Finite Group Databasebeta

← C67×#60333 #30739584

Subgroups — C67×#60333 #30739584

Order 17152.

1926 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×2−1+4×C67C211yesmaximal, characteristic
2C67×#2991C214yesmaximal, complemented
3C67×#2989C216yesmaximal, complemented
4C67×#2984C214yesmaximal, complemented
5C67×#2982C211yesmaximal, characteristic, complemented
6C67×#2971C214yesmaximal, complemented
7C67×#2965C216yesmaximal, complemented
8Q8×M16×C67C214yesmaximal, complemented
9C2×C8×Q8×C67C211yesmaximal, characteristic, complemented
10C2×2−1+4×C67C411yescharacteristic
11#520×C67C414yes—
12#520×C67C416yes—
13#519×C67C414yes—
14#519×C67C411yescharacteristic
15C4×#41×C67C2216yes—
16C4×#41×C67C2214yes—
17C2×C4×Q8×C67C2211yescharacteristic
18C2×C4×Q8×C67C2214yes—
19#446×C67C2216yes—
20#446×C67C22112yes—
21#446×C67C2216yes—
22C8×Q8×C67C2214yes—
23C8×Q8×C67C2214yes—
24C8×Q8×C67C2214yes—
25C8×Q8×C67C2214yes—
26#436×C67C2216yes—
27#436×C67C2216yes—
28#436×C67C22112yes—
29#435×C67C2214yes—
30#435×C67C22112yes—
31#432×C67C22112yes—
32#431×C67C2216yes—
33#431×C67C22112yes—
34#423×C67C22112yes—
35C2×C4⋊C8×C67C2213yes—
36C4×M16×C67C22112yes—
37C2×C4×C8×C67C2213yes—
382−1+4×C67—28no—
39C2×#41×C67C2×C414yes—
40C2×#41×C67C2×C416yes—
41C22×Q8×C67C2×C414yes—
42C22×Q8×C67C2×C411yescharacteristic
43C2×M16×C67C2314yes—
44C2×M16×C67C23112yes—
45C22×C8×C67C2311yescharacteristic
46C22×C8×C67C2313yes—
47C4×Q8×C67C2×C4112yes—
48C4×Q8×C67C2×C414yes—
49C4×Q8×C67C2×C414yes—
50C4×Q8×C67C2×C416yes—
51C4×Q8×C67C2×C414yes—
52C4×Q8×C67C2314yes—
53C4×Q8×C67C2316yes—
54C4×D4×C67C2×C4112yes—
55C4×D4×C67C2×C416yes—
56C4×D4×C67C2×C4112yes—
57C4×D4×C67C2314yes—
58C4×D4×C67C2316yes—
59#117×C67C2×C4112yes—
60#117×C67C2×C4112yes—
61#117×C67C2×C416yes—
62C2×C4⋊C4×C67C2×C4112yes—
63C2×C4⋊C4×C67C2×C413yes—
64C2×C42×C67C2313yes—
65C2×C42×C67C23112yes—
66C4⋊C8×C67C23112yes—
67C4⋊C8×C67C23112yes—
68C4⋊C8×C67C2316yes—
69C4⋊C8×C67C2314yes—
70C4⋊C8×C67C2316yes—
71C22⋊C8×C67C2314yes—
72C22⋊C8×C67C2314yes—
73C22⋊C8×C67C23112yes—
74#97×C67C2314yes—
75#97×C67C23112yes—
76C4×C8×C67C2316yes—
77C4×C8×C67C2316yes—
78C4×C8×C67C23112yes—
79#41×C67—216no—
80#41×C67—224no—
81C2×Q8×C67—216no—
82C2×Q8×C67M1618yescomplemented
83C2×Q8×C67C22×C416yes—
84C2×Q8×C67C22×C414yes—
85C2×D4×C67C22×C414yes—
86C2×D4×C67C22×C416yes—
87C22×C4×C67C22×C4112yes—
88C22×C4×C67C22×C4112yes—
89C22×C4×C67C22×C413yes—
90C22×C4×C67C22×C413yes—
91C22×C4×C67C2411yescharacteristic
92C22×C4×C67C2414yes—
93M16×C67—28no—
94M16×C67—412no—
95C2×C8×C67—22no—
96C2×C8×C67—43no—
97C2×C8×C67C2416yes—
98C2×C8×C67C2414yes—
99C2×C8×C67C2411yescharacteristic
100C2×C8×C67C2414yes—
101C2×C8×C67C2411yescharacteristic
102C4⋊C4×C67C22×C4112yes—
103C4⋊C4×C67C22×C4112yes—
104C4⋊C4×C67C22×C416yes—
105C4⋊C4×C67C22×C414yes—
106C4⋊C4×C67C22×C416yes—
107C22⋊C4×C67C22×C414yes—
108C22⋊C4×C67C22×C4112yes—
109C22⋊C4×C67C22×C414yes—
110C42×C67C22×C4112yes—
111C42×C67C22×C416yes—
112C42×C67C22×C4112yes—
