Finite Group Databasebeta

← #17023401

Subgroups — #17023401

Order 4352.

712 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 179 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#14539945C213yesmaximal, complemented
2#14539727C213yesmaximal
3#14528437C213yesmaximal, complemented
4#14528406C213yesmaximal, complemented
5#14528396C211yesmaximal, characteristic, complemented
6#14528371C211yesmaximal, characteristic, complemented
7C17×#1454C211yesmaximal, characteristic, complemented
8#11763366C2213yes—
9C2×#10587560C2211yescharacteristic
10#11763214C2213yescomplemented
11#11763207C2213yescomplemented
12#11763182C2213yes—
13#11763182C2213yes—
14#11763160C2213yescomplemented
15#11763160C2213yescomplemented
16#11763156C2213yes—
17#11763156C2213yes—
18#11763151—26no—
19#11763117—26no—
20#11763010—26no—
21#11762962—26no—
22C17×#431C2213yes—
23C17×#322C2213yescomplemented
24C82×C17C2211yescharacteristic, complemented
25C22×D68—23no—
26C2×C42×C17C2311yescharacteristic
27#10587612—23no—
28#10587610—23no—
29#10587610—23no—
30C2×#63219—26no—
31C2×#63219—26no—
32#10587560—23no—
33#10587560C2311yescharacteristic
34#10587560C2311yescharacteristic
35#10587556—23no—
36#10587555C2313yes—
37#10587555C2313yes—
38C4×Dic34—23no—
39C4×Dic34—23no—
40C17×C4⋊C8D416yes—
41C17×C22⋊C8D416yes—
42C4×C8×C17C2313yes—
43C4×C8×C17C2313yes—
44#10587492—26no—
45#10587492—26no—
46#10587489—26no—
47#10587488—26no—
48#10587487—26no—
49#10587487—26no—
50C23×D17—41no—
51C22×C4×C17C2×D413yes—
52C2×D68—23no—
53C2×D68—23no—
54C2×D68—43no—
55C2×D68—86no—
56C2×Dic34—23no—
57C2×Dic34—23no—
58C2×C8×C17C2×D416yes—
59C2×C8×C17C2×D416yes—
60C42×C17C2×D413yes—
61C42×C17C2411yescharacteristic
62C22⋊Dic17—43no—
63C4⋊Dic17—23no—
64C4⋊Dic17—23no—
65C4⋊Dic17—23no—
66C4⋊Dic17—23no—
67#63225—43no—
68#63225—43no—
69C4×Dic17—43no—
70#63219—86no—
71#63219—86no—
72#19733—171nomaximal, Sylow
73C23×C17#12711yescharacteristic
74C22×D17—41no—
75C22×D17—41no—
76C22×D17—86no—
77C2×C4×C17—26no—
78C2×C4×C17#12713yes—
79C2×C4×C17C22×D413yes—
80C2×Dic17—43no—
81C2×Dic17—43no—
82D68—46no—
83D68—46no—
84D68—86no—
85Dic34—46no—
86Dic34—46no—
87C8×C17—26no—
88C8×C17—26no—
89#3231—173no—
90#3208—173no—
91#1719—173no—
92#1714—173no—
93#1711—171no—
94#1705—171no—
95#1454D1711yescharacteristic, complemented
96C22×C17—23no—
97C22×C17C2×#12711yescharacteristic
98C2×D17—83no—
99C2×D17—83no—
100C2×D17—164no—
101C4×C17—23no—
102C4×C17#57716yes—
103Dic17—83no—
104Dic17—83no—
105#533—173no—
106C2×#127—171no—
107#500—173no—
108#498—173no—
109#492—173no—
110#492—173no—
111#487—173no—
112#487—173no—
113#486—173no—
114#486—173no—
115#485—346no—
116#479—346no—
117#460—346no—
118#450—346no—
119#431C2×D1713yes—
120#322C2×D1713yescomplemented
121C82C2×D1711yescharacteristic, complemented
122C2×C17—41no—
123C2×C17#315413yes—
124D17—161no—
125D17—161no—
126C22×D4—343no—
127C2×SD16—346no—
128C2×SD16—346no—
129C4⋊Q8—173no—
130C4⋊Q8—173no—
131#127—171no—
132#127—171no—
133#127—343no—
134#125—343no—
135C22⋊Q8—343no—
136C22⋊Q8—343no—
137C4×Q8—343no—
138C4×Q8—343no—
139#117—343no—
140C2×C42C22×D1711yescharacteristic
141#107—346no—
142#106—346no—
143C4⋊C8D6816yes—
144#103—346no—
145#103—346no—
146#102—346no—
147#102—346no—
148C22⋊C8D6816yes—
149C4×C8C22×D1713yes—
150C4×C8C22×D1713yes—
151C17#1973311yescharacteristic, complemented, Sylow
152C24—681no—
153C2×Q8—343no—
154C2×Q8—343no—
155C2×D4—343no—
156C2×D4—343no—
157C2×D4—683no—
158C2×D4—1366no—
159C22×C4C2×D6813yes—
160SD16—1366no—
161SD16—1366no—
162C2×C8C2×D6816yes—
163C2×C8C2×D6816yes—
164C4⋊C4—343no—
165C4⋊C4—343no—
166C4⋊C4—343no—
167C4⋊C4—343no—
168C4⋊C4—683no—
169C4⋊C4—683no—
170C22⋊C4—683no—
171C42—683no—
172C42C2×D6813yes—
173C42C23×D1711yescharacteristic
174C23—681no—
175C23—681no—
176C23—1366no—
177C23#1058756011yescharacteristic
178Q8—686no—
179Q8—686no—
180D4—686no—
181D4—686no—
182D4—1366no—
183C2×C4—26no—
184C2×C4—683no—
185C2×C4—683no—
186C2×C4#1058756013yes—
187C2×C4C22×D6813yes—
188C8—26no—
189C8—26no—
190C22—23no—
191C22—2724no—
192C22—1363no—
193C22—1363no—
194C22C2×#1058756011yescharacteristic
195C4—23no—
196C4—1363no—
197C4—1363no—
198C4#1176362116yes—
199C2—41no—
200C2—2721no—
201C2—2721no—
202C2#1453926013yes—
subgroupG/H
centerC22C2×#10587560
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#1454C2
socleC22×C17C2×#127
prime radicalC2×#10587560C22
hypercenter#1454D17
hypocenterC17#19733
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1454

Complements

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

NH ≅ G/Ntype
1. #14539945C2semidirect
3. #14528437C2semidirect
4. #14528406C2semidirect
5. #14528396C2semidirect
6. #14528371C2semidirect
7. C17×#1454C2semidirect
10. #11763214C22semidirect
11. #11763207C22semidirect
14. #11763160C22semidirect
15. #11763160C22semidirect
23. C17×#322C22semidirect
24. C82×C17C22semidirect
95. #1454D17semidirect
120. #322C2×D17semidirect
121. C82C2×D17semidirect
151. C17#19733semidirect

Maximal subgroups

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

subgroupindex
#145399452
#145397272
#145284372
#145284062
#145283962
#145283712
C17×#14542
#1973317