C67×#1517C211yesmaximal, characteristic, complemented
5C67×#1480C211yesmaximal, characteristic, complemented
6C67×#1301C211yesmaximal, characteristic, complemented
7C67×#1286C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C67—21no—
9#532×C67C2211yescharacteristic
10C2×C4×D4×C67C2211yescharacteristic
11#463×C67—22no—
12#461×C67C412yescomplemented
13#459×C67C412yescomplemented
14C8×D4×C67—22no—
15C8×D4×C67—22no—
16#431×C67C2211yescharacteristic
17#431×C67C2211yescharacteristic
18#423×C67C2211yescharacteristic
19C2×C4⋊C8×C67C2211yescharacteristic
20#417×C67—21no—
21C2×#102×C67—21no—
22C2×C22⋊C8×C67—22no—
23#382×C67C2211yescharacteristic
24#340×C67—21no—
25#327×C67—22no—
26D4⋊C8×C67—22no—
27C2×#41×C67—21no—
28C22×D4×C67D411yescharacteristic
29C23×C4×C67—21no—
30AGL(1,ℤ/8)×C67—42no—
31AGL(1,ℤ/8)×C67—42no—
32C2×SD16×C67—22no—
33C2×D8×C67—22no—
34C2×M16×C67—21no—
35C22×C8×C67—24no—
36C22×C8×C67—21no—
37C4⋊Q8×C67C2×C411yescharacteristic
38#127×C67C2×C411yescharacteristic
39#124×C67—21no—
40#121×C67—21no—
41#121×C67—21no—
42C4×D4×C67—21no—
43C4×D4×C67—21no—
44C4×D4×C67—21no—
45C4×D4×C67—21no—
46C4×D4×C67C2×C412yes—
47C4×D4×C67D412yes—
48C2×C4⋊C4×C67—21no—
49C2×C4⋊C4×C67D411yescharacteristic
50C2×C22⋊C4×C67—21no—
51C2×C42×C67C2311yescharacteristic
52C4⋊C8×C67—21no—
53C4⋊C8×C67—21no—
54C4⋊C8×C67—21no—
55C4⋊C8×C67—21no—
56C4⋊C8×C67—21no—
57C4⋊C8×C67C2×C412yes—
58#103×C67—22no—
59#102×C67—22no—
60#102×C67—42no—
61C22⋊C8×C67—24no—
62C22⋊C8×C67—21no—
63C22⋊C8×C67D412yes—
64C4×C8×C67—21no—
65C4×C8×C67D412yes—
66#95×C67—21no—
67#95×C67—21no—
68C24×C67—21no—
69#41×C67—41no—
70#41×C67—41no—
71#41×C67—41no—
72#41×C67—41no—
73C2×Q8×C67—21no—
74C2×D4×C67—21no—
75C2×D4×C67—21no—
76C2×D4×C67—21no—
77C2×D4×C67—21no—
78C2×D4×C67—21no—
79C2×D4×C67—21no—
80C2×D4×C67—41no—
81C2×D4×C67C22⋊C412yes—
82C2×D4×C67D812yes—
83C2×D4×C67SD1612yes—
84C22×C4×C67—21no—
85C22×C4×C67—22no—
86C22×C4×C67—22no—
87C22×C4×C67—41no—
88C22×C4×C67—41no—
89C22×C4×C67—82no—
90C22×C4×C67C2×D411yescharacteristic
91C22×C4×C67C2×D411yescharacteristic
92C22×C4×C67#4111yescharacteristic
93SD16×C67—44no—
94D8×C67—44no—
95M16×C67—41no—
96M16×C67—41no—
97C2×C8×C67—22no—
98C2×C8×C67—22no—
99C2×C8×C67—22no—
100C2×C8×C67—41no—
101C2×C8×C67—41no—
102C2×C8×C67—41no—
103C2×C8×C67—84no—
104C4⋊C4×C67—21no—
105C4⋊C4×C67—21no—
106C4⋊C4×C67—21no—
107C4⋊C4×C67—41no—
108C4⋊C4×C67—41no—
109C4⋊C4×C67C22⋊C412yes—
110C22⋊C4×C67—22no—
111C22⋊C4×C67—22no—
112C22⋊C4×C67—41no—
113C22⋊C4×C67—41no—
114C42×C67C22×C411yescharacteristic
115C42×C67C2×D411yescharacteristic
116C42×C67C2×D411yescharacteristic
117C42×C67#4111yescharacteristic
118C23×C67—22no—
119C23×C67—22no—
120C23×C67—22no—
121C23×C67—41no—
122C23×C67—81no—
123C23×C67—81no—
124C23×C67#12111yescharacteristic
125Q8×C67—41no—
126Q8×C67—41no—
127D4×C67—22no—
128D4×C67—22no—
129D4×C67—22no—
130D4×C67—42no—
131D4×C67—41no—
132D4×C67—41no—
133D4×C67—81no—
134D4×C67—81no—
