Finite Group Databasebeta

← #30765006

Subgroups — #30765006

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#22558157C214yesmaximal
3C67×#2296C211yesmaximal, characteristic
4#16863449—22no—
5#16863122—22no—
6#16863120C2214yes—
7C2×#117×C67C411yescharacteristic
8C2×#117×C67C2211yescharacteristic
9C22×C42×C67C411yescharacteristic
10#389×C67—22no—
11#385×C67C412yes—
12#385×C67C2212yes—
13C4×C4⋊C4×C67—22no—
14C4⋊(C67⋊C8)—42no—
15C22⋊(C67⋊C8)—24no—
16C4×C67⋊C8—42no—
17C23×C4×C67C2×C411yescharacteristic
18C23×C4×C67C2×C412yes—
19#117×C67—22no—
20#117×C67—41no—
21#117×C67—41no—
22#117×C67C814yescomplemented
23C2×C4⋊C4×C67—21no—
24C2×C4⋊C4×C67—22no—
25C2×C4⋊C4×C67—22no—
26C2×C4⋊C4×C67D412yes—
27C2×C22⋊C4×C67C2×C412yes—
28C2×C22⋊C4×C67C2×C411yescharacteristic
29C2×C22⋊C4×C67C2311yescharacteristic
30C2×C42×C67—21no—
31C2×C42×C67—21no—
32C2×C42×C67—21no—
33C2×C42×C67—41no—
34C2×C42×C67—41no—
35C2×C42×C67D412yes—
36#95×C67—22no—
37#95×C67—22no—
38C2×C67⋊C8—44no—
39C24×C67C22×C411yescharacteristic
40C22×C4×C67—21no—
41C22×C4×C67—21no—
42C22×C4×C67—22no—
43C22×C4×C67—22no—
44C22×C4×C67—22no—
45C22×C4×C67—44no—
46C22×C4×C67—42no—
47C22×C4×C67C22⋊C411yescharacteristic
48C22×C4×C67C22⋊C412yes—
49C22×C4×C67C22⋊C411yescharacteristic
50C22×C4×C67C2×C812yes—
51C22×C4×C67M1612yes—
52C22×C4×C67C2×D412yes—
53C4⋊C4×C67—24no—
54C4⋊C4×C67—42no—
55C4⋊C4×C67—44no—
56C4⋊C4×C67—44no—
57C22⋊C4×C67—22no—
58C22⋊C4×C67—41no—
59C22⋊C4×C67—41no—
60C22⋊C4×C67C2×C814yes—
61C42×C67—24no—
62C42×C67—22no—
63C42×C67—22no—
64C42×C67—42no—
65C42×C67—41no—
66C42×C67—41no—
67C23×C67—21no—
68C23×C67—21no—
69C23×C67—42no—
70C23×C67C2×C22⋊C411yescharacteristic
71C23×C67C22×C811yescharacteristic
72C23×C67C2×M1611yescharacteristic
73C2×C4×C67—21no—
74C2×C4×C67—21no—
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—24no—
80C2×C4×C67—24no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—42no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—44no—
87C2×C4×C67—42no—
88C2×C4×C67—42no—
89C2×C4×C67—42no—
90C2×C4×C67—44no—
91C2×C4×C67C22⋊C812yes—
92C2×C4×C67C22⋊C812yes—
93C2×C4×C67#9911yescharacteristic
94C2×C4×C67#9911yescharacteristic
95C2×C4×C67#10112yes—
96C67⋊C8—84no—
97C22×C67—22no—
98C22×C67—21no—
99C22×C67—21no—
100C22×C67—41no—
101C22×C67—41no—
102C22×C67—42no—
103C22×C67—42no—
104C22×C67C2×C22⋊C811yescharacteristic
105C22×C67C2×#9911yescharacteristic
106C22×C67C2×#10111yescharacteristic
107C4×C67—22no—
108C4×C67—44no—
109C4×C67—42no—
110C4×C67—42no—
111C4×C67—44no—
112C4×C67#32412yes—
113C4×C67#32412yes—
114#11452—671nomaximal, Sylow
115C2×C67—21no—
116C2×C67—21no—
117C2×C67—41no—
118C2×C67—41no—
119C2×C67C2×#32411yescharacteristic
120C2×C67#212012yes—
121#2296D6711yescharacteristic
122#1458—672no—
123#1326—674no—
124C67#1145211yescharacteristic, complemented, Sylow
125C2×#117Dic6711yescharacteristic
126C2×#117C2×D6711yescharacteristic
127C22×C42Dic6711yescharacteristic
128#431—1342no—
129#389—22no—
130#385Dic6712yes—
131#385C2×D6712yes—
132C4×C4⋊C4—22no—
133#324—1342no—
134#323—674no—
135C23×C4C2×Dic6711yescharacteristic
136C23×C4C2×Dic6712yes—
137#117—22no—
138#117—41no—
139#117—41no—
140#117C67⋊C814yescomplemented
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C22⋊D6712yes—
145C2×C22⋊C4C2×Dic6712yes—
146C2×C22⋊C4C2×Dic6711yescharacteristic
147C2×C22⋊C4C22×D6711yescharacteristic
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42C22⋊D6712yes—
154C4⋊C8—2682no—
155C22⋊C8—1344no—
156C4×C8—2682no—
157#95—22no—
158#95—22no—
159C24C22×Dic6711yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4C22⋊Dic6711yescharacteristic
168C22×C4C22⋊Dic6712yes—
169C22×C4C22⋊Dic6711yescharacteristic
170C22×C4C2×C67⋊C812yes—
171C22×C4#1176161212yes—
172C22×C4C2×C22⋊D6712yes—
173C2×C8—2684no—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C22⋊C4—22no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4C2×C67⋊C814yes—
182C42—24no—
183C42—22no—
184C42—22no—
185C42—42no—
186C42—41no—
187C42—41no—
188C23—21no—
189C23—21no—
190C23—42no—
191C23C2×C22⋊Dic6711yescharacteristic
192C23C22×C67⋊C811yescharacteristic
193C23C2×#1176161211yescharacteristic
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4C22⋊(C67⋊C8)12yes—
213C2×C4C22⋊(C67⋊C8)12yes—
214C2×C4#1452395811yescharacteristic
215C2×C4#1452395811yescharacteristic
216C2×C4#1452396212yes—
217C8—5364no—
218C22—22no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22C2×C22⋊(C67⋊C8)11yescharacteristic
226C22C2×#1452395811yescharacteristic
227C22C2×#1452396211yescharacteristic
228C4—22no—
229C4—44no—
230C4—42no—
231C4—44no—
232C4—42no—
233C4#1686312212yes—
234C4#1686312212yes—
235C2—21no—
236C2—21no—
237C2—41no—
238C2—41no—
239C2C2×#1686312211yescharacteristic
240C2#2256185012yes—
subgroupG/H
centerC2×C4#14523958
commutatorC23×C67C22×C8
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2296C2
socleC22×C67C2×#99
prime radicalC24×C67C22×C4
cosocleC2×C22⋊C4×C67C23
hypercenter#2296D67
hypocenterC67#11452
LCSG ▷ C23×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2296

Complements

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

NH ≅ G/Ntype
22. #117×C67C8semidirect
124. C67#11452semidirect
140. #117C67⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225585232
#225581572
C67×#22962
#1145267