Finite Group Databasebeta

← D17×#396 #14532003

Subgroups — D17×#396 #14532003

Order 2176.

908 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 203 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
1C2×C4⋊C4×D17C213yesmaximal, complemented
2C2×C22⋊C4×D17C213yesmaximal
3#11762733C211yesmaximal, characteristic, complemented
4#11762732C213yesmaximal, complemented
5#11762731C213yesmaximal, complemented
6#95×D17C211yesmaximal, characteristic, complemented
7C17×#396C211yesmaximal, characteristic, complemented
8C24×D17C2211yescharacteristic
9C22×C4×D17C2213yes—
10C22×C4×D17C2213yes—
11C2×C4⋊C4×C17C2213yescomplemented
12C2×C22⋊C4×C17C2213yes—
13C2×C22⋊Dic17C2213yes—
14C2×#63227C2213yes—
15C2×#63227C2213yes—
16C2×C4⋊Dic17C2213yescomplemented
17C2×#63225C2213yescomplemented
18C2×#63225C2213yescomplemented
19C4⋊C4×D17—212no—
20C22⋊C4×D17—212no—
21C17×#95C2211yescharacteristic, complemented
22#10587502C2213yescomplemented
23C24×C17C2311yescharacteristic
24C23×D17—26no—
25C23×D17—26no—
26C23×D17Q814yes—
27C23×D17C2311yescharacteristic
28C23×D17C2311yescharacteristic
29C22×C4×C17C2313yes—
30C22×C4×C17C2313yes—
31C22×Dic17C2313yes—
32C22×Dic17C2313yes—
33C2×C4×D17—26no—
34C2×C4×D17—26no—
35C2×C4×D17—26no—
36C2×C4×D17—26no—
37C2×C4×D17—26no—
38C2×C4×D17D4112yes—
39C4⋊C4×C17—26no—
40C22⋊C4×C17—26no—
41C22⋊Dic17—26no—
42#63227—26no—
43#63227—26no—
44C4⋊Dic17—26no—
45#63225—26no—
46#63225—26no—
47C23×C17—23no—
48C23×C17—23no—
49C23×C17C2×Q812yes—
50C23×C17C2411yescharacteristic
51C22×D17—23no—
52C22×D17—23no—
53C22×D17—224no—
54C22×D17—412no—
55C22×D17—44no—
56C22×D17C2×D416yes—
57C22×D17C2×Q812yes—
58C22×D17C2×Q812yes—
59C22×D17#4116yes—
60C2×C4×C17—23no—
61C2×C4×C17—23no—
62C2×C4×C17—23no—
63C2×C4×C17—23no—
64C2×C4×C17—23no—
65C2×C4×C17C2×D416yes—
66C2×Dic17—23no—
67C2×Dic17—23no—
68C2×Dic17—23no—
69C2×Dic17—23no—
70C2×Dic17—23no—
71C2×Dic17C2×D416yes—
72C4×D17—224no—
73C4×D17—412no—
74C2×#396—171nomaximal, Sylow
75C22×C17—26no—
76C22×C17—43no—
77C22×C17—41no—
78C22×C17C22×D413yes—
79C22×C17C22×Q811yescharacteristic
80C22×C17C2×#4113yes—
81C2×D17—26no—
82C2×D17—43no—
83C2×D17—41no—
84C2×D17—416no—
85C2×D17#121112yes—
86C2×D17C22⋊Q8112yes—
87C2×D17#12614yes—
88C4×C17—26no—
89C4×C17—43no—
90Dic17—26no—
91Dic17—43no—
92C22×C4⋊C4—173no—
93C22×C22⋊C4—173no—
94#396—173no—
95#396—173no—
96#396—171no—
97#396D1711yescharacteristic, direct product factor
98C2×#95—171no—
99C2×C17—42no—
100C2×C17C2×#12113yes—
101C2×C17C2×C22⋊Q813yes—
102C2×C17C2×#12611yescharacteristic
103D17—42no—
104D17#39618yesdirect product factor
105C25—171no—
106C23×C4—173no—
107C23×C4—173no—
108C2×C4⋊C4—173no—
109C2×C4⋊C4—173no—
110C2×C4⋊C4—173no—
111C2×C4⋊C4—3412no—
112C2×C4⋊C4C2×D1713yescomplemented
113C2×C22⋊C4—173no—
114C2×C22⋊C4—173no—
115C2×C22⋊C4—173no—
116C2×C22⋊C4—3412no—
117C2×C22⋊C4C2×D1713yes—
118#95—173no—
119#95C2×D1711yescharacteristic, complemented
120C17C2×#39611yescharacteristic, complemented, Sylow
121C24—171no—
122C24—174no—
123C24—171no—
124C24—346no—
125C24—346no—
126C24C22×D1711yescharacteristic
127C22×C4—173no—
128C22×C4—1712no—
129C22×C4—173no—
130C22×C4—346no—
131C22×C4—346no—
132C22×C4—346no—
133C22×C4—346no—
134C22×C4—346no—
135C22×C4C22×D1713yes—
136C22×C4C22×D1713yes—
137C4⋊C4—26no—
138C4⋊C4—346no—
139C4⋊C4—346no—
140C4⋊C4—346no—
141C22⋊C4—26no—
142C22⋊C4—346no—
143C22⋊C4—346no—
144C22⋊C4—346no—
145C23—23no—
146C23—23no—
147C23—176no—
148C23—172no—
149C23—176no—
150C23—172no—
151C23—3424no—
152C23—343no—
153C23—343no—
154C23—6812no—
155C23—684no—
156C23Q8×D1712yes—
157C23C23×D1711yescharacteristic
158C2×C4—23no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—23no—
162C2×C4—23no—
163C2×C4—176no—
164C2×C4—3424no—
165C2×C4—343no—
166C2×C4—343no—
167C2×C4—343no—
168C2×C4—343no—
169C2×C4—343no—
170C2×C4—6812no—
171C2×C4D4×D1716yes—
172C22—26no—
173C22—43no—
174C22—41no—
175C22—1712no—
176C22—1712no—
177C22—174no—
178C22—346no—
179C22—6816no—
180C22—683no—
181C22—681no—
182C22C2×D4×D1713yes—
183C22C2×Q8×D1711yescharacteristic
184C22#41×D1713yes—
185C4—26no—
186C4—43no—
187C4—346no—
188C4—683no—
189C2—42no—
190C2—178no—
191C2—682no—
192C2#121×D1713yes—
193C2C22⋊Q8×D1713yes—
194C2#126×D1711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#396C2
prime radicalC24×D17C22
hypercenter#396D17
hypocenterC17C2×#396
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D17C2semidirect
3. #11762733C2semidirect
4. #11762732C2semidirect
5. #11762731C2semidirect
6. #95×D17C2semidirect
7. C17×#396C2semidirect
11. C2×C4⋊C4×C17C22semidirect
16. C2×C4⋊Dic17C22semidirect
17. C2×#63225C22semidirect
18. C2×#63225C22semidirect
21. C17×#95C22semidirect
22. #10587502C22semidirect
97. #396D17direct (N × H)
104. D17#396direct (N × H)
112. C2×C4⋊C4C2×D17semidirect
119. #95C2×D17semidirect
120. C17C2×#396semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D172
C2×C22⋊C4×D172
#117627332
#117627322
#117627312
#95×D172
C17×#3962
C2×#39617