135D4×C67—81no—
136D4×C67#10214yes—
137C2×C4×C67—22no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—44no—
141C2×C4×C67—41no—
142C2×C4×C67—82no—
143C2×C4×C67—81no—
144C2×C4×C67—81no—
145C2×C4×C67—81no—
146C2×C4×C67—81no—
147C2×C4×C67—81no—
148C2×C4×C67—81no—
149C2×C4×C67C2×C22⋊C411yescharacteristic
150C2×C4×C67C4×D411yescharacteristic
151C2×C4×C67C4×D411yescharacteristic
152C2×C4×C67#12011yescharacteristic
153C2×C4×C67#12111yescharacteristic
154C2×C4×C67#12311yescharacteristic
155C2×C4×C67C2×D811yescharacteristic
156C2×C4×C67C2×SD1611yescharacteristic
157C2×C4×C67#13511yescharacteristic
158C2×C4×C67#13511yescharacteristic
159C8×C67—42no—
160C8×C67—42no—
161C8×C67—42no—
162C8×C67—81no—
163C22×C67—21no—
164C22×C67—21no—
165C22×C67—24no—
166C22×C67—42no—
167C22×C67—42no—
168C22×C67—82no—
169C22×C67—82no—
170C22×C67—81no—
171C22×C67—81no—
172C22×C67—81no—
173C22×C67#38611yescharacteristic
174C22×C67#46511yescharacteristic
175C22×C67#46611yescharacteristic
176C4×C67—21no—
177C4×C67—21no—
178C4×C67—21no—
179C4×C67—21no—
180C4×C67—42no—
181C4×C67—81no—
182C4×C67—161no—
183C4×C67C2×#10211yescharacteristic
184C4×C67#41611yescharacteristic
185C4×C67#46511yescharacteristic
186C4×C67#46611yescharacteristic
187#9305C6711yesmaximal, characteristic, direct product factor, Sylow
188C2×C67—21no—
189C2×C67—21no—
190C2×C67—44no—
191C2×C67—161no—
192C2×C67#188111yescharacteristic
193C2×C67#189311yescharacteristic
194C2×C67#190411yescharacteristic
195#3036C2×C6711yescharacteristic
196#2972C2×C6711yescharacteristic, complemented
197#1804C2×C6711yescharacteristic, complemented
198#1517C2×C6711yescharacteristic, complemented
199#1480C2×C6711yescharacteristic, complemented
200#1301C2×C6711yescharacteristic, complemented
201#1286C2×C6711yescharacteristic, complemented
202C67#930511yescharacteristic, direct product factor, Sylow
203C2×AGL(1,ℤ/8)—21no—
204#532C22×C6711yescharacteristic
205C2×C4×D4C22×C6711yescharacteristic
206#463—22no—
207#461C4×C6712yescomplemented
208#459C4×C6712yescomplemented
209C8×D4—22no—
210C8×D4—22no—
211#431C22×C6711yescharacteristic
212#431C22×C6711yescharacteristic
213#423C22×C6711yescharacteristic
214C2×C4⋊C8C22×C6711yescharacteristic
215#417—21no—
216C2×#102—21no—
217C2×C22⋊C8—22no—
218#382C22×C6711yescharacteristic
219#340—21no—
220#327—22no—
221D4⋊C8—22no—
222C2×#41—21no—
223C22×D4D4×C6711yescharacteristic
224C23×C4—21no—
225AGL(1,ℤ/8)—42no—
226AGL(1,ℤ/8)—42no—
227C2×SD16—22no—
228C2×D8—22no—
229C2×M16—21no—
230C22×C8—24no—
231C22×C8—21no—
232C4⋊Q8C2×C4×C6711yescharacteristic
233#127C2×C4×C6711yescharacteristic
234#124—21no—
235#121—21no—
236#121—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4C2×C4×C6712yes—
242C4×D4D4×C6712yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4D4×C6711yescharacteristic
245C2×C22⋊C4—21no—
246C2×C42C23×C6711yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8C2×C4×C6712yes—
253#103—22no—
254#102—22no—