113C42×C67C2416yes—
114C42×C67C2414yes—
115C23×C67C23×C414yes—
116C23×C67C23×C411yescharacteristic
117Q8×C67—212no—
118Q8×C67C2×M16116yes—
119D4×C67—28no—
120D4×C67—212no—
121C2×C4×C67—224no—
122C2×C4×C67—21no—
123C2×C4×C67—24no—
124C2×C4×C67—21no—
125C2×C4×C67—24no—
126C2×C4×C67—43no—
127C2×C4×C67—412no—
128C2×C4×C67C2×M16112yes—
129C2×C4×C67C23×C411yescharacteristic
130C2×C4×C67C23×C414yes—
131C2×C4×C67C23×C416yes—
132C2×C4×C67C23×C414yes—
133C2×C4×C67C23×C414yes—
134C2×C4×C67C23×C416yes—
135C2×C4×C67C2511yescharacteristic
136C8×C67—21no—
137C8×C67—24no—
138C8×C67—26no—
139C8×C67—24no—
140C8×C672−1+412yescomplemented
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—24no—
144C22×C67—24no—
145C22×C67—24no—
146C22×C67C22×M1612yes—
147C22×C67C24×C411yescharacteristic
148C4×C67—21no—
149C4×C67—24no—
150C4×C67—24no—
151C4×C67—26no—
152C4×C67—24no—
153C4×C67C22×M16112yes—
154C4×C67C2×2−1+411yescharacteristic
155C4×C67#58511yescharacteristic
156#60333C6711yesmaximal, characteristic, direct product factor, Sylow
157C2×C67—22no—
158C2×C67—44no—
159C2×C67C4×2−1+411yescharacteristic
160C2×C67C23×M1611yescharacteristic
161C2×C67#357311yescharacteristic
162C4×2−1+4C2×C6711yescharacteristic
163#2991C2×C6714yescomplemented
164#2989C2×C6716yescomplemented
165#2984C2×C6714yescomplemented
166#2982C2×C6711yescharacteristic, complemented
167#2971C2×C6714yescomplemented
168#2965C2×C6716yescomplemented
169Q8×M16C2×C6714yescomplemented
170C2×C8×Q8C2×C6711yescharacteristic, complemented
171C67#6033311yescharacteristic, direct product factor, Sylow
172C2×2−1+4C4×C6711yescharacteristic
173#520C4×C6714yes—
174#520C4×C6716yes—
175#519C4×C6714yes—
176#519C4×C6711yescharacteristic
177C4×#41C22×C6716yes—
178C4×#41C22×C6714yes—
179C2×C4×Q8C22×C6711yescharacteristic
180C2×C4×Q8C22×C6714yes—
181#446C22×C6716yes—
182#446C22×C67112yes—
183#446C22×C6716yes—
184C8×Q8C22×C6714yes—
185C8×Q8C22×C6714yes—
186C8×Q8C22×C6714yes—
187C8×Q8C22×C6714yes—
188#436C22×C6716yes—
189#436C22×C6716yes—
190#436C22×C67112yes—
191#435C22×C6714yes—
192#435C22×C67112yes—
193#432C22×C67112yes—
194#431C22×C6716yes—
195#431C22×C67112yes—
196#423C22×C67112yes—
197C2×C4⋊C8C22×C6713yes—
198C4×M16C22×C67112yes—
199C2×C4×C8C22×C6713yes—
2002−1+4—28no—
201C2×#41C2×C4×C6714yes—
202C2×#41C2×C4×C6716yes—
203C22×Q8C2×C4×C6714yes—
204C22×Q8C2×C4×C6711yescharacteristic
205C2×M16C23×C6714yes—
206C2×M16C23×C67112yes—
207C22×C8C23×C6711yescharacteristic
208C22×C8C23×C6713yes—
209C4×Q8C2×C4×C67112yes—
210C4×Q8C2×C4×C6714yes—
211C4×Q8C2×C4×C6714yes—
212C4×Q8C2×C4×C6716yes—
213C4×Q8C2×C4×C6714yes—
214C4×Q8C23×C6714yes—
215C4×Q8C23×C6716yes—
216C4×D4C2×C4×C67112yes—
217C4×D4C2×C4×C6716yes—
218C4×D4C2×C4×C67112yes—
219C4×D4C23×C6714yes—
220C4×D4C23×C6716yes—
221#117C2×C4×C67112yes—
222#117C2×C4×C67112yes—
223#117C2×C4×C6716yes—
224C2×C4⋊C4C2×C4×C67112yes—
225C2×C4⋊C4C2×C4×C6713yes—
226C2×C42C23×C6713yes—
227C2×C42C23×C67112yes—
228C4⋊C8C23×C67112yes—
