Finite Group Databasebeta

← C67×#32878 #30712129

Subgroups — C67×#32878 #30712129

Order 17152.

6020 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 6020 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, complemented
2C2×#447×C67C216yesmaximal, complemented
3C67×#2847C211yesmaximal, 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×D8×C67C2214yescomplemented
12C2×#121×C67C2213yescomplemented
13C2×#120×C67—21no—
14#447×C67—224no—
15C2×#102×C67C22112yescomplemented
16C2×C22⋊C8×C67C2216yescomplemented
17#386×C67—23no—
18#380×C67C2213yescomplemented
19C25×C67—224no—
20C25×C67—26no—
21C25×C67—412no—
22C25×C67—44no—
23C25×C67D411yescharacteristic
24C22×D4×C67—2128no—
25C22×D4×C67—224no—
26C22×D4×C67—248no—
27C22×D4×C67—248no—
28C22×D4×C67—23no—
29C22×D4×C67—26no—
30C22×D4×C67—212no—
31C22×D4×C67D4124yescomplemented
32C22×D4×C67C2314yes—
33C22×D4×C67C2311yescharacteristic, complemented
34C23×C4×C67—212no—
35C23×C4×C67D413yes—
36C23×C4×C67C2313yes—
37C2×D8×C67—224no—
38C22×C8×C67C2314yescomplemented
39#121×C67—212no—
40#120×C67—44no—
41C2×C4⋊C4×C67C2313yescomplemented
42C2×C22⋊C4×C67—23no—
43C2×C22⋊C4×C67—23no—
44#102×C67—224no—
45C22⋊C8×C67—212no—
46#95×C67—23no—
47C24×C67—26no—
48C24×C67—26no—
49C24×C67—264no—
50C24×C67—248no—
51C24×C67—224no—
52C24×C67—24no—
53C24×C67—21no—
54C24×C67—496no—
55C24×C67—424no—
56C24×C67—496no—
57C24×C67—412no—
58C24×C67—412no—
59C24×C67D814yescomplemented
60C24×C67C2×D413yes—
61C2×D4×C67—2192no—
62C2×D4×C67—248no—
63C2×D4×C67—296no—
64C2×D4×C67—296no—
65C2×D4×C67—224no—
66C2×D4×C67—212no—
67C2×D4×C67—26no—
68C2×D4×C67—46no—
69C2×D4×C67—46no—
70C2×D4×C67—46no—
71C2×D4×C67C2×D4124yes—
72C22×C4×C67—232no—
73C22×C4×C67—26no—
74C22×C4×C67—212no—
75C22×C4×C67—212no—
76C22×C4×C67—23no—
77C22×C4×C67C2×D413yes—
78C22×C4×C67C2×D4112yes—
79C22×C4×C67C2411yescharacteristic
80D8×C67—416no—
81C2×C8×C67—212no—
82C4⋊C4×C67—26no—
83C22⋊C4×C67—46no—
84C22⋊C4×C67—46no—
85C23×C67—28no—
86C23×C67—224no—
87C23×C67—26no—
88C23×C67—26no—
89C23×C67—224no—
90C23×C67—296no—
91C23×C67—224no—
92C23×C67—4256no—
93C23×C67—496no—
94C23×C67—448no—
95C23×C67—496no—
96C23×C67—44no—
97C23×C67—412no—
98C23×C67—448no—
99C23×C67—43no—
100C23×C67—41no—
101C23×C67—43no—
102C23×C67#12016yes—
103C23×C67C2×D816yes—
104C23×C67AGL(1,ℤ/8)16yes—
105C23×C67C22×D411yescharacteristic
106D4×C67—296no—
107D4×C67—224no—
108D4×C67—44no—
109D4×C67—812no—
110D4×C67#120116yescomplemented
111C2×C4×C67—224no—
112C2×C4×C67—26no—
113C2×C4×C67—224no—
114C2×C4×C67—43no—
115C2×C4×C67—43no—
116C2×C4×C67—43no—
117C2×C4×C67#12016yes—
118C2×C4×C67C22×D416yes—
119C8×C67—44no—
120C22×C67—232no—
121C22×C67—26no—
122C22×C67—212no—
123C22×C67—26no—
124C22×C67—212no—
125C22×C67—216no—
126C22×C67—4192no—
127C22×C67—424no—
128C22×C67—424no—
129C22×C67—81no—
130C22×C67—83no—
131C22×C67—83no—
132C22×C67#447112yescomplemented
133C22×C67C2×#12013yes—
134C22×C67C22×D811yescharacteristic
135C22×C67C2×AGL(1,ℤ/8)13yes—
136C4×C67—26no—
137C4×C67—83no—
138C4×C67C2×#12014yes—
139#32878C6711yesmaximal, characteristic, direct product factor, Sylow
140C2×C67—212no—
141C2×C67—416no—
142C2×C67—161no—
143C2×C67#284611yescharacteristic
144C2×C67C2×#44713yes—
145C2×C67#300313yes—
146C24×D4C2×C6711yescharacteristic, complemented
147C2×#447C2×C6716yescomplemented
148#2847C2×C6711yescharacteristic, complemented
149#1786C2×C6716yescomplemented
150#1779C2×C6711yescharacteristic, complemented
151C67#3287811yescharacteristic, direct product factor, Sylow
152C26—21no—
153C23×D4—224no—
154C23×D4—23no—
155C23×D4C22×C6716yes—
156C24×C4C22×C6711yescharacteristic
157C22×D8C22×C6714yescomplemented
158C2×#121C22×C6713yescomplemented
