Finite Group Databasebeta

← C67×#61375 #30740626

Subgroups — C67×#61375 #30740626

Order 17152.

1706 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3504C212yesmaximal, complemented
2C67×#3390C214yesmaximal, complemented
3C67×#3389C214yesmaximal, complemented
4C67×#3381C218yesmaximal, complemented
5C67×#3163C212yesmaximal, complemented
6C2×#499×C67C211yesmaximal, characteristic, complemented
7C67×#3083C214yesmaximal, complemented
8C67×#3080C212yesmaximal, complemented
9C67×#2960C211yesmaximal, characteristic, complemented
10C67×#2952C212yesmaximal
11C4×#135×C67C211yesmaximal, characteristic
12C2×#135×C67C2211yescharacteristic
13#557×C67C2218yes—
14#554×C67—28no—
15#554×C67C2218yes—
16#544×C67—24no—
17#544×C67C2214yes—
18#533×C67—22no—
19#533×C67C2211yescharacteristic
20C2×#125×C67—22no—
21C2×#125×C67C2212yes—
22#520×C67C2214yes—
23C4×#41×C67C2212yes—
24#501×C67C2218yes—
25#499×C67C2214yes—
26#499×C67C2214yes—
27#499×C67C2214yes—
28#498×C67C2214yes—
29#479×C67C2218yes—
30#479×C67C2218yes—
31#478×C67C2218yes—
32#478×C67C2218yes—
33#477×C67C2218yes—
34#474×C67C2218yes—
35#446×C67C2214yes—
36C8×Q8×C67C2214yes—
37#442×C67C2214yes—
38#441×C67C2214yes—
39C4×Q16×C67C2212yes—
40C4×SD16×C67C2214yes—
41C4×SD16×C67C2218yes—
42C4×D8×C67C2212yes—
43#428×C67C2214yes—
44#427×C67C2211yescharacteristic
45C2×#107×C67C2212yes—
46C2×#106×C67C2212yes—
47#424×C67C2212yes—
48#423×C67C2211yescharacteristic
49#417×C67C2214yes—
50#416×C67C2212yes—
51#405×C67C2212yes—
52C2×C4×C8×C67C2211yescharacteristic
53C2×#41×C67C2312yes—
54#135×C67Q818yescomplemented
55C2×Q16×C67C2311yescharacteristic
56C2×SD16×C67C2312yes—
57C2×D8×C67C2311yescharacteristic
58C2×M16×C67C2312yes—
59C22×C8×C67C2311yescharacteristic
60C22×C8×C67C2311yescharacteristic
61C4⋊Q8×C67—28no—
62C4⋊Q8×C67—24no—
63C4⋊Q8×C67D412yes—
64C4⋊Q8×C67C2318yes—
65#125×C67—24no—
66#125×C67—28no—
67#125×C67—28no—
68#125×C67—22no—
69#125×C67—22no—
70#125×C67D412yes—
71#125×C67C2318yes—
72#125×C67C2318yes—
73C22⋊Q8×C67—28no—
74C22⋊Q8×C67—28no—
75C22⋊Q8×C67—28no—
76C22⋊Q8×C67—28no—
77C22⋊Q8×C67D414yes—
78C4×Q8×C67—28no—
79C4×Q8×C67C2318yes—
80C4×Q8×C67C2314yes—
81C4×Q8×C67C2314yes—
82C4×D4×C67—24no—
83C4×D4×C67—28no—
84C4×D4×C67C2318yes—
85C4×D4×C67C2314yes—
86#117×C67—24no—
87#117×C67—22no—
88#117×C67C2312yes—
89#117×C67C2312yes—
90C2×C4⋊C4×C67—24no—
91C2×C4⋊C4×C67—24no—
92C2×C4⋊C4×C67—28no—
93C2×C4⋊C4×C67C2314yes—
94C2×C42×C67—22no—
95C2×C42×C67C2311yescharacteristic
96#107×C67C2314yes—
97#107×C67C2318yes—
98#107×C67C2314yes—
99#107×C67C2312yes—
100#106×C67C2318yes—
