Finite Group Databasebeta

← #30765228

Subgroups — #30765228

Order 17152.

490 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 107 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#22558153C214yesmaximal, complemented
3C2×#382×C67C211yesmaximal, characteristic
4#16863360—22no—
5#16863122—22no—
6#16863122C2214yes—
7C22×C4⋊C4×C67C411yescharacteristic
8C22×C4⋊C4×C67C2211yescharacteristic
9C22×C42×C67C411yescharacteristic
10#382×C67—22no—
11#382×C67C412yes—
12#382×C67C2212yes—
13C2×#95×C67—22no—
14C2×#11761612—42no—
15C22⋊(C67⋊C8)—24no—
16C22⋊(C67⋊C8)—42no—
17C23×C4×C67—21no—
18C23×C4×C67—21no—
19C23×C4×C67C2×C411yescharacteristic
20C23×C4×C67D412yes—
21C2×C4⋊C4×C67—21no—
22C2×C4⋊C4×C67—41no—
23C2×C4⋊C4×C67—41no—
24C2×C4⋊C4×C67—41no—
25C2×C4⋊C4×C67—41no—
26C2×C4⋊C4×C67C2×C412yes—
27C2×C4⋊C4×C67C2×C411yescharacteristic
28C2×C4⋊C4×C67D412yes—
29C2×C4⋊C4×C67C2311yescharacteristic
30C2×C42×C67—21no—
31C2×C42×C67—24no—
32C2×C42×C67C2×C412yes—
33#95×C67—24no—
34#95×C67—24no—
35#11761612—84no—
36C2×C67⋊C8—44no—
37C24×C67C22⋊C411yescharacteristic
38C22×C4×C67—21no—
39C22×C4×C67—21no—
40C22×C4×C67—22no—
41C22×C4×C67—22no—
42C22×C4×C67—24no—
43C22×C4×C67—22no—
44C22×C4×C67—22no—
45C22×C4×C67—22no—
46C22×C4×C67—22no—
47C22×C4×C67—42no—
48C22×C4×C67—42no—
49C22×C4×C67—44no—
50C22×C4×C67—44no—
51C22×C4×C67—41no—
52C22×C4×C67—41no—
53C22×C4×C67C22⋊C411yescharacteristic
54C22×C4×C67C22⋊C412yes—
55C22×C4×C67C22×C411yescharacteristic
56C22×C4×C67C2×D412yes—
57C4⋊C4×C67—41no—
58C4⋊C4×C67—42no—
59C4⋊C4×C67—41no—
60C4⋊C4×C67—42no—
61C4⋊C4×C67—42no—
62C42×C67—24no—
63C42×C67—22no—
64C42×C67M1614yescomplemented
65C23×C67—22no—
66C23×C67—22no—
67C23×C67—21no—
68C23×C67—21no—
69C23×C67#9911yescharacteristic
70C23×C67#9911yescharacteristic
71C23×C67C2×C22⋊C411yescharacteristic
72C2×C4×C67—24no—
73C2×C4×C67—24no—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—24no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—48no—
82C2×C4×C67—42no—
83C2×C4×C67—42no—
84C2×C4×C67—44no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67—44no—
88C2×C4×C67—42no—
89C2×C4×C67—82no—
90C2×C4×C67—82no—
91C2×C4×C67—82no—
92C2×C4×C67#10112yes—
93C2×C4×C67C2×M1612yes—
94C67⋊C8—84no—
95C22×C67—24no—
96C22×C67—24no—
97C22×C67—21no—
98C22×C67—21no—
99C22×C67—21no—
100C22×C67—21no—
101C22×C67—21no—
102C22×C67—21no—
103C22×C67#35412yes—
104C22×C67#35512yes—
105C22×C67#40711yescharacteristic
106C22×C67C2×#9911yescharacteristic
107C22×C67C2×#10111yescharacteristic
108C4×C67—22no—
109C4×C67—22no—
110C4×C67—44no—
111C4×C67—42no—
112C4×C67—42no—
113C4×C67—82no—
114#11511—671nomaximal, Sylow
115C2×C67—21no—
116C2×C67—22no—
117C2×C67—22no—
118C2×C67—21no—
119C2×C67#146311yescharacteristic
120C2×C67C2×#35411yescharacteristic
121C2×C67C2×#35511yescharacteristic
122C2×#382D6711yescharacteristic
123#1463—672no—
124#1324—674no—
125C67#1151111yescharacteristic, complemented, Sylow
126C22×C4⋊C4Dic6711yescharacteristic
127C22×C4⋊C4C2×D6711yescharacteristic
128C22×C42Dic6711yescharacteristic
129#407—1342no—
130#382—22no—
131#382Dic6712yes—
132#382C2×D6712yes—
133C2×#95—22no—
134#324—674no—
135#324—1342no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C2×Dic6711yescharacteristic
139C23×C4C22⋊D6712yes—
140C2×M16—2682no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C2×Dic6712yes—
147C2×C4⋊C4C2×Dic6711yescharacteristic
148C2×C4⋊C4C22⋊D6712yes—
149C2×C4⋊C4C22×D6711yescharacteristic
150C2×C42—21no—
151C2×C42—24no—
152C2×C42C2×Dic6712yes—
153C22⋊C8—2682no—
154C22⋊C8—1344no—
155#95—24no—
156#95—24no—
157C24C22⋊Dic6711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4C22⋊Dic6711yescharacteristic
174C22×C4C22⋊Dic6712yes—
175C22×C4C22×Dic6711yescharacteristic
176C22×C4C2×C22⋊D6712yes—
177M16—5364no—
178C2×C8—2684no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C42—24no—
185C42—22no—
186C42#1176161214yescomplemented
187C23—21no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23#1452395811yescharacteristic
192C23#1452395811yescharacteristic
193C23C2×C22⋊Dic6711yescharacteristic
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—48no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4#1452396212yes—
215C2×C4C2×#1176161212yes—
216C8—5364no—
217C22—24no—
218C22—24no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22#1686319112yes—
226C22#1686319312yes—
227C22#1686336011yescharacteristic
228C22C2×#1452395811yescharacteristic
229C22C2×#1452396211yescharacteristic
230C4—22no—
231C4—22no—
232C4—44no—
233C4—42no—
234C4—42no—
235C4—82no—
236C2—21no—
237C2—21no—
238C2—22no—
239C2—22no—
240C2#2255853911yescharacteristic
241C2C2×#1686319111yescharacteristic
242C2C2×#1686319311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC2×#382×C67C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenterC2×#382D67
hypocenterC67#11511
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. #22558153C2semidirect
64. C42×C67M16semidirect
125. C67#11511semidirect
186. C42#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585392
#225581532
C2×#382×C672
#1151167