Finite Group Databasebeta

← C67×#60218 #30739469

Subgroups — C67×#60218 #30739469

Order 17152.

1950 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3480C211yesmaximal, characteristic
2C67×#2971C212yesmaximal, complemented
3C67×#2969C216yesmaximal, complemented
4C67×#2966C216yesmaximal, complemented
5C8×#41×C67C212yesmaximal, complemented
6Q8×M16×C67C211yesmaximal, characteristic, complemented
7C2×C8×Q8×C67C211yesmaximal, characteristic, complemented
8C67×#2935C213yesmaximal, complemented
9C67×#2931C213yesmaximal, complemented
10C67×#2925C213yesmaximal, complemented
11C4×#131×C67C213yesmaximal, complemented
12C2×#131×C67C2213yes—
13C2×#131×C67C2213yes—
14#559×C67C413yes—
15#557×C67C411yescharacteristic
16#556×C67C413yes—
17#553×C67C416yes—
18#550×C67C411yescharacteristic
19#548×C67C412yes—
20#532×C67C2213yes—
21#529×C67C2216yes—
22C4×#41×C67C2212yes—
23C4×#41×C67C2213yes—
24C2×C4×Q8×C67C2211yescharacteristic
25#446×C67C2216yes—
26C8×Q8×C67C2211yescharacteristic
27C8×Q8×C67C2213yes—
28C8×Q8×C67C2213yes—
29C8×Q8×C67C2211yescharacteristic
30C8×Q8×C67C2212yes—
31#436×C67C2216yes—
32#436×C67C2216yes—
33#436×C67C2216yes—
34#435×C67C22112yes—
35C8×D4×C67C2216yes—
36C8×D4×C67C2216yes—
37C8×D4×C67C2216yes—
38#433×C67C22112yes—
39#432×C67C2216yes—
40#431×C67C2216yes—
41#423×C67C2213yes—
42C2×C4⋊C8×C67C2213yes—
43#408×C67C2216yes—
44#405×C67C2211yescharacteristic
45#405×C67C2213yes—
46#405×C67C2216yes—
47#405×C67C2216yes—
48C4×M16×C67C2213yes—
49C4×M16×C67C2216yes—
50C2×C4×C8×C67C2213yes—
51C2×C4×C8×C67C2216yes—
52C2×#41×C67C2×C412yes—
53C2×#41×C67C2313yes—
54C22×Q8×C67C2×C411yescharacteristic
55#131×C67—212no—
56#131×C67—212no—
57C2×M16×C67C2313yes—
58C2×M16×C67C2316yes—
59C2×M16×C67C2316yes—
60C2×M16×C67C2311yescharacteristic
61C22×C8×C67C2313yes—
62C22×C8×C67C2316yes—
63C22×C8×C67C2316yes—
64C22×C8×C67C2311yescharacteristic
65C4⋊Q8×C67C2×C413yes—
66#127×C67C2×C413yes—
67#126×C67C2×C4112yes—
68#125×C67C2×C416yes—
69#124×C67C2×C416yes—
70#124×C67C2×C416yes—
71#123×C67C2×C4112yes—
72C22⋊Q8×C67C2×C416yes—
73#121×C67C2×C416yes—
74#121×C67C2×C416yes—
75#121×C67C2×C416yes—
76C4×Q8×C67C2×C411yescharacteristic
77C4×Q8×C67C2×C413yes—
78C4×Q8×C67C2×C412yes—
79C4×Q8×C67C2×C413yes—
80C4×Q8×C67C2×C413yes—
81C4×Q8×C67C2311yescharacteristic
82C4×Q8×C67C2313yes—
83C4×D4×C67C2×C416yes—
84C4×D4×C67C2×C413yes—
85C4×D4×C67C2×C416yes—
86C4×D4×C67C2×C416yes—
