Finite Group Databasebeta

← #14537047

Subgroups — #14537047

Order 2176.

548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 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#11762746C211yesmaximal, characteristic, complemented
2#11762744C213yesmaximal, complemented
3#11762733C211yesmaximal, characteristic, complemented
4#11762732C213yesmaximal, complemented
5#11762731C213yesmaximal, complemented
6#11762664C213yesmaximal
7C17×#396C211yesmaximal, characteristic, complemented
8C24×D17C2211yescharacteristic
9C2×C4⋊C4×C17C2213yescomplemented
10C2×C22⋊C4×C17C2213yes—
11C2×C22⋊Dic17C2213yes—
12C2×#63227C2213yes—
13C2×#63227C2213yes—
14C2×#63227C2213yes—
15C2×#63227C2213yes—
16C2×C4⋊Dic17C2213yescomplemented
17C2×#63225C2213yescomplemented
18C2×#63225C2213yescomplemented
19C17×#95C2211yescharacteristic, complemented
20#10587502C2213yescomplemented
21C24×C17C2311yescharacteristic
22C23×D17—43no—
23C23×D17—43no—
24C23×D17Q814yes—
25C23×D17C2311yescharacteristic
26C23×D17C2311yescharacteristic
27C22×C4×C17C2313yes—
28C22×C4×C17C2313yes—
29C22×Dic17C2313yes—
30C22×Dic17C2313yes—
31C4⋊C4×C17—43no—
32C22⋊C4×C17—43no—
33C22⋊Dic17—43no—
34#63227—43no—
35#63227—43no—
36#63227—43no—
37#63227—43no—
38C4⋊Dic17—43no—
39#63225—43no—
40#63225—43no—
41C23×C17—23no—
42C23×C17—23no—
43C23×C17C2×Q812yes—
44C23×C17C2411yescharacteristic
45C22×D17—23no—
46C22×D17—23no—
47C22×D17—23no—
48C22×D17—23no—
49C22×D17—412no—
50C22×D17—412no—
51C22×D17—44no—
52C22×D17C2×Q812yes—
53C22×D17C2×Q812yes—
54C2×C4×C17—23no—
55C2×C4×C17—23no—
56C2×C4×C17—23no—
57C2×C4×C17—23no—
58C2×C4×C17—23no—
59C2×C4×C17—23no—
60C2×Dic17—23no—
61C2×Dic17—23no—
62C2×Dic17—23no—
63C2×Dic17—23no—
64C2×Dic17—23no—
65C2×Dic17—23no—
66#2842—171nomaximal, Sylow
67C22×C17—26no—
68C22×C17—43no—
69C22×C17—41no—
70C22×C17C22×Q811yescharacteristic
71C22×C17H(5,𝔽2)13yes—
72C22×C17H(5,𝔽2)13yes—
73C2×D17—26no—
74C2×D17—26no—
75C2×D17—416no—
76C2×D17—43no—
77C2×D17—43no—
78C2×D17—41no—
79C2×D17—41no—
80C4×C17—43no—
81C4×C17—43no—
82Dic17—43no—
83Dic17—43no—
84#399—173no—
85#399—171no—
86#396—173no—
87#396—173no—
88#396—171no—
89#396D1711yescharacteristic, complemented
90#379—173no—
91C2×C17—42no—
92C2×C17#54313yes—
93C2×C17#56013yes—
94C2×C17#56011yescharacteristic
95D17—42no—
96D17—42no—
97C25—171no—
98C2×C4⋊C4—173no—
99C2×C4⋊C4—173no—
100C2×C4⋊C4—173no—
101C2×C4⋊C4C2×D1713yescomplemented
102C2×C22⋊C4—173no—
103C2×C22⋊C4—173no—
104C2×C22⋊C4—173no—
105C2×C22⋊C4—173no—
106C2×C22⋊C4—173no—
107C2×C22⋊C4C2×D1713yes—
108#95—173no—
109#95C2×D1711yescharacteristic, complemented
110C17#284211yescharacteristic, complemented, Sylow
111C24—171no—
112C24—171no—
113C24—174no—
114C24—683no—
115C24—683no—
116C24C22×D1711yescharacteristic
117C22×C4—173no—
118C22×C4—173no—
119C22×C4C22×D1713yes—
120C22×C4C22×D1713yes—
121C4⋊C4—43no—
122C4⋊C4—683no—
123C4⋊C4—683no—
124C4⋊C4—683no—
125C22⋊C4—43no—
126C22⋊C4—683no—
127C22⋊C4—683no—
128C22⋊C4—683no—
129C22⋊C4—683no—
130C22⋊C4—683no—
131C23—23no—
132C23—23no—
133C23—172no—
134C23—172no—
135C23—343no—
136C23—343no—
137C23—343no—
138C23—343no—
139C23—6812no—
140C23—6812no—
141C23—684no—
142C23Q8×D1712yes—
143C23C23×D1711yescharacteristic
144C2×C4—23no—
145C2×C4—23no—
146C2×C4—23no—
147C2×C4—23no—
148C2×C4—23no—
149C2×C4—23no—
150C2×C4—343no—
151C2×C4—343no—
152C2×C4—343no—
153C2×C4—343no—
154C2×C4—343no—
155C2×C4—343no—
156C22—26no—
157C22—43no—
158C22—41no—
159C22—346no—
160C22—346no—
161C22—6816no—
162C22—683no—
163C22—683no—
164C22—681no—
165C22—681no—
166C22#1058769113yes—
167C22C2×Q8×D1711yescharacteristic
168C22#1058769613yes—
169C4—43no—
170C4—43no—
171C4—683no—
172C4—683no—
173C2—42no—
174C2—682no—
175C2—682no—
176C2#1176343413yes—
177C2#1176354713yes—
178C2#1176355011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#396C2
prime radicalC24×D17C22
hypercenter#396D17
hypocenterC17#2842
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #11762746C2semidirect
2. #11762744C2semidirect
3. #11762733C2semidirect
4. #11762732C2semidirect
5. #11762731C2semidirect
7. C17×#396C2semidirect
9. C2×C4⋊C4×C17C22semidirect
16. C2×C4⋊Dic17C22semidirect
17. C2×#63225C22semidirect
18. C2×#63225C22semidirect
19. C17×#95C22semidirect
20. #10587502C22semidirect
89. #396D17semidirect
101. C2×C4⋊C4C2×D17semidirect
109. #95C2×D17semidirect
110. C17#2842semidirect

Maximal subgroups

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

subgroupindex
#117627462
#117627442
#117627332
#117627322
#117627312
#117626642
C17×#3962
#284217