229C4⋊C8C23×C67112yes—
230C4⋊C8C23×C6716yes—
231C4⋊C8C23×C6714yes—
232C4⋊C8C23×C6716yes—
233C22⋊C8C23×C6714yes—
234C22⋊C8C23×C6714yes—
235C22⋊C8C23×C67112yes—
236#97C23×C6714yes—
237#97C23×C67112yes—
238C4×C8C23×C6716yes—
239C4×C8C23×C6716yes—
240C4×C8C23×C67112yes—
241#41—216no—
242#41—224no—
243C2×Q8—216no—
244C2×Q8M16×C6718yescomplemented
245C2×Q8C22×C4×C6716yes—
246C2×Q8C22×C4×C6714yes—
247C2×D4C22×C4×C6714yes—
248C2×D4C22×C4×C6716yes—
249C22×C4C22×C4×C67112yes—
250C22×C4C22×C4×C67112yes—
251C22×C4C22×C4×C6713yes—
252C22×C4C22×C4×C6713yes—
253C22×C4C24×C6711yescharacteristic
254C22×C4C24×C6714yes—
255M16—28no—
256M16—412no—
257C2×C8—22no—
258C2×C8—43no—
259C2×C8C24×C6711yescharacteristic
260C2×C8C24×C6714yes—
261C2×C8C24×C6714yes—
262C2×C8C24×C6716yes—
263C2×C8C24×C6711yescharacteristic
264C4⋊C4C22×C4×C67112yes—
265C4⋊C4C22×C4×C67112yes—
266C4⋊C4C22×C4×C6716yes—
267C4⋊C4C22×C4×C6714yes—
268C4⋊C4C22×C4×C6716yes—
269C22⋊C4C22×C4×C6714yes—
270C22⋊C4C22×C4×C67112yes—
271C22⋊C4C22×C4×C6714yes—
272C42C22×C4×C67112yes—
273C42C22×C4×C6716yes—
274C42C22×C4×C67112yes—
275C42C24×C6716yes—
276C42C24×C6714yes—
277C23C23×C4×C6714yes—
278C23C23×C4×C6711yescharacteristic
279Q8—212no—
280Q8C2×M16×C67116yes—
281D4—28no—
282D4—212no—
283C2×C4—224no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—24no—
288C2×C4—43no—
289C2×C4—412no—
290C2×C4C2×M16×C67112yes—
291C2×C4C23×C4×C6711yescharacteristic
292C2×C4C23×C4×C6714yes—
293C2×C4C23×C4×C6716yes—
294C2×C4C23×C4×C6714yes—
295C2×C4C23×C4×C6714yes—
296C2×C4C23×C4×C6716yes—
297C2×C4C25×C6711yescharacteristic
298C8—21no—
299C8—24no—
300C8—26no—
301C8—24no—
302C82−1+4×C6712yescomplemented
303C22—21no—
304C22—21no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22C22×M16×C6712yes—
309C22C24×C4×C6711yescharacteristic
310C4—24no—
311C4—26no—
312C4—24no—
313C4—21no—
314C4—24no—
315C4C22×M16×C67112yes—
316C4C2×2−1+4×C6711yescharacteristic
317C4#585×C6711yescharacteristic
318C2—22no—
319C2—44no—
320C2C4×2−1+4×C6711yescharacteristic
321C2C23×M16×C6711yescharacteristic
322C2C67×#357311yescharacteristic
subgroupG/H
centerC2×C4×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC2×2−1+4×C67C4
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×#2989C2semidirect
4. C67×#2984C2semidirect
5. C67×#2982C2semidirect
6. C67×#2971C2semidirect
7. C67×#2965C2semidirect
8. Q8×M16×C67C2semidirect
9. C2×C8×Q8×C67C2semidirect
82. C2×Q8×C67M16semidirect
140. C8×C672−1+4semidirect
156. #60333C67direct (N × H)
163. #2991C2×C67semidirect
164. #2989C2×C67semidirect
165. #2984C2×C67semidirect
166. #2982C2×C67semidirect
167. #2971C2×C67semidirect
168. #2965C2×C67semidirect
169. Q8×M16C2×C67semidirect
170. C2×C8×Q8C2×C67semidirect
171. C67#60333direct (N × H)
244. C2×Q8M16×C67semidirect
302. C82−1+4×C67semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+4×C672
C67×#29912
C67×#29892
C67×#29842
C67×#29822
C67×#29712
C67×#29652
Q8×M16×C672
C2×C8×Q8×C672
#6033367