101#106×C67C2312yes—
102#106×C67C2314yes—
103#106×C67C2314yes—
104C4⋊C8×C67C2318yes—
105C4⋊C8×C67C2314yes—
106#103×C67C2314yes—
107#103×C67C2318yes—
108#102×C67C2314yes—
109#102×C67C2318yes—
110#97×C67C2314yes—
111C4×C8×C67C2314yes—
112C4×C8×C67C2312yes—
113C4×C8×C67C2312yes—
114#41×C67—24no—
115#41×C67C2×Q818yes—
116C2×Q8×C67—28no—
117C2×Q8×C67C2×D414yes—
118C2×Q8×C67C2412yes—
119C2×D4×C67—22no—
120C2×D4×C67C2412yes—
121C22×C4×C67—22no—
122C22×C4×C67—24no—
123C22×C4×C67—24no—
124C22×C4×C67—28no—
125C22×C4×C67C2×D412yes—
126C22×C4×C67C2411yescharacteristic
127Q16×C67C2×Q814yes—
128SD16×C67C2×Q818yes—
129D8×C67C2×Q814yes—
130M16×C67—24no—
131C2×C8×C67—22no—
132C2×C8×C67C2×Q814yes—
133C2×C8×C67C2414yes—
134C2×C8×C67C2412yes—
135C2×C8×C67C2411yescharacteristic
136C2×C8×C67C2411yescharacteristic
137C4⋊C4×C67—28no—
138C4⋊C4×C67—24no—
139C4⋊C4×C67—28no—
140C4⋊C4×C67—28no—
141C4⋊C4×C67—28no—
142C4⋊C4×C67—28no—
143C4⋊C4×C67—24no—
144C4⋊C4×C67—28no—
145C4⋊C4×C67C2×D414yes—
146C4⋊C4×C67C2×D418yes—
147C4⋊C4×C67C2×D414yes—
148C4⋊C4×C67C2418yes—
149C4⋊C4×C67C2414yes—
150C22⋊C4×C67—22no—
151C22⋊C4×C67—24no—
152C22⋊C4×C67—28no—
153C22⋊C4×C67C2×D414yes—
154C42×C67—22no—
155C42×C67—24no—
156C42×C67—28no—
157C42×C67C2×D411yescharacteristic
158C42×C67C2×D411yescharacteristic
159C42×C67C2412yes—
160C42×C67C2414yes—
161C23×C67—22no—
162C23×C67C22×D411yescharacteristic
163Q8×C67—22no—
164Q8×C67—44no—
165Q8×C67—88no—
166Q8×C67C22×Q814yes—
167D4×C67—28no—
168D4×C67—22no—
169D4×C67C22×Q814yes—
170C2×C4×C67—22no—
171C2×C4×C67—28no—
172C2×C4×C67—24no—
173C2×C4×C67—28no—
174C2×C4×C67—42no—
175C2×C4×C67—84no—
176C2×C4×C67—88no—
177C2×C4×C67—84no—
178C2×C4×C67C22×D412yes—
179C2×C4×C67C22×D418yes—
180C2×C4×C67C22×D412yes—
181C2×C4×C67C22×D411yescharacteristic
182C2×C4×C67C22×Q812yes—
183C2×C4×C672−1+412yes—
184C2×C4×C67C2511yescharacteristic
185C8×C67—22no—
186C8×C67—24no—
187C8×C67C22×Q812yes—
188C8×C67C22×Q812yes—
189C22×C67—22no—
190C22×C67—24no—
191C22×C67—44no—
192C22×C67D4×Q812yes—
193C22×C67C23×D411yescharacteristic
194C4×C67—22no—
195C4×C67—22no—
196C4×C67—28no—
197C4×C67—24no—
198C4×C67—48no—
199C4×C67—44no—
200C4×C67D4×Q812yes—
201C4×C67C23×Q811yescharacteristic
202C4×C67C2×2−1+411yescharacteristic
203#61375C6711yesmaximal, characteristic, direct product factor, Sylow
204C2×C67—22no—
205C2×C67—44no—
206C2×C67C2×D4×Q811yescharacteristic
207C2×C67#358412yes—
208#3504C2×C6712yescomplemented
209#3390C2×C6714yescomplemented
210#3389C2×C6714yescomplemented
211#3381C2×C6718yescomplemented
212#3163C2×C6712yescomplemented
