Finite Group Databasebeta

← C67×#60328 #30739579

Subgroups — C67×#60328 #30739579

Order 17152.

1920 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×#3483C211yesmaximal, characteristic
2C67×#2991C211yesmaximal, characteristic, complemented
3C67×#2990C213yesmaximal, complemented
4C67×#2988C212yesmaximal, complemented
5C67×#2981C216yesmaximal, complemented
6C67×#2979C213yesmaximal, complemented
7C67×#2971C212yesmaximal, complemented
8C67×#2965C213yesmaximal, complemented
9C8×#41×C67C211yesmaximal, characteristic, complemented
10C67×#2935C216yesmaximal, complemented
11C67×#2931C213yesmaximal, complemented
12C2×#131×C67C2216yes—
13Q82×C67C411yescharacteristic
14#555×C67C416yes—
15#555×C67C413yes—
16#550×C67C411yescharacteristic
17#550×C67C412yes—
18#550×C67C413yes—
19#532×C67C2213yes—
20#532×C67C2216yes—
21C4×#41×C67C2211yescharacteristic
22C4×#41×C67C2213yes—
23C4×#41×C67C2212yes—
24#446×C67C2211yescharacteristic
25#446×C67C2213yes—
26C8×Q8×C67C2211yescharacteristic
27C8×Q8×C67C2211yescharacteristic
28C8×Q8×C67C2212yes—
29#436×C67C2213yes—
30#436×C67C2216yes—
31#436×C67C2216yes—
32#436×C67C2213yes—
33#436×C67C2216yes—
34#435×C67C2216yes—
35#435×C67C22112yes—
36#435×C67C2216yes—
37C8×D4×C67C2216yes—
38C8×D4×C67C2216yes—
39C8×D4×C67C2213yes—
40C8×D4×C67C2216yes—
41C8×D4×C67C2213yes—
42#432×C67C2216yes—
43#431×C67C2213yes—
44#431×C67C2213yes—
45#431×C67C2216yes—
46#423×C67C2213yes—
47C2×C4⋊C8×C67C2216yes—
48#408×C67C2216yes—
49#408×C67C22112yes—
50C4×M16×C67C2216yes—
51C2×C4×C8×C67C2213yes—
52C2×#41×C67C2×C411yescharacteristic
53C2×#41×C67C2×C412yes—
54C2×#41×C67C2×C413yes—
55C2×#41×C67C2316yes—
56#131×C67—412no—
57C2×M16×C67C2316yes—
58C2×M16×C67C2316yes—
59C2×M16×C67C2316yes—
60C22×C8×C67C2313yes—
61C22×C8×C67C23112yes—
62C22×C8×C67C2313yes—
63C4⋊Q8×C67C2×C413yes—
64C4⋊Q8×C67C2×C416yes—
65#127×C67C2×C413yes—
66#127×C67C2×C416yes—
67#124×C67C2×C416yes—
68#124×C67C2×C4112yes—
69#121×C67C2×C416yes—
70#121×C67C2×C416yes—
71#121×C67C2×C416yes—
72#121×C67C2×C416yes—
73#121×C67C2×C4112yes—
74C4×Q8×C67C2×C413yes—
75C4×Q8×C67C2×C411yescharacteristic
76C4×Q8×C67C2×C412yes—
77C4×Q8×C67C2311yescharacteristic
78C4×Q8×C67C2311yescharacteristic
79C4×D4×C67C2×C413yes—
80C4×D4×C67C2×C413yes—
81C4×D4×C67C2×C416yes—
82C4×D4×C67C2×C416yes—
83C4×D4×C67C2313yes—
84C4×D4×C67C2313yes—
85C4×D4×C67C2316yes—
86C4×D4×C67C2316yes—
87#117×C67C2×C413yes—
88#117×C67C2×C413yes—
89#117×C67C2×C416yes—
90#117×C67C2×C416yes—
91C2×C42×C67C2313yes—
92C2×C42×C67C2316yes—
93C4⋊C8×C67C2313yes—
94C4⋊C8×C67C2316yes—
95C4⋊C8×C67C2316yes—
96C4⋊C8×C67C2313yes—
97C4⋊C8×C67C2312yes—
98C4⋊C8×C67C2311yescharacteristic
99C4⋊C8×C67C2313yes—
100C22⋊C8×C67C2316yes—
101C22⋊C8×C67C2316yes—
102C22⋊C8×C67C2316yes—
103C22⋊C8×C67C2316yes—
104C22⋊C8×C67C23112yes—
105#97×C67C2312yes—
106#97×C67C2316yes—
107C4×C8×C67C2311yescharacteristic
108C4×C8×C67C2313yes—
109C4×C8×C67C2313yes—
