Finite Group Databasebeta

← C67×#7204 #30686455

Subgroups — C67×#7204 #30686455

Order 17152.

380 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C67×#1869C211yesmaximal, characteristic
2C4×#99×C67C211yesmaximal, characteristic
3C67×#1460C211yesmaximal, characteristic
4C23×C8×C67C411yescharacteristic, complemented
5C2×C4×D4×C67C2211yescharacteristic
6#431×C67—21no—
7C2×#99×C67C412yes—
8C2×C22⋊C8×C67—21no—
9C2×C22⋊C8×C67C411yescharacteristic, complemented
10C4×C22⋊C4×C67—21no—
11#342×C67—21no—
12#336×C67—21no—
13#323×C67—21no—
14#323×C67C412yescomplemented
15C22×D4×C67C2×C411yescharacteristic
16C23×C4×C67C2×C411yescharacteristic
17C23×C4×C67D411yescharacteristic
18C22×C8×C67—21no—
19C22×C8×C67—21no—
20C22×C8×C67—42no—
21C22×C8×C67—41no—
22C4×D4×C67—41no—
23C4×D4×C67—41no—
24C2×C4⋊C4×C67Q811yescharacteristic
25C2×C22⋊C4×C67—22no—
26C2×C22⋊C4×C67C2×C411yescharacteristic
27C2×C22⋊C4×C67D411yescharacteristic
28C2×C42×C67—41no—
29C2×C42×C67D411yescharacteristic
30C4⋊C8×C67—41no—
31#99×C67—22no—
32#99×C67C814yescomplemented
33C22⋊C8×C67—22no—
34C22⋊C8×C67—42no—
35C22⋊C8×C67—41no—
36C4×C8×C67—41no—
37#95×C67—41no—
38C24×C67C4211yescharacteristic
39C24×C67C22⋊C411yescharacteristic
40C2×D4×C67—41no—
41C2×D4×C67—41no—
42C2×D4×C67C2×C812yes—
43C2×D4×C67M1612yes—
44C22×C4×C67—21no—
45C22×C4×C67—21no—
46C22×C4×C67—21no—
47C22×C4×C67—21no—
48C22×C4×C67—41no—
49C22×C4×C67—41no—
50C22×C4×C67—41no—
51C22×C4×C67—41no—
52C22×C4×C67—42no—
53C22×C4×C67C22⋊C411yescharacteristic
54C22×C4×C67C22⋊C411yescharacteristic
55C22×C4×C67C4⋊C411yescharacteristic
56C22×C4×C67C4⋊C411yescharacteristic
57C22×C4×C67C4⋊C411yescharacteristic
58C2×C8×C67—22no—
59C2×C8×C67—41no—
60C2×C8×C67—42no—
61C2×C8×C67—41no—
62C2×C8×C67—42no—
63C2×C8×C67—42no—
64C2×C8×C67—41no—
65C2×C8×C67—41no—
66C2×C8×C67—41no—
67C4⋊C4×C67—41no—
68C22⋊C4×C67—24no—
69C22⋊C4×C67—41no—
70C22⋊C4×C67—41no—
71C22⋊C4×C67—42no—
72C42×C67—21no—
73C42×C67—21no—
74C42×C67—81no—
75C42×C67—81no—
76C23×C67—21no—
77C23×C67—21no—
78C23×C67—21no—
79C23×C67—21no—
80C23×C67—41no—
81C23×C67—41no—
82C23×C67—41no—
83C23×C67—41no—
84C23×C67#9511yescharacteristic
85C23×C67C4×C811yescharacteristic
86C23×C67#9711yescharacteristic
87C23×C67C22⋊C812yes—
88D4×C67—42no—
89D4×C67—42no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—42no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—44no—
106C2×C4×C67—81no—
107C2×C4×C67—82no—
108C2×C4×C67—82no—
109C2×C4×C67#9911yescharacteristic
110C2×C4×C67#9911yescharacteristic
111C2×C4×C67C4⋊C812yes—
112C8×C67—42no—
113C8×C67—81no—
114C8×C67—82no—
115C22×C67—21no—
116C22×C67—21no—
117C22×C67—22no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67—41no—
121C22×C67—41no—
122C22×C67—42no—
123C22×C67—42no—
124C22×C67—41no—
125C22×C67—41no—
126C22×C67—41no—
127C22×C67—41no—
128C22×C67—81no—
129C22×C67#33611yescharacteristic
130C22×C67#34211yescharacteristic
131C22×C67#34411yescharacteristic
132C4×C67—22no—
133C4×C67—21no—
134C4×C67—41no—
135C4×C67—41no—
136C4×C67—41no—
137C4×C67—84no—
138C4×C67AΣL(1,𝔽16)11yescharacteristic
139C4×C67#35211yescharacteristic
140#7204C6711yesmaximal, characteristic, direct product factor, Sylow
141C2×C67—21no—
142C2×C67—21no—
143C2×C67—42no—
