Finite Group Databasebeta

← C67×#8763 #30688014

Subgroups — C67×#8763 #30688014

Order 17152.

472 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2353C211yesmaximal, characteristic
2C67×#1799C211yesmaximal, characteristic
3C67×#1459C211yesmaximal, characteristic
4C67×#1294C212yesmaximal, complemented
5C67×#1289C212yesmaximal, complemented
6C2×C4×D4×C67C2211yescharacteristic
7C2×#99×C67—21no—
8C2×C22⋊C8×C67C2212yes—
9#388×C67—21no—
10#387×C67—21no—
11#386×C67—21no—
12#383×C67C411yescharacteristic, complemented
13#381×C67C411yescharacteristic, complemented
14#380×C67C2212yes—
15#380×C67C2211yescharacteristic
16C4×C22⋊C4×C67—21no—
17C4×C22⋊C4×C67C411yescharacteristic, complemented
18#376×C67C411yescharacteristic, complemented
19C2×#95×C67C2211yescharacteristic
20#342×C67—21no—
21#324×C67—21no—
22#323×C67—21no—
23C22×D4×C67D411yescharacteristic
24C23×C4×C67—21no—
25C23×C4×C67C2×C411yescharacteristic
26C23×C4×C67C2311yescharacteristic
27C22×C8×C67—22no—
28C4×D4×C67—22no—
29C4×D4×C67—22no—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67D411yescharacteristic
33C2×C22⋊C4×C67—21no—
34C2×C22⋊C4×C67—21no—
35C2×C22⋊C4×C67—22no—
36C2×C22⋊C4×C67—21no—
37C2×C22⋊C4×C67D411yescharacteristic
38C2×C22⋊C4×C67D411yescharacteristic
39C2×C42×C67—21no—
40C2×C42×C67—21no—
41C2×C42×C67C2×C411yescharacteristic
42#99×C67—42no—
43C22⋊C8×C67—22no—
44C22⋊C8×C67—22no—
45#95×C67—21no—
46#95×C67—21no—
47#95×C67—21no—
48#95×C67—21no—
49#95×C67—21no—
50#95×C67—22no—
51#95×C67C2×C411yescharacteristic
52#95×C67C2×C411yescharacteristic
53#95×C67C2×C411yescharacteristic
54#95×C67C2×C411yescharacteristic
55C24×C67C22⋊C411yescharacteristic
56C24×C67C2×D411yescharacteristic
57C2×D4×C67—22no—
58C2×D4×C67—22no—
59C2×D4×C67—21no—
60C2×D4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—42no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67—81no—
71C22×C4×C67—81no—
72C22×C4×C67—81no—
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22×C411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
78C22×C4×C67#4112yes—
79C22×C4×C67#4112yes—
80C2×C8×C67—42no—
81C2×C8×C67—82no—
82C4⋊C4×C67—22no—
83C4⋊C4×C67—81no—
84C4⋊C4×C67—82no—
85C22⋊C4×C67—22no—
86C22⋊C4×C67—22no—
87C22⋊C4×C67—42no—
88C22⋊C4×C67—41no—
89C22⋊C4×C67—41no—
90C22⋊C4×C67—42no—
91C22⋊C4×C67—42no—
92C22⋊C4×C67—81no—
93C42×C67—21no—
94C42×C67—21no—
95C42×C67—42no—
96C42×C67—81no—
97C23×C67—21no—
98C23×C67—21no—
99C23×C67—21no—
100C23×C67—41no—
101C23×C67—41no—
102C23×C67—41no—
103C23×C67—41no—
104C23×C67C2×C22⋊C411yescharacteristic
105C23×C67#12312yes—
106C23×C67#13512yes—
107C23×C67AGL(1,ℤ/8)11yescharacteristic
108C23×C67#13711yescharacteristic
109D4×C67—42no—
110D4×C67—42no—
111C2×C4×C67—22no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—41no—
118C2×C4×C67—42no—
119C2×C4×C67—42no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—81no—
135C2×C4×C67—81no—
136C2×C4×C67—84no—
137C2×C4×C67—82no—
138C2×C4×C67—82no—
139C2×C4×C67—82no—
140C2×C4×C67#11711yescharacteristic
141C2×C4×C67#11711yescharacteristic
142C2×C4×C67#12312yes—
143C8×C67—82no—
144C22×C67—21no—
145C22×C67—22no—
146C22×C67—21no—
147C22×C67—22no—
148C22×C67—42no—
149C22×C67—41no—
150C22×C67—41no—
151C22×C67—41no—
152C22×C67—41no—
153C22×C67—41no—
154C22×C67—81no—
155C22×C67—81no—
156C22×C67#38111yescharacteristic
157C22×C67#41611yescharacteristic
158C22×C67#41711yescharacteristic
159C22×C67#48212yes—
160C22×C67#48512yes—
161C4×C67—22no—
162C4×C67—21no—
163C4×C67—21no—
164C4×C67—42no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67—81no—
169C4×C67—81no—
170C4×C67—81no—
171C4×C67—162no—
