Finite Group Databasebeta

← #17023650

Subgroups — #17023650

Order 4352.

736 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 151 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#14539528C213yesmaximal, complemented
2#14539155C213yesmaximal
3#14528445C213yesmaximal, complemented
4#14528441C213yesmaximal, complemented
5#14528396C211yesmaximal, characteristic, complemented
6#14527086C211yesmaximal, characteristic, complemented
7C17×#1711C211yesmaximal, characteristic, complemented
8#11763364C2213yes—
9#127×D17C2211yescharacteristic
10#11763205C2213yescomplemented
11#11763188C2213yescomplemented
12#11763172—26no—
13#11763160C2213yescomplemented
14#11763157C2213yes—
15#11763156C2213yes—
16#11763155C2213yes—
17#11763066—26no—
18#11762913—26no—
19#11762812—26no—
20C4×#63218C2213yes—
21#11762530C2213yescomplemented
22C17×#487C2213yescomplemented
23C17×#486C2213yes—
24C82×C17C2211yescharacteristic, complemented
25C2×D4×D17—23no—
26C17×C4⋊Q8C2313yes—
27C17×#127C2311yescharacteristic
28#10587632—23no—
29Q8×Dic17—23no—
30C2×#63231—26no—
31#10587626—23no—
32C2×#63218D416yes—
33#10587583—23no—
34#10587578—23no—
35#10587576—23no—
36#10587573—23no—
37#10587560C2311yescharacteristic
38C42×D17C2311yescharacteristic
39#10587555C2313yes—
40C17×#103—26no—
41C17×#102—26no—
42C4×C8×C17C2313yes—
43#10587503—26no—
44#10587492—26no—
45#10587487—26no—
46#10587486D416yes—
47#10587480—26no—
48#10587479—26no—
49C4×#3622C2313yes—
50C23×D17—41no—
51C2×Q8×C17—23no—
52C2×D4×C17—23no—
53C2×C22⋊D17—43no—
54D4×D17—412no—
55C2×D68—23no—
56C2×C4×D17C2×D413yes—
57C2×Dic34—23no—
58C2×C8×C17C2×D416yes—
59C4⋊C4×C17—23no—
60C4⋊C4×C17—23no—
61C42×C17C2411yescharacteristic
62#63231—412no—
63#63227—43no—
64C4⋊Dic17—23no—
65C4⋊Dic17—23no—
66C4⋊Dic17—43no—
67#63225—43no—
68C4×Dic17—43no—
69C4×Dic17C2×D413yes—
70C2×#3622C2×D416yes—
71#63218—46no—
72#19753—171nomaximal, Sylow
73C23×C17—41no—
74C22×D17—41no—
75C22×D17—86no—
76C22×D17#12711yescharacteristic
77Q8×C17—46no—
78D4×C17—46no—
79C2×C4×C17—43no—
80C2×C4×C17C22×D413yes—
81C22⋊D17—86no—
82C2×Dic17—43no—
83C2×Dic17#12713yes—
84D68—46no—
85C4×D17—26no—
86Dic34—46no—
87C8×C17—26no—
88#3622—26no—
89#3185—173no—
90#3138—173no—
91#1720—173no—
92#1720—173no—
93#1711—171no—
94#1711D1711yescharacteristic, complemented
95#1455—171no—
96C22×C17—83no—
97C22×C17C2×#12711yescharacteristic
98C2×D17—23no—
99C2×D17—83no—
100C2×D17—164no—
101C4×C17—83no—
102C4×C17#57516yes—
103Dic17—23no—
104Dic17—83no—
105#533—173no—
106C2×#127—171no—
107#498—173no—
108#494—173no—
109#490—346no—
110#487—173no—
111#487C2×D1713yescomplemented
112#486—173no—
113#486—173no—
114#486—173no—
115#486C2×D1713yes—
116#470—346no—
117#441—346no—
118#418—346no—
119C4×M16—173no—
120#322—173no—
121C82C2×D1711yescharacteristic, complemented
122C2×C17—161no—
123C2×C17#315313yes—
124D17—41no—
125D17—161no—
126C22×D4—343no—
127C2×Q16—346no—
128C2×M16—176no—
129C4⋊Q8—173no—
130C4⋊Q8C22×D1713yes—
131#127—171no—
132#127—343no—
133#127C22×D1711yescharacteristic
134#125—343no—
135C22⋊Q8—343no—
136C22⋊Q8—343no—
137C4×Q8—343no—
138C4×Q8—343no—
139#117—343no—
140C2×C42—171no—
141#106—346no—
142#103—26no—
143#103—346no—
144#102—26no—
145#102—346no—
146#102—346no—
147#102—346no—
148#97—176no—
149C4×C8—173no—
150C4×C8C22×D1713yes—
151C17#1975311yescharacteristic, complemented, Sylow
152C24—681no—
153C2×Q8—23no—
154C2×Q8—343no—
155C2×D4—23no—
156C2×D4—343no—
157C2×D4—683no—
158C2×D4—6812no—
159C22×C4—173no—
160Q16—6812no—
161M16—686no—
162C2×C8—176no—
163C2×C8D4×D1716yes—
164C4⋊C4—23no—
165C4⋊C4—23no—
166C4⋊C4—343no—
167C4⋊C4—343no—
168C4⋊C4—683no—
169C4⋊C4—683no—
170C22⋊C4—683no—
171C42—173no—
172C42—683no—
173C42C23×D1711yescharacteristic
174C23—41no—
175C23—171no—
176C23—681no—
177C23—1366no—
178Q8—46no—
179Q8—686no—
180D4—46no—
181D4—686no—
182D4—1366no—
183C2×C4—43no—
184C2×C4—173no—
185C2×C4—346no—
186C2×C4—683no—
187C2×C4C2×D4×D1713yes—
188C8—26no—
189C8—346no—
190C22—83no—
191C22—2724no—
192C22—343no—
193C22—1363no—
194C22#127×D1711yescharacteristic
195C4—83no—
196C4—343no—
197C4—1363no—
198C4#1176360616yes—
199C2—161no—
200C2—2721no—
201C2—681no—
202C2#1453924513yes—
subgroupG/H
centerC22#127×D17
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#1711C2
socleC22×C17C2×#127
prime radical#127×D17C22
hypercenter#1711D17
hypocenterC17#19753
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #1711

Complements

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

NH ≅ G/Ntype
1. #14539528C2semidirect
3. #14528445C2semidirect
4. #14528441C2semidirect
5. #14528396C2semidirect
6. #14527086C2semidirect
7. C17×#1711C2semidirect
10. #11763205C22semidirect
11. #11763188C22semidirect
13. #11763160C22semidirect
21. #11762530C22semidirect
22. C17×#487C22semidirect
24. C82×C17C22semidirect
94. #1711D17semidirect
111. #487C2×D17semidirect
121. C82C2×D17semidirect
151. C17#19753semidirect

Maximal subgroups

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

subgroupindex
#145395282
#145391552
#145284452
#145284412
#145283962
#145270862
C17×#17112
#1975317