Finite Group Databasebeta

← C67×#19397 #30698648

Subgroups — C67×#19397 #30698648

Order 17152.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3206C211yesmaximal, characteristic
2C2×#475×C67C211yesmaximal, characteristic, complemented
3C67×#3037C211yesmaximal, characteristic
4C67×#2982C211yesmaximal, characteristic, complemented
5C67×#1680C214yesmaximal, complemented
6C67×#1666C214yesmaximal, complemented
7C67×#1570C211yesmaximal, characteristic, complemented
8C67×#1549C211yesmaximal, characteristic, complemented
9C67×#1540C211yesmaximal, characteristic
10C2×#137×C67—21no—
11#532×C67C2211yescharacteristic
12C2×C4⋊Q8×C67C2211yescharacteristic
13C2×C4×Q8×C67C2211yescharacteristic
14#492×C67C2212yes—
15#487×C67C2212yes—
16#475×C67—22no—
17#475×C67C2214yes—
18#464×C67—22no—
19#462×C67C2212yes—
20#461×C67—22no—
21#460×C67C2212yes—
22#448×C67—22no—
23C8×Q8×C67—22no—
24C8×Q8×C67C2214yes—
25#431×C67—22no—
26#431×C67C2211yescharacteristic
27#431×C67C2211yescharacteristic
28C2×#106×C67—21no—
29#423×C67C2211yescharacteristic
30C2×C4⋊C8×C67C2211yescharacteristic
31#417×C67—21no—
32C2×#103×C67—21no—
33#334×C67C2214yes—
34#332×C67C2212yes—
35#332×C67C2214yes—
36#331×C67C2212yes—
37C2×#41×C67—21no—
38C22×Q8×C67—21no—
39C22×Q8×C67D411yescharacteristic
40#137×C67—44no—
41C2×Q16×C67—22no—
42C2×SD16×C67—22no—
43C2×SD16×C67—24no—
44C2×M16×C67—21no—
45C22×C8×C67—21no—
46C4⋊Q8×C67—21no—
47C4⋊Q8×C67—21no—
48C4⋊Q8×C67—21no—
49C4⋊Q8×C67C2311yescharacteristic
50C4⋊Q8×C67C2312yes—
51#127×C67C2311yescharacteristic
52#124×C67—21no—
53#121×C67—21no—
54#121×C67—21no—
55C4×Q8×C67—21no—
56C4×Q8×C67—21no—
57C4×Q8×C67D412yes—
58C4×Q8×C67C2312yes—
59C4×D4×C67—21no—
60C4×D4×C67—21no—
61C2×C4⋊C4×C67—21no—
62C2×C4⋊C4×C67—21no—
63C2×C4⋊C4×C67—21no—
64C2×C4⋊C4×C67D411yescharacteristic
65C2×C42×C67—21no—
66C2×C42×C67C2311yescharacteristic
67#106×C67—24no—
68C4⋊C8×C67—24no—
69C4⋊C8×C67—21no—
70C4⋊C8×C67—21no—
71C4⋊C8×C67D412yes—
72C4⋊C8×C67C2312yes—
73C4⋊C8×C67C2312yes—
74C4⋊C8×C67C2312yes—
75#103×C67—24no—
76#103×C67—24no—
77#103×C67—22no—
78#102×C67—22no—
79C22⋊C8×C67—21no—
80C22⋊C8×C67D412yes—
81C4×C8×C67—24no—
82C4×C8×C67—21no—
83C4×C8×C67C2312yes—
84#41×C67—42no—
85#41×C67—42no—
86C2×Q8×C67—21no—
87C2×Q8×C67—22no—
88C2×Q8×C67—21no—
89C2×Q8×C67—22no—
90C2×Q8×C67—41no—
91C2×Q8×C67—82no—
92C2×Q8×C67SD1614yescomplemented
93C2×Q8×C67C2×D412yes—
94C2×D4×C67—21no—
95C2×D4×C67—21no—
96C2×D4×C67—21no—
97C2×D4×C67—41no—
98C22×C4×C67—21no—
99C22×C4×C67—21no—
100C22×C4×C67—41no—
101C22×C4×C67—41no—
102C22×C4×C67C2×D411yescharacteristic
103C22×C4×C67C2×D411yescharacteristic
104C22×C4×C67#4111yescharacteristic
105Q16×C67—44no—
106SD16×C67—44no—
107SD16×C67—84no—
108M16×C67—42no—
109C2×C8×C67—22no—
110C2×C8×C67—22no—
111C2×C8×C67—22no—
112C2×C8×C67—42no—
113C2×C8×C67C2×D414yes—
114C4⋊C4×C67—22no—
115C4⋊C4×C67—21no—
116C4⋊C4×C67—22no—
117C4⋊C4×C67—21no—
118C4⋊C4×C67—22no—
119C4⋊C4×C67—22no—
120C4⋊C4×C67—21no—
121C4⋊C4×C67—41no—
122C4⋊C4×C67—41no—