172#8763C6711yesmaximal, characteristic, direct product factor, Sylow
173C2×C67—21no—
174C2×C67—21no—
175C2×C67—22no—
176C2×C67—22no—
177C2×C67—81no—
178C2×C67#182511yescharacteristic
179C2×C67#182711yescharacteristic
180C2×C67#182811yescharacteristic
181#2353C2×C6711yescharacteristic
182#1799C2×C6711yescharacteristic
183#1459C2×C6711yescharacteristic
184#1294C2×C6712yescomplemented
185#1289C2×C6712yescomplemented
186C67#876311yescharacteristic, direct product factor, Sylow
187C2×C4×D4C22×C6711yescharacteristic
188C2×#99—21no—
189C2×C22⋊C8C22×C6712yes—
190#388—21no—
191#387—21no—
192#386—21no—
193#383C4×C6711yescharacteristic, complemented
194#381C4×C6711yescharacteristic, complemented
195#380C22×C6712yes—
196#380C22×C6711yescharacteristic
197C4×C22⋊C4—21no—
198C4×C22⋊C4C4×C6711yescharacteristic, complemented
199#376C4×C6711yescharacteristic, complemented
200C2×#95C22×C6711yescharacteristic
201#342—21no—
202#324—21no—
203#323—21no—
204C22×D4D4×C6711yescharacteristic
205C23×C4—21no—
206C23×C4C2×C4×C6711yescharacteristic
207C23×C4C23×C6711yescharacteristic
208C22×C8—22no—
209C4×D4—22no—
210C4×D4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4D4×C6711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4D4×C6711yescharacteristic
219C2×C22⋊C4D4×C6711yescharacteristic
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C2×C4×C6711yescharacteristic
223#99—42no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95C2×C4×C6711yescharacteristic
233#95C2×C4×C6711yescharacteristic
234#95C2×C4×C6711yescharacteristic
235#95C2×C4×C6711yescharacteristic
236C24C22⋊C4×C6711yescharacteristic
237C24C2×D4×C6711yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C22⋊C4×C6711yescharacteristic
255C22×C4C22⋊C4×C6711yescharacteristic
256C22×C4C22⋊C4×C6711yescharacteristic
257C22×C4C22×C4×C6711yescharacteristic
258C22×C4C2×D4×C6711yescharacteristic
259C22×C4#41×C6712yes—
260C22×C4#41×C6712yes—
261C2×C8—42no—
262C2×C8—82no—
263C4⋊C4—22no—
264C4⋊C4—81no—
265C4⋊C4—82no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—81no—
274C42—21no—
275C42—21no—
276C42—42no—
277C42—81no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23C2×C22⋊C4×C6711yescharacteristic
286C23#123×C6712yes—
287C23#135×C6712yes—
288C23AGL(1,ℤ/8)×C6711yescharacteristic
289C23#137×C6711yescharacteristic
290D4—42no—
291D4—42no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—84no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4#117×C6711yescharacteristic
322C2×C4#117×C6711yescharacteristic
323C2×C4#123×C6712yes—
324C8—82no—
325C22—21no—
326C22—22no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#381×C6711yescharacteristic
338C22#416×C6711yescharacteristic
339C22#417×C6711yescharacteristic
340C22#482×C6712yes—
341C22#485×C6712yes—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—162no—
353C2—21no—
354C2—21no—
355C2—22no—
356C2—22no—
357C2—81no—
358C2C67×#182511yescharacteristic
359C2C67×#182711yescharacteristic
360C2C67×#182811yescharacteristic
subgroupG/H
center, socleC22×C67#381
commutatorC22×C4C22×C4×C67
Frattini, cosocleC23×C4C23×C67
prime radicalC22×D4×C67D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C67×#1294C2semidirect
5. C67×#1289C2semidirect
12. #383×C67C4semidirect
13. #381×C67C4semidirect
17. C4×C22⋊C4×C67C4semidirect
18. #376×C67C4semidirect
172. #8763C67direct (N × H)
184. #1294C2×C67semidirect
185. #1289C2×C67semidirect
186. C67#8763direct (N × H)
193. #383C4×C67semidirect
194. #381C4×C67semidirect
198. C4×C22⋊C4C4×C67semidirect
199. #376C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#23532
C67×#17992
C67×#14592
C67×#12942
C67×#12892
#876367