144C2×C67—41no—
145C2×C67—41no—
146C2×C67#128211yescharacteristic
147C2×C67#139011yescharacteristic
148C2×C67#139111yescharacteristic
149#1869C2×C6711yescharacteristic
150C4×#99C2×C6711yescharacteristic
151#1460C2×C6711yescharacteristic
152C67#720411yescharacteristic, direct product factor, Sylow
153C23×C8C4×C6711yescharacteristic, complemented
154C2×C4×D4C22×C6711yescharacteristic
155#431—21no—
156C2×#99C4×C6712yes—
157C2×C22⋊C8—21no—
158C2×C22⋊C8C4×C6711yescharacteristic, complemented
159C4×C22⋊C4—21no—
160#342—21no—
161#336—21no—
162#323—21no—
163#323C4×C6712yescomplemented
164C22×D4C2×C4×C6711yescharacteristic
165C23×C4C2×C4×C6711yescharacteristic
166C23×C4D4×C6711yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—42no—
170C22×C8—41no—
171C4×D4—41no—
172C4×D4—41no—
173C2×C4⋊C4Q8×C6711yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4C2×C4×C6711yescharacteristic
176C2×C22⋊C4D4×C6711yescharacteristic
177C2×C42—41no—
178C2×C42D4×C6711yescharacteristic
179C4⋊C8—41no—
180#99—22no—
181#99C8×C6714yescomplemented
182C22⋊C8—22no—
183C22⋊C8—42no—
184C22⋊C8—41no—
185C4×C8—41no—
186#95—41no—
187C24C42×C6711yescharacteristic
188C24C22⋊C4×C6711yescharacteristic
189C2×D4—41no—
190C2×D4—41no—
191C2×D4C2×C8×C6712yes—
192C2×D4M16×C6712yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4C22⋊C4×C6711yescharacteristic
203C22×C4C22⋊C4×C6711yescharacteristic
204C22×C4C4⋊C4×C6711yescharacteristic
205C22×C4C4⋊C4×C6711yescharacteristic
206C22×C4C4⋊C4×C6711yescharacteristic
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C4⋊C4—41no—
217C22⋊C4—24no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—42no—
221C42—21no—
222C42—21no—
223C42—81no—
224C42—81no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23#95×C6711yescharacteristic
234C23C4×C8×C6711yescharacteristic
235C23#97×C6711yescharacteristic
236C23C22⋊C8×C6712yes—
237D4—42no—
238D4—42no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#99×C6711yescharacteristic
259C2×C4#99×C6711yescharacteristic
260C2×C4C4⋊C8×C6712yes—
261C8—42no—
262C8—81no—
263C8—82no—
264C22—21no—
265C22—21no—
266C22—22no—
267C22—21no—
268C22—21no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22#336×C6711yescharacteristic
279C22#342×C6711yescharacteristic
280C22#344×C6711yescharacteristic
281C4—22no—
282C4—21no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—84no—
287C4AΣL(1,𝔽16)×C6711yescharacteristic
288C4#352×C6711yescharacteristic
289C2—21no—
290C2—21no—
291C2—42no—
292C2—41no—
293C2—41no—
294C2C67×#128211yescharacteristic
295C2C67×#139011yescharacteristic
296C2C67×#139111yescharacteristic
subgroupG/H
centerC2×C4×C67#99
commutatorC23C4×C8×C67
Frattini, cosocleC2×C4×D4C22×C67
socleC22×C67#342
prime radicalC22×D4×C67C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ C22×C4×C67 ◁ C2×C4×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×C8×C67C4semidirect
9. C2×C22⋊C8×C67C4semidirect
14. #323×C67C4semidirect
32. #99×C67C8semidirect
140. #7204C67direct (N × H)
152. C67#7204direct (N × H)
153. C23×C8C4×C67semidirect
158. C2×C22⋊C8C4×C67semidirect
163. #323C4×C67semidirect
181. #99C8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#18692
C4×#99×C672
C67×#14602
#720467