213C2×#499C2×C6711yescharacteristic, complemented
214#3083C2×C6714yescomplemented
215#3080C2×C6712yescomplemented
216#2960C2×C6711yescharacteristic, complemented
217#2952C2×C6712yes—
218C4×#135C2×C6711yescharacteristic
219C67#6137511yescharacteristic, direct product factor, Sylow
220C2×#135C22×C6711yescharacteristic
221#557C22×C6718yes—
222#554—28no—
223#554C22×C6718yes—
224#544—24no—
225#544C22×C6714yes—
226#533—22no—
227#533C22×C6711yescharacteristic
228C2×#125—22no—
229C2×#125C22×C6712yes—
230#520C22×C6714yes—
231C4×#41C22×C6712yes—
232#501C22×C6718yes—
233#499C22×C6714yes—
234#499C22×C6714yes—
235#499C22×C6714yes—
236#498C22×C6714yes—
237#479C22×C6718yes—
238#479C22×C6718yes—
239#478C22×C6718yes—
240#478C22×C6718yes—
241#477C22×C6718yes—
242#474C22×C6718yes—
243#446C22×C6714yes—
244C8×Q8C22×C6714yes—
245#442C22×C6714yes—
246#441C22×C6714yes—
247C4×Q16C22×C6712yes—
248C4×SD16C22×C6714yes—
249C4×SD16C22×C6718yes—
250C4×D8C22×C6712yes—
251#428C22×C6714yes—
252#427C22×C6711yescharacteristic
253C2×#107C22×C6712yes—
254C2×#106C22×C6712yes—
255#424C22×C6712yes—
256#423C22×C6711yescharacteristic
257#417C22×C6714yes—
258#416C22×C6712yes—
259#405C22×C6712yes—
260C2×C4×C8C22×C6711yescharacteristic
261C2×#41C23×C6712yes—
262#135Q8×C6718yescomplemented
263C2×Q16C23×C6711yescharacteristic
264C2×SD16C23×C6712yes—
265C2×D8C23×C6711yescharacteristic
266C2×M16C23×C6712yes—
267C22×C8C23×C6711yescharacteristic
268C22×C8C23×C6711yescharacteristic
269C4⋊Q8—28no—
270C4⋊Q8—24no—
271C4⋊Q8D4×C6712yes—
272C4⋊Q8C23×C6718yes—
273#125—24no—
274#125—28no—
275#125—28no—
276#125—22no—
277#125—22no—
278#125D4×C6712yes—
279#125C23×C6718yes—
280#125C23×C6718yes—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8D4×C6714yes—
286C4×Q8—28no—
287C4×Q8C23×C6714yes—
288C4×Q8C23×C6718yes—
289C4×Q8C23×C6714yes—
290C4×D4—24no—
291C4×D4—28no—
292C4×D4C23×C6714yes—
293C4×D4C23×C6718yes—
294#117—24no—
295#117—22no—
296#117C23×C6712yes—
297#117C23×C6712yes—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4C23×C6714yes—
302C2×C42—22no—
303C2×C42C23×C6711yescharacteristic
304#107C23×C6714yes—
305#107C23×C6714yes—
306#107C23×C6712yes—
307#107C23×C6718yes—
308#106C23×C6714yes—
309#106C23×C6714yes—
310#106C23×C6712yes—
311#106C23×C6718yes—
312C4⋊C8C23×C6714yes—
313C4⋊C8C23×C6718yes—
314#103C23×C6714yes—
315#103C23×C6718yes—
316#102C23×C6714yes—
317#102C23×C6718yes—
318#97C23×C6714yes—
319C4×C8C23×C6712yes—
320C4×C8C23×C6712yes—
321C4×C8C23×C6714yes—
322#41—24no—
323#41C2×Q8×C6718yes—
324C2×Q8—28no—
325C2×Q8C2×D4×C6714yes—
326C2×Q8C24×C6712yes—
327C2×D4—22no—
