Finite Group Databasebeta

← #14536779

Subgroups — #14536779

Order 2176.

712 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587639C211yesmaximal, characteristic, complemented
2#11762737C213yesmaximal, complemented
3#11762724C213yesmaximal, complemented
4#11762723C213yesmaximal, complemented
5#11762717C213yesmaximal, complemented
6#11762665C211yesmaximal, characteristic, complemented
7C2×C17×#120C211yesmaximal, characteristic, complemented
8C25×C17C2211yescharacteristic
9C22×D4×C17C2213yes—
10C22×C22⋊D17C2213yes—
11C22×C22⋊D17C2211yescharacteristic, complemented
12C17×#120—24no—
13C2×C22⋊C4×C17C2213yes—
14#10587639—24no—
15C2×C22⋊Dic17C2213yes—
16C2×C22⋊Dic17C2213yes—
17C2×C22⋊Dic17C2213yes—
18C2×C22⋊Dic17C2213yescomplemented
19C2×#63227C2213yescomplemented
20C2×#63225C2213yescomplemented
21#10587502C2213yescomplemented
22#10587502C2213yescomplemented
23C24×C17—43no—
24C24×C17—43no—
25C24×C17D414yescomplemented
26C24×C17C2311yescharacteristic
27C24×C17C2313yes—
28C23×D17C2311yescharacteristic
29C2×D4×C17—26no—
30C2×D4×C17—43no—
31C2×D4×C17—43no—
32C22×C4×C17C2313yes—
33C2×C22⋊D17—26no—
34C2×C22⋊D17—43no—
35C2×C22⋊D17—43no—
36C2×C22⋊D17—43no—
37C22×Dic17C2313yes—
38C22×Dic17C2311yescharacteristic
39C22×Dic17C2313yes—
40C22⋊C4×C17—26no—
41C22⋊Dic17—26no—
42C22⋊Dic17—43no—
43C22⋊Dic17—43no—
44C22⋊Dic17—43no—
45#63227—43no—
46#63225—43no—
47C23×C17—23no—
48C23×C17—23no—
49C23×C17—23no—
50C23×C17—23no—
51C23×C17—23no—
52C23×C17—23no—
53C23×C17—21no—
54C23×C17—23no—
55C23×C17—23no—
56C23×C17—412no—
57C23×C17—412no—
58C23×C17—44no—
59C23×C17C2×D416yes—
60C23×C17C2411yescharacteristic
61C22×D17—21no—
62C22×D17—23no—
63C22×D17—23no—
64D4×C17—412no—
65C2×C4×C17—23no—
66C2×C4×C17—23no—
67C2×C4×C17—23no—
68C22⋊D17—412no—
69C22⋊D17—44no—
70C2×Dic17—23no—
71C2×Dic17—23no—
72C2×Dic17—23no—
73C2×Dic17—23no—
74C2×Dic17—23no—
75C2×Dic17—23no—
76C2×Dic17—23no—
77#2812—171nomaximal, Sylow
78C22×C17—26no—
79C22×C17—26no—
80C22×C17—26no—
81C22×C17—416no—
82C22×C17—43no—
83C22×C17—43no—
84C22×C17—43no—
85C22×C17—43no—
86C22×C17—43no—
87C22×C17—43no—
88C22×C17—41no—
89C22×C17C22×D411yescharacteristic
90C22×C17H(5,𝔽2)13yes—
91C22×C17H(5,𝔽2)13yes—
92C2×D17—43no—
93C2×D17—43no—
94C2×D17—41no—
95C4×C17—43no—
96Dic17—43no—
97Dic17—43no—
98Dic17—41no—
99C2×#120—171no—
100C2×#120D1711yescharacteristic, complemented
101#397—173no—
102#394—173no—
103#394—173no—
104#392—173no—
105#379—171no—
106C2×C17—46no—
107C2×C17—81no—
108C2×C17#53413yes—
109C2×C17#56013yes—
110C2×C17H(3,𝔽4)11yescharacteristic
111D17—81no—
112C25C2×D1711yescharacteristic
113C22×D4—173no—
114C22×D4—171no—
115C22×D4C2×D1713yes—
116#120—24no—
117#120—344no—
118C2×C4⋊C4—173no—
119C2×C22⋊C4—173no—
120C2×C22⋊C4—173no—
121C2×C22⋊C4—173no—
122C2×C22⋊C4—173no—
123C2×C22⋊C4—173no—
124C2×C22⋊C4C2×D1713yes—
125#95—173no—
126#95—173no—
127C17#281211yescharacteristic, complemented, Sylow
128C24—43no—
129C24—43no—
130C24—171no—
131C24C22⋊D1714yescomplemented
132C24C22×D1711yescharacteristic
133C24C22×D1713yes—
134C2×D4—26no—
135C2×D4—43no—
136C2×D4—43no—
137C2×D4—346no—
138C2×D4—683no—
139C2×D4—683no—
140C2×D4—683no—
141C22×C4—171no—
142C22×C4—173no—
143C22×C4—173no—
144C22×C4C22×D1713yes—
145C4⋊C4—683no—
146C22⋊C4—26no—
147C22⋊C4—346no—
148C22⋊C4—683no—
149C22⋊C4—683no—
150C22⋊C4—683no—
151C22⋊C4—683no—
152C23—23no—
153C23—23no—
154C23—23no—
155C23—23no—
156C23—23no—
157C23—23no—
158C23—23no—
159C23—23no—
160C23—21no—
161C23—412no—
162C23—412no—
163C23—44no—
164C23—343no—
165C23—343no—
166C23—341no—
167C23C2×C22⋊D1716yes—
168C23C23×D1711yescharacteristic
169D4—412no—
170D4—6812no—
171D4—684no—
172C2×C4—23no—
173C2×C4—23no—
174C2×C4—23no—
175C2×C4—343no—
176C2×C4—343no—
177C2×C4—343no—
178C2×C4—343no—
179C2×C4—343no—
180C2×C4—343no—
181C2×C4—343no—
182C22—26no—
183C22—26no—
184C22—26no—
185C22—416no—
186C22—43no—
187C22—43no—
188C22—43no—
189C22—43no—
190C22—43no—
191C22—41no—
192C22—43no—
193C22—683no—
194C22—681no—
195C22—683no—
196C22#1058769113yes—
197C22#1058769113yes—
198C22C22×C22⋊D1711yescharacteristic
199C4—43no—
200C4—683no—
201C4—681no—
202C4—683no—
203C2—46no—
204C2—81no—
205C2—1361no—
206C2#1176336813yes—
207C2#1176354813yes—
208C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#120C2
hypercenterC2×#120D17
hypocenterC17#2812
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#10587639C2semidirect
2. #11762737C2semidirect
3. #11762724C2semidirect
4. #11762723C2semidirect
5. #11762717C2semidirect
6. #11762665C2semidirect
7. C2×C17×#120C2semidirect
11. C22×C22⋊D17C22semidirect
18. C2×C22⋊Dic17C22semidirect
19. C2×#63227C22semidirect
20. C2×#63225C22semidirect
21. #10587502C22semidirect
22. #10587502C22semidirect
25. C24×C17D4semidirect
100. C2×#120D17semidirect
127. C17#2812semidirect
131. C24C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876392
#117627372
#117627242
#117627232
#117627172
#117626652
C2×C17×#1202
#281217