Finite Group Databasebeta

← #30764989

Subgroups — #30764989

Order 17152.

462 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#22558523C212yesmaximal
2#22558155C212yesmaximal, complemented
3#22558153C212yesmaximal, complemented
4C67×#2285C211yesmaximal, characteristic
5#16863449—22no—
6#16863122C2214yes—
7#16863120—22no—
8C2×#117×C67C411yescharacteristic
9C2×#117×C67C2211yescharacteristic
10C22×C42×C67C411yescharacteristic
11#384×C67C411yescharacteristic
12#384×C67C2211yescharacteristic
13#382×C67C411yescharacteristic
14#382×C67C2211yescharacteristic
15#380×C67—22no—
16C4×C4⋊C4×C67—22no—
17C4⋊(C67⋊C8)—42no—
18C22⋊(C67⋊C8)—24no—
19C4×C67⋊C8—42no—
20C23×C4×C67—21no—
21C23×C4×C67C2×C411yescharacteristic
22#117×C67—41no—
23#117×C67—41no—
24#117×C67—41no—
25#117×C67—41no—
26C2×C4⋊C4×C67—22no—
27C2×C4⋊C4×C67—22no—
28C2×C4⋊C4×C67C2×C411yescharacteristic
29C2×C4⋊C4×C67C2×C411yescharacteristic
30C2×C4⋊C4×C67C2×C411yescharacteristic
31C2×C4⋊C4×C67C2311yescharacteristic
32C2×C22⋊C4×C67—21no—
33C2×C22⋊C4×C67D412yes—
34C2×C42×C67—21no—
35C2×C42×C67—22no—
36C2×C42×C67—21no—
37C2×C42×C67C814yescomplemented
38C2×C42×C67C2×C411yescharacteristic
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67D412yes—
41#95×C67—22no—
42#95×C67—22no—
43C2×C67⋊C8—44no—
44C24×C67C22⋊C411yescharacteristic
45C22×C4×C67—21no—
46C22×C4×C67—21no—
47C22×C4×C67—21no—
48C22×C4×C67—22no—
49C22×C4×C67—24no—
50C22×C4×C67—21no—
51C22×C4×C67—21no—
52C22×C4×C67—21no—
53C22×C4×C67—42no—
54C22×C4×C67—42no—
55C22×C4×C67C22⋊C412yes—
56C22×C4×C67C22⋊C411yescharacteristic
57C22×C4×C67C2×C812yes—
58C22×C4×C67M1612yes—
59C22×C4×C67C22×C411yescharacteristic
60C22×C4×C67C2×D412yes—
61C4⋊C4×C67—41no—
62C4⋊C4×C67—41no—
63C4⋊C4×C67—41no—
64C4⋊C4×C67—41no—
65C4⋊C4×C67—44no—
66C4⋊C4×C67—44no—
67C22⋊C4×C67—42no—
68C22⋊C4×C67—42no—
69C42×C67—24no—
70C42×C67—22no—
71C42×C67—22no—
72C42×C67—21no—
73C42×C67—21no—
74C42×C67—41no—
75C42×C67—41no—
76C42×C67C2×C812yes—
77C42×C67C2×C812yes—
78C23×C67—21no—
79C23×C67—21no—
80C23×C67—41no—
81C23×C67—41no—
82C23×C67C22⋊C812yes—
83C23×C67C2×C22⋊C411yescharacteristic
84C2×C4×C67—22no—
85C2×C4×C67—22no—
86C2×C4×C67—22no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—22no—
92C2×C4×C67—22no—
93C2×C4×C67—22no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—48no—
97C2×C4×C67—41no—
98C2×C4×C67—41no—
99C2×C4×C67—42no—
100C2×C4×C67—42no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67C22⋊C812yes—
106C2×C4×C67#9911yescharacteristic
107C2×C4×C67#9911yescharacteristic
108C2×C4×C67#10112yes—
109C2×C4×C67C22×C811yescharacteristic
110C2×C4×C67C2×M1611yescharacteristic
111C67⋊C8—84no—
112C22×C67—21no—
113C22×C67—21no—
114C22×C67—22no—
115C22×C67—42no—
116C22×C67—42no—
117C22×C67—41no—
118C22×C67—41no—
119C22×C67C2×C22⋊C811yescharacteristic
120C22×C67C2×#9911yescharacteristic
121C22×C67C2×#10111yescharacteristic
122C4×C67—24no—
123C4×C67—21no—
124C4×C67—21no—
125C4×C67—42no—
126C4×C67—42no—
127C4×C67—42no—
128C4×C67—81no—
129C4×C67—81no—
130C4×C67#32412yes—