328C2×D4C24×C6712yes—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4C2×D4×C6712yes—
334C22×C4C24×C6711yescharacteristic
335Q16C2×Q8×C6714yes—
336SD16C2×Q8×C6718yes—
337D8C2×Q8×C6714yes—
338M16—24no—
339C2×C8—22no—
340C2×C8C2×Q8×C6714yes—
341C2×C8C24×C6712yes—
342C2×C8C24×C6714yes—
343C2×C8C24×C6711yescharacteristic
344C2×C8C24×C6711yescharacteristic
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4C2×D4×C6714yes—
354C4⋊C4C2×D4×C6718yes—
355C4⋊C4C2×D4×C6714yes—
356C4⋊C4C24×C6718yes—
357C4⋊C4C24×C6714yes—
358C22⋊C4—22no—
359C22⋊C4—24no—
360C22⋊C4—28no—
361C22⋊C4C2×D4×C6714yes—
362C42—22no—
363C42—24no—
364C42—28no—
365C42C2×D4×C6711yescharacteristic
366C42C2×D4×C6711yescharacteristic
367C42C24×C6712yes—
368C42C24×C6714yes—
369C23—22no—
370C23C22×D4×C6711yescharacteristic
371Q8—22no—
372Q8—44no—
373Q8—88no—
374Q8C22×Q8×C6714yes—
375D4—28no—
376D4—22no—
377D4C22×Q8×C6714yes—
378C2×C4—22no—
379C2×C4—28no—
380C2×C4—24no—
381C2×C4—28no—
382C2×C4—42no—
383C2×C4—84no—
384C2×C4—88no—
385C2×C4—84no—
386C2×C4C22×D4×C6712yes—
387C2×C4C22×D4×C6718yes—
388C2×C4C22×D4×C6712yes—
389C2×C4C22×D4×C6711yescharacteristic
390C2×C4C22×Q8×C6712yes—
391C2×C42−1+4×C6712yes—
392C2×C4C25×C6711yescharacteristic
393C8—22no—
394C8—24no—
395C8C22×Q8×C6712yes—
396C8C22×Q8×C6712yes—
397C22—22no—
398C22—24no—
399C22—44no—
400C22D4×Q8×C6712yes—
401C22C23×D4×C6711yescharacteristic
402C4—22no—
403C4—22no—
404C4—28no—
405C4—24no—
406C4—48no—
407C4—44no—
408C4D4×Q8×C6712yes—
409C4C23×Q8×C6711yescharacteristic
410C4C2×2−1+4×C6711yescharacteristic
411C2—22no—
412C2—44no—
413C2C2×D4×Q8×C6711yescharacteristic
414C2C67×#358412yes—
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
prime radicalC2×#135×C67C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ #533×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3504C2semidirect
2. C67×#3390C2semidirect
3. C67×#3389C2semidirect
4. C67×#3381C2semidirect
5. C67×#3163C2semidirect
6. C2×#499×C67C2semidirect
7. C67×#3083C2semidirect
8. C67×#3080C2semidirect
9. C67×#2960C2semidirect
54. #135×C67Q8semidirect
203. #61375C67direct (N × H)
208. #3504C2×C67semidirect
209. #3390C2×C67semidirect
210. #3389C2×C67semidirect
211. #3381C2×C67semidirect
212. #3163C2×C67semidirect
213. C2×#499C2×C67semidirect
214. #3083C2×C67semidirect
215. #3080C2×C67semidirect
216. #2960C2×C67semidirect
219. C67#61375direct (N × H)
262. #135Q8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#35042
C67×#33902
C67×#33892
C67×#33812
C67×#31632
C2×#499×C672
C67×#30832
C67×#30802
C67×#29602
C67×#29522
C4×#135×C672
#6137567