Finite Group Databasebeta

← #30765331

Subgroups — #30765331

Order 17152.

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

normal subgroups only

This group has 348 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×#2929C211yesmaximal, characteristic
5#16863217—22no—
6#16863178C2214yes—
7#16863174—22no—
8C2×#131×C67C411yescharacteristic
9C23×C8×C67C411yescharacteristic
10#525×C67C411yescharacteristic
11#435×C67—22no—
12C8×D4×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×C67C2×C411yescharacteristic
23C2×M16×C67C2311yescharacteristic
24C22×C8×C67—21no—
25C22×C8×C67—21no—
26C22×C8×C67—21no—
27C22×C8×C67—42no—
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—22no—
47C2×D4×C67—21no—
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—41no—
59C2×C8×C67—22no—
60C2×C8×C67—21no—
61C2×C8×C67—21no—
62C2×C8×C67—42no—
63C2×C8×C67—42no—
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—42no—
79C23×C67C2×M1611yescharacteristic
80Q8×C67—41no—
81D4×C67—41no—
82D4×C67—44no—
83D4×C67—42no—
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—22no—
103C22×C67—21no—
104C22×C67—41no—
105C22×C67—41no—
106C22×C67—42no—
107C22×C67—42no—
108C22×C67—42no—
109C22×C67—41no—
110C22×C67—41no—
111C22×C67#40711yescharacteristic
112C4×C67—21no—
113C4×C67—42no—
114C4×C67—41no—
115C4×C67—41no—
116C4×C67#41011yescharacteristic
117C4×C67C2×#10011yescharacteristic
118#11537—671nomaximal, Sylow
119C2×C67—21no—
120C2×C67—42no—
121C2×C67—41no—
122C2×C67—81no—
123C2×C67#146611yescharacteristic
124#2929D6711yescharacteristic
125#2166—672no—
126#1321—672no—
127#1320—672no—
128C67#1153711yescharacteristic, complemented, Sylow
129C2×#131Dic6711yescharacteristic
130C23×C8Dic6711yescharacteristic
131#525Dic6711yescharacteristic
132#435—22no—
133C8×D4—22no—
134#424Dic6711yescharacteristic
135#408C2×D6711yescharacteristic
136#407C2×D6711yescharacteristic
137#405C2×D6711yescharacteristic
138#364—1342no—
139#349—674no—
140#347—1342no—
141C2×#41C2×Dic6711yescharacteristic
142C23×C4C2×Dic6711yescharacteristic
143#131—42no—
144C2×M16—21no—
145C2×M16C2×Dic6711yescharacteristic
146C2×M16C22×D6711yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—42no—
151C22×C8C2×Dic6711yescharacteristic
152C22×C8C2×Dic6711yescharacteristic
153#123—21no—
154C22⋊Q8—21no—
155#121—21no—
156#120—21no—
157C4×D4—22no—
158#117C2×Dic6711yescharacteristic
159M32—1344no—
160C4⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163#97C22⋊D6712yes—
164C4×C8C22⋊D6712yes—
165C24#1176161211yescharacteristic
166#41—41no—
167#41—41no—
168C2×Q8#1176161211yescharacteristic
169C2×D4—22no—
170C2×D4—21no—
171C2×D4#1176161211yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4#1176161211yescharacteristic
178C22×C4C22×Dic6711yescharacteristic
179M16—41no—
180M16—42no—
181M16—41no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8C22⋊Dic6711yescharacteristic
190C2×C8C22⋊Dic6711yescharacteristic
191C2×C8C2×C22⋊D6712yes—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C42C22⋊Dic6712yes—
198C16—2684no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—42no—
203C23C2×#1176161211yescharacteristic
204Q8—41no—
205D4—41no—
206D4—42no—
207D4—44no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4C2×C22⋊Dic6711yescharacteristic
220C2×C4C2×#1176161211yescharacteristic
221C8—22no—
222C8—21no—
223C8—41no—
224C8—41no—
225C8#1452396012yes—
226C22—22no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22#1686336011yescharacteristic
236C4—21no—
237C4—42no—
238C4—41no—
239C4—41no—
240C4#1686337011yescharacteristic
241C4C2×#1452396011yescharacteristic
242C2—21no—
243C2—42no—
244C2—41no—
245C2—81no—
246C2#2255855011yescharacteristic
subgroupG/H
centerC4#16863370
commutatorC22×C4×C67C22×C4
FrattiniC2×M16C22×D67
FittingC67×#2929C2
socleC2×C67#1466
prime radical#525×C67C4
cosocleC2×M16×C67C23
hypercenter#2929D67
hypocenterC67#11537
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #22562025C2semidirect
2. #22558147C2semidirect
3. #22558145C2semidirect
128. C67#11537semidirect

Maximal subgroups

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

subgroupindex
#225620252
#225581472
#225581452
C67×#29292
#1153767