123C4⋊C4×C67C2×D412yes—
124C22⋊C4×C67—41no—
125C22⋊C4×C67—41no—
126C42×C67—22no—
127C42×C67—22no—
128C42×C67C2×D411yescharacteristic
129C42×C67C2×D411yescharacteristic
130C42×C67#4111yescharacteristic
131C42×C67C2411yescharacteristic
132C23×C67—41no—
133C23×C67#12111yescharacteristic
134Q8×C67—22no—
135Q8×C67—24no—
136Q8×C67—41no—
137Q8×C67—41no—
138Q8×C67—44no—
139Q8×C67—82no—
140Q8×C67C2×SD1614yes—
141D4×C67—41no—
142D4×C67—41no—
143D4×C67—42no—
144D4×C67—81no—
145D4×C67—82no—
146C2×C4×C67—22no—
147C2×C4×C67—24no—
148C2×C4×C67—22no—
149C2×C4×C67—22no—
150C2×C4×C67—41no—
151C2×C4×C67—41no—
152C2×C4×C67—42no—
153C2×C4×C67—81no—
154C2×C4×C67—82no—
155C2×C4×C67—82no—
156C2×C4×C67#12111yescharacteristic
157C2×C4×C67#12111yescharacteristic
158C2×C4×C67#12111yescharacteristic
159C2×C4×C67C2×SD1612yes—
160C2×C4×C67#13512yes—
161C2×C4×C67C22×D411yescharacteristic
162C2×C4×C67C22×D411yescharacteristic
163C2×C4×C67C2×#4111yescharacteristic
164C8×C67—24no—
165C8×C67—42no—
166C8×C67—42no—
167C8×C67—42no—
168C22×C67—21no—
169C22×C67—21no—
170C22×C67—81no—
171C22×C67—81no—
172C22×C67—81no—
173C22×C67#46512yes—
174C22×C67C2×#12111yescharacteristic
175C4×C67—21no—
176C4×C67—24no—
177C4×C67—21no—
178C4×C67—21no—
179C4×C67—42no—
180C4×C67—82no—
181C4×C67—81no—
182C4×C67#46512yes—
183C4×C67C22×SD1611yescharacteristic
184C4×C67C2×#13511yescharacteristic
185C4×C67#57712yes—
186#19397C6711yesmaximal, characteristic, direct product factor, Sylow
187C2×C67—22no—
188C2×C67—161no—
189C2×C67#304311yescharacteristic
190C2×C67C2×#46511yescharacteristic
191C2×C67#306911yescharacteristic
192#3206C2×C6711yescharacteristic
193C2×#475C2×C6711yescharacteristic, complemented
194#3037C2×C6711yescharacteristic
195#2982C2×C6711yescharacteristic, complemented
196#1680C2×C6714yescomplemented
197#1666C2×C6714yescomplemented
198#1570C2×C6711yescharacteristic, complemented
199#1549C2×C6711yescharacteristic, complemented
200#1540C2×C6711yescharacteristic
201C67#1939711yescharacteristic, direct product factor, Sylow
202C2×#137—21no—
203#532C22×C6711yescharacteristic
204C2×C4⋊Q8C22×C6711yescharacteristic
205C2×C4×Q8C22×C6711yescharacteristic
206#492C22×C6712yes—
207#487C22×C6712yes—
208#475—22no—
209#475C22×C6714yes—
210#464—22no—
211#462C22×C6712yes—
212#461—22no—
213#460C22×C6712yes—
214#448—22no—
215C8×Q8—22no—
216C8×Q8C22×C6714yes—
217#431—22no—
218#431C22×C6711yescharacteristic
219#431C22×C6711yescharacteristic
220C2×#106—21no—
221#423C22×C6711yescharacteristic
222C2×C4⋊C8C22×C6711yescharacteristic
223#417—21no—
224C2×#103—21no—
225#334C22×C6714yes—
226#332C22×C6712yes—
227#332C22×C6714yes—
228#331C22×C6712yes—
229C2×#41—21no—
230C22×Q8—21no—
231C22×Q8D4×C6711yescharacteristic
232#137—44no—
233C2×Q16—22no—
234C2×SD16—22no—
235C2×SD16—24no—
236C2×M16—21no—
237C22×C8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8C23×C6712yes—
242C4⋊Q8C23×C6711yescharacteristic
243#127C23×C6711yescharacteristic
244#124—21no—
245#121—21no—
246#121—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8D4×C6712yes—