255#102—42no—
256C22⋊C8—24no—
257C22⋊C8—21no—
258C22⋊C8D4×C6712yes—
259C4×C8—21no—
260C4×C8D4×C6712yes—
261#95—21no—
262#95—21no—
263C24—21no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4C22⋊C4×C6712yes—
277C2×D4D8×C6712yes—
278C2×D4SD16×C6712yes—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4C2×D4×C6711yescharacteristic
286C22×C4C2×D4×C6711yescharacteristic
287C22×C4#41×C6711yescharacteristic
288SD16—44no—
289D8—44no—
290M16—41no—
291M16—41no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—84no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4C22⋊C4×C6712yes—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42C22×C4×C6711yescharacteristic
310C42C2×D4×C6711yescharacteristic
311C42C2×D4×C6711yescharacteristic
312C42#41×C6711yescharacteristic
313C23—22no—
314C23—22no—
315C23—22no—
316C23—41no—
317C23—81no—
318C23—81no—
319C23#121×C6711yescharacteristic
320Q8—41no—
321Q8—41no—
322D4—22no—
323D4—22no—
324D4—22no—
325D4—42no—
326D4—41no—
327D4—41no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4#102×C6714yes—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—41no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4C2×C22⋊C4×C6711yescharacteristic
345C2×C4C4×D4×C6711yescharacteristic
346C2×C4C4×D4×C6711yescharacteristic
347C2×C4#120×C6711yescharacteristic
348C2×C4#121×C6711yescharacteristic
349C2×C4#123×C6711yescharacteristic
350C2×C4C2×D8×C6711yescharacteristic
351C2×C4C2×SD16×C6711yescharacteristic
352C2×C4#135×C6711yescharacteristic
353C2×C4#135×C6711yescharacteristic
354C8—42no—
355C8—42no—
356C8—42no—
357C8—81no—
358C22—21no—
359C22—21no—
360C22—24no—
361C22—42no—
362C22—42no—
363C22—82no—
364C22—82no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22#386×C6711yescharacteristic
369C22#465×C6711yescharacteristic
370C22#466×C6711yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—161no—
378C4C2×#102×C6711yescharacteristic
379C4#416×C6711yescharacteristic
380C4#465×C6711yescharacteristic
381C4#466×C6711yescharacteristic
382C2—21no—
383C2—21no—
384C2—44no—
385C2—161no—
386C2C67×#188111yescharacteristic
387C2C67×#189311yescharacteristic
388C2C67×#190411yescharacteristic
subgroupG/H
center, socleC22×C67#386
commutatorC42C22×C4×C67
Frattini, cosocleC2×C42C23×C67
prime radicalC67×#3036C2
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×#2972C2semidirect
3. C67×#1804C2semidirect
4. C67×#1517C2semidirect
5. C67×#1480C2semidirect
6. C67×#1301C2semidirect
7. C67×#1286C2semidirect
12. #461×C67C4semidirect
13. #459×C67C4semidirect
187. #9305C67direct (N × H)
196. #2972C2×C67semidirect
197. #1804C2×C67semidirect
198. #1517C2×C67semidirect
199. #1480C2×C67semidirect
200. #1301C2×C67semidirect
201. #1286C2×C67semidirect
202. C67#9305direct (N × H)
207. #461C4×C67semidirect
208. #459C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#30362
C67×#29722
C67×#18042
C67×#15172
C67×#14802
C67×#13012
C67×#12862
#930567