Finite Group Databasebeta

← C67×#10520 #30689771

Subgroups — C67×#10520 #30689771

Order 17152.

440 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3246C211yesmaximal, characteristic
2C67×#1994C211yesmaximal, characteristic, complemented
3C67×#1962C211yesmaximal, characteristic
4C67×#1842C211yesmaximal, characteristic, complemented
5C67×#1550C211yesmaximal, characteristic, complemented
6C67×#1547C211yesmaximal, characteristic
7C67×#1292C211yesmaximal, characteristic, complemented
8#533×C67C2211yescharacteristic
9#532×C67C2211yescharacteristic
10#476×C67—21no—
11#475×C67—21no—
12#465×C67—22no—
13#461×C67—21no—
14#460×C67—21no—
15#423×C67C2211yescharacteristic
16#423×C67C2211yescharacteristic
17C2×C4⋊C8×C67—22no—
18C2×C4⋊C8×C67C2211yescharacteristic
19C2×C4⋊C8×C67C2211yescharacteristic
20C2×#104×C67—21no—
21C2×#104×C67—21no—
22C43×C67C2211yescharacteristic
23#339×C67—21no—
24#339×C67—21no—
25#333×C67—21no—
26#332×C67—21no—
27#332×C67—21no—
28#331×C67—21no—
29C2×#41×C67—21no—
30C2×SD16×C67—22no—
31C2×M16×C67—21no—
32C2×M16×C67—21no—
33C22×C8×C67—21no—
34C22×C8×C67D412yes—
35C4⋊Q8×C67D411yescharacteristic
36C4⋊Q8×C67D411yescharacteristic
37C4⋊Q8×C67D411yescharacteristic
38#127×C67D411yescharacteristic
39#125×C67—21no—
40#124×C67—21no—
41C22⋊Q8×C67—21no—
42C22⋊Q8×C67—21no—
43#121×C67—21no—
44#121×C67—21no—
45C4×Q8×C67—21no—
46C4×Q8×C67—21no—
47C4×D4×C67—21no—
48C4×D4×C67—21no—
49#117×C67—21no—
50C2×C42×C67—21no—
51C2×C42×C67—21no—
52C2×C42×C67D411yescharacteristic
53C2×C42×C67D411yescharacteristic
54C2×C42×C67C2311yescharacteristic
55#106×C67—22no—
56C4⋊C8×C67—21no—
57C4⋊C8×C67—21no—
58C4⋊C8×C67—21no—
59C4⋊C8×C67—21no—
60C4⋊C8×C67—21no—
61C4⋊C8×C67—21no—
62C4⋊C8×C67—21no—
63C4⋊C8×C67—21no—
64C4⋊C8×C67—44no—
65#104×C67—42no—
66#104×C67—42no—
67#103×C67—22no—
68#102×C67—22no—
69#41×C67—41no—
70#41×C67—41no—
71#41×C67—41no—
72#41×C67—41no—
73C2×Q8×C67—21no—
74C2×Q8×C67—21no—
75C2×Q8×C67—21no—
76C2×D4×C67—21no—
77C2×D4×C67—21no—
78C2×D4×C67—21no—
79C2×D4×C67—41no—
80C22×C4×C67—21no—
81C22×C4×C67—21no—
82C22×C4×C67—41no—
83C22×C4×C67C2×D411yescharacteristic
84C22×C4×C67C2×D411yescharacteristic
85C22×C4×C67#4111yescharacteristic
86SD16×C67—82no—
87M16×C67—41no—
88M16×C67—41no—
89M16×C67—41no—
90M16×C67—41no—
91C2×C8×C67—22no—
92C2×C8×C67—42no—
93C2×C8×C67—42no—
94C2×C8×C67—41no—
95C2×C8×C67—41no—
96C2×C8×C67—41no—
97C4⋊C4×C67—21no—
98C4⋊C4×C67—21no—
99C4⋊C4×C67—21no—
100C4⋊C4×C67—21no—
101C4⋊C4×C67—21no—
102C4⋊C4×C67—21no—
103C4⋊C4×C67—41no—
104C4⋊C4×C67—41no—
105C22⋊C4×C67—41no—
106C22⋊C4×C67—41no—
107C22⋊C4×C67—41no—
108C42×C67—22no—
109C42×C67—22no—
110C42×C67—21no—
111C42×C67—21no—
112C42×C67—21no—
113C42×C67—42no—
114C42×C67—41no—
115C42×C67SD1612yescomplemented
116C42×C67C2×D411yescharacteristic
117C42×C67C2×D411yescharacteristic
118C42×C67#4111yescharacteristic
