Finite Group Databasebeta

← C67×#32881 #30712132

Subgroups — C67×#32881 #30712132

Order 17152.

5942 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 5942 conjugacy classes of subgroups — too many to draw. Only its 438 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×D4×C67C211yesmaximal, characteristic
2C2×#450×C67C216yesmaximal, complemented
3C67×#2848C211yesmaximal, characteristic, complemented
4C67×#1786C216yesmaximal, complemented
5C67×#1779C211yesmaximal, characteristic, complemented
6C26×C67—21no—
7C23×D4×C67—224no—
8C23×D4×C67—23no—
9C23×D4×C67C2216yes—
10C24×C4×C67C2211yescharacteristic
11C22×SD16×C67C2214yescomplemented
12C2×C22⋊Q8×C67C2213yescomplemented
13#450×C67—224no—
14C2×#102×C67C22112yescomplemented
15C2×C22⋊C8×C67C2216yescomplemented
16#385×C67—23no—
17#380×C67C2213yescomplemented
18#379×C67—21no—
19C25×C67—224no—
20C25×C67—26no—
21C25×C67—412no—
22C25×C67—44no—
23C25×C67D411yescharacteristic
24C22×Q8×C67C2311yescharacteristic, complemented
25C22×D4×C67—2128no—
26C22×D4×C67—224no—
27C22×D4×C67—248no—
28C22×D4×C67—248no—
29C22×D4×C67—26no—
30C22×D4×C67—212no—
31C22×D4×C67D4124yes—
32C22×D4×C67C2314yes—
33C23×C4×C67—212no—
34C23×C4×C67D413yes—
35C23×C4×C67C2313yes—
36C2×SD16×C67—224no—
37C22×C8×C67C2314yescomplemented
38C22⋊Q8×C67—212no—
39C2×C4⋊C4×C67—23no—
40C2×C4⋊C4×C67C2313yescomplemented
41C2×C22⋊C4×C67—23no—
42C2×C22⋊C4×C67—23no—
43#102×C67—224no—
44C22⋊C8×C67—212no—
45#95×C67—23no—
46C24×C67—26no—
47C24×C67—26no—
48C24×C67—264no—
49C24×C67—248no—
50C24×C67—224no—
51C24×C67—24no—
52C24×C67—496no—
53C24×C67—424no—
54C24×C67—496no—
55C24×C67—412no—
56C24×C67—412no—
57C24×C67SD1614yescomplemented
58C24×C67C2×D413yes—
59C2×Q8×C67—26no—
60C2×D4×C67—2192no—
61C2×D4×C67—248no—
62C2×D4×C67—296no—
63C2×D4×C67—296no—
64C2×D4×C67—224no—
65C2×D4×C67—212no—
66C2×D4×C67C2×D4124yes—
67C22×C4×C67—232no—
68C22×C4×C67—26no—
69C22×C4×C67—212no—
70C22×C4×C67—212no—
71C22×C4×C67—21no—
72C22×C4×C67—23no—
73C22×C4×C67C2×D413yes—
74C22×C4×C67C2×D4112yes—
75C22×C4×C67C2411yescharacteristic
76SD16×C67—416no—
77C2×C8×C67—212no—
78C4⋊C4×C67—26no—
79C4⋊C4×C67—46no—
80C22⋊C4×C67—46no—
81C22⋊C4×C67—46no—
82C23×C67—28no—
83C23×C67—224no—
84C23×C67—26no—
85C23×C67—26no—
86C23×C67—224no—
87C23×C67—296no—
88C23×C67—224no—
89C23×C67—4256no—
90C23×C67—496no—
91C23×C67—448no—
92C23×C67—496no—
93C23×C67—44no—
94C23×C67—412no—
95C23×C67—448no—
96C23×C67#12016yes—
97C23×C67C2×SD1616yes—
98C23×C67AGL(1,ℤ/8)16yes—
99C23×C67C22×D411yescharacteristic
100Q8×C67—44no—
101D4×C67—296no—
102D4×C67—224no—
103D4×C67#120116yes—
104C2×C4×C67—224no—
105C2×C4×C67—26no—
106C2×C4×C67—224no—
107C2×C4×C67—43no—
108C2×C4×C67—43no—
109C2×C4×C67—46no—
110C2×C4×C67#12016yes—
111C2×C4×C67C22×D416yes—
112C8×C67—44no—
113C22×C67—232no—
114C22×C67—26no—
115C22×C67—212no—
116C22×C67—26no—
117C22×C67—212no—
118C22×C67—216no—
119C22×C67—4192no—
120C22×C67—424no—
121C22×C67—424no—
122C22×C67#450112yescomplemented
123C22×C67C2×#12013yes—
124C22×C67C22×SD1611yescharacteristic
125C22×C67C2×AGL(1,ℤ/8)13yes—
126C4×C67—26no—
127C4×C67—81no—
128C4×C67—83no—
129C4×C67C2×#12014yes—
130#32881C6711yesmaximal, characteristic, direct product factor, Sylow
131C2×C67—212no—
132C2×C67—416no—
133C2×C67#284611yescharacteristic
134C2×C67C2×#45013yes—
135C2×C67#300313yes—
136C24×D4C2×C6711yescharacteristic
137C2×#450C2×C6716yescomplemented
138#2848C2×C6711yescharacteristic, complemented
139#1786C2×C6716yescomplemented
140#1779C2×C6711yescharacteristic, complemented
141C67#3288111yescharacteristic, direct product factor, Sylow
142C26—21no—
143C23×D4—224no—
144C23×D4—23no—
