Finite Group Databasebeta

← #30749327

Subgroups — #30749327

Order 17152.

384 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 384 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#22558065C214yesmaximal, complemented
3C67×#2298C211yesmaximal, characteristic
4#16863453C2214yes—
5#16863122—22no—
6#16863120—22no—
7C2×#117×C67C2211yescharacteristic
8#388×C67—22no—
9#387×C67—22no—
10C4×C4⋊C4×C67C2212yes—
11C4×C22⋊C4×C67C412yes—
12#376×C67C411yescharacteristic
13C43×C67C411yescharacteristic
14C4⋊(C67⋊C8)—24no—
15C22⋊(C67⋊C8)—24no—
16#14523954—24no—
17C23×C4×C67C2×C411yescharacteristic
18#117×C67—24no—
19C2×C4⋊C4×C67—22no—
20C2×C4⋊C4×C67D412yes—
21C2×C22⋊C4×C67—22no—
22C2×C22⋊C4×C67D412yes—
23C2×C42×C67—22no—
24C2×C42×C67C2×C411yescharacteristic
25C2×C42×C67C2×C412yes—
26C2×C42×C67C2×C412yes—
27C2×C42×C67C2311yescharacteristic
28#95×C67—22no—
29#95×C67—22no—
30#95×C67—22no—
31C2×C67⋊C8—44no—
32C24×C67C22⋊C411yescharacteristic
33C22×C4×C67—22no—
34C22×C4×C67—22no—
35C22×C4×C67—22no—
36C22×C4×C67—42no—
37C22×C4×C67C22⋊C411yescharacteristic
38C22×C4×C67C22⋊C412yes—
39C22×C4×C67C22×C411yescharacteristic
40C22×C4×C67C2×D412yes—
41C4⋊C4×C67—24no—
42C4⋊C4×C67—44no—
43C22⋊C4×C67—24no—
44C22⋊C4×C67—44no—
45C42×C67—22no—
46C42×C67—42no—
47C42×C67—42no—
48C42×C67—44no—
49C42×C67#4114yes—
50C23×C67—21no—
51C23×C67—22no—
52C23×C67—42no—
53C23×C67C2×C22⋊C411yescharacteristic
54C2×C4×C67—22no—
55C2×C4×C67—24no—
56C2×C4×C67—22no—
57C2×C4×C67—21no—
58C2×C4×C67—22no—
59C2×C4×C67—42no—
60C2×C4×C67—44no—
61C2×C4×C67—42no—
62C2×C4×C67—42no—
63C2×C4×C67—42no—
64C2×C4×C67—42no—
65C2×C4×C67—42no—
66C2×C4×C67—42no—
67C2×C4×C67#11712yes—
68C2×C4×C67#12314yes—
69C67⋊C8—84no—
70C22×C67—22no—
71C22×C67—41no—
72C22×C67—42no—
73C22×C67—82no—
74C22×C67#38111yescharacteristic
75C22×C67#41312yes—
76C4×C67—22no—
77C4×C67—24no—
78C4×C67—44no—
79C4×C67—41no—
80C4×C67—42no—
81C4×C67—82no—
82#8745—671nomaximal, Sylow
83C2×C67—22no—
84C2×C67—81no—
85C2×C67#182211yescharacteristic
86C2×C67#182712yes—
87#2298D6711yescharacteristic
88#1464—672no—
89#1285—674no—
90C67#874511yescharacteristic, complemented, Sylow
91C2×#117C2×D6711yescharacteristic
92#432—674no—
93#388—22no—
94#387—22no—
95C4×C4⋊C4C2×D6712yes—
96C4×C22⋊C4Dic6712yes—
97#376Dic6711yescharacteristic
98C43Dic6711yescharacteristic
99#324—1342no—
100#323—1342no—
101C23×C4C2×Dic6711yescharacteristic
102#117—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4C22⋊D6712yes—
105C2×C22⋊C4—22no—
106C2×C22⋊C4C22⋊D6712yes—
107C2×C42—22no—
108C2×C42C2×Dic6711yescharacteristic
109C2×C42C2×Dic6712yes—
110C2×C42C2×Dic6712yes—
111C2×C42C22×D6711yescharacteristic
112C4⋊C8—1344no—
113C22⋊C8—1344no—
114#97—1344no—
115#95—22no—
116#95—22no—
117#95—22no—
118C24C22⋊Dic6711yescharacteristic
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4C22⋊Dic6711yescharacteristic
124C22×C4C22⋊Dic6712yes—
125C22×C4C22×Dic6711yescharacteristic
126C22×C4C2×C22⋊D6712yes—
127C2×C8—2684no—
128C4⋊C4—24no—
129C4⋊C4—44no—
130C22⋊C4—24no—
131C22⋊C4—44no—
132C42—22no—
133C42—42no—
134C42—42no—
135C42—44no—
136C42#1176162814yes—
137C23—21no—
138C23—22no—
139C23—42no—
140C23C2×C22⋊Dic6711yescharacteristic
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4#1452400212yes—
155C2×C4#1452402614yes—
156C8—5364no—
157C22—22no—
158C22—41no—
159C22—42no—
160C22—82no—
161C22#1686325911yescharacteristic
162C22#1686338012yes—
163C4—22no—
164C4—24no—
165C4—44no—
166C4—41no—
167C4—42no—
168C4—82no—
169C2—22no—
170C2—81no—
171C2#2256029211yescharacteristic
172C2#2256031312yes—
subgroupG/H
centerC22#16863259
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2298C2
socleC22×C67#381
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2298D67
hypocenterC67#8745
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2298

Complements

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

NH ≅ G/Ntype
2. #22558065C2semidirect
90. C67#8745semidirect

Maximal subgroups

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

subgroupindex
#225585432
#225580652
C67×#22982
#874567