119C42×C67#4111yescharacteristic
120C23×C67—41no—
121C23×C67#12111yescharacteristic
122Q8×C67—41no—
123Q8×C67—41no—
124Q8×C67—41no—
125Q8×C67—41no—
126Q8×C67—81no—
127D4×C67—41no—
128D4×C67—41no—
129D4×C67—41no—
130D4×C67—41no—
131D4×C67—81no—
132D4×C67—81no—
133D4×C67—81no—
134C2×C4×C67—22no—
135C2×C4×C67—22no—
136C2×C4×C67—22no—
137C2×C4×C67—21no—
138C2×C4×C67—21no—
139C2×C4×C67—22no—
140C2×C4×C67—41no—
141C2×C4×C67—41no—
142C2×C4×C67—41no—
143C2×C4×C67—41no—
144C2×C4×C67—41no—
145C2×C4×C67—41no—
146C2×C4×C67—81no—
147C2×C4×C67—81no—
148C2×C4×C67#12011yescharacteristic
149C2×C4×C67#12111yescharacteristic
150C2×C4×C67#12111yescharacteristic
151C2×C4×C67#12311yescharacteristic
152C2×C4×C67#12311yescharacteristic
153C2×C4×C67#12411yescharacteristic
154C2×C4×C67C2×SD1611yescharacteristic
155C2×C4×C67#13511yescharacteristic
156C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C67#13711yescharacteristic
158C8×C67—42no—
159C8×C67—81no—
160C8×C67—81no—
161C8×C67—81no—
162C22×C67—21no—
163C22×C67—21no—
164C22×C67—81no—
165C22×C67—81no—
166C22×C67—81no—
167C22×C67#39411yescharacteristic
168C22×C67#46511yescharacteristic
169C22×C67#46811yescharacteristic
170C4×C67—24no—
171C4×C67—21no—
172C4×C67—21no—
173C4×C67—21no—
174C4×C67—42no—
175C4×C67—81no—
176C4×C67—81no—
177C4×C67—81no—
178C4×C67#45311yescharacteristic
179C4×C67#45611yescharacteristic
180C4×C67#46011yescharacteristic
181C4×C67#46111yescharacteristic
182C4×C67#46311yescharacteristic
183C4×C67#46411yescharacteristic
184#10520C6711yesmaximal, characteristic, direct product factor, Sylow
185C2×C67—21no—
186C2×C67—21no—
187C2×C67—161no—
188C2×C67#203911yescharacteristic
189C2×C67#205011yescharacteristic
190C2×C67#205511yescharacteristic
191#3246C2×C6711yescharacteristic
192#1994C2×C6711yescharacteristic, complemented
193#1962C2×C6711yescharacteristic
194#1842C2×C6711yescharacteristic, complemented
195#1550C2×C6711yescharacteristic, complemented
196#1547C2×C6711yescharacteristic
197#1292C2×C6711yescharacteristic, complemented
198C67#1052011yescharacteristic, direct product factor, Sylow
199#533C22×C6711yescharacteristic
200#532C22×C6711yescharacteristic
201#476—21no—
202#475—21no—
203#465—22no—
204#461—21no—
205#460—21no—
206#423C22×C6711yescharacteristic
207#423C22×C6711yescharacteristic
208C2×C4⋊C8—22no—
209C2×C4⋊C8C22×C6711yescharacteristic
210C2×C4⋊C8C22×C6711yescharacteristic
211C2×#104—21no—
212C2×#104—21no—
213C43C22×C6711yescharacteristic
214#339—21no—
215#339—21no—
216#333—21no—
217#332—21no—
218#332—21no—
219#331—21no—
220C2×#41—21no—
221C2×SD16—22no—
222C2×M16—21no—
223C2×M16—21no—
224C22×C8—21no—
225C22×C8D4×C6712yes—
226C4⋊Q8D4×C6711yescharacteristic
227C4⋊Q8D4×C6711yescharacteristic
228C4⋊Q8D4×C6711yescharacteristic
229#127D4×C6711yescharacteristic
230#125—21no—