131C4×C67#32412yes—
132#11447—671nomaximal, Sylow
133C2×C67—21no—
134C2×C67—21no—
135C2×C67—42no—
136C2×C67C2×#32411yescharacteristic
137C2×C67#212611yescharacteristic
138C2×C67#213211yescharacteristic
139#2285D6711yescharacteristic
140#1458—672no—
141#1325—672no—
142#1324—672no—
143C67#1144711yescharacteristic, complemented, Sylow
144C2×#117Dic6711yescharacteristic
145C2×#117C2×D6711yescharacteristic
146C22×C42Dic6711yescharacteristic
147#431—1342no—
148#384Dic6711yescharacteristic
149#384C2×D6711yescharacteristic
150#382Dic6711yescharacteristic
151#382C2×D6711yescharacteristic
152#380—22no—
153C4×C4⋊C4—22no—
154#324—674no—
155#323—1342no—
156C23×C4—21no—
157C23×C4C2×Dic6711yescharacteristic
158#117—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C2×Dic6711yescharacteristic
165C2×C4⋊C4C2×Dic6711yescharacteristic
166C2×C4⋊C4C2×Dic6711yescharacteristic
167C2×C4⋊C4C22×D6711yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4C22⋊D6712yes—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42C67⋊C814yescomplemented
174C2×C42C2×Dic6711yescharacteristic
175C2×C42C2×Dic6711yescharacteristic
176C2×C42C22⋊D6712yes—
177C4⋊C8—2682no—
178C22⋊C8—1344no—
179C4×C8—2682no—
180#95—22no—
181#95—22no—
182C24C22⋊Dic6711yescharacteristic
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4C22⋊Dic6712yes—
194C22×C4C22⋊Dic6711yescharacteristic
195C22×C4C2×C67⋊C812yes—
196C22×C4#1176161212yes—
197C22×C4C22×Dic6711yescharacteristic
198C22×C4C2×C22⋊D6712yes—
199C2×C8—2684no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—24no—
209C42—22no—
210C42—22no—
211C42—21no—
212C42—21no—
213C42—41no—
214C42—41no—
215C42C2×C67⋊C812yes—
216C42C2×C67⋊C812yes—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23C22⋊(C67⋊C8)12yes—
222C23C2×C22⋊Dic6711yescharacteristic
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—48no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4C22⋊(C67⋊C8)12yes—
245C2×C4#1452395811yescharacteristic
246C2×C4#1452395811yescharacteristic
247C2×C4#1452396212yes—
248C2×C4C22×C67⋊C811yescharacteristic
249C2×C4C2×#1176161211yescharacteristic
250C8—5364no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—41no—
258C22C2×C22⋊(C67⋊C8)11yescharacteristic
259C22C2×#1452395811yescharacteristic
260C22C2×#1452396211yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—42no—
267C4—81no—
268C4—81no—
269C4#1686312212yes—
270C4#1686312212yes—
271C2—21no—
272C2—21no—
273C2—42no—
274C2C2×#1686312211yescharacteristic
275C2#2256187411yescharacteristic
276C2#2256189411yescharacteristic
subgroupG/H
centerC2×C4#14523958
commutatorC2×C4×C67C22×C8
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2285C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2285D67
hypocenterC67#11447
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #22558155C2semidirect
3. #22558153C2semidirect
37. C2×C42×C67C8semidirect
143. C67#11447semidirect
173. C2×C42C67⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225585232
#225581552
#225581532
C67×#22852
#1144767