Finite Group Databasebeta

← #30765205

Subgroups — #30765205

Order 17152.

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

normal subgroups only

This group has 510 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#22558543C212yesmaximal
2#22558139C214yesmaximal
3C67×#2310C211yesmaximal, characteristic
4#16863453—22no—
5#16863122—22no—
6#16863120C2214yes—
7C23×Q8×C67C411yescharacteristic
8C2×#117×C67C411yescharacteristic
9C2×#117×C67C2211yescharacteristic
10#391×C67—22no—
11#391×C67—22no—
12#381×C67C412yes—
13#381×C67C2212yes—
14C4⋊(C67⋊C8)—42no—
15C22⋊(C67⋊C8)—24no—
16#14523954—42no—
17C22×Q8×C67—21no—
18C22×Q8×C67—21no—
19C22×Q8×C67—24no—
20C22×Q8×C67—41no—
21C22×Q8×C67—41no—
22C22×Q8×C67—42no—
23C23×C4×C67C2×C411yescharacteristic
24C23×C4×C67C2×C412yes—
25#117×C67—41no—
26#117×C67—41no—
27#117×C67—42no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67D412yes—
30C2×C22⋊C4×C67C2×C412yes—
31C2×C22⋊C4×C67C2×C411yescharacteristic
32C2×C22⋊C4×C67C2311yescharacteristic
33C2×C42×C67—21no—
34C2×C42×C67D412yes—
35#95×C67—24no—
36#95×C67—24no—
37C2×C67⋊C8—44no—
38C24×C67C22×C411yescharacteristic
39C2×Q8×C67—24no—
40C2×Q8×C67—24no—
41C2×Q8×C67—48no—
42C2×Q8×C67—48no—
43C2×Q8×C67—42no—
44C2×Q8×C67—42no—
45C2×Q8×C67—41no—
46C2×Q8×C67—41no—
47C2×Q8×C67—42no—
48C22×C4×C67—22no—
49C22×C4×C67—21no—
50C22×C4×C67—21no—
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67—22no—
54C22×C4×C67—22no—
55C22×C4×C67—42no—
56C22×C4×C67—42no—
57C22×C4×C67C22⋊C411yescharacteristic
58C22×C4×C67C22⋊C412yes—
59C22×C4×C67C22⋊C411yescharacteristic
60C22×C4×C67M1614yes—
61C22×C4×C67C2×D412yes—
62C4⋊C4×C67—42no—
63C4⋊C4×C67—42no—
64C22⋊C4×C67—41no—
65C22⋊C4×C67—41no—
66C22⋊C4×C67—42no—
67C42×C67—22no—
68C42×C67—22no—
69C42×C67—42no—
70C23×C67—21no—
71C23×C67—21no—
72C23×C67—42no—
73C23×C67C2×C22⋊C411yescharacteristic
74C23×C67C2×M1612yes—
75Q8×C67—44no—
76Q8×C67—44no—
77Q8×C67—48no—
78C2×C4×C67—24no—
79C2×C4×C67—24no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—24no—
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—42no—
93C2×C4×C67—44no—
94C2×C4×C67—42no—
95C2×C4×C67#9911yescharacteristic
96C2×C4×C67#9911yescharacteristic
97C2×C4×C67#10012yes—
98C67⋊C8—84no—
99C22×C67—22no—
100C22×C67—22no—
101C22×C67—42no—
102C22×C67—42no—
103C22×C67—42no—
104C22×C67#40711yescharacteristic
105C22×C67C2×#9911yescharacteristic
106C22×C67C2×#10011yescharacteristic
107C4×C67—21no—
108C4×C67—21no—
109C4×C67—22no—
110C4×C67—42no—
111C4×C67—42no—
112C4×C67—44no—
113C4×C67—82no—
114#11504—671nomaximal, Sylow
115C2×C67—21no—
116C2×C67—21no—
117C2×C67—42no—
118C2×C67#146211yescharacteristic
119C2×C67#212312yes—
120#2310D6711yescharacteristic
121#1464—672no—
122#1317—674no—
123C67#1150411yescharacteristic, complemented, Sylow
124C23×Q8Dic6711yescharacteristic
125C2×#117Dic6711yescharacteristic
126C2×#117C2×D6711yescharacteristic
127#432—1342no—
128#391—22no—
129#391—22no—
130#381Dic6712yes—
131#381C2×D6712yes—
132#324—1342no—
133#323—674no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—42no—
140C23×C4C2×Dic6711yescharacteristic
141C23×C4C2×Dic6712yes—
142#117—41no—
143#117—41no—
144#117—42no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4C22⋊D6712yes—
147C2×C22⋊C4C2×Dic6712yes—
148C2×C22⋊C4C2×Dic6711yescharacteristic
149C2×C22⋊C4C22×D6711yescharacteristic
150C2×C42—21no—
151C2×C42C22⋊D6712yes—
152C4⋊C8—2682no—
153C22⋊C8—1344no—
154#97—2682no—
155#95—24no—
156#95—24no—
157C24C22×Dic6711yescharacteristic
158C2×Q8—24no—
159C2×Q8—24no—
160C2×Q8—48no—
161C2×Q8—48no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—42no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4C22⋊Dic6711yescharacteristic
177C22×C4C22⋊Dic6712yes—
178C22×C4C22⋊Dic6711yescharacteristic
179C22×C4#1176161214yes—
180C22×C4C2×C22⋊D6712yes—
181C2×C8—2684no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—42no—
187C42—22no—
188C42—22no—
189C42—42no—
190C23—21no—
191C23—21no—
192C23—42no—
193C23C2×C22⋊Dic6711yescharacteristic
194C23C2×#1176161212yes—
195Q8—44no—
196Q8—44no—
197Q8—48no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4#1452395811yescharacteristic
216C2×C4#1452395811yescharacteristic
217C2×C4#1452396012yes—
218C8—5364no—
219C22—22no—
220C22—22no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22#1686336011yescharacteristic
225C22C2×#1452395811yescharacteristic
226C22C2×#1452396011yescharacteristic
227C4—21no—
228C4—21no—
229C4—22no—
230C4—42no—
231C4—42no—
232C4—44no—
233C4—82no—
234C2—21no—
235C2—21no—
236C2—42no—
237C2#2255853611yescharacteristic
238C2#2256186312yes—
subgroupG/H
centerC22C2×#14523958
commutator, prime radicalC24×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2310C2
socleC22×C67C2×#99
cosocleC2×C22⋊C4×C67C23
hypercenter#2310D67
hypocenterC67#11504
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
123. C67#11504semidirect

Maximal subgroups

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

subgroupindex
#225585432
#225581392
C67×#23102
#1150467