87C4×D4×C67C2×C413yes—
88C4×D4×C67C2316yes—
89C4×D4×C67C2316yes—
90#117×C67C2×C416yes—
91#117×C67C2311yescharacteristic
92#117×C67C2316yes—
93C2×C4⋊C4×C67C2×C413yes—
94C2×C42×C67C2313yes—
95C2×C42×C67C2316yes—
96C4⋊C8×C67C2313yes—
97C4⋊C8×C67C2313yes—
98C4⋊C8×C67C2316yes—
99C4⋊C8×C67C2316yes—
100C4⋊C8×C67C2316yes—
101C22⋊C8×C67C23112yes—
102C22⋊C8×C67C23112yes—
103#97×C67C2312yes—
104#97×C67C2316yes—
105#97×C67C2316yes—
106C4×C8×C67C2316yes—
107C4×C8×C67C2313yes—
108C4×C8×C67C2316yes—
109C4×C8×C67C2313yes—
110C4×C8×C67C2312yes—
111C4×C8×C67C2313yes—
112C4×C8×C67C2313yes—
113#41×C67—28no—
114#41×C67—26no—
115#41×C67—26no—
116C2×Q8×C67—24no—
117C2×Q8×C67C22×C413yes—
118C2×Q8×C67C22×C413yes—
119C2×Q8×C67C22×C411yescharacteristic
120C2×D4×C67C22×C416yes—
121C2×D4×C67C22×C413yes—
122C2×D4×C67C22×C416yes—
123C2×D4×C67C22×C416yes—
124C22×C4×C67C22×C413yes—
125C22×C4×C67C22×C416yes—
126C22×C4×C67C22×C413yes—
127C22×C4×C67C22×C416yes—
128C22×C4×C67C2416yes—
129C22×C4×C67C2411yescharacteristic
130M16×C67—26no—
131M16×C67—212no—
132M16×C67—212no—
133M16×C67#4114yescomplemented
134C2×C8×C67—26no—
135C2×C8×C67—212no—
136C2×C8×C67—212no—
137C2×C8×C67#4114yescomplemented
138C2×C8×C67C2413yes—
139C2×C8×C67C2413yes—
140C2×C8×C67C2416yes—
141C2×C8×C67C2412yes—
142C2×C8×C67C2411yescharacteristic
143C2×C8×C67C2411yescharacteristic
144C4⋊C4×C67C22×C413yes—
145C4⋊C4×C67C22×C413yes—
146C4⋊C4×C67C22×C416yes—
147C4⋊C4×C67C22×C416yes—
148C4⋊C4×C67C22×C411yescharacteristic
149C4⋊C4×C67C22×C411yescharacteristic
150C4⋊C4×C67C22×C416yes—
151C4⋊C4×C67C22×C416yes—
152C4⋊C4×C67C22×C416yes—
153C22⋊C4×C67C22×C4112yes—
154C22⋊C4×C67C22×C412yes—
155C22⋊C4×C67C22×C416yes—
156C22⋊C4×C67C22×C416yes—
157C22⋊C4×C67C22×C4112yes—
158C42×C67C22×C416yes—
159C42×C67C22×C416yes—
160C42×C67C22×C413yes—
161C42×C67C22×C413yes—
162C42×C67C2413yes—
163C42×C67C2413yes—
164C42×C67C2412yes—
165C23×C67C23×C416yes—
166C23×C67C23×C411yescharacteristic
167Q8×C67—26no—
168Q8×C67—26no—
169Q8×C67#13114yescomplemented
170D4×C67—212no—
171D4×C67—26no—
172D4×C67—212no—
173D4×C67—46no—
174C2×C4×C67—26no—
175C2×C4×C67—212no—
176C2×C4×C67—26no—
177C2×C4×C67—26no—
178C2×C4×C67—46no—
179C2×C4×C67—43no—
180C2×C4×C67C23×C412yes—
181C2×C4×C67C23×C413yes—
182C2×C4×C67C23×C412yes—
183C2×C4×C67C23×C416yes—
184C2×C4×C67C23×C413yes—
185C2×C4×C67C23×C413yes—
186C2×C4×C67C23×C411yescharacteristic
