Finite Group Databasebeta

← #30765335

Subgroups — #30765335

Order 17152.

334 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 91 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#22562025C212yesmaximal, complemented
2#22558147C212yesmaximal, complemented
3#22558145C212yesmaximal, complemented
4C67×#2930C211yesmaximal, characteristic
5#16863217—22no—
6#16863178C2214yes—
7#16863174—22no—
8C2×#131×C67C411yescharacteristic
9C22×M16×C67C411yescharacteristic
10#525×C67C411yescharacteristic
11#436×C67—22no—
12#435×C67—22no—
13#424×C67C411yescharacteristic
14#408×C67C2211yescharacteristic
15#407×C67C2211yescharacteristic
16#405×C67C2211yescharacteristic
17#14523983—24no—
18C2×#41×C67C2×C411yescharacteristic
19C23×C4×C67C2×C411yescharacteristic
20#131×C67—42no—
21C2×M16×C67—21no—
22C2×M16×C67—21no—
23C2×M16×C67—21no—
24C2×M16×C67—42no—
25C2×M16×C67C2×C411yescharacteristic
26C2×M16×C67C2311yescharacteristic
27C22×C8×C67—21no—
28C22×C8×C67C2×C411yescharacteristic
29C22×C8×C67C2×C411yescharacteristic
30#123×C67—21no—
31C22⋊Q8×C67—21no—
32#121×C67—21no—
33#120×C67—21no—
34C4×D4×C67—22no—
35#117×C67C2×C411yescharacteristic
36C4⋊C8×C67—22no—
37C22⋊C8×C67—22no—
38C22⋊C8×C67—22no—
39#97×C67D412yes—
40C4×C8×C67D412yes—
41C67⋊C16—44no—
42C24×C67M1611yescharacteristic
43#41×C67—41no—
44#41×C67—41no—
45C2×Q8×C67M1611yescharacteristic
46C2×D4×C67—21no—
47C2×D4×C67—22no—
48C2×D4×C67M1611yescharacteristic
49C22×C4×C67—21no—
50C22×C4×C67—21no—
51C22×C4×C67—21no—
52C22×C4×C67—41no—
53C22×C4×C67—41no—
54C22×C4×C67M1611yescharacteristic
55C22×C4×C67C22×C411yescharacteristic
56M16×C67—41no—
57M16×C67—42no—
58M16×C67—42no—
59M16×C67—42no—
60M16×C67—41no—
61C2×C8×C67—22no—
62C2×C8×C67—21no—
63C2×C8×C67—21no—
64C2×C8×C67—42no—
65C2×C8×C67—41no—
66C2×C8×C67C22⋊C411yescharacteristic
67C2×C8×C67C22⋊C411yescharacteristic
68C2×C8×C67C2×D412yes—
69C4⋊C4×C67—22no—
70C4⋊C4×C67—21no—
71C22⋊C4×C67—22no—
72C22⋊C4×C67—22no—
73C22⋊C4×C67—21no—
74C42×C67C22⋊C412yes—
75C23×C67—21no—
76C23×C67—21no—
77C23×C67—41no—
78C23×C67—81no—
79C23×C67C2×M1611yescharacteristic
80Q8×C67—41no—
81D4×C67—41no—
82D4×C67—42no—
83D4×C67—82no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—22no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—42no—
90C2×C4×C67—41no—
91C2×C4×C67—41no—
92C2×C4×C67—41no—
93C2×C4×C67—41no—
94C2×C4×C67—41no—
95C2×C4×C67C2×C22⋊C411yescharacteristic
96C2×C4×C67C2×M1611yescharacteristic
97C8×C67—22no—
98C8×C67—21no—
99C8×C67—41no—
100C8×C67—41no—
101C8×C67#10012yes—
102C22×C67—21no—
103C22×C67—41no—
104C22×C67—41no—
105C22×C67—42no—
106C22×C67—41no—
107C22×C67—41no—
108C22×C67—41no—
109C22×C67—82no—
110C22×C67#40711yescharacteristic
111C4×C67—21no—
112C4×C67—42no—
113C4×C67—41no—
114C4×C67—41no—
115C4×C67#41011yescharacteristic
116C4×C67C2×#10011yescharacteristic
117#11538—671nomaximal, Sylow
118C2×C67—21no—
119C2×C67—41no—
120C2×C67—81no—
121C2×C67—81no—
122C2×C67#146611yescharacteristic
123#2930D6711yescharacteristic
124#2166—672no—
125#1321—672no—
126#1320—672no—
127C67#1153811yescharacteristic, complemented, Sylow
128C2×#131Dic6711yescharacteristic
129C22×M16Dic6711yescharacteristic
130#525Dic6711yescharacteristic
131#436—22no—
132#435—22no—
133#424Dic6711yescharacteristic
134#408C2×D6711yescharacteristic
135#407C2×D6711yescharacteristic
136#405C2×D6711yescharacteristic
137#364—1342no—
138#349—674no—
139#347—1342no—
140C2×#41C2×Dic6711yescharacteristic
141C23×C4C2×Dic6711yescharacteristic
142#131—42no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—42no—
147C2×M16C2×Dic6711yescharacteristic
148C2×M16C22×D6711yescharacteristic
149C22×C8—21no—
150C22×C8C2×Dic6711yescharacteristic
151C22×C8C2×Dic6711yescharacteristic
152#123—21no—
153C22⋊Q8—21no—
154#121—21no—
155#120—21no—
156C4×D4—22no—
157#117C2×Dic6711yescharacteristic
158M32—1344no—
159C4⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162#97C22⋊D6712yes—
163C4×C8C22⋊D6712yes—
164C24#1176161211yescharacteristic
165#41—41no—
166#41—41no—
167C2×Q8#1176161211yescharacteristic
168C2×D4—21no—
169C2×D4—22no—
170C2×D4#1176161211yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4#1176161211yescharacteristic
177C22×C4C22×Dic6711yescharacteristic
178M16—41no—
179M16—42no—
180M16—42no—
181M16—42no—
182M16—41no—
183C2×C8—22no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8C22⋊Dic6711yescharacteristic
189C2×C8C22⋊Dic6711yescharacteristic
190C2×C8C2×C22⋊D6712yes—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—21no—
196C42C22⋊Dic6712yes—
197C16—2684no—
198C23—21no—
199C23—21no—
200C23—41no—
201C23—81no—
202C23C2×#1176161211yescharacteristic
203Q8—41no—
204D4—41no—
205D4—42no—
206D4—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4C2×C22⋊Dic6711yescharacteristic
219C2×C4C2×#1176161211yescharacteristic
220C8—22no—
221C8—21no—
222C8—41no—
223C8—41no—
224C8#1452396012yes—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—82no—
233C22#1686336011yescharacteristic
234C4—21no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4#1686337011yescharacteristic
239C4C2×#1452396011yescharacteristic
240C2—21no—
241C2—41no—
242C2—81no—
243C2—81no—
244C2#2255855011yescharacteristic
subgroupG/H
centerC4#16863370
commutatorC22×C4×C67C22×C4
FrattiniC2×M16C22×D67
FittingC67×#2930C2
socleC2×C67#1466
prime radical#525×C67C4
cosocleC2×M16×C67C23
hypercenter#2930D67
hypocenterC67#11538
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ #2930

Complements

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

NH ≅ G/Ntype
1. #22562025C2semidirect
2. #22558147C2semidirect
3. #22558145C2semidirect
127. C67#11538semidirect

Maximal subgroups

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

subgroupindex
#225620252
#225581472
#225581452
C67×#29302
#1153867