Finite Group Databasebeta

← C67×#34973 #30714224

Subgroups — C67×#34973 #30714224

Order 17152.

1716 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2524C214yesmaximal, complemented
2C67×#2519C214yesmaximal
3C67×#2469C214yesmaximal
4C67×#2462C214yesmaximal, complemented
5C67×#2402C211yesmaximal, characteristic, complemented
6C67×#2399C211yesmaximal, characteristic, complemented
7C67×#2396C212yesmaximal
8C4×#125×C67C214yesmaximal, complemented
9C4×#126×C67C214yesmaximal
10C67×#2298C211yesmaximal, characteristic
11C42×Q8×C67C212yesmaximal, complemented
12C2×#126×C67C2214yes—
13C2×#125×C67C2214yes—
14C2×C4×Q8×C67C2212yes—
15C2×C4×Q8×C67C2214yes—
16C2×#117×C67C2211yescharacteristic
17#400×C67C2212yes—
18#400×C67C2214yes—
19#399×C67C2212yes—
20#398×C67C2214yes—
21#398×C67C2214yes—
22#398×C67C2214yes—
23#396×C67C2214yes—
24#395×C67C2212yes—
25#395×C67C2214yes—
26#393×C67C2212yes—
27#389×C67C2214yes—
28#389×C67C2214yes—
29#388×C67C2214yes—
30#388×C67C2214yes—
31#388×C67C2214yes—
32#388×C67C2214yes—
33#387×C67C2214yes—
34#387×C67C2214yes—
35#387×C67C2214yes—
36#387×C67C2214yes—
37#387×C67C2214yes—
38#387×C67C2214yes—
39#387×C67C2214yes—
40#387×C67C2214yes—
41#383×C67C2214yes—
42#382×C67C2214yes—
43C4×C4⋊C4×C67C2214yes—
44C4×C4⋊C4×C67C2214yes—
45C4×C4⋊C4×C67C2214yes—
46C4×C4⋊C4×C67C2214yes—
47C4×C4⋊C4×C67C2212yes—
48C4×C4⋊C4×C67C2212yes—
49C4×C4⋊C4×C67C2214yes—
50C4×C4⋊C4×C67C2214yes—
51C4×C4⋊C4×C67C2214yes—
52C4×C22⋊C4×C67C2212yes—
53C4×C22⋊C4×C67C2214yes—
54#376×C67C2211yescharacteristic
55C43×C67C2211yescharacteristic
56C43×C67C2214yes—
57C22×Q8×C67C2312yes—
58C23×C4×C67C2311yescharacteristic
59#126×C67—216no—
60#125×C67—216no—
61C4×Q8×C67—216no—
62C4×Q8×C67—28no—
63#117×C67—24no—
64C2×C4⋊C4×C67C2312yes—
65C2×C4⋊C4×C67C2314yes—
66C2×C4⋊C4×C67C2314yes—
67C2×C4⋊C4×C67C2314yes—
68C2×C4⋊C4×C67C2312yes—
69C2×C4⋊C4×C67C2314yes—
70C2×C4⋊C4×C67C2314yes—
71C2×C4⋊C4×C67C2314yes—
72C2×C4⋊C4×C67C2314yes—
73C2×C4⋊C4×C67C2314yes—
74C2×C4⋊C4×C67C2314yes—
75C2×C4⋊C4×C67C2314yes—
76C2×C4⋊C4×C67C2314yes—
77C2×C22⋊C4×C67C2312yes—
78C2×C22⋊C4×C67C2314yes—
79C2×C22⋊C4×C67C2314yes—
80C2×C22⋊C4×C67C2314yes—
81C2×C42×C67C2314yes—
82C2×C42×C67C2314yes—
83C2×C42×C67C2314yes—
84C2×C42×C67C2314yes—
85C2×C42×C67C2314yes—
86C2×C42×C67C2311yescharacteristic
87C2×C42×C67C2314yes—
88C2×C42×C67C2312yes—
89C2×C42×C67C2314yes—
90C2×C42×C67C2312yes—
91C2×C42×C67C2311yescharacteristic
92#95×C67C2314yes—
93#95×C67C2314yes—
94#95×C67C2314yes—
95#95×C67C2314yes—
96#95×C67C2314yes—
97#95×C67C2314yes—
98#95×C67C2312yes—
99#95×C67C2312yes—
100#95×C67C2314yes—
101#95×C67C2314yes—
102#95×C67C2314yes—
103#95×C67C2314yes—
104#95×C67C2314yes—
105#95×C67C2314yes—
106#95×C67C2314yes—
107C24×C67C2411yescharacteristic
108C2×Q8×C67—28no—
109C2×Q8×C67—24no—
110C22×C4×C67—22no—
111C22×C4×C67—24no—
112C22×C4×C67C2414yes—
113C22×C4×C67C2414yes—
114C22×C4×C67C2412yes—
115C22×C4×C67C2414yes—
116C22×C4×C67C2411yescharacteristic
117C22×C4×C67C2414yes—
118C22×C4×C67C2414yes—
119C22×C4×C67C2414yes—
120C22×C4×C67C2412yes—
121C22×C4×C67C2411yescharacteristic
122C4⋊C4×C67—24no—
123C4⋊C4×C67—28no—
124C4⋊C4×C67—28no—