187C2×C4×C67C23×C413yes—
188C2×C4×C67C2×#4112yes—
189C2×C4×C67C2×#4112yes—
190C2×C4×C67C2511yescharacteristic
191C8×C67—23no—
192C8×C67—23no—
193C8×C67—26no—
194C8×C67C2×#4112yes—
195C8×C67C2×#4114yes—
196C8×C67C2×#4112yes—
197C22×C67—21no—
198C22×C67—21no—
199C22×C67—26no—
200C22×C67—26no—
201C22×C67—26no—
202C22×C67C4×#4112yes—
203C22×C67C24×C411yescharacteristic
204C4×C67—26no—
205C4×C67—22no—
206C4×C67—23no—
207C4×C67—22no—
208C4×C67—23no—
209C4×C67—23no—
210C4×C67C4×#4112yes—
211C4×C67C2×#13116yes—
212C4×C67C22×#4111yescharacteristic
213C4×C67C22×#4111yescharacteristic
214#60218C6711yesmaximal, characteristic, direct product factor, Sylow
215C2×C67—22no—
216C2×C67—46no—
217C2×C67C2×C4×#4111yescharacteristic
218C2×C67C22×#13111yescharacteristic
219C2×C67#357311yescharacteristic
220#3480C2×C6711yescharacteristic
221#2971C2×C6712yescomplemented
222#2969C2×C6716yescomplemented
223#2966C2×C6716yescomplemented
224C8×#41C2×C6712yescomplemented
225Q8×M16C2×C6711yescharacteristic, complemented
226C2×C8×Q8C2×C6711yescharacteristic, complemented
227#2935C2×C6713yescomplemented
228#2931C2×C6713yescomplemented
229#2925C2×C6713yescomplemented
230C4×#131C2×C6713yescomplemented
231C67#6021811yescharacteristic, direct product factor, Sylow
232C2×#131C22×C6713yes—
233C2×#131C22×C6713yes—
234#559C4×C6713yes—
235#557C4×C6711yescharacteristic
236#556C4×C6713yes—
237#553C4×C6716yes—
238#550C4×C6711yescharacteristic
239#548C4×C6712yes—
240#532C22×C6713yes—
241#529C22×C6716yes—
242C4×#41C22×C6712yes—
243C4×#41C22×C6713yes—
244C2×C4×Q8C22×C6711yescharacteristic
245#446C22×C6716yes—
246C8×Q8C22×C6711yescharacteristic
247C8×Q8C22×C6713yes—
248C8×Q8C22×C6713yes—
249C8×Q8C22×C6711yescharacteristic
250C8×Q8C22×C6712yes—
251#436C22×C6716yes—
252#436C22×C6716yes—
253#436C22×C6716yes—
254#435C22×C67112yes—
255C8×D4C22×C6716yes—
256C8×D4C22×C6716yes—
257C8×D4C22×C6716yes—
258#433C22×C67112yes—
259#432C22×C6716yes—
260#431C22×C6716yes—
261#423C22×C6713yes—
262C2×C4⋊C8C22×C6713yes—
263#408C22×C6716yes—
264#405C22×C6711yescharacteristic
265#405C22×C6713yes—
266#405C22×C6716yes—
267#405C22×C6716yes—
268C4×M16C22×C6713yes—
269C4×M16C22×C6716yes—
270C2×C4×C8C22×C6713yes—
271C2×C4×C8C22×C6716yes—
272C2×#41C2×C4×C6712yes—
273C2×#41C23×C6713yes—
274C22×Q8C2×C4×C6711yescharacteristic
275#131—212no—
276#131—212no—
277C2×M16C23×C6713yes—
278C2×M16C23×C6716yes—
279C2×M16C23×C6716yes—