110C4×C8×C67C2313yes—
111C4×C8×C67C2316yes—
112#41×C67—24no—
113#41×C67—28no—
114#41×C67—46no—
115#41×C67—46no—
116#41×C67—46no—
117C2×Q8×C67C22×C411yescharacteristic
118C2×Q8×C67C22×C416yes—
119C2×Q8×C67C22×C411yescharacteristic
120C2×D4×C67C22×C413yes—
121C2×D4×C67C22×C416yes—
122C2×D4×C67C22×C413yes—
123C2×D4×C67C22×C416yes—
124C2×D4×C67C22×C4112yes—
125C2×D4×C67C22×C416yes—
126C22×C4×C67C22×C413yes—
127C22×C4×C67C22×C416yes—
128C22×C4×C67C22×C413yes—
129C22×C4×C67C22×C416yes—
130C22×C4×C67C2413yes—
131C22×C4×C67C2416yes—
132M16×C67—212no—
133M16×C67—46no—
134M16×C67—46no—
135C2×C8×C67—26no—
136C2×C8×C67—43no—
137C2×C8×C67—412no—
138C2×C8×C67C2413yes—
139C2×C8×C67C2416yes—
140C2×C8×C67C2411yescharacteristic
141C2×C8×C67C2413yes—
142C2×C8×C67C2412yes—
143C2×C8×C67C2411yescharacteristic
144C4⋊C4×C67C22×C412yes—
145C4⋊C4×C67C22×C413yes—
146C4⋊C4×C67C22×C411yescharacteristic
147C4⋊C4×C67C22×C416yes—
148C4⋊C4×C67C22×C413yes—
149C4⋊C4×C67C22×C416yes—
150C4⋊C4×C67C22×C413yes—
151C22⋊C4×C67C22×C416yes—
152C22⋊C4×C67C22×C416yes—
153C22⋊C4×C67C22×C4112yes—
154C22⋊C4×C67C22×C416yes—
155C22⋊C4×C67C22×C416yes—
156C42×C67C22×C413yes—
157C42×C67C22×C416yes—
158C42×C67C22×C416yes—
159C42×C67C22×C413yes—
160C42×C67C2413yes—
161C42×C67C2411yescharacteristic
162C42×C67C2412yes—
163C23×C67C23×C413yes—
164C23×C67C23×C416yes—
165Q8×C67—22no—
166Q8×C67—46no—
167Q8×C67#13114yescomplemented
168D4×C67—26no—
169D4×C67—212no—
170D4×C67—26no—
171D4×C67—46no—
172D4×C67—412no—
173D4×C67—46no—
174C2×C4×C67—26no—
175C2×C4×C67—212no—
176C2×C4×C67—26no—
177C2×C4×C67—23no—
178C2×C4×C67—26no—
179C2×C4×C67—23no—
180C2×C4×C67—26no—
181C2×C4×C67—46no—
182C2×C4×C67C23×C411yescharacteristic
183C2×C4×C67C23×C412yes—
184C2×C4×C67C23×C413yes—
185C2×C4×C67C23×C412yes—
186C2×C4×C67C23×C413yes—
187C2×C4×C67C23×C416yes—
188C2×C4×C67C23×C413yes—
189C2×C4×C67C23×C411yescharacteristic
190C2×C4×C67C2511yescharacteristic
191C8×C67—23no—
192C8×C67—26no—
193C8×C67—21no—
194C8×C67—23no—
195C8×C67—22no—
196C8×C67H(5,𝔽2)12yes—
197C22×C67—23no—
198C22×C67—26no—
199C22×C67—23no—
200C22×C67—26no—
201C22×C67—23no—
202C22×C67—26no—
203C22×C67C24×C411yescharacteristic
204C4×C67—23no—
205C4×C67—26no—
206C4×C67—21no—
207C4×C67—22no—
208C4×C67—23no—
209C4×C67—22no—
210C4×C67C2×#13116yes—
211C4×C67#56812yes—
212C4×C67C2×H(5,𝔽2)11yescharacteristic
213C4×C67#58511yescharacteristic
214#60328C6711yesmaximal, characteristic, direct product factor, Sylow
215C2×C67—43no—
216C2×C67—46no—
217C2×C67C4×H(5,𝔽2)11yescharacteristic
218C2×C67C22×#13111yescharacteristic
219C2×C67C2×#56811yescharacteristic
220#3483C2×C6711yescharacteristic
221#2991C2×C6711yescharacteristic, complemented
222#2990C2×C6713yescomplemented
223#2988C2×C6712yescomplemented
224#2981C2×C6716yescomplemented
225#2979C2×C6713yescomplemented
226#2971C2×C6712yescomplemented
227#2965C2×C6713yescomplemented
