Finite Group Databasebeta

← #30765225

Subgroups — #30765225

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#22558143C214yesmaximal, complemented
3C67×#2310C211yesmaximal, characteristic
4#16863453—22no—
5#16863122C2214yes—
6#16863120—22no—
7C23×Q8×C67C411yescharacteristic
8C2×#117×C67C411yescharacteristic
9C2×#117×C67C2211yescharacteristic
10#391×C67—22no—
11#391×C67C412yes—
12#391×C67C2212yes—
13#381×C67—22no—
14C4⋊(C67⋊C8)—42no—
15C22⋊(C67⋊C8)—24no—
16#14523954—42no—
17C22×Q8×C67—24no—
18C22×Q8×C67—21no—
19C22×Q8×C67—42no—
20C22×Q8×C67—42no—
21C22×Q8×C67C2×C412yes—
22C23×C4×C67—21no—
23C23×C4×C67C2×C411yescharacteristic
24#117×C67—42no—
25#117×C67—42no—
26C2×C4⋊C4×C67—21no—
27C2×C4⋊C4×C67C2×C411yescharacteristic
28C2×C4⋊C4×C67C2311yescharacteristic
29C2×C22⋊C4×C67—21no—
30C2×C22⋊C4×C67D412yes—
31C2×C42×C67C2×C412yes—
32C2×C42×C67D412yes—
33#95×C67—24no—
34#95×C67—24no—
35C2×C67⋊C8—44no—
36C24×C67C22⋊C411yescharacteristic
37C2×Q8×C67—24no—
38C2×Q8×C67—22no—
39C2×Q8×C67—416no—
40C2×Q8×C67—42no—
41C2×Q8×C67—42no—
42C2×Q8×C67—42no—
43C2×Q8×C67—42no—
44C2×Q8×C67M1614yes—
45C22×C4×C67—22no—
46C22×C4×C67—22no—
47C22×C4×C67—22no—
48C22×C4×C67—24no—
49C22×C4×C67—22no—
50C22×C4×C67—22no—
51C22×C4×C67—42no—
52C22×C4×C67—42no—
53C22×C4×C67C22⋊C412yes—
54C22×C4×C67C22⋊C411yescharacteristic
55C22×C4×C67C22×C411yescharacteristic
56C22×C4×C67C2×D412yes—
57C4⋊C4×C67—41no—
58C4⋊C4×C67—41no—
59C4⋊C4×C67—42no—
60C22⋊C4×C67—42no—
61C22⋊C4×C67—42no—
62C42×C67—22no—
63C42×C67—22no—
64C42×C67—42no—
65C23×C67—21no—
66C23×C67—21no—
67C23×C67—41no—
68C23×C67—41no—
69C23×C67#10012yes—
70C23×C67C2×C22⋊C411yescharacteristic
71Q8×C67—44no—
72Q8×C67—48no—
73Q8×C67—44no—
74C2×C4×C67—28no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—24no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—48no—
83C2×C4×C67—42no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67—42no—
88C2×C4×C67—42no—
89C2×C4×C67—42no—
90C2×C4×C67#9912yes—
91C2×C4×C67C2×M1612yes—
92C67⋊C8—84no—
93C22×C67—21no—
94C22×C67—21no—
95C22×C67—22no—
96C22×C67—42no—
97C22×C67—42no—
98C22×C67—41no—
99C22×C67—41no—
100C22×C67#40711yescharacteristic
101C22×C67C2×#9911yescharacteristic
102C22×C67C2×#10011yescharacteristic
103C4×C67—22no—
104C4×C67—22no—
105C4×C67—42no—
106C4×C67—42no—
107C4×C67—44no—
108C4×C67—82no—
109#11510—671nomaximal, Sylow
110C2×C67—21no—
111C2×C67—21no—
112C2×C67—42no—
113C2×C67#146211yescharacteristic
114C2×C67#212911yescharacteristic
115C2×C67#213511yescharacteristic
116#2310D6711yescharacteristic
117#1464—672no—
118#1319—674no—
119C67#1151011yescharacteristic, complemented, Sylow
120C23×Q8Dic6711yescharacteristic
121C2×#117Dic6711yescharacteristic
122C2×#117C2×D6711yescharacteristic
123#432—1342no—
124#391—22no—
125#391Dic6712yes—
126#391C2×D6712yes—
127#381—22no—
128#324—674no—
129#323—1342no—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8C2×Dic6712yes—
135C23×C4—21no—
136C23×C4C2×Dic6711yescharacteristic
137#117—42no—
138#117—42no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4C2×Dic6711yescharacteristic
141C2×C4⋊C4C22×D6711yescharacteristic
142C2×C22⋊C4—21no—
143C2×C22⋊C4C22⋊D6712yes—
144C2×C42C2×Dic6712yes—
145C2×C42C22⋊D6712yes—
146C4⋊C8—2682no—
147C22⋊C8—1344no—
148#97—2682no—
149#95—24no—
150#95—24no—
151C24C22⋊Dic6711yescharacteristic
152C2×Q8—24no—
153C2×Q8—22no—
154C2×Q8—416no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8#1176161214yes—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C22⋊Dic6712yes—
169C22×C4C22⋊Dic6711yescharacteristic
170C22×C4C22×Dic6711yescharacteristic
171C22×C4C2×C22⋊D6712yes—
172C2×C8—2684no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42—22no—
179C42—22no—
180C42—42no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23#1452396012yes—
186C23C2×C22⋊Dic6711yescharacteristic
187Q8—44no—
188Q8—48no—
189Q8—44no—
190C2×C4—28no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—48no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4#1452395812yes—
207C2×C4C2×#1176161212yes—
208C8—5364no—
209C22—21no—
210C22—21no—
211C22—22no—
212C22—42no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22#1686336011yescharacteristic
217C22C2×#1452395811yescharacteristic
218C22C2×#1452396011yescharacteristic
219C4—22no—
220C4—22no—
221C4—44no—
222C4—42no—
223C4—42no—
224C4—82no—
225C2—21no—
226C2—21no—
227C2—42no—
228C2#2255853611yescharacteristic
229C2#2256188411yescharacteristic
230C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2310C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2310D67
hypocenterC67#11510
LCSG ▷ C22×C4×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
2. #22558143C2semidirect
119. C67#11510semidirect

Maximal subgroups

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

subgroupindex
#225585432
#225581432
C67×#23102
#1151067