280C2×M16C23×C6711yescharacteristic
281C22×C8C23×C6713yes—
282C22×C8C23×C6716yes—
283C22×C8C23×C6716yes—
284C22×C8C23×C6711yescharacteristic
285C4⋊Q8C2×C4×C6713yes—
286#127C2×C4×C6713yes—
287#126C2×C4×C67112yes—
288#125C2×C4×C6716yes—
289#124C2×C4×C6716yes—
290#124C2×C4×C6716yes—
291#123C2×C4×C67112yes—
292C22⋊Q8C2×C4×C6716yes—
293#121C2×C4×C6716yes—
294#121C2×C4×C6716yes—
295#121C2×C4×C6716yes—
296C4×Q8C2×C4×C6711yescharacteristic
297C4×Q8C2×C4×C6713yes—
298C4×Q8C2×C4×C6712yes—
299C4×Q8C2×C4×C6713yes—
300C4×Q8C2×C4×C6713yes—
301C4×Q8C23×C6711yescharacteristic
302C4×Q8C23×C6713yes—
303C4×D4C2×C4×C6716yes—
304C4×D4C2×C4×C6713yes—
305C4×D4C2×C4×C6716yes—
306C4×D4C2×C4×C6716yes—
307C4×D4C2×C4×C6713yes—
308C4×D4C23×C6716yes—
309C4×D4C23×C6716yes—
310#117C2×C4×C6716yes—
311#117C23×C6711yescharacteristic
312#117C23×C6716yes—
313C2×C4⋊C4C2×C4×C6713yes—
314C2×C42C23×C6713yes—
315C2×C42C23×C6716yes—
316C4⋊C8C23×C6713yes—
317C4⋊C8C23×C6713yes—
318C4⋊C8C23×C6716yes—
319C4⋊C8C23×C6716yes—
320C4⋊C8C23×C6716yes—
321C22⋊C8C23×C67112yes—
322C22⋊C8C23×C67112yes—
323#97C23×C6712yes—
324#97C23×C6716yes—
325#97C23×C6716yes—
326C4×C8C23×C6712yes—
327C4×C8C23×C6713yes—
328C4×C8C23×C6716yes—
329C4×C8C23×C6713yes—
330C4×C8C23×C6716yes—
331C4×C8C23×C6713yes—
332C4×C8C23×C6713yes—
333#41—28no—
334#41—26no—
335#41—26no—
336C2×Q8—24no—
337C2×Q8C22×C4×C6713yes—
338C2×Q8C22×C4×C6713yes—
339C2×Q8C22×C4×C6711yescharacteristic
340C2×D4C22×C4×C6716yes—
341C2×D4C22×C4×C6713yes—
342C2×D4C22×C4×C6716yes—
343C2×D4C22×C4×C6716yes—
344C22×C4C22×C4×C6713yes—
345C22×C4C22×C4×C6716yes—
346C22×C4C22×C4×C6713yes—
347C22×C4C22×C4×C6716yes—
348C22×C4C24×C6716yes—
349C22×C4C24×C6711yescharacteristic
350M16—26no—
351M16—212no—
352M16—212no—
353M16#41×C6714yescomplemented
354C2×C8—26no—
355C2×C8—212no—
356C2×C8—212no—
357C2×C8#41×C6714yescomplemented
358C2×C8C24×C6713yes—
359C2×C8C24×C6716yes—
360C2×C8C24×C6713yes—
361C2×C8C24×C6711yescharacteristic
362C2×C8C24×C6711yescharacteristic
363C2×C8C24×C6712yes—
364C4⋊C4C22×C4×C6713yes—
365C4⋊C4C22×C4×C6713yes—
366C4⋊C4C22×C4×C6716yes—
367C4⋊C4C22×C4×C6716yes—
368C4⋊C4C22×C4×C6711yescharacteristic
369C4⋊C4C22×C4×C6711yescharacteristic
370C4⋊C4C22×C4×C6716yes—
371C4⋊C4C22×C4×C6716yes—
372C4⋊C4C22×C4×C6716yes—
373C22⋊C4C22×C4×C67112yes—
374C22⋊C4C22×C4×C6712yes—
375C22⋊C4C22×C4×C6716yes—
376C22⋊C4C22×C4×C6716yes—
377C22⋊C4C22×C4×C67112yes—