228C8×#41C2×C6711yescharacteristic, complemented
229#2935C2×C6716yescomplemented
230#2931C2×C6713yescomplemented
231C67#6032811yescharacteristic, direct product factor, Sylow
232C2×#131C22×C6716yes—
233Q82C4×C6711yescharacteristic
234#555C4×C6716yes—
235#555C4×C6713yes—
236#550C4×C6711yescharacteristic
237#550C4×C6712yes—
238#550C4×C6713yes—
239#532C22×C6713yes—
240#532C22×C6716yes—
241C4×#41C22×C6711yescharacteristic
242C4×#41C22×C6713yes—
243C4×#41C22×C6712yes—
244#446C22×C6711yescharacteristic
245#446C22×C6713yes—
246C8×Q8C22×C6711yescharacteristic
247C8×Q8C22×C6711yescharacteristic
248C8×Q8C22×C6712yes—
249#436C22×C6713yes—
250#436C22×C6716yes—
251#436C22×C6716yes—
252#436C22×C6713yes—
253#436C22×C6716yes—
254#435C22×C6716yes—
255#435C22×C67112yes—
256#435C22×C6716yes—
257C8×D4C22×C6716yes—
258C8×D4C22×C6716yes—
259C8×D4C22×C6713yes—
260C8×D4C22×C6716yes—
261C8×D4C22×C6713yes—
262#432C22×C6716yes—
263#431C22×C6713yes—
264#431C22×C6713yes—
265#431C22×C6716yes—
266#423C22×C6713yes—
267C2×C4⋊C8C22×C6716yes—
268#408C22×C6716yes—
269#408C22×C67112yes—
270C4×M16C22×C6716yes—
271C2×C4×C8C22×C6713yes—
272C2×#41C2×C4×C6711yescharacteristic
273C2×#41C2×C4×C6712yes—
274C2×#41C2×C4×C6713yes—
275C2×#41C23×C6716yes—
276#131—412no—
277C2×M16C23×C6716yes—
278C2×M16C23×C6716yes—
279C2×M16C23×C6716yes—
280C22×C8C23×C6713yes—
281C22×C8C23×C67112yes—
282C22×C8C23×C6713yes—
283C4⋊Q8C2×C4×C6713yes—
284C4⋊Q8C2×C4×C6716yes—
285#127C2×C4×C6713yes—
286#127C2×C4×C6716yes—
287#124C2×C4×C6716yes—
288#124C2×C4×C67112yes—
289#121C2×C4×C6716yes—
290#121C2×C4×C6716yes—
291#121C2×C4×C6716yes—
292#121C2×C4×C6716yes—
293#121C2×C4×C67112yes—
294C4×Q8C2×C4×C6713yes—
295C4×Q8C2×C4×C6711yescharacteristic
296C4×Q8C2×C4×C6712yes—
297C4×Q8C23×C6711yescharacteristic
298C4×Q8C23×C6711yescharacteristic
299C4×D4C2×C4×C6713yes—
300C4×D4C2×C4×C6713yes—
301C4×D4C2×C4×C6716yes—
302C4×D4C2×C4×C6716yes—
303C4×D4C23×C6713yes—
304C4×D4C23×C6713yes—
305C4×D4C23×C6716yes—
306C4×D4C23×C6716yes—
307#117C2×C4×C6713yes—
308#117C2×C4×C6713yes—
309#117C2×C4×C6716yes—
310#117C2×C4×C6716yes—
311C2×C42C23×C6713yes—
312C2×C42C23×C6716yes—
313C4⋊C8C23×C6713yes—
314C4⋊C8C23×C6716yes—
315C4⋊C8C23×C6716yes—
316C4⋊C8C23×C6713yes—
317C4⋊C8C23×C6712yes—
318C4⋊C8C23×C6711yescharacteristic
319C4⋊C8C23×C6713yes—
320C22⋊C8C23×C6716yes—
321C22⋊C8C23×C6716yes—
322C22⋊C8C23×C6716yes—
323C22⋊C8C23×C6716yes—
324C22⋊C8C23×C67112yes—
325#97C23×C6712yes—
326#97C23×C6716yes—
327C4×C8C23×C6711yescharacteristic
328C4×C8C23×C6713yes—
329C4×C8C23×C6713yes—
330C4×C8C23×C6713yes—
331C4×C8C23×C6716yes—
332#41—24no—
333#41—28no—
334#41—46no—
335#41—46no—
336#41—46no—
337C2×Q8C22×C4×C6711yescharacteristic
338C2×Q8C22×C4×C6716yes—
339C2×Q8C22×C4×C6711yescharacteristic
340C2×D4C22×C4×C6713yes—
341C2×D4C22×C4×C6716yes—
342C2×D4C22×C4×C6713yes—
343C2×D4C22×C4×C6716yes—
344C2×D4C22×C4×C67112yes—
345C2×D4C22×C4×C6716yes—
346C22×C4C22×C4×C6713yes—
