Finite Group Databasebeta

← #14529324

Subgroups — #14529324

Order 2176.

380 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 85 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#11763368C211yesmaximal, characteristic
2#11763284C211yesmaximal, characteristic, complemented
3C2×#10587477C212yesmaximal, complemented
4#11762576C212yesmaximal, complemented
5C2×C17×#99C211yesmaximal, characteristic, complemented
6C22×D4×C17C2211yescharacteristic
7C22×C22⋊D17C2211yescharacteristic
8C2×C22⋊C4×C17C2212yes—
9#10587639—21no—
10C2×C22⋊Dic17—21no—
11C2×C22⋊Dic17C2211yescharacteristic
12C2×C22⋊Dic17C2212yes—
13#10587625—22no—
14#10587623C412yescomplemented
15#10587621C412yescomplemented
16D4×Dic17—24no—
17#10587615—21no—
18#10587612—21no—
19C17×#99—22no—
20#10587477—24no—
21C24×C17D411yescharacteristic
22C24×C17C2311yescharacteristic
23C23×D17C2×C411yescharacteristic, complemented
24C2×D4×C17—22no—
25C2×D4×C17—22no—
26C2×D4×C17C2×C412yes—
27C2×D4×C17D412yes—
28C22×C4×C17—22no—
29C22×C4×C17D411yescharacteristic
30C2×C22⋊D17—22no—
31C2×C22⋊D17—21no—
32C2×C22⋊D17—21no—
33C2×C22⋊D17—22no—
34C2×C22⋊D17—22no—
35C22×Dic17—22no—
36C22×Dic17—22no—
37C22×Dic17—21no—
38C22×Dic17C2×C411yescharacteristic, complemented
39C22⋊C4×C17—22no—
40C22⋊C4×C17—22no—
41C22⋊Dic17—22no—
42C22⋊Dic17—22no—
43C22⋊Dic17—22no—
44C22⋊Dic17—21no—
45C22⋊Dic17C2×C412yescomplemented
46C22⋊Dic17D414yescomplemented
47#63227—22no—
48C4⋊Dic17—22no—
49#63225—22no—
50C4×Dic17—22no—
51C23×C17—21no—
52C23×C17—21no—
53C23×C17—81no—
54C23×C17—81no—
55C23×C17C22⋊C412yes—
56C23×C17C22×C411yescharacteristic
57C23×C17C2×D411yescharacteristic
58C23×C17C2×D412yes—
59C23×C17C2×D411yescharacteristic
60C23×C17#4112yes—
61C22×D17—21no—
62C22×D17—22no—
63C22×D17—81no—
64D4×C17—24no—
65D4×C17—24no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—42no—
69C2×C4×C17—82no—
70C2×C4×C17C22⋊C412yes—
71C22⋊D17—41no—
72C22⋊D17—42no—
73C22⋊D17—41no—
74C22⋊D17—82no—
75C2×Dic17—22no—
76C2×Dic17—21no—
77C2×Dic17—22no—
78C2×Dic17—24no—
79C2×Dic17—82no—
80C2×Dic17—82no—
81C2×Dic17—81no—
82#1913—171nomaximal, Sylow
83C22×C17—22no—
84C22×C17—42no—
85C22×C17—41no—
86C22×C17—42no—
87C22×C17—41no—
88C22×C17—41no—
89C22×C17—42no—
90C22×C17—42no—
91C22×C17—81no—
92C22×C17C2×C22⋊C411yescharacteristic
93C22×C17C4×D412yes—
94C22×C17#12011yescharacteristic
95C22×C17#12112yes—
96C22×C17#12311yescharacteristic
97C2×D17—42no—
98C2×D17—41no—
99C2×D17—82no—
100C4×C17—22no—
101C4×C17—82no—
102Dic17—44no—
103Dic17—42no—
104Dic17—41no—
105#534—171no—
106#518—171no—
107C2×#99—172no—
108C2×#99D1711yescharacteristic, complemented
109#342—172no—
110C2×C17—21no—
111C2×C17—21no—
112C2×C17—22no—
113C2×C17—22no—
114C2×C17—21no—
115C2×C17—42no—
116C2×C17#38611yescharacteristic
117D17—81no—
118C22×D4—171no—
119C22×D4C2×D1711yescharacteristic
120#123—172no—
121#123—341no—
122#121—342no—
123#120—172no—
124#120—341no—
125C4×D4—344no—
126#117—341no—
127C2×C22⋊C4—172no—
128C2×C22⋊C4—171no—
129C2×C22⋊C4—341no—
130C2×C22⋊C4C2×D1712yes—
131#99—22no—
132#99—344no—
133C17#191311yescharacteristic, complemented, Sylow
134C24—171no—
135C24D6811yescharacteristic
136C24C22×D1711yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—341no—
140C2×D4—342no—
141C2×D4—342no—
142C2×D4—341no—
143C2×D4—342no—
144C2×D4C4×D1712yes—
145C2×D4C22⋊D1712yes—
146C22×C4—22no—
147C22×C4—171no—
148C22×C4—342no—
149C22×C4—342no—
150C22×C4—341no—
151C22×C4D6811yescharacteristic
152C4⋊C4—342no—
153C4⋊C4—342no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—172no—
157C22⋊C4—174no—
158C22⋊C4—342no—
159C22⋊C4—342no—
160C22⋊C4—341no—
161C22⋊C4—342no—
162C22⋊C4—342no—
163C42—342no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—341no—
169C23—342no—
170C23—1361no—
171C23#6322712yes—
172C23C2×C4×D1711yescharacteristic
173C23C2×D6811yescharacteristic
174C23D4×D1712yes—
175C23#6325412yes—
176C23C2×C22⋊D1711yescharacteristic
177D4—24no—
178D4—24no—
179D4—681no—
180D4—681no—
181D4—682no—
182D4—1362no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—342no—
188C2×C4—344no—
189C2×C4—342no—
190C2×C4—341no—
191C2×C4—1362no—
192C2×C4—1362no—
193C2×C4—1361no—
194C2×C4#6322712yes—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—682no—
205C22—681no—
206C22—1362no—
207C22#1058756712yes—
208C22#1058756811yescharacteristic
209C22#1058757211yescharacteristic
210C22C2×#6322711yescharacteristic
211C22#1058762512yes—
212C4—22no—
213C4—82no—
214C4—684no—
215C4—682no—
216C4—681no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—1361no—
224C2#1176268611yescharacteristic
subgroupG/H
centerC2#11762686
commutatorC23×C17C22×C4
FrattiniC24C22×D17
FittingC2×C17×#99C2
socleC2×C17#386
prime radical#11763368C2
cosocleC24×C17C23
hypercenterC2×#99D17
hypocenterC17#1913
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #11763284C2semidirect
3. C2×#10587477C2semidirect
4. #11762576C2semidirect
5. C2×C17×#99C2semidirect
14. #10587623C4semidirect
15. #10587621C4semidirect
23. C23×D17C2×C4semidirect
38. C22×Dic17C2×C4semidirect
45. C22⋊Dic17C2×C4semidirect
46. C22⋊Dic17D4semidirect
108. C2×#99D17semidirect
133. C17#1913semidirect

Maximal subgroups

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

subgroupindex
#117633682
#117632842
C2×#105874772
#117625762
C2×C17×#992
#191317