Finite Group Databasebeta

← #30765239

Subgroups — #30765239

Order 17152.

434 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 99 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#22558539C212yesmaximal
2#22558155C212yesmaximal, complemented
3#22558153C212yesmaximal, complemented
4C67×#2317C211yesmaximal, characteristic
5#16863360—22no—
6#16863122—22no—
7#16863122C2214yes—
8C2×#117×C67C411yescharacteristic
9C2×#117×C67C411yescharacteristic
10C22×C4⋊C4×C67C2211yescharacteristic
11#389×C67—22no—
12#385×C67—22no—
13#384×C67C411yescharacteristic
14#384×C67C2211yescharacteristic
15#382×C67C411yescharacteristic
16#382×C67C2211yescharacteristic
17C2×#11761612—42no—
18C22⋊(C67⋊C8)—24no—
19C22⋊(C67⋊C8)—42no—
20C23×C4×C67C2×C411yescharacteristic
21C23×C4×C67D412yes—
22#117×C67—22no—
23#117×C67—22no—
24#117×C67—42no—
25C2×C4⋊C4×C67—21no—
26C2×C4⋊C4×C67—22no—
27C2×C4⋊C4×C67—22no—
28C2×C4⋊C4×C67—42no—
29C2×C4⋊C4×C67C2×C411yescharacteristic
30C2×C4⋊C4×C67C2×C411yescharacteristic
31C2×C4⋊C4×C67C2×C411yescharacteristic
32C2×C4⋊C4×C67D412yes—
33C2×C4⋊C4×C67C2311yescharacteristic
34C2×C22⋊C4×C67—21no—
35C2×C22⋊C4×C67—21no—
36C2×C42×C67—21no—
37C2×C42×C67C2×C411yescharacteristic
38C2×C42×C67C2×C411yescharacteristic
39#95×C67—22no—
40#95×C67—22no—
41#11761612—84no—
42C2×C67⋊C8—44no—
43C24×C67C22⋊C411yescharacteristic
44C22×C4×C67—22no—
45C22×C4×C67—22no—
46C22×C4×C67—21no—
47C22×C4×C67—21no—
48C22×C4×C67—22no—
49C22×C4×C67—21no—
50C22×C4×C67—21no—
51C22×C4×C67—22no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67—82no—
55C22×C4×C67C22⋊C411yescharacteristic
56C22×C4×C67C22⋊C412yes—
57C22×C4×C67C22×C411yescharacteristic
58C22×C4×C67C2×D412yes—
59C4⋊C4×C67—24no—
60C4⋊C4×C67—41no—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—41no—
63C4⋊C4×C67—41no—
64C4⋊C4×C67—41no—
65C4⋊C4×C67—82no—
66C4⋊C4×C67—82no—
67C22⋊C4×C67—24no—
68C22⋊C4×C67—42no—
69C42×C67—21no—
70C42×C67—21no—
71C42×C67—42no—
72C42×C67M1612yescomplemented
73C42×C67M1612yescomplemented
74C23×C67—21no—
75C23×C67—21no—
76C23×C67—41no—
77C23×C67—41no—
78C23×C67#9912yes—
79C23×C67C2×C22⋊C411yescharacteristic
80C2×C4×C67—24no—
81C2×C4×C67—24no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—22no—
87C2×C4×C67—22no—
88C2×C4×C67—22no—
89C2×C4×C67—22no—
90C2×C4×C67—41no—
91C2×C4×C67—41no—
92C2×C4×C67—41no—
93C2×C4×C67—41no—
94C2×C4×C67—44no—
95C2×C4×C67—42no—
96C2×C4×C67—42no—
97C2×C4×C67—42no—
98C2×C4×C67—42no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—82no—
102C2×C4×C67#10112yes—
103C2×C4×C67C2×M1611yescharacteristic
104C2×C4×C67C2×M1611yescharacteristic
105C67⋊C8—84no—
106C22×C67—21no—
107C22×C67—21no—
108C22×C67—22no—
109C22×C67—22no—
110C22×C67—22no—
111C22×C67—42no—
112C22×C67—42no—
113C22×C67#40711yescharacteristic
114C22×C67C2×#9911yescharacteristic
115C22×C67C2×#10111yescharacteristic
116C4×C67—22no—
117C4×C67—21no—
118C4×C67—21no—
119C4×C67—42no—
120C4×C67—42no—
121C4×C67—42no—
122C4×C67—42no—
123C4×C67—81no—
124C4×C67—81no—
125#11514—671nomaximal, Sylow
126C2×C67—21no—
127C2×C67—21no—
128C2×C67—42no—
129C2×C67#146311yescharacteristic
130C2×C67#212711yescharacteristic
131C2×C67#213311yescharacteristic
132#2317D6711yescharacteristic
133#1463—672no—
134#1325—672no—
135#1324—672no—
136C67#1151411yescharacteristic, complemented, Sylow
137C2×#117Dic6711yescharacteristic
138C2×#117Dic6711yescharacteristic
139C22×C4⋊C4C2×D6711yescharacteristic
140#407—1342no—
141#389—22no—
142#385—22no—
143#384Dic6711yescharacteristic
144#384C2×D6711yescharacteristic
145#382Dic6711yescharacteristic
146#382C2×D6711yescharacteristic
147#324—674no—
148#324—1342no—
149C23×C4C2×Dic6711yescharacteristic
150C23×C4C22⋊D6712yes—
151C2×M16—2682no—
152#117—22no—
153#117—22no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C2×Dic6711yescharacteristic
160C2×C4⋊C4C2×Dic6711yescharacteristic
161C2×C4⋊C4C2×Dic6711yescharacteristic
162C2×C4⋊C4C22⋊D6712yes—
163C2×C4⋊C4C22×D6711yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42—21no—
167C2×C42C2×Dic6711yescharacteristic
168C2×C42C2×Dic6711yescharacteristic
169C22⋊C8—2682no—
170C22⋊C8—1344no—
171#95—22no—
172#95—22no—
173C24C22⋊Dic6711yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—82no—
185C22×C4C22⋊Dic6711yescharacteristic
186C22×C4C22⋊Dic6712yes—
187C22×C4C22×Dic6711yescharacteristic
188C22×C4C2×C22⋊D6712yes—
189M16—5364no—
190C2×C8—2684no—
191C4⋊C4—24no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C42—21no—
202C42—21no—
203C42—42no—
204C42#1176161212yescomplemented
205C42#1176161212yescomplemented
206C23—21no—
207C23—21no—
208C23—41no—
209C23—41no—
210C23#1452395812yes—
211C23C2×C22⋊Dic6711yescharacteristic
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—44no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4#1452396212yes—
235C2×C4C2×#1176161211yescharacteristic
236C2×C4C2×#1176161211yescharacteristic
237C8—5364no—
238C22—21no—
239C22—21no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22#1686336011yescharacteristic
246C22C2×#1452395811yescharacteristic
247C22C2×#1452396211yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C2—21no—
258C2—21no—
259C2—42no—
260C2#2255853911yescharacteristic
261C2#2256187811yescharacteristic
262C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2317C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2317D67
hypocenterC67#11514
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #22558155C2semidirect
3. #22558153C2semidirect
72. C42×C67M16semidirect
73. C42×C67M16semidirect
136. C67#11514semidirect
204. C42#11761612semidirect
205. C42#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585392
#225581552
#225581532
C67×#23172
#1151467