159C2×#120—21no—
160#447—224no—
161C2×#102C22×C67112yescomplemented
162C2×C22⋊C8C22×C6716yescomplemented
163#386—23no—
164#380C22×C6713yescomplemented
165C25—224no—
166C25—26no—
167C25—412no—
168C25—44no—
169C25D4×C6711yescharacteristic
170C22×D4—2128no—
171C22×D4—224no—
172C22×D4—248no—
173C22×D4—248no—
174C22×D4—23no—
175C22×D4—26no—
176C22×D4—212no—
177C22×D4D4×C67124yescomplemented
178C22×D4C23×C6714yes—
179C22×D4C23×C6711yescharacteristic, complemented
180C23×C4—212no—
181C23×C4D4×C6713yes—
182C23×C4C23×C6713yes—
183C2×D8—224no—
184C22×C8C23×C6714yescomplemented
185#121—212no—
186#120—44no—
187C2×C4⋊C4C23×C6713yescomplemented
188C2×C22⋊C4—23no—
189C2×C22⋊C4—23no—
190#102—224no—
191C22⋊C8—212no—
192#95—23no—
193C24—26no—
194C24—26no—
195C24—264no—
196C24—248no—
197C24—224no—
198C24—24no—
199C24—21no—
200C24—496no—
201C24—424no—
202C24—496no—
203C24—412no—
204C24—412no—
205C24D8×C6714yescomplemented
206C24C2×D4×C6713yes—
207C2×D4—2192no—
208C2×D4—248no—
209C2×D4—296no—
210C2×D4—296no—
211C2×D4—224no—
212C2×D4—212no—
213C2×D4—26no—
214C2×D4—46no—
215C2×D4—46no—
216C2×D4—46no—
217C2×D4C2×D4×C67124yes—
218C22×C4—232no—
219C22×C4—26no—
220C22×C4—212no—
221C22×C4—212no—
222C22×C4—23no—
223C22×C4C2×D4×C6713yes—
224C22×C4C2×D4×C67112yes—
225C22×C4C24×C6711yescharacteristic
226D8—416no—
227C2×C8—212no—
228C4⋊C4—26no—
229C22⋊C4—46no—
230C22⋊C4—46no—
231C23—28no—
232C23—224no—
233C23—26no—
234C23—26no—
235C23—224no—
236C23—296no—
237C23—224no—
238C23—4256no—
239C23—496no—
240C23—448no—
241C23—496no—
242C23—44no—
243C23—412no—
244C23—448no—
245C23—43no—
246C23—41no—
247C23—43no—
248C23#120×C6716yes—
249C23C2×D8×C6716yes—
250C23AGL(1,ℤ/8)×C6716yes—
251C23C22×D4×C6711yescharacteristic
252D4—296no—
253D4—224no—
254D4—44no—
255D4—812no—
256D4#120×C67116yescomplemented
257C2×C4—224no—
258C2×C4—26no—
259C2×C4—224no—
260C2×C4—43no—
261C2×C4—43no—
262C2×C4—43no—
263C2×C4#120×C6716yes—
264C2×C4C22×D4×C6716yes—
265C8—44no—
266C22—232no—
267C22—26no—
268C22—212no—
269C22—26no—
270C22—212no—
271C22—216no—
272C22—4192no—
273C22—424no—
274C22—424no—
275C22—81no—
276C22—83no—
277C22—83no—
278C22#447×C67112yescomplemented
279C22C2×#120×C6713yes—
280C22C22×D8×C6711yescharacteristic
281C22C2×AGL(1,ℤ/8)×C6713yes—
282C4—26no—
283C4—83no—
284C4C2×#120×C6714yes—
285C2—212no—
286C2—416no—
287C2—161no—
288C2C67×#284611yescharacteristic
289C2C2×#447×C6713yes—
290C2C67×#300313yes—
subgroupG/H
center, socleC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
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
1. C24×D4×C67C2semidirect
2. C2×#447×C67C2semidirect
3. C67×#2847C2semidirect
4. C67×#1786C2semidirect
5. C67×#1779C2semidirect
11. C22×D8×C67C22semidirect
12. C2×#121×C67C22semidirect
15. C2×#102×C67C22semidirect
16. C2×C22⋊C8×C67C22semidirect
18. #380×C67C22semidirect
31. C22×D4×C67D4semidirect
33. C22×D4×C67C23semidirect
38. C22×C8×C67C23semidirect
41. C2×C4⋊C4×C67C23semidirect
59. C24×C67D8semidirect
110. D4×C67#120semidirect
132. C22×C67#447semidirect
139. #32878C67direct (N × H)
146. C24×D4C2×C67semidirect
147. C2×#447C2×C67semidirect
148. #2847C2×C67semidirect
149. #1786C2×C67semidirect
150. #1779C2×C67semidirect
151. C67#32878direct (N × H)
157. C22×D8C22×C67semidirect
158. C2×#121C22×C67semidirect
161. C2×#102C22×C67semidirect
162. C2×C22⋊C8C22×C67semidirect
164. #380C22×C67semidirect
177. C22×D4D4×C67semidirect
179. C22×D4C23×C67semidirect
184. C22×C8C23×C67semidirect
187. C2×C4⋊C4C23×C67semidirect
205. C24D8×C67semidirect
256. D4#120×C67semidirect
278. C22#447×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×#447×C672
C67×#28472
C67×#17862
C67×#17792
#3287867