Finite Group Databasebeta

← #30765263

Subgroups — #30765263

Order 17152.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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#22562021C212yesmaximal, complemented
2#22558160C212yesmaximal, complemented
3#22558145C212yesmaximal
4C67×#2920C211yesmaximal, characteristic
5#16863217—22no—
6#16863178C2214yes—
7#16863174—22no—
8C22×C42×C67C411yescharacteristic
9#432×C67—22no—
10#431×C67—22no—
11#423×C67C411yescharacteristic
12#423×C67C2211yescharacteristic
13#407×C67C411yescharacteristic
14#407×C67C2211yescharacteristic
15C4×M16×C67C411yescharacteristic
16C4×M16×C67C2211yescharacteristic
17#14523983—24no—
18C23×C4×C67—21no—
19C23×C4×C67C2×C411yescharacteristic
20C2×M16×C67C2×C411yescharacteristic
21C2×M16×C67C2×C411yescharacteristic
22C2×M16×C67C2×C411yescharacteristic
23C2×M16×C67C2311yescharacteristic
24C2×C42×C67—21no—
25C2×C42×C67—21no—
26C2×C42×C67—21no—
27C2×C42×C67—21no—
28C2×C42×C67—21no—
29C2×C42×C67C2×C411yescharacteristic
30C2×C42×C67C2×C411yescharacteristic
31C4⋊C8×C67—22no—
32C4⋊C8×C67—22no—
33C22⋊C8×C67—22no—
34C22⋊C8×C67—22no—
35#97×C67—21no—
36#97×C67D412yes—
37C4×C8×C67—21no—
38C4×C8×C67D412yes—
39C67⋊C16—44no—
40C24×C67M1611yescharacteristic
41C22×C4×C67—21no—
42C22×C4×C67—21no—
43C22×C4×C67—21no—
44C22×C4×C67—21no—
45C22×C4×C67—21no—
46C22×C4×C67—21no—
47C22×C4×C67—21no—
48C22×C4×C67—21no—
49C22×C4×C67—44no—
50C22×C4×C67—41no—
51C22×C4×C67—41no—
52C22×C4×C67M1611yescharacteristic
53C22×C4×C67C22×C411yescharacteristic
54M16×C67—41no—
55M16×C67—41no—
56M16×C67—41no—
57M16×C67—41no—
58C2×C8×C67—21no—
59C2×C8×C67—21no—
60C2×C8×C67C22⋊C412yes—
61C2×C8×C67C2×D412yes—
62C42×C67—21no—
63C42×C67—21no—
64C42×C67—21no—
65C42×C67—21no—
66C42×C67—21no—
67C42×C67—21no—
68C42×C67C22⋊C411yescharacteristic
69C42×C67C22⋊C411yescharacteristic
70C42×C67M1611yescharacteristic
71C42×C67M1611yescharacteristic
72C23×C67—21no—
73C23×C67—42no—
74C23×C67—41no—
75C23×C67C2×M1611yescharacteristic
76C2×C4×C67—24no—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
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—41no—
91C2×C4×C67—41no—
92C2×C4×C67—41no—
93C2×C4×C67C2×C22⋊C411yescharacteristic
94C2×C4×C67C2×M1611yescharacteristic
95C8×C67—22no—
96C8×C67—22no—
97C8×C67—41no—
98C8×C67—41no—
99C22×C67—22no—
100C22×C67—21no—
101C22×C67—41no—
102C22×C67—41no—
103C22×C67—42no—
104C22×C67—42no—
105C22×C67—41no—
106C22×C67#40711yescharacteristic
107C4×C67—22no—
108C4×C67—22no—
109C4×C67—21no—
110C4×C67—41no—
111C4×C67—41no—
112C4×C67—41no—
113C4×C67#41011yescharacteristic
114C4×C67#41311yescharacteristic
115#11521—671nomaximal, Sylow
116C2×C67—21no—
117C2×C67—42no—
118C2×C67—41no—
119C2×C67#146411yescharacteristic
120#2920D6711yescharacteristic
121#2165—672no—
122#1327—672no—
123#1320—672no—
124C67#1152111yescharacteristic, complemented, Sylow
125C22×C42Dic6711yescharacteristic
126#432—22no—
127#431—22no—
128#423Dic6711yescharacteristic
129#423C2×D6711yescharacteristic
130#407Dic6711yescharacteristic
131#407C2×D6711yescharacteristic
132C4×M16Dic6711yescharacteristic
133C4×M16C2×D6711yescharacteristic
134#364—1342no—
135#349—674no—
136#347—1342no—
137C23×C4—21no—
138C23×C4C2×Dic6711yescharacteristic
139C2×M16C2×Dic6711yescharacteristic
140C2×M16C2×Dic6711yescharacteristic
141C2×M16C2×Dic6711yescharacteristic
142C2×M16C22×D6711yescharacteristic
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42C2×Dic6711yescharacteristic
149C2×C42C2×Dic6711yescharacteristic
150M32—1344no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155#97—21no—
156#97C22⋊D6712yes—
157C4×C8—21no—
158C4×C8C22⋊D6712yes—
159C24#1176161211yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—44no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4#1176161211yescharacteristic
172C22×C4C22×Dic6711yescharacteristic
173M16—41no—
174M16—41no—
175M16—41no—
176M16—41no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8C22⋊Dic6712yes—
180C2×C8C2×C22⋊D6712yes—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42C22⋊Dic6711yescharacteristic
188C42C22⋊Dic6711yescharacteristic
189C42#1176161211yescharacteristic
190C42#1176161211yescharacteristic
191C16—2684no—
192C23—21no—
193C23—42no—
194C23—41no—
195C23C2×#1176161211yescharacteristic
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4C2×C22⋊Dic6711yescharacteristic
214C2×C4C2×#1176161211yescharacteristic
215C8—22no—
216C8—22no—
217C8—41no—
218C8—41no—
219C22—22no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22#1686336011yescharacteristic
227C4—22no—
228C4—22no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4#1686337011yescharacteristic
234C4#1686338011yescharacteristic
235C2—21no—
236C2—42no—
237C2—41no—
238C2#2255854311yescharacteristic
subgroupG/H
centerC4#16863370
commutatorC22×C4×C67C22×C4
FrattiniC2×M16C22×D67
FittingC67×#2920C2
socleC2×C67#1464
prime radicalC24×C67M16
cosocleC2×M16×C67C23
hypercenter#2920D67
hypocenterC67#11521
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2920

Complements

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

NH ≅ G/Ntype
1. #22562021C2semidirect
2. #22558160C2semidirect
124. C67#11521semidirect

Maximal subgroups

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

subgroupindex
#225620212
#225581602
#225581452
C67×#29202
#1152167