125C4⋊C4×C67—28no—
126C4⋊C4×C67—24no—
127C4⋊C4×C67—28no—
128C4⋊C4×C67—28no—
129C4⋊C4×C67—28no—
130C4⋊C4×C67—28no—
131C4⋊C4×C67—28no—
132C4⋊C4×C67—28no—
133C4⋊C4×C67—28no—
134C4⋊C4×C67#41116yes—
135C22⋊C4×C67—24no—
136C22⋊C4×C67—28no—
137C22⋊C4×C67—28no—
138C22⋊C4×C67—28no—
139C42×C67—22no—
140C42×C67—28no—
141C42×C67—28no—
142C42×C67—28no—
143C42×C67—28no—
144C42×C67—28no—
145C42×C67—28no—
146C42×C67—24no—
147C42×C67—28no—
148C42×C67—24no—
149C42×C67#4114yes—
150C23×C67—24no—
151C23×C67—21no—
152C23×C67—22no—
153C23×C67C2511yescharacteristic
154Q8×C67—216no—
155C2×C4×C67—24no—
156C2×C4×C67—24no—
157C2×C4×C67—24no—
158C2×C4×C67—24no—
159C2×C4×C67—24no—
160C2×C4×C67—24no—
161C2×C4×C67—24no—
162C2×C4×C67—22no—
163C2×C4×C67—24no—
164C2×C4×C67—24no—
165C2×C4×C67—24no—
166C2×C4×C67—24no—
167C2×C4×C67—24no—
168C2×C4×C67—24no—
169C2×C4×C67—24no—
170C2×C4×C67—24no—
171C2×C4×C67—21no—
172C2×C4×C67—24no—
173C2×C4×C67—24no—
174C2×C4×C67—22no—
175C2×C4×C67—44no—
176C2×C4×C67C2×#4118yes—
177C2×C4×C67C2×#4118yes—
178C2×C4×C67C2×#4118yes—
179C2×C4×C67C2×#4114yes—
180C2×C4×C67C2×#4112yes—
181C2×C4×C672−1+414yes—
182C22×C67—42no—
183C22×C67—41no—
184C22×C67—44no—
185C22×C67C22×#4111yescharacteristic
186C22×C67C22×#4114yes—
187C22×C67C2×2−1+412yes—
188C4×C67—28no—
189C4×C67—24no—
190C4×C67—28no—
191C4×C67—28no—
192C4×C67—44no—
193C4×C67—42no—
194C4×C67—41no—
195C4×C67—44no—
196C4×C67—44no—
197C4×C67#54114yes—
198#34973C6711yesmaximal, characteristic, direct product factor, Sylow
199C2×C67—81no—
200C2×C67#344412yes—
201C2×C67C2×#54111yescharacteristic
202C2×C67#348214yes—
203#2524C2×C6714yescomplemented
204#2519C2×C6714yes—
205#2469C2×C6714yes—
206#2462C2×C6714yescomplemented
207#2402C2×C6711yescharacteristic, complemented
208#2399C2×C6711yescharacteristic, complemented
209#2396C2×C6712yes—
210C4×#125C2×C6714yescomplemented
211C4×#126C2×C6714yes—
212#2298C2×C6711yescharacteristic
213C42×Q8C2×C6712yescomplemented
214C67#3497311yescharacteristic, direct product factor, Sylow
215C2×#126C22×C6714yes—
216C2×#125C22×C6714yes—
217C2×C4×Q8C22×C6712yes—
218C2×C4×Q8C22×C6714yes—
219C2×#117C22×C6711yescharacteristic
220#400C22×C6712yes—
221#400C22×C6714yes—
222#399C22×C6712yes—
223#398C22×C6714yes—
224#398C22×C6714yes—
225#398C22×C6714yes—
226#396C22×C6714yes—
227#395C22×C6712yes—
228#395C22×C6714yes—
229#393C22×C6712yes—
230#389C22×C6714yes—
231#389C22×C6714yes—
232#388C22×C6714yes—
233#388C22×C6714yes—
234#388C22×C6714yes—
235#388C22×C6714yes—
236#387C22×C6714yes—
237#387C22×C6714yes—
238#387C22×C6714yes—
239#387C22×C6714yes—
240#387C22×C6714yes—
241#387C22×C6714yes—
242#387C22×C6714yes—
243#387C22×C6714yes—
244#383C22×C6714yes—
245#382C22×C6714yes—
246C4×C4⋊C4C22×C6714yes—
247C4×C4⋊C4C22×C6714yes—
248C4×C4⋊C4C22×C6714yes—
249C4×C4⋊C4C22×C6714yes—
250C4×C4⋊C4C22×C6712yes—
251C4×C4⋊C4C22×C6712yes—
252C4×C4⋊C4C22×C6714yes—
253C4×C4⋊C4C22×C6714yes—
254C4×C4⋊C4C22×C6714yes—
255C4×C22⋊C4C22×C6714yes—
256C4×C22⋊C4C22×C6712yes—
257#376C22×C6711yescharacteristic
258C43C22×C6714yes—
259C43C22×C6711yescharacteristic
260C22×Q8C23×C6712yes—
261C23×C4C23×C6711yescharacteristic
262#126—216no—
