Finite Group Databasebeta

← #17949122

Subgroups — #17949122

Order 4352.

324 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 36 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#14543716C212yesmaximal
2#14528707C211yesmaximal, characteristic
3#11763705—22no—
4C23×Dic34C411yescharacteristic
5C2×#10587495—22no—
6#11762773C2211yescharacteristic
7C17×#407C411yescharacteristic
8C23×C4×C17C2×C411yescharacteristic
9C23×Dic17C812yes—
10C22×Dic34—21no—
11C22×Dic34—21no—
12C22×Dic34—44no—
13C22×Dic34—41no—
14C22×Dic34—41no—
15C2×M16×C17—21no—
16C2×#63223D412yes—
17#10587538—42no—
18C17×C22⋊C8—41no—
19C22⋊#3622—41no—
20#10587495—44no—
21C24×C17C2×C811yescharacteristic
22C22×C4×C17—21no—
23C22×C4×C17—21no—
24C22×C4×C17—42no—
25C22×C4×C17C22⋊C411yescharacteristic
26C22×C4×C17M1611yescharacteristic
27C22×Dic17—22no—
28C22×Dic17—44no—
29C22×Dic17—42no—
30C2×Dic34—24no—
31C2×Dic34—44no—
32C2×Dic34—42no—
33C2×Dic34—42no—
34C2×Dic34—48no—
35C2×Dic34—44no—
36C2×Dic34—41no—
37C2×Dic34—41no—
38C2×Dic34—44no—
39M16×C17—42no—
40C2×C8×C17—41no—
41#63223—42no—
42C2×#3622—22no—
43#63216—82no—
44#7341—171nomaximal, Sylow
45C23×C17—21no—
46C23×C17—41no—
47C23×C17—42no—
48C23×C17C22⋊C811yescharacteristic
49C2×C4×C17—22no—
50C2×C4×C17—21no—
51C2×C4×C17—42no—
52C2×C4×C17—42no—
53C2×C4×C17—41no—
54C2×C4×C17#9911yescharacteristic
55C2×C4×C17#10011yescharacteristic
56C2×Dic17—24no—
57C2×Dic17—42no—
58C2×Dic17—44no—
59C2×Dic17—42no—
60C2×Dic17—44no—
61Dic34—44no—
62Dic34—48no—
63Dic34—44no—
64C8×C17—81no—
65#3622—42no—
66#1784—171no—
67#1320—172no—
68C22×C17—22no—
69C22×C17—21no—
70C22×C17—41no—
71C22×C17—41no—
72C22×C17—42no—
73C22×C17—42no—
74C22×C17—41no—
75C22×C17#32311yescharacteristic
76C4×C17—21no—
77C4×C17—41no—
78C4×C17AΣL(1,𝔽16)12yes—
79Dic17—44no—
80C23×Q8—171no—
81C2×#101—342no—
82#407—171no—
83#407#59511yescharacteristic
84#349—342no—
85C2×C17—21no—
86C2×C17—42no—
87C2×C17—41no—
88C2×C17#131611yescharacteristic
89C22×Q8—341no—
90C22×Q8—341no—
91C22×Q8—681no—
92C22×Q8—681no—
93C22×Q8—684no—
94C23×C4—172no—
95C23×C4C2×#59511yescharacteristic
96C2×M16—21no—
97C2×M16—172no—
98M32—682no—
99#101—684no—
100C22⋊C8—41no—
101C22⋊C8—681no—
102C17#734111yescharacteristic, complemented, Sylow
103C24#6324211yescharacteristic
104C2×Q8—344no—
105C2×Q8—682no—
106C2×Q8—682no—
107C2×Q8—684no—
108C2×Q8—684no—
109C2×Q8—688no—
110C2×Q8—684no—
111C2×Q8—681no—
112C2×Q8—681no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—342no—
117C22×C4—684no—
118C22×C4—682no—
119C22×C4#6324311yescharacteristic
120C22×C4C22⋊#59511yescharacteristic
121M16—42no—
122M16—682no—
123C2×C8—41no—
124C2×C8—342no—
125C16—1362no—
126C23—21no—
127C23—41no—
128C23—42no—
129C23#1058754311yescharacteristic
130Q8—684no—
131Q8—688no—
132Q8—684no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—344no—
139C2×C4—682no—
140C2×C4—684no—
141C2×C4—682no—
142C2×C4—684no—
143C2×C4#1058753911yescharacteristic
144C2×C4#1058755311yescharacteristic
145C8—81no—
146C8—682no—
147C22—22no—
148C22—21no—
149C22—41no—
150C22—41no—
151C22—42no—
152C22—42no—
153C22—41no—
154C22#1176365311yescharacteristic
155C4—21no—
156C4—41no—
157C4—684no—
158C4#1176371012yes—
159C2—21no—
160C2—42no—
161C2—41no—
162C2#1454370911yescharacteristic
subgroupG/H
centerC2#14543709
commutator, prime radicalC24×C17C2×C8
FrattiniC23×C4C2×#595
FittingC17×#407C4
socleC2×C17#1316
cosocle#11762773C22
hypercenter#407#595
hypocenterC17#7341
LCSG ▷ C24×C17 ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
102. C17#7341semidirect

Maximal subgroups

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

subgroupindex
#145437162
#145287072
#734117