Finite Group Databasebeta

← #17240747

Subgroups — #17240747

Order 4352.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 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#14537048C213yesmaximal
2#14537044C2112yesmaximal, complemented
3#14532051C2112yesmaximal, complemented
4#14531162C213yesmaximal
5C17×#2840C211yesmaximal, characteristic
6C24×Dic17C2211yescharacteristic
7C2×Q8×Dic17C2214yescomplemented
8C2×#10587563C2216yes—
9#11762749C22112yescomplemented
10#11762745C22112yes—
11#11762743C22124yescomplemented
12#11762732C22124yes—
13#11762728C22112yescomplemented
14#11762670C2216yes—
15#11762667C22112yes—
16#11762665C2213yes—
17#11762660C22112yescomplemented
18C22⋊C4×Dic17C22112yes—
19C17×#393C22112yes—
20C17×#379C2213yes—
21C25×C17C2311yescharacteristic
22C22×Q8×C17C2314yes—
23C23×Dic17—44no—
24C23×Dic17Q818yescomplemented
25C23×Dic17C2313yes—
26C23×Dic17C2313yes—
27C2×C22⋊C4×C17C23112yes—
28C2×C22⋊C4×C17C2316yes—
29C2×C22⋊Dic17C2316yes—
30C2×C22⋊Dic17C23112yes—
31Q8×Dic17—48no—
32C2×C4⋊Dic17C23112yes—
33C2×#63225C23124yes—
34C2×#63225C23112yes—
35C2×C4×Dic17C23112yes—
36#10587563—412no—
37C17×#95C23112yes—
38#10587502C23112yes—
39#10587502C23124yes—
40C24×C17—42no—
41C24×C17—44no—
42C24×C17C2×Q814yes—
43C24×C17C2413yes—
44C2×Q8×C17—48no—
45C2×Q8×C17—44no—
46C22×C4×C17C24112yes—
47C22×Dic17—212no—
48C22×Dic17—416no—
49C22×Dic17—46no—
50C22×Dic17C2×Q8112yes—
51C22×Dic17C2×Q8112yes—
52C22×Dic17C24112yes—
53C22×Dic17C2413yes—
54C22×Dic17C2411yescharacteristic
55C22⋊C4×C17—46no—
56C22⋊C4×C17—412no—
57C22⋊Dic17—46no—
58C22⋊Dic17—412no—
59C4⋊Dic17—412no—
60#63225—412no—
61#63225—424no—
62C4×Dic17—224no—
63#36674—171nomaximal, Sylow
64C23×C17—26no—
65C23×C17—212no—
66C23×C17—44no—
67C23×C17—48no—
68C23×C17—416no—
69C23×C17C22×Q816yes—
70C23×C17C2511yescharacteristic
71Q8×C17—416no—
72C2×C4×C17—212no—
73C2×C4×C17—224no—
74C2×Dic17—26no—
75C2×Dic17—224no—
76C2×Dic17—212no—
77C2×Dic17—224no—
78C2×Dic17—412no—
79C2×Dic17C22×Q816yes—
80C2×Dic17C22×Q812yes—
81C2×Dic17H(5,𝔽2)14yes—
82#2842—173no—
83#2841—1712no—
84#2840D1711yescharacteristic
85#2396—1712no—
86#2280—173no—
87C22×C17—26no—
88C22×C17—212no—
89C22×C17—416no—
90C22×C17—412no—
91C22×C17C23×Q811yescharacteristic
92C22×C17C2×H(5,𝔽2)12yes—
93C22×C17#58514yes—
94C4×C17—412no—
95Dic17—26no—
96Dic17—412no—
97Dic17#54314yes—
98C24×C4—171no—
99C2×C4×Q8—174no—
100C2×#117—176no—
101#400—1712no—
102#399—1712no—
103#398—1724no—
104#396—1724no—
105#395—1712no—
106#393C2×D17112yes—
107#381—176no—
108#380—1712no—
109#379—173no—
110#379C2×D1713yes—
111C4×C4⋊C4—1712no—
112C4×C22⋊C4—1712no—
113C2×C17—46no—
114C2×C17C2×#54311yescharacteristic
115C2×C17#345812yes—
116C2×C17#352914yes—
117C25C22×D1711yescharacteristic
118C22×Q8C22×D1714yes—
119C23×C4—173no—
120C23×C4—178no—
121C23×C4—173no—
122C23×C4—684no—
123C4×Q8—688no—
124#117—6812no—
125C2×C4⋊C4—1712no—
126C2×C4⋊C4—1712no—
127C2×C4⋊C4—1724no—
128C2×C22⋊C4—176no—
129C2×C22⋊C4—1712no—
130C2×C22⋊C4C22×D1716yes—
131C2×C22⋊C4C22×D17112yes—
132C2×C42—1712no—
133#95—1724no—
134#95—1712no—
135#95C22×D17112yes—
136C17#3667411yescharacteristic, complemented, Sylow
137C24—42no—
138C24—44no—
139C24Q8×D1714yes—
140C24C23×D1713yes—
141C2×Q8—44no—
142C2×Q8—48no—
143C22×C4—171no—
144C22×C4—1712no—
145C22×C4—173no—
146C22×C4—1712no—
147C22×C4—1712no—
148C22×C4—3412no—
149C22×C4—6816no—
150C22×C4—686no—
151C22×C4C23×D17112yes—
152C4⋊C4—6812no—
153C4⋊C4—6812no—
154C4⋊C4—6824no—
155C22⋊C4—412no—
156C22⋊C4—46no—
157C22⋊C4—6812no—
158C22⋊C4—686no—
159C42—3424no—
160C23—26no—
161C23—212no—
162C23—44no—
163C23—48no—
164C23—416no—
165C23C2×Q8×D1716yes—
166C23C24×D1711yescharacteristic
167Q8—416no—
168C2×C4—212no—
169C2×C4—224no—
170C2×C4—172no—
171C2×C4—174no—
172C2×C4—176no—
173C2×C4—3424no—
174C2×C4—346no—
175C2×C4—3412no—
176C2×C4—3424no—
177C2×C4—6812no—
178C22—26no—
179C22—212no—
180C22—416no—
181C22—412no—
182C22C22×Q8×D1711yescharacteristic
183C22H(5,𝔽2)×D1712yes—
184C22#1176364414yes—
185C4—412no—
186C4—174no—
187C4—346no—
188C4—6812no—
189C2—46no—
190C2D17×#54311yescharacteristic
191C2#1454218612yes—
192C2#1454292414yes—
subgroupG/H
center, FrattiniC23C24×D17
commutator, socle, cosocleC23×C17C25
FittingC17×#2840C2
prime radicalC25×C17C23
hypercenter#2840D17
hypocenterC17#36674
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #14537044C2semidirect
3. #14532051C2semidirect
7. C2×Q8×Dic17C22semidirect
9. #11762749C22semidirect
11. #11762743C22semidirect
13. #11762728C22semidirect
17. #11762660C22semidirect
24. C23×Dic17Q8semidirect
136. C17#36674semidirect

Maximal subgroups

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

subgroupindex
#145370482
#145370442
#145320512
#145311622
C17×#28402
#3667417