250C4×Q8C23×C6712yes—
251C4×D4—21no—
252C4×D4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4D4×C6711yescharacteristic
257C2×C42—21no—
258C2×C42C23×C6711yescharacteristic
259#106—24no—
260C4⋊C8—24no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8D4×C6712yes—
264C4⋊C8C23×C6712yes—
265C4⋊C8C23×C6712yes—
266C4⋊C8C23×C6712yes—
267#103—24no—
268#103—24no—
269#103—22no—
270#102—22no—
271C22⋊C8—21no—
272C22⋊C8D4×C6712yes—
273C4×C8—24no—
274C4×C8—21no—
275C4×C8C23×C6712yes—
276#41—42no—
277#41—42no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—82no—
284C2×Q8SD16×C6714yescomplemented
285C2×Q8C2×D4×C6712yes—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C2×D4×C6711yescharacteristic
295C22×C4C2×D4×C6711yescharacteristic
296C22×C4#41×C6711yescharacteristic
297Q16—44no—
298SD16—44no—
299SD16—84no—
300M16—42no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—42no—
305C2×C8C2×D4×C6714yes—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4C2×D4×C6712yes—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—22no—
319C42—22no—
320C42C2×D4×C6711yescharacteristic
321C42C2×D4×C6711yescharacteristic
322C42#41×C6711yescharacteristic
323C42C24×C6711yescharacteristic
324C23—41no—
325C23#121×C6711yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—41no—
329Q8—41no—
330Q8—44no—
331Q8—82no—
332Q8C2×SD16×C6714yes—
333D4—41no—
334D4—41no—
335D4—42no—
336D4—81no—
337D4—82no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4#121×C6711yescharacteristic
349C2×C4#121×C6711yescharacteristic
350C2×C4#121×C6711yescharacteristic
351C2×C4C2×SD16×C6712yes—
352C2×C4#135×C6712yes—
353C2×C4C22×D4×C6711yescharacteristic
354C2×C4C22×D4×C6711yescharacteristic
355C2×C4C2×#41×C6711yescharacteristic
356C8—24no—
357C8—42no—
358C8—42no—
359C8—42no—
360C22—21no—
361C22—21no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22#465×C6712yes—
366C22C2×#121×C6711yescharacteristic
367C4—21no—
368C4—24no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—82no—
373C4—81no—
374C4#465×C6712yes—
375C4C22×SD16×C6711yescharacteristic
376C4C2×#135×C6711yescharacteristic
377C4#577×C6712yes—
378C2—22no—
379C2—161no—
380C2C67×#304311yescharacteristic
381C2C2×#465×C6711yescharacteristic
382C2C67×#306911yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC42C24×C67
prime radical#532×C67C22
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. C2×#475×C67C2semidirect
4. C67×#2982C2semidirect
5. C67×#1680C2semidirect
6. C67×#1666C2semidirect
7. C67×#1570C2semidirect
8. C67×#1549C2semidirect
92. C2×Q8×C67SD16semidirect
186. #19397C67direct (N × H)
193. C2×#475C2×C67semidirect
195. #2982C2×C67semidirect
196. #1680C2×C67semidirect
197. #1666C2×C67semidirect
198. #1570C2×C67semidirect
199. #1549C2×C67semidirect
201. C67#19397direct (N × H)
284. C2×Q8SD16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32062
C2×#475×C672
C67×#30372
C67×#29822
C67×#16802
C67×#16662
C67×#15702
C67×#15492
C67×#15402
#1939767