263#125—216no—
264C4×Q8—216no—
265C4×Q8—28no—
266#117—24no—
267C2×C4⋊C4C23×C6712yes—
268C2×C4⋊C4C23×C6714yes—
269C2×C4⋊C4C23×C6714yes—
270C2×C4⋊C4C23×C6714yes—
271C2×C4⋊C4C23×C6712yes—
272C2×C4⋊C4C23×C6714yes—
273C2×C4⋊C4C23×C6714yes—
274C2×C4⋊C4C23×C6714yes—
275C2×C4⋊C4C23×C6714yes—
276C2×C4⋊C4C23×C6714yes—
277C2×C4⋊C4C23×C6714yes—
278C2×C4⋊C4C23×C6714yes—
279C2×C4⋊C4C23×C6714yes—
280C2×C22⋊C4C23×C6712yes—
281C2×C22⋊C4C23×C6714yes—
282C2×C22⋊C4C23×C6714yes—
283C2×C22⋊C4C23×C6714yes—
284C2×C42C23×C6711yescharacteristic
285C2×C42C23×C6714yes—
286C2×C42C23×C6714yes—
287C2×C42C23×C6714yes—
288C2×C42C23×C6714yes—
289C2×C42C23×C6714yes—
290C2×C42C23×C6714yes—
291C2×C42C23×C6712yes—
292C2×C42C23×C6714yes—
293C2×C42C23×C6712yes—
294C2×C42C23×C6711yescharacteristic
295#95C23×C6714yes—
296#95C23×C6714yes—
297#95C23×C6714yes—
298#95C23×C6714yes—
299#95C23×C6714yes—
300#95C23×C6714yes—
301#95C23×C6712yes—
302#95C23×C6712yes—
303#95C23×C6714yes—
304#95C23×C6714yes—
305#95C23×C6714yes—
306#95C23×C6714yes—
307#95C23×C6714yes—
308#95C23×C6714yes—
309#95C23×C6714yes—
310C24C24×C6711yescharacteristic
311C2×Q8—28no—
312C2×Q8—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4C24×C6711yescharacteristic
316C22×C4C24×C6714yes—
317C22×C4C24×C6714yes—
318C22×C4C24×C6714yes—
319C22×C4C24×C6712yes—
320C22×C4C24×C6714yes—
321C22×C4C24×C6714yes—
322C22×C4C24×C6714yes—
323C22×C4C24×C6712yes—
324C22×C4C24×C6711yescharacteristic
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—24no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—28no—
337C4⋊C4#41×C67116yes—
338C22⋊C4—24no—
339C22⋊C4—28no—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C42—22no—
343C42—28no—
344C42—28no—
345C42—28no—
346C42—28no—
347C42—28no—
348C42—28no—
349C42—24no—
350C42—28no—
351C42—24no—
352C42#41×C6714yes—
353C23—24no—
354C23—21no—
355C23—22no—
356C23C25×C6711yescharacteristic
357Q8—216no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—44no—
379C2×C4C2×#41×C6718yes—
380C2×C4C2×#41×C6718yes—
381C2×C4C2×#41×C6718yes—
382C2×C4C2×#41×C6714yes—
383C2×C4C2×#41×C6712yes—
384C2×C42−1+4×C6714yes—
385C22—42no—
386C22—41no—
387C22—44no—
388C22C22×#41×C6711yescharacteristic
389C22C22×#41×C6714yes—
390C22C2×2−1+4×C6712yes—
391C4—28no—
392C4—24no—
393C4—28no—
394C4—28no—
395C4—44no—
396C4—42no—
397C4—41no—
398C4—44no—
399C4—44no—
400C4#541×C6714yes—
401C2—81no—
402C2C67×#344412yes—
403C2C2×#541×C6711yescharacteristic
404C2C67×#348214yes—
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
prime radicalC24×C67C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2524C2semidirect
4. C67×#2462C2semidirect
5. C67×#2402C2semidirect
6. C67×#2399C2semidirect
8. C4×#125×C67C2semidirect
11. C42×Q8×C67C2semidirect
198. #34973C67direct (N × H)
203. #2524C2×C67semidirect
206. #2462C2×C67semidirect
207. #2402C2×C67semidirect
208. #2399C2×C67semidirect
210. C4×#125C2×C67semidirect
213. C42×Q8C2×C67semidirect
214. C67#34973direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#25242
C67×#25192
C67×#24692
C67×#24622
C67×#24022
C67×#23992
C67×#23962
C4×#125×C672
C4×#126×C672
C67×#22982
C42×Q8×C672
#3497367