Finite Group Databasebeta

← #16933707

Subgroups — #16933707

Order 4352.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 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#14539107C211yesmaximal, characteristic
2#14528915C211yesmaximal, characteristic, complemented
3#14527349C212yesmaximal, complemented
4#14527345C212yesmaximal
5C17×#1450C211yesmaximal, characteristic, complemented
6#11763366C2211yescharacteristic
7#11763359C2211yescharacteristic
8#11763164—21no—
9#11763164—21no—
10#11763160C412yescomplemented
11#11763156C412yescomplemented
12#11762806—22no—
13#11762675C2212yes—
14#11762572—22no—
15#11762547—24no—
16#11762544—24no—
17C17×#431C2212yes—
18C2×C4×C8×C17C2211yescharacteristic
19C17×#322—21no—
20C82×C17—21no—
21C2×#63252—21no—
22C22×C8×C17D412yes—
23C2×C42×C17C2311yescharacteristic
24#10587616—22no—
25#10587612—21no—
26C2×C4⋊Dic17—22no—
27#10587610—22no—
28#10587561—21no—
29#10587560C2×C411yescharacteristic
30C4×D68—22no—
31#10587556—21no—
32#10587555C2×C411yescharacteristic
33#10587555C2×C412yes—
34C4×Dic34—22no—
35C17×C4⋊C8—22no—
36C17×C22⋊C8D414yes—
37C4×C8×C17—22no—
38C4×C8×C17C2×C412yes—
39C4×C8×C17D411yescharacteristic
40C4×C8×C17D411yescharacteristic
41#10587502—22no—
42#10587502—22no—
43#10587492—22no—
44#10587492—22no—
45#10587487—22no—
46#10587487—22no—
47C22×C4×C17C2×D412yes—
48C22×C4×C17C2×D411yescharacteristic
49C2×C22⋊D17—41no—
50C22×Dic17—42no—
51#63252—44no—
52C2×D68—21no—
53C2×D68—22no—
54C2×C4×D17—41no—
55C2×Dic34—21no—
56C2×Dic34—22no—
57C2×C8×C17—24no—
58C2×C8×C17—24no—
59C2×C8×C17C22⋊C412yes—
60C2×C8×C17C22⋊C412yes—
61C42×C17C22×C411yescharacteristic
62C42×C17C2×D411yescharacteristic
63C42×C17#4112yes—
64C22⋊Dic17—41no—
65#63227—42no—
66C4⋊Dic17—22no—
67C4⋊Dic17—22no—
68C4⋊Dic17—22no—
69C4⋊Dic17—44no—
70#63225—42no—
71C4×Dic17—41no—
72#9757—171nomaximal, Sylow
73C23×C17#12711yescharacteristic
74C22×D17—41no—
75C2×C4×C17—42no—
76C2×C4×C17C2×C22⋊C411yescharacteristic
77C2×C4×C17C4×D412yes—
78C2×C4×C17#12112yes—
79C2×C4×C17#12411yescharacteristic
80C2×C4×C17#13514yes—
81C22⋊D17—82no—
82C2×Dic17—42no—
83C2×Dic17—41no—
84C2×Dic17—84no—
85C2×Dic17—82no—
86D68—42no—
87D68—82no—
88C4×D17—82no—
89Dic34—42no—
90Dic34—82no—
91C8×C17—22no—
92C8×C17—22no—
93C8×C17—44no—
94#3131—171no—
95#1835—171no—
96#1518—172no—
97#1517—172no—
98#1450D1711yescharacteristic, complemented
99C22×C17—22no—
100C22×C17#39011yescharacteristic
101C22×C17#49512yes—
102C2×D17—81no—
103C2×D17—82no—
104C4×C17—22no—
105C4×C17—22no—
106C4×C17#41612yes—
107C4×C17#48812yes—
108Dic17—81no—
109Dic17—82no—
110Dic17—162no—
111#533—171no—
112#532—171no—
113#488—341no—
114#488—341no—
115#487—172no—
116#486—172no—
117#431C2×D1712yes—
118#416—342no—
119C2×C4×C8C2×D1711yescharacteristic
120#382—172no—
121#340—342no—
122#328—344no—
123#327—344no—
124#322—21no—
125C82—21no—
126C2×C17—22no—
127C2×C17#195811yescharacteristic
128C2×C17#197612yes—
129D17—161no—
130C2×#41—341no—
131C22×C8D6812yes—
132C4⋊Q8—171no—
133C4⋊Q8—172no—
134#127—171no—
135#125—341no—
136#124—341no—
137C22⋊Q8—342no—
138#121—342no—
139C4×Q8—342no—
140C4×D4—342no—
141#117—341no—
142C2×C4⋊C4—342no—
143C2×C42C22×D1711yescharacteristic
144C4⋊C8—22no—
145#103—342no—
146#103—342no—
147#102—342no—
148#102—342no—
149C22⋊C8D6814yes—
150C4×C8—22no—
151C4×C8C4×D1712yes—
152C4×C8C22⋊D1711yescharacteristic
153C4×C8C22⋊D1711yescharacteristic
154#95—342no—
155#95—342no—
156C17#975711yescharacteristic, complemented, Sylow
157#41—684no—
158C2×Q8—342no—
159C2×Q8—341no—
160C2×D4—342no—
161C2×D4—341no—
162C2×D4—681no—
163C22×C4—681no—
164C22×C4—682no—
165C22×C4C2×D6812yes—
166C22×C4C2×D6811yescharacteristic
167C2×C8—24no—
168C2×C8—24no—
169C2×C8#6322712yes—
170C2×C8#6322712yes—
171C4⋊C4—342no—
172C4⋊C4—342no—
173C4⋊C4—342no—
174C4⋊C4—682no—
175C4⋊C4—684no—
176C22⋊C4—681no—
177C22⋊C4—682no—
178C42—681no—
179C42C2×C4×D1711yescharacteristic
180C42#6325212yes—
181C42C2×C22⋊D1711yescharacteristic
182C23—681no—
183C23#1058756011yescharacteristic
184Q8—682no—
185Q8—1362no—
186D4—682no—
187D4—1362no—
188D4—1362no—
189C2×C4—42no—
190C2×C4—682no—
191C2×C4—681no—
192C2×C4—1364no—
193C2×C4—1362no—
194C2×C4—1362no—
195C2×C4C4×D6812yes—
196C2×C4#1058756111yescharacteristic
197C2×C4#1058759014yes—
198C2×C4C2×#6322711yescharacteristic
199C2×C4#1058761612yes—
200C8—22no—
201C8—22no—
202C8—44no—
203C22—22no—
204C22—1361no—
205C22—1362no—
206C22#1176271111yescharacteristic
207C22#1176319312yes—
208C4—22no—
209C4—22no—
210C4—2722no—
211C4—1361no—
212C4—1362no—
213C4#1176280612yes—
214C4#1176316412yes—
215C2—22no—
216C2—2721no—
217C2#1452959511yescharacteristic
218C2#1452968212yes—
subgroupG/H
centerC22#11762711
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1450C2
socleC22×C17#390
prime radical#11763359C22
cosocleC2×C42×C17C23
hypercenter#1450D17
hypocenterC17#9757
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #14528915C2semidirect
3. #14527349C2semidirect
5. C17×#1450C2semidirect
10. #11763160C4semidirect
11. #11763156C4semidirect
98. #1450D17semidirect
156. C17#9757semidirect

Maximal subgroups

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

subgroupindex
#145391072
#145289152
#145273492
#145273452
C17×#14502
#975717