Finite Group Databasebeta

← #14544059

Subgroups — #14544059

Order 2176.

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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#11763780C211yesmaximal, characteristic, complemented
2#11763777C211yesmaximal, characteristic, complemented
3#11763652C214yesmaximal, complemented
4C2×#10587624C211yesmaximal, characteristic
5C22×D4×C17C411yescharacteristic
6C22×C22⋊D17C412yes—
7C22×C4×D17C2211yescharacteristic
8C2×#63243—21no—
9C2×#63242—21no—
10C2×C22⋊Dic17C2212yes—
11#10587624—21no—
12#10587624—21no—
13#10587624—21no—
14#10587624—21no—
15C2×C4⋊Dic17C411yescharacteristic
16C22⋊#3624C2212yes—
17C22⋊#3624C2212yes—
18#10587543—21no—
19#10587543—21no—
20C24×C17C2×C412yes—
21C23×D17C2×C411yescharacteristic
22C2×D4×C17—21no—
23C2×D4×C17—21no—
24C2×D4×C17—42no—
25C22×C4×C17C2×C411yescharacteristic
26C2×C22⋊D17—22no—
27C2×C22⋊D17—22no—
28C2×C22⋊D17—44no—
29C22×Dic17C2×C412yes—
30C22×Dic17C2311yescharacteristic
31C2×C4×D17—42no—
32C2×C4×D17D411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×#3624—22no—
37C2×#3624—22no—
38#63243—42no—
39#63242—42no—
40C22⋊Dic17—22no—
41C22⋊Dic17—22no—
42C4⋊Dic17—21no—
43C4⋊Dic17—21no—
44C23×C17—22no—
45C23×C17—24no—
46C23×C17—42no—
47C23×C17—42no—
48C23×C17C22×C411yescharacteristic
49C22×D17—22no—
50C22×D17—22no—
51C22×D17—41no—
52C22×D17C22⋊C411yescharacteristic
53C22×D17C22⋊C411yescharacteristic
54D4×C17—42no—
55D4×C17—42no—
56C2×C4×C17—22no—
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22⋊C411yescharacteristic
59C22⋊D17—44no—
60C22⋊D17—44no—
61C2×Dic17—22no—
62C2×Dic17—41no—
63C2×Dic17—42no—
64C2×Dic17C2×D411yescharacteristic
65C2×Dic17C2×D411yescharacteristic
66C4×D17—22no—
67C4×D17—22no—
68C4×D17—41no—
69C4×D17—41no—
70#3624—42no—
71#3624—42no—
72#1471—171nomaximal, Sylow
73C22×C17—21no—
74C22×C17—21no—
75C22×C17—42no—
76C22×C17—42no—
77C22×C17—42no—
78C22×C17—84no—
79C22×C17C2×C22⋊C411yescharacteristic
80C22×C17#13112yes—
81C2×D17—21no—
82C2×D17—21no—
83C2×D17—21no—
84C2×D17—21no—
85C2×D17—21no—
86C2×D17—44no—
87C2×D17#10012yes—
88C4×C17—21no—
89C4×C17#9912yes—
90Dic17—21no—
91Dic17—21no—
92Dic17—42no—
93C2×#121—171no—
94#407—171no—
95C2×C22⋊C8—171no—
96#323—174no—
97C2×C17—22no—
98C2×C17—82no—
99C2×C17#40811yescharacteristic
100C2×C17C2×#9911yescharacteristic
101C2×C17C2×#10011yescharacteristic
102D17—22no—
103D17—41no—
104C22×D4—172no—
105C22×D4#59511yescharacteristic
106C23×C4—171no—
107C2×M16—341no—
108C22×C8—341no—
109#121—341no—
110#121—341no—
111#121—341no—
112#121—341no—
113C2×C4⋊C4—171no—
114C2×C22⋊C4—172no—
115C22⋊C8—172no—
116C22⋊C8—172no—
117C22⋊C8—341no—
118C22⋊C8—341no—
119C17#147111yescharacteristic, complemented, Sylow
120C24—171no—
121C24C2×#59512yes—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—42no—
125C2×D4—342no—
126C2×D4—342no—
127C2×D4—684no—
128C22×C4—171no—
129C22×C4—171no—
130C22×C4—171no—
131C22×C4—171no—
132C22×C4—171no—
133C22×C4—172no—
134C22×C4—682no—
135C22×C4C2×#59511yescharacteristic
136M16—682no—
137C2×C8—342no—
138C2×C8—342no—
139C2×C8—682no—
140C4⋊C4—341no—
141C4⋊C4—341no—
142C22⋊C4—342no—
143C22⋊C4—342no—
144C23—22no—
145C23—24no—
146C23—42no—
147C23—42no—
148C23—171no—
149C23—171no—
150C23—342no—
151C23—342no—
152C23—681no—
153C23C22×#59511yescharacteristic
154D4—42no—
155D4—42no—
156D4—684no—
157D4—684no—
158C2×C4—22no—
159C2×C4—171no—
160C2×C4—171no—
161C2×C4—342no—
162C2×C4—342no—
163C2×C4—342no—
164C2×C4—681no—
165C2×C4—681no—
166C2×C4—681no—
167C2×C4—682no—
168C2×C4C22⋊#59511yescharacteristic
169C2×C4C22⋊#59511yescharacteristic
170C8—682no—
171C8—682no—
172C22—21no—
173C22—21no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—84no—
178C22—172no—
179C22—341no—
180C22—341no—
181C22—341no—
182C22—341no—
183C22—341no—
184C22—684no—
185C22#1058767812yes—
186C22C2×C22⋊#59511yescharacteristic
187C4—21no—
188C4—341no—
189C4—341no—
190C4—682no—
191C4#1058755112yes—
192C2—22no—
193C2—82no—
194C2—342no—
195C2—681no—
196C2#1176378311yescharacteristic
197C2C2×#1058755111yescharacteristic
198C2D17⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC22×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×#10587624C2
cosocleC22×Dic17C23
hypercenterC22×D4#595
hypocenterC17#1471
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #11763780C2semidirect
2. #11763777C2semidirect
3. #11763652C2semidirect
119. C17#1471semidirect

Maximal subgroups

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

subgroupindex
#117637802
#117637772
#117636522
C2×#105876242
#147117