347C22×C4C22×C4×C6716yes—
348C22×C4C22×C4×C6713yes—
349C22×C4C22×C4×C6716yes—
350C22×C4C24×C6713yes—
351C22×C4C24×C6716yes—
352M16—212no—
353M16—46no—
354M16—46no—
355C2×C8—26no—
356C2×C8—412no—
357C2×C8—43no—
358C2×C8C24×C6711yescharacteristic
359C2×C8C24×C6712yes—
360C2×C8C24×C6713yes—
361C2×C8C24×C6713yes—
362C2×C8C24×C6716yes—
363C2×C8C24×C6711yescharacteristic
364C4⋊C4C22×C4×C6712yes—
365C4⋊C4C22×C4×C6713yes—
366C4⋊C4C22×C4×C6711yescharacteristic
367C4⋊C4C22×C4×C6716yes—
368C4⋊C4C22×C4×C6713yes—
369C4⋊C4C22×C4×C6716yes—
370C4⋊C4C22×C4×C6713yes—
371C22⋊C4C22×C4×C6716yes—
372C22⋊C4C22×C4×C6716yes—
373C22⋊C4C22×C4×C67112yes—
374C22⋊C4C22×C4×C6716yes—
375C22⋊C4C22×C4×C6716yes—
376C42C22×C4×C6713yes—
377C42C22×C4×C6716yes—
378C42C22×C4×C6716yes—
379C42C22×C4×C6713yes—
380C42C24×C6713yes—
381C42C24×C6711yescharacteristic
382C42C24×C6712yes—
383C23C23×C4×C6713yes—
384C23C23×C4×C6716yes—
385Q8—22no—
386Q8—46no—
387Q8#131×C6714yescomplemented
388D4—26no—
389D4—212no—
390D4—26no—
391D4—46no—
392D4—46no—
393D4—412no—
394C2×C4—26no—
395C2×C4—212no—
396C2×C4—26no—
397C2×C4—23no—
398C2×C4—26no—
399C2×C4—23no—
400C2×C4—26no—
401C2×C4—46no—
402C2×C4C23×C4×C6711yescharacteristic
403C2×C4C23×C4×C6712yes—
404C2×C4C23×C4×C6713yes—
405C2×C4C23×C4×C6712yes—
406C2×C4C23×C4×C6713yes—
407C2×C4C23×C4×C6716yes—
408C2×C4C23×C4×C6713yes—
409C2×C4C23×C4×C6711yescharacteristic
410C2×C4C25×C6711yescharacteristic
411C8—23no—
412C8—26no—
413C8—21no—
414C8—23no—
415C8—22no—
416C8H(5,𝔽2)×C6712yes—
417C22—23no—
418C22—26no—
419C22—23no—
420C22—26no—
421C22—23no—
422C22—26no—
423C22C24×C4×C6711yescharacteristic
424C4—21no—
425C4—22no—
426C4—23no—
427C4—22no—
428C4—23no—
429C4—26no—
430C4C2×#131×C6716yes—
431C4#568×C6712yes—
432C4C2×H(5,𝔽2)×C6711yescharacteristic
433C4#585×C6711yescharacteristic
434C2—43no—
435C2—46no—
436C2C4×H(5,𝔽2)×C6711yescharacteristic
437C2C22×#131×C6711yescharacteristic
438C2C2×#568×C6711yescharacteristic
subgroupG/H
centerC2×C4×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC67×#3483C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2991C2semidirect
3. C67×#2990C2semidirect
4. C67×#2988C2semidirect
5. C67×#2981C2semidirect
6. C67×#2979C2semidirect
7. C67×#2971C2semidirect
8. C67×#2965C2semidirect
9. C8×#41×C67C2semidirect
10. C67×#2935C2semidirect
11. C67×#2931C2semidirect
167. Q8×C67#131semidirect
214. #60328C67direct (N × H)
221. #2991C2×C67semidirect
222. #2990C2×C67semidirect
223. #2988C2×C67semidirect
224. #2981C2×C67semidirect
225. #2979C2×C67semidirect
226. #2971C2×C67semidirect
227. #2965C2×C67semidirect
228. C8×#41C2×C67semidirect
229. #2935C2×C67semidirect
230. #2931C2×C67semidirect
231. C67#60328direct (N × H)
387. Q8#131×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34832
C67×#29912
C67×#29902
C67×#29882
C67×#29812
C67×#29792
C67×#29712
C67×#29652
C8×#41×C672
C67×#29352
C67×#29312
#6032867