145C23×D4C22×C6716yes—
146C24×C4C22×C6711yescharacteristic
147C22×SD16C22×C6714yescomplemented
148C2×C22⋊Q8C22×C6713yescomplemented
149#450—224no—
150C2×#102C22×C67112yescomplemented
151C2×C22⋊C8C22×C6716yescomplemented
152#385—23no—
153#380C22×C6713yescomplemented
154#379—21no—
155C25—224no—
156C25—26no—
157C25—412no—
158C25—44no—
159C25D4×C6711yescharacteristic
160C22×Q8C23×C6711yescharacteristic, complemented
161C22×D4—2128no—
162C22×D4—224no—
163C22×D4—248no—
164C22×D4—248no—
165C22×D4—26no—
166C22×D4—212no—
167C22×D4D4×C67124yes—
168C22×D4C23×C6714yes—
169C23×C4—212no—
170C23×C4D4×C6713yes—
171C23×C4C23×C6713yes—
172C2×SD16—224no—
173C22×C8C23×C6714yescomplemented
174C22⋊Q8—212no—
175C2×C4⋊C4—23no—
176C2×C4⋊C4C23×C6713yescomplemented
177C2×C22⋊C4—23no—
178C2×C22⋊C4—23no—
179#102—224no—
180C22⋊C8—212no—
181#95—23no—
182C24—26no—
183C24—26no—
184C24—264no—
185C24—248no—
186C24—224no—
187C24—24no—
188C24—496no—
189C24—496no—
190C24—424no—
191C24—412no—
192C24—412no—
193C24SD16×C6714yescomplemented
194C24C2×D4×C6713yes—
195C2×Q8—26no—
196C2×D4—2192no—
197C2×D4—248no—
198C2×D4—296no—
199C2×D4—296no—
200C2×D4—224no—
201C2×D4—212no—
202C2×D4C2×D4×C67124yes—
203C22×C4—232no—
204C22×C4—26no—
205C22×C4—212no—
206C22×C4—212no—
207C22×C4—23no—
208C22×C4—21no—
209C22×C4C2×D4×C6713yes—
210C22×C4C2×D4×C67112yes—
211C22×C4C24×C6711yescharacteristic
212SD16—416no—
213C2×C8—212no—
214C4⋊C4—26no—
215C4⋊C4—46no—
216C22⋊C4—46no—
217C22⋊C4—46no—
218C23—28no—
219C23—224no—
220C23—224no—
221C23—26no—
222C23—26no—
223C23—296no—
224C23—224no—
225C23—4256no—
226C23—496no—
227C23—448no—
228C23—496no—
229C23—448no—
230C23—44no—
231C23—412no—
232C23#120×C6716yes—
233C23C2×SD16×C6716yes—
234C23AGL(1,ℤ/8)×C6716yes—
235C23C22×D4×C6711yescharacteristic
236Q8—44no—
237D4—296no—
238D4—224no—
239D4#120×C67116yes—
240C2×C4—224no—
241C2×C4—26no—
242C2×C4—224no—
243C2×C4—43no—
244C2×C4—43no—
245C2×C4—46no—
246C2×C4#120×C6716yes—
247C2×C4C22×D4×C6716yes—
248C8—44no—
249C22—232no—
250C22—26no—
251C22—212no—
252C22—26no—
253C22—212no—
254C22—216no—
255C22—4192no—
256C22—424no—
257C22—424no—
258C22#450×C67112yescomplemented
259C22C2×#120×C6713yes—
260C22C22×SD16×C6711yescharacteristic
261C22C2×AGL(1,ℤ/8)×C6713yes—
262C4—26no—
263C4—81no—
264C4—83no—
265C4C2×#120×C6714yes—
266C2—212no—
267C2—416no—
268C2C67×#284611yescharacteristic
269C2C2×#450×C6713yes—
270C2C67×#300313yes—
subgroupG/H
center, socleC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC24×D4×C67C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C24×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#450×C67C2semidirect
3. C67×#2848C2semidirect
4. C67×#1786C2semidirect
5. C67×#1779C2semidirect
11. C22×SD16×C67C22semidirect
12. C2×C22⋊Q8×C67C22semidirect
14. C2×#102×C67C22semidirect
15. C2×C22⋊C8×C67C22semidirect
17. #380×C67C22semidirect
24. C22×Q8×C67C23semidirect
37. C22×C8×C67C23semidirect
40. C2×C4⋊C4×C67C23semidirect
57. C24×C67SD16semidirect
122. C22×C67#450semidirect
130. #32881C67direct (N × H)
137. C2×#450C2×C67semidirect
138. #2848C2×C67semidirect
139. #1786C2×C67semidirect
140. #1779C2×C67semidirect
141. C67#32881direct (N × H)
147. C22×SD16C22×C67semidirect
148. C2×C22⋊Q8C22×C67semidirect
150. C2×#102C22×C67semidirect
151. C2×C22⋊C8C22×C67semidirect
153. #380C22×C67semidirect
160. C22×Q8C23×C67semidirect
173. C22×C8C23×C67semidirect
176. C2×C4⋊C4C23×C67semidirect
193. C24SD16×C67semidirect
258. C22#450×C67semidirect

Maximal subgroups

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

subgroupindex
C24×D4×C672
C2×#450×C672
C67×#28482
C67×#17862
C67×#17792
#3288167