Finite Group Databasebeta

← #30767367

Subgroups — #30767367

Order 17152.

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

normal subgroups only

This group has 360 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#22558250C212yesmaximal, complemented
3#22558248C212yesmaximal, complemented
4C67×#3152C211yesmaximal, characteristic
5#16863217C2214yes—
6#16863178—22no—
7#16863174—22no—
8C2×#137×C67—22no—
9#532×C67C411yescharacteristic
10C2×C4⋊Q8×C67C411yescharacteristic
11#497×C67—22no—
12#494×C67C411yescharacteristic
13#494×C67C2211yescharacteristic
14#492×C67C411yescharacteristic
15#492×C67C2211yescharacteristic
16C4×M16×C67C2211yescharacteristic
17#14523983—24no—
18C2×#41×C67—21no—
19C22×Q8×C67—21no—
20#137×C67—48no—
21C2×Q16×C67—22no—
22C2×Q16×C67—22no—
23C2×SD16×C67—22no—
24C2×SD16×C67—22no—
25C2×M16×C67D412yes—
26C4⋊Q8×C67—21no—
27C4⋊Q8×C67—41no—
28C4⋊Q8×C67C2×C411yescharacteristic
29C4⋊Q8×C67C2×C411yescharacteristic
30C4⋊Q8×C67C2×C411yescharacteristic
31#127×C67C2×C411yescharacteristic
32#124×C67—21no—
33#121×C67—41no—
34C4×D4×C67—41no—
35C2×C4⋊C4×C67—41no—
36C2×C42×C67C2×C411yescharacteristic
37#97×C67D412yes—
38C4×C8×C67C2×C411yescharacteristic
39C4×C8×C67C2311yescharacteristic
40C67⋊C16—44no—
41#41×C67—44no—
42C2×Q8×C67—21no—
43C2×Q8×C67—21no—
44C2×Q8×C67—21no—
45C2×Q8×C67—44no—
46C2×Q8×C67—41no—
47C2×D4×C67—21no—
48C2×D4×C67—41no—
49C2×D4×C67—41no—
50C22×C4×C67—21no—
51C22×C4×C67—41no—
52C22×C4×C67—41no—
53C22×C4×C67C22⋊C411yescharacteristic
54Q16×C67—44no—
55Q16×C67—44no—
56SD16×C67—44no—
57SD16×C67—44no—
58M16×C67—24no—
59C2×C8×C67C22⋊C412yes—
60C2×C8×C67C2×D412yes—
61C4⋊C4×C67—41no—
62C4⋊C4×C67—41no—
63C4⋊C4×C67—41no—
64C4⋊C4×C67—42no—
65C22⋊C4×C67—42no—
66C42×C67—21no—
67C42×C67C22⋊C411yescharacteristic
68C42×C67C22×C411yescharacteristic
69C23×C67—41no—
70C23×C67#10011yescharacteristic
71Q8×C67—42no—
72Q8×C67—42no—
73Q8×C67—42no—
74Q8×C67—82no—
75D4×C67—42no—
76D4×C67—82no—
77D4×C67—82no—
78C2×C4×C67—21no—
79C2×C4×C67—22no—
80C2×C4×C67—21no—
81C2×C4×C67—42no—
82C2×C4×C67—41no—
83C2×C4×C67—41no—
84C2×C4×C67—41no—
85C2×C4×C67—82no—
86C2×C4×C67—82no—
87C2×C4×C67#10011yescharacteristic
88C2×C4×C67C2×C22⋊C411yescharacteristic
89C8×C67—22no—
90C8×C67AGL(1,ℤ/8)14yes—
91C22×C67—21no—
92C22×C67—41no—
93C22×C67—82no—
94C22×C67—81no—
95C22×C67C2×#10011yescharacteristic
96C4×C67—22no—
97C4×C67—21no—
98C4×C67—41no—
99C4×C67—81no—
100C4×C67—81no—
101C4×C67—81no—
102C4×C67#41812yes—
103#11993—671nomaximal, Sylow
104C2×C67—21no—
105C2×C67—41no—
106C2×C67—161no—
107C2×C67#155511yescharacteristic
108#3152D6711yescharacteristic
109#2165—672no—
110#1359—672no—
111#1358—672no—
112C67#1199311yescharacteristic, complemented, Sylow
113C2×#137—22no—
114#532Dic6711yescharacteristic
115C2×C4⋊Q8Dic6711yescharacteristic
116#497—22no—
117#494Dic6711yescharacteristic
118#494C2×D6711yescharacteristic
119#492Dic6711yescharacteristic
120#492C2×D6711yescharacteristic
121C4×M16C2×D6711yescharacteristic
122#364—674no—
123#349—1342no—
124#347—1342no—
125C2×#41—21no—
126C22×Q8—21no—
127#137—48no—
128C2×Q16—22no—
129C2×Q16—22no—
130C2×SD16—22no—
131C2×SD16—22no—
132C2×M16C22⋊D6712yes—
133C4⋊Q8—21no—
134C4⋊Q8—41no—
135C4⋊Q8C2×Dic6711yescharacteristic
136C4⋊Q8C2×Dic6711yescharacteristic
137C4⋊Q8C2×Dic6711yescharacteristic
138#127C2×Dic6711yescharacteristic
139#124—21no—
140#121—41no—
141C4×D4—41no—
142C2×C4⋊C4—41no—
143C2×C42C2×Dic6711yescharacteristic
144M32—1344no—
145#97C22⋊D6712yes—
146C4×C8C2×Dic6711yescharacteristic
147C4×C8C22×D6711yescharacteristic
148#41—44no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—44no—
153C2×Q8—41no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C22⋊Dic6711yescharacteristic
161Q16—44no—
162Q16—44no—
163SD16—44no—
164SD16—44no—
165M16—24no—
166C2×C8C22⋊Dic6712yes—
167C2×C8C2×C22⋊D6712yes—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C22⋊C4—42no—
173C42—21no—
174C42C22⋊Dic6711yescharacteristic
175C42C22×Dic6711yescharacteristic
176C16—2684no—
177C23—41no—
178C23#1452396011yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—42no—
182Q8—82no—
183D4—42no—
184D4—82no—
185D4—82no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4#1452396011yescharacteristic
196C2×C4C2×C22⋊Dic6711yescharacteristic
197C8—22no—
198C8#1452406914yes—
199C22—21no—
200C22—41no—
201C22—82no—
202C22—81no—
203C22C2×#1452396011yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—81no—
210C4#1686340112yes—
211C2—21no—
212C2—41no—
213C2—161no—
214C2#2255892311yescharacteristic
subgroupG/H
centerC2#22558923
commutatorC42×C67C22×C4
FrattiniC4×C8C22×D67
FittingC67×#3152C2
socleC2×C67#1555
prime radical#532×C67C4
cosocleC4×C8×C67C23
hypercenter#3152D67
hypocenterC67#11993
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. #22562021C2semidirect
2. #22558250C2semidirect
3. #22558248C2semidirect
112. C67#11993semidirect

Maximal subgroups

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

subgroupindex
#225620212
#225582502
#225582482
C67×#31522
#1199367