231#124—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234#121—21no—
235#121—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×D4—21no—
239C4×D4—21no—
240#117—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42D4×C6711yescharacteristic
244C2×C42D4×C6711yescharacteristic
245C2×C42C23×C6711yescharacteristic
246#106—22no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—44no—
256#104—42no—
257#104—42no—
258#103—22no—
259#102—22no—
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4C2×D4×C6711yescharacteristic
275C22×C4C2×D4×C6711yescharacteristic
276C22×C4#41×C6711yescharacteristic
277SD16—82no—
278M16—41no—
279M16—41no—
280M16—41no—
281M16—41no—
282C2×C8—22no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C42—22no—
300C42—22no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—42no—
305C42—41no—
306C42SD16×C6712yescomplemented
307C42C2×D4×C6711yescharacteristic
308C42C2×D4×C6711yescharacteristic
309C42#41×C6711yescharacteristic
310C42#41×C6711yescharacteristic
311C23—41no—
312C23#121×C6711yescharacteristic
313Q8—41no—
314Q8—41no—
315Q8—41no—
316Q8—41no—
317Q8—81no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—81no—
323D4—81no—
324D4—81no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4#120×C6711yescharacteristic
340C2×C4#121×C6711yescharacteristic
341C2×C4#121×C6711yescharacteristic
342C2×C4#123×C6711yescharacteristic
343C2×C4#123×C6711yescharacteristic
344C2×C4#124×C6711yescharacteristic
345C2×C4C2×SD16×C6711yescharacteristic
346C2×C4#135×C6711yescharacteristic
347C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
348C2×C4#137×C6711yescharacteristic
349C8—42no—
350C8—81no—
351C8—81no—
352C8—81no—
353C22—21no—
354C22—21no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22#394×C6711yescharacteristic
359C22#465×C6711yescharacteristic
360C22#468×C6711yescharacteristic
361C4—24no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4#453×C6711yescharacteristic
370C4#456×C6711yescharacteristic
371C4#460×C6711yescharacteristic
372C4#461×C6711yescharacteristic
373C4#463×C6711yescharacteristic
374C4#464×C6711yescharacteristic
375C2—21no—
376C2—21no—
377C2—161no—
378C2C67×#203911yescharacteristic
379C2C67×#205011yescharacteristic
380C2C67×#205511yescharacteristic
subgroupG/H
center, socleC22×C67#394
commutator, Frattini, cosocleC2×C42C23×C67
prime radical#532×C67C22
LCSG ▷ C2×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×#1994C2semidirect
4. C67×#1842C2semidirect
5. C67×#1550C2semidirect
7. C67×#1292C2semidirect
115. C42×C67SD16semidirect
184. #10520C67direct (N × H)
192. #1994C2×C67semidirect
194. #1842C2×C67semidirect
195. #1550C2×C67semidirect
197. #1292C2×C67semidirect
198. C67#10520direct (N × H)
306. C42SD16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32462
C67×#19942
C67×#19622
C67×#18422
C67×#15502
C67×#15472
C67×#12922
#1052067