Finite Group Databasebeta

← #14545323

Subgroups — #14545323

Order 2176.

688 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 152 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#11763761C213yesmaximal, complemented
2#11763759C216yesmaximal, complemented
3#11763757C213yesmaximal, complemented
4#11763750C212yesmaximal, complemented
5#120×D17C211yesmaximal, characteristic
6C24×D17C2211yescharacteristic
7C23×#595C2211yescharacteristic, complemented
8C2×D4×D17C2213yes—
9C2×C22⋊#595C2216yes—
10C2×C22⋊#595C2216yes—
11C2×C22⋊#595C2213yescomplemented
12C2×C4⋊#595C2213yescomplemented
13C2×C4×#595C2213yescomplemented
14C17×#120C411yescharacteristic, complemented
15#10587639C411yescharacteristic, complemented
16#10587623C413yescomplemented
17#10587568C413yescomplemented
18C22⋊C4×D17C2213yes—
19(C2×#595)⋊C4C2216yescomplemented
20C24×C17C2×C411yescharacteristic, complemented
21C23×D17—43no—
22C23×D17—43no—
23C23×D17C2×C413yescomplemented
24C23×D17C2313yes—
25C23×D17C2311yescharacteristic
26C22×#595—43no—
27C22×#595C2316yes—
28C22×#595C2312yes—
29C2×D4×C17C2×C413yes—
30C2×C22⋊D17C2×C413yes—
31C2×C22⋊D17C2×C413yes—
32D4×D17—43no—
33D4×D17—43no—
34C2×D68C2×C413yes—
35C2×C4×D17C2313yes—
36C22⋊#595—46no—
37C22⋊#595—46no—
38C22⋊#595—43no—
39C4⋊#595—43no—
40C4×#595—43no—
41C22⋊C4×C17C2×C413yescomplemented
42C22⋊Dic17C2×C413yescomplemented
43#63227C2×C416yescomplemented
44C23×C17—43no—
45C23×C17C22×C413yes—
46C23×C17C22×C411yescharacteristic
47C22×D17—23no—
48C22×D17—26no—
49C22×D17—26no—
50C22×D17—23no—
51C22×D17—23no—
52C22×D17—23no—
53C22×D17—46no—
54C22×D17—43no—
55C22×D17—412no—
56C22×D17—44no—
57C22×D17C22×C411yescharacteristic
58C22×D17C22×C413yes—
59C22×D17C2411yescharacteristic
60C2×#595—26no—
61C2×#595—26no—
62C2×#595—26no—
63C2×#595—26no—
64C2×#595—44no—
65D4×C17—43no—
66C2×C4×C17C22×C413yes—
67C22⋊D17—43no—
68C22⋊D17—43no—
69C2×Dic17C22×C413yes—
70D68—43no—
71C4×D17—23no—
72C4×D17—23no—
73#2375—171nomaximal, Sylow
74C22×C17—23no—
75C22×C17—23no—
76C22×C17—26no—
77C22×C17—44no—
78C22×C17C23×C411yescharacteristic
79C2×D17—23no—
80C2×D17—23no—
81C2×D17—26no—
82C2×D17—412no—
83C2×D17—43no—
84C2×D17—43no—
85C2×D17—46no—
86C2×D17—41no—
87C2×D17—43no—
88C2×D17H(5,𝔽2)13yes—
89C2×D17H(5,𝔽2)13yes—
90#595—46no—
91#595—42no—
92C4×C17—23no—
93Dic17—23no—
94C2×#120—171no—
95#390—173no—
96#388—176no—
97#386—173no—
98#379—172no—
99C2×C17—43no—
100C2×C17—41no—
101C2×C17#51813yes—
102D17—41no—
103D17—43no—
104D17#56013yes—
105D17H(3,𝔽4)11yescharacteristic
106C25—171no—
107C22×D4—173no—
108C23×C4—171no—
109#120—171no—
110#120—173no—
111#120—173no—
112#120#59511yescharacteristic, complemented
113C2×C4⋊C4—173no—
114C2×C22⋊C4—173no—
115C2×C22⋊C4—176no—
116C2×C22⋊C4—176no—
117C2×C22⋊C4—173no—
118C2×C42—173no—
119#95—176no—
120C17#237511yescharacteristic, complemented, Sylow
121C24—173no—
122C24—171no—
123C24—173no—
124C24—683no—
125C24—683no—
126C24C2×#59511yescharacteristic, complemented
127C2×D4—173no—
128C2×D4—173no—
129C2×D4—173no—
130C2×D4—683no—
131C2×D4—683no—
132C2×D4C2×#59513yes—
133C22×C4—173no—
134C22×C4—172no—
135C22×C4—176no—
136C22×C4—683no—
137C4⋊C4—683no—
138C22⋊C4—173no—
139C22⋊C4—176no—
140C22⋊C4—686no—
141C22⋊C4—683no—
142C22⋊C4—686no—
143C22⋊C4C2×#59513yescomplemented
144C42—683no—
145C23—43no—
146C23—171no—
147C23—173no—
148C23—171no—
149C23—343no—
150C23—346no—
151C23—346no—
152C23—343no—
153C23—343no—
154C23—343no—
155C23—683no—
156C23—686no—
157C23—6812no—
158C23—684no—
159C23C22×#59513yes—
160C23C22×#59511yescharacteristic
161D4—43no—
162D4—683no—
163D4—683no—
164D4—683no—
165C2×C4—173no—
166C2×C4—343no—
167C2×C4—343no—
168C2×C4—346no—
169C2×C4—346no—
170C2×C4—346no—
171C2×C4—346no—
172C2×C4—684no—
173C2×C4C22×#59513yes—
174C22—23no—
175C22—23no—
176C22—26no—
177C22—44no—
178C22—173no—
179C22—173no—
180C22—343no—
181C22—343no—
182C22—346no—
183C22—681no—
184C22—683no—
185C22—686no—
186C22—683no—
187C22—683no—
188C22—6812no—
189C22C23×#59511yescharacteristic
190C4—23no—
191C4—343no—
192C4—682no—
193C4—686no—
194C2—43no—
195C2—41no—
196C2—173no—
197C2—171no—
198C2—683no—
199C2—681no—
200C2#1176387613yes—
subgroupG/H
center, FrattiniC22C23×#595
commutator, socleC22×C17C23×C4
FittingC17×#120C4
prime radical#120×D17C2
cosocleC22×D17C24
hypercenter#120#595
hypocenterC17#2375
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #11763761C2semidirect
2. #11763759C2semidirect
3. #11763757C2semidirect
4. #11763750C2semidirect
7. C23×#595C22semidirect
11. C2×C22⋊#595C22semidirect
12. C2×C4⋊#595C22semidirect
13. C2×C4×#595C22semidirect
14. C17×#120C4semidirect
15. #10587639C4semidirect
16. #10587623C4semidirect
17. #10587568C4semidirect
19. (C2×#595)⋊C4C22semidirect
20. C24×C17C2×C4semidirect
23. C23×D17C2×C4semidirect
41. C22⋊C4×C17C2×C4semidirect
42. C22⋊Dic17C2×C4semidirect
43. #63227C2×C4semidirect
112. #120#595semidirect
120. C17#2375semidirect
126. C24C2×#595semidirect
143. C22⋊C4C2×#595semidirect

Maximal subgroups

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

subgroupindex
#117637612
#117637592
#117637572
#117637502
#120×D172
#237517