378C42C22×C4×C6716yes—
379C42C22×C4×C6716yes—
380C42C22×C4×C6713yes—
381C42C22×C4×C6713yes—
382C42C24×C6712yes—
383C42C24×C6713yes—
384C42C24×C6713yes—
385C23C23×C4×C6716yes—
386C23C23×C4×C6711yescharacteristic
387Q8—26no—
388Q8—26no—
389Q8#131×C6714yescomplemented
390D4—212no—
391D4—26no—
392D4—212no—
393D4—46no—
394C2×C4—26no—
395C2×C4—212no—
396C2×C4—26no—
397C2×C4—26no—
398C2×C4—46no—
399C2×C4—43no—
400C2×C4C23×C4×C6712yes—
401C2×C4C23×C4×C6713yes—
402C2×C4C23×C4×C6712yes—
403C2×C4C23×C4×C6716yes—
404C2×C4C23×C4×C6713yes—
405C2×C4C23×C4×C6713yes—
406C2×C4C23×C4×C6711yescharacteristic
407C2×C4C23×C4×C6713yes—
408C2×C4C2×#41×C6712yes—
409C2×C4C2×#41×C6712yes—
410C2×C4C25×C6711yescharacteristic
411C8—23no—
412C8—23no—
413C8—26no—
414C8C2×#41×C6712yes—
415C8C2×#41×C6714yes—
416C8C2×#41×C6712yes—
417C22—21no—
418C22—21no—
419C22—26no—
420C22—26no—
421C22—26no—
422C22C4×#41×C6712yes—
423C22C24×C4×C6711yescharacteristic
424C4—22no—
425C4—23no—
426C4—22no—
427C4—23no—
428C4—23no—
429C4—26no—
430C4C4×#41×C6712yes—
431C4C2×#131×C6716yes—
432C4C22×#41×C6711yescharacteristic
433C4C22×#41×C6711yescharacteristic
434C2—22no—
435C2—46no—
436C2C2×C4×#41×C6711yescharacteristic
437C2C22×#131×C6711yescharacteristic
438C2C67×#357311yescharacteristic
subgroupG/H
centerC2×C8×C67C24
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC67×#3480C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C8×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2971C2semidirect
3. C67×#2969C2semidirect
4. C67×#2966C2semidirect
5. C8×#41×C67C2semidirect
6. Q8×M16×C67C2semidirect
7. C2×C8×Q8×C67C2semidirect
8. C67×#2935C2semidirect
9. C67×#2931C2semidirect
10. C67×#2925C2semidirect
11. C4×#131×C67C2semidirect
133. M16×C67#41semidirect
137. C2×C8×C67#41semidirect
169. Q8×C67#131semidirect
214. #60218C67direct (N × H)
221. #2971C2×C67semidirect
222. #2969C2×C67semidirect
223. #2966C2×C67semidirect
224. C8×#41C2×C67semidirect
225. Q8×M16C2×C67semidirect
226. C2×C8×Q8C2×C67semidirect
227. #2935C2×C67semidirect
228. #2931C2×C67semidirect
229. #2925C2×C67semidirect
230. C4×#131C2×C67semidirect
231. C67#60218direct (N × H)
353. M16#41×C67semidirect
357. C2×C8#41×C67semidirect
389. Q8#131×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34802
C67×#29712
C67×#29692
C67×#29662
C8×#41×C672
Q8×M16×C672
C2×C8×Q8×C672
C67×#29352
C67×#29312
C67×#29252
C4×#131×C672
#6021867