Finite Group Databasebeta

← #16933708

Subgroups — #16933708

Order 4352.

400 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 119 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#14528915C211yesmaximal, characteristic, complemented
2#14527348C212yesmaximal, complemented
3#14527344C212yesmaximal
4#14527067C211yesmaximal, characteristic, complemented
5C17×#3131C211yesmaximal, characteristic
6#11762862C2212yes—
7#11762675C2212yes—
8#11762572—22no—
9#11762546—24no—
10#11762543—24no—
11#11762530—21no—
12C8×#3622—21no—
13C17×#533C2211yescharacteristic
14C17×#532C2211yescharacteristic
15C17×#488—21no—
16C17×#488—21no—
17C17×#487C412yescomplemented
18C17×#486C412yescomplemented
19C17×#416—22no—
20C2×C4×C8×C17C2211yescharacteristic
21C2×#41×C17—21no—
22C22×C8×C17D412yes—
23C17×C4⋊Q8C2×C411yescharacteristic
24C17×C4⋊Q8C2×C412yes—
25C17×#127C2×C411yescharacteristic
26C17×#125—21no—
27C17×#124—21no—
28C17×C22⋊Q8—22no—
29C17×#121—22no—
30C4×Q8×C17—22no—
31C4×D4×C17—22no—
32C17×#117—21no—
33C2×C42×C17C2311yescharacteristic
34C2×C4⋊Dic17—22no—
35C17×#103—22no—
36C17×#103—22no—
37C17×#102—22no—
38C17×#102—22no—
39C4×C8×C17C2×C412yes—
40C4×C8×C17D411yescharacteristic
41C4×C8×C17D411yescharacteristic
42#10587502—22no—
43#10587502—22no—
44C22⋊#3622D414yes—
45C4⋊#3622—22no—
46C4×#3622—22no—
47#41×C17—44no—
48C2×Q8×C17—21no—
49C2×Q8×C17—22no—
50C2×D4×C17—21no—
51C2×D4×C17—22no—
52C2×D4×C17—41no—
53C22×C4×C17—41no—
54C22×C4×C17C2×D412yes—
55C22×C4×C17C2×D411yescharacteristic
56C22×Dic17—42no—
57C2×C8×C17—24no—
58C2×C8×C17C22⋊C412yes—
59C2×C8×C17C22⋊C412yes—
60C4⋊C4×C17—22no—
61C4⋊C4×C17—22no—
62C4⋊C4×C17—22no—
63C4⋊C4×C17—42no—
64C22⋊C4×C17—41no—
65C22⋊C4×C17—42no—
66C42×C17—41no—
67C42×C17C22×C411yescharacteristic
68C42×C17C2×D411yescharacteristic
69C42×C17#4112yes—
70C4⋊Dic17—44no—
71C2×#3622—24no—
72#9757—171nomaximal, Sylow
73C23×C17—41no—
74C23×C17#12711yescharacteristic
75Q8×C17—42no—
76Q8×C17—82no—
77D4×C17—42no—
78D4×C17—82no—
79D4×C17—82no—
80C2×C4×C17—42no—
81C2×C4×C17—42no—
82C2×C4×C17—41no—
83C2×C4×C17—82no—
84C2×C4×C17C2×C22⋊C411yescharacteristic
85C2×C4×C17C4×D412yes—
86C2×C4×C17#12112yes—
87C2×C4×C17#12411yescharacteristic
88C2×C4×C17#13514yes—
89C2×Dic17—84no—
90C2×Dic17—82no—
91C8×C17—22no—
92C8×C17—22no—
93#3622—44no—
94#3131D1711yescharacteristic
95#1835—171no—
96#1518—172no—
97#1517—172no—
98#1450—171no—
99C22×C17—22no—
100C22×C17—81no—
101C22×C17—82no—
102C22×C17#39011yescharacteristic
103C22×C17#49512yes—
104C4×C17—22no—
105C4×C17—22no—
106C4×C17—81no—
107C4×C17—82no—
108C4×C17#41612yes—
109C4×C17#48812yes—
110Dic17—162no—
111#533C2×D1711yescharacteristic
112#532C2×D1711yescharacteristic
113#488—21no—
114#488—21no—
115#487Dic1712yescomplemented
116#486Dic1712yescomplemented
117#431—172no—
118#416—22no—
119C2×C4×C8C2×D1711yescharacteristic
120#382—172no—
121#340—342no—
122#328—344no—
123#327—344no—
124#322—341no—
125C82—341no—
126C2×C17—22no—
127C2×C17—161no—
128C2×C17#195811yescharacteristic
129C2×C17#197612yes—
130C2×#41—21no—
131C22×C8C22⋊D1712yes—
132C4⋊Q8C2×Dic1711yescharacteristic
133C4⋊Q8C2×Dic1712yes—
134#127C2×Dic1711yescharacteristic
135#125—21no—
136#124—21no—
137C22⋊Q8—22no—
138#121—22no—
139C4×Q8—22no—
140C4×D4—22no—
141#117—21no—
142C2×C4⋊C4—342no—
143C2×C42C22×D1711yescharacteristic
144C4⋊C8—342no—
145#103—22no—
146#103—22no—
147#102—22no—
148#102—22no—
149C22⋊C8—174no—
150C4×C8—342no—
151C4×C8C2×Dic1712yes—
152C4×C8C22⋊D1711yescharacteristic
153C4×C8C22⋊D1711yescharacteristic
154#95—342no—
155#95—342no—
156C17#975711yescharacteristic, complemented, Sylow
157#41—44no—
158C2×Q8—22no—
159C2×Q8—21no—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—41no—
163C22×C4—41no—
164C22×C4—682no—
165C22×C4D4×D1712yes—
166C22×C4C2×C22⋊D1711yescharacteristic
167C2×C8—24no—
168C2×C8—344no—
169C2×C8C22⋊Dic1712yes—
170C2×C8C22⋊Dic1712yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—684no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C42—41no—
179C42#6325412yes—
180C42C22×Dic1711yescharacteristic
181C42C2×C22⋊D1711yescharacteristic
182C23—41no—
183C23#1058762611yescharacteristic
184Q8—42no—
185Q8—82no—
186D4—42no—
187D4—82no—
188D4—82no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—1364no—
194C2×C4—1362no—
195C2×C4D4×Dic1712yes—
196C2×C4#1058762211yescharacteristic
197C2×C4#1058762412yes—
198C2×C4#1058763614yes—
199C2×C4C2×C22⋊Dic1711yescharacteristic
200C8—22no—
201C8—22no—
202C8—684no—
203C22—22no—
204C22—81no—
205C22—82no—
206C22#1176271211yescharacteristic
207C22#1176319112yes—
208C4—22no—
209C4—22no—
210C4—81no—
211C4—82no—
212C4—2722no—
213C4#1176280512yes—
214C4#1176316312yes—
215C2—22no—
216C2—161no—
217C2#1452959611yescharacteristic
218C2#1452968312yes—
subgroupG/H
centerC22#11762712
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#3131C2
socleC22×C17#390
prime radicalC17×#532C22
cosocleC2×C42×C17C23
hypercenter#3131D17
hypocenterC17#9757
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #14528915C2semidirect
2. #14527348C2semidirect
4. #14527067C2semidirect
17. C17×#487C4semidirect
18. C17×#486C4semidirect
115. #487Dic17semidirect
116. #486Dic17semidirect
156. C17#9757semidirect

Maximal subgroups

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

subgroupindex
#145289152
#145273482
#145273442
#145270672
C17×#31312
#975717