Finite Group Databasebeta

← #30765231

Subgroups — #30765231

Order 17152.

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

normal subgroups only

This group has 578 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#22558155C214yesmaximal, complemented
3C2×#384×C67C211yesmaximal, characteristic
4#16863360—22no—
5#16863122—22no—
6#16863122C2214yes—
7C22×C4⋊C4×C67—22no—
8C22×C4⋊C4×C67C411yescharacteristic
9C22×C4⋊C4×C67C2211yescharacteristic
10C22×C42×C67C411yescharacteristic
11#384×C67—22no—
12#384×C67C412yes—
13#384×C67C2212yes—
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—24no—
22C2×C4⋊C4×C67—21no—
23C2×C4⋊C4×C67—24no—
24C2×C4⋊C4×C67—41no—
25C2×C4⋊C4×C67—41no—
26C2×C4⋊C4×C67—41no—
27C2×C4⋊C4×C67—41no—
28C2×C4⋊C4×C67—48no—
29C2×C4⋊C4×C67C2×C412yes—
30C2×C4⋊C4×C67C2×C411yescharacteristic
31C2×C4⋊C4×C67D412yes—
32C2×C4⋊C4×C67C2311yescharacteristic
33C2×C42×C67—21no—
34C2×C42×C67—24no—
35C2×C42×C67C2×C412yes—
36#11761612—84no—
37C2×C67⋊C8—44no—
38C24×C67C22⋊C411yescharacteristic
39C22×C4×C67—24no—
40C22×C4×C67—24no—
41C22×C4×C67—24no—
42C22×C4×C67—21no—
43C22×C4×C67—21no—
44C22×C4×C67—22no—
45C22×C4×C67—22no—
46C22×C4×C67—22no—
47C22×C4×C67—22no—
48C22×C4×C67—22no—
49C22×C4×C67—22no—
50C22×C4×C67—44no—
51C22×C4×C67—44no—
52C22×C4×C67—41no—
53C22×C4×C67—41no—
54C22×C4×C67C22⋊C411yescharacteristic
55C22×C4×C67C22⋊C412yes—
56C22×C4×C67C22×C411yescharacteristic
57C22×C4×C67C2×D412yes—
58C4⋊C4×C67—41no—
59C4⋊C4×C67—42no—
60C4⋊C4×C67—41no—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—48no—
63C4⋊C4×C67—48no—
64C4⋊C4×C67—42no—
65C42×C67—24no—
66C42×C67—22no—
67C42×C67M1614yescomplemented
68C23×C67—22no—
69C23×C67—22no—
70C23×C67—21no—
71C23×C67—21no—
72C23×C67#9911yescharacteristic
73C23×C67#9911yescharacteristic
74C23×C67C2×C22⋊C411yescharacteristic
75C2×C4×C67—24no—
76C2×C4×C67—216no—
77C2×C4×C67—24no—
78C2×C4×C67—24no—
79C2×C4×C67—24no—
80C2×C4×C67—21no—
81C2×C4×C67—24no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—22no—
86C2×C4×C67—22no—
87C2×C4×C67—44no—
88C2×C4×C67—42no—
89C2×C4×C67—42no—
90C2×C4×C67—44no—
91C2×C4×C67—42no—
92C2×C4×C67—82no—
93C2×C4×C67—82no—
94C2×C4×C67—82no—
95C2×C4×C67#10112yes—
96C2×C4×C67C2×M1612yes—
97C67⋊C8—84no—
98C22×C67—24no—
99C22×C67—24no—
100C22×C67—21no—
101C22×C67—21no—
102C22×C67—21no—
103C22×C67—21no—
104C22×C67—21no—
105C22×C67—21no—
106C22×C67#35312yes—
107C22×C67#35612yes—
108C22×C67#40711yescharacteristic
109C22×C67C2×#9911yescharacteristic
110C22×C67C2×#10111yescharacteristic
111C4×C67—28no—
112C4×C67—22no—
113C4×C67—22no—
114C4×C67—42no—
115C4×C67—42no—
116C4×C67—82no—
117#11512—671nomaximal, Sylow
118C2×C67—21no—
119C2×C67—22no—
120C2×C67—22no—
121C2×C67—21no—
122C2×C67#146311yescharacteristic
123C2×C67C2×#35311yescharacteristic
124C2×C67C2×#35611yescharacteristic
125C2×#384D6711yescharacteristic
126#1463—672no—
127#1325—674no—
128C67#1151211yescharacteristic, complemented, Sylow
129C22×C4⋊C4—22no—
130C22×C4⋊C4Dic6711yescharacteristic
131C22×C4⋊C4C2×D6711yescharacteristic
132C22×C42Dic6711yescharacteristic
133#407—1342no—
134#384—22no—
135#384Dic6712yes—
136#384C2×D6712yes—
137#324—674no—
138#324—1342no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C2×Dic6711yescharacteristic
142C23×C4C22⋊D6712yes—
143C2×M16—2682no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—48no—
152C2×C4⋊C4C2×Dic6712yes—
153C2×C4⋊C4C2×Dic6711yescharacteristic
154C2×C4⋊C4C22⋊D6712yes—
155C2×C4⋊C4C22×D6711yescharacteristic
156C2×C42—21no—
157C2×C42—24no—
158C2×C42C2×Dic6712yes—
159C22⋊C8—2682no—
160C22⋊C8—1344no—
161C24C22⋊Dic6711yescharacteristic
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4C22⋊Dic6711yescharacteristic
178C22×C4C22⋊Dic6712yes—
179C22×C4C22×Dic6711yescharacteristic
180C22×C4C2×C22⋊D6712yes—
181M16—5364no—
182C2×C8—2684no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—41no—
186C4⋊C4—42no—
187C4⋊C4—48no—
188C4⋊C4—48no—
189C4⋊C4—42no—
190C42—24no—
191C42—22no—
192C42#1176161214yescomplemented
193C23—21no—
194C23—21no—
195C23—22no—
196C23—22no—
197C23#1452395811yescharacteristic
198C23#1452395811yescharacteristic
199C23C2×C22⋊Dic6711yescharacteristic
200C2×C4—24no—
201C2×C4—216no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4#1452396212yes—
221C2×C4C2×#1176161212yes—
222C8—5364no—
223C22—24no—
224C22—24no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22#1686318912yes—
232C22#1686319512yes—
233C22#1686336011yescharacteristic
234C22C2×#1452395811yescharacteristic
235C22C2×#1452396211yescharacteristic
236C4—28no—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—42no—
241C4—82no—
242C2—21no—
243C2—21no—
244C2—22no—
245C2—22no—
246C2#2255853911yescharacteristic
247C2C2×#1686318911yescharacteristic
248C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC2×#384×C67C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenterC2×#384D67
hypocenterC67#11512
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. #22558155C2semidirect
67. C42×C67M16semidirect
128. C67#11512semidirect
192. C42#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585392
#225581552
C2×#384×C672
#1151267