Finite Group Databasebeta

← #14534492

Subgroups — #14534492

Order 2176.

388 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 141 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#11762748C211yesmaximal, characteristic
2#11762742C212yesmaximal
3#11762729C211yesmaximal, characteristic
4#11762713C212yesmaximal
5#11762706C212yesmaximal
6#11762696C212yesmaximal
7#11762692C212yesmaximal
8#11762655C212yesmaximal
9C17×#382C211yesmaximal, characteristic
10C22×Dic34C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C42×C17C2211yescharacteristic
14C2×C4⋊Dic17C2212yes—
15C2×C4⋊Dic17C2211yescharacteristic
16C2×#63225C2212yes—
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×C4×Dic17C2212yes—
20C2×C4×Dic17C2212yes—
21C17×#95C2212yes—
22C17×#95C2212yes—
23#10587502C2211yescharacteristic
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C22×C4×C17C2311yescharacteristic
32C22×C4×C17C2311yescharacteristic
33C22×C4×C17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2312yes—
36C22×Dic17C2312yes—
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C2×Dic34—24no—
41C2×Dic34—41no—
42C4⋊C4×C17—41no—
43C4⋊C4×C17—41no—
44C42×C17Q814yescomplemented
45C4⋊Dic17—24no—
46C4⋊Dic17—41no—
47#63225—24no—
48#63225—24no—
49#63225—42no—
50C4×Dic17—24no—
51C4×Dic17—24no—
52C23×C17C2411yescharacteristic
53C2×C4×C17—22no—
54C2×C4×C17—22no—
55C2×C4×C17—21no—
56C2×C4×C17—21no—
57C2×C4×C17—22no—
58C2×C4×C17—22no—
59C2×C4×C17—22no—
60C2×C4×C17—22no—
61C2×C4×C17—21no—
62C2×C4×C17—22no—
63C2×C4×C17—21no—
64C2×C4×C17C2×Q814yes—
65C2×C4×C17C2×Q812yes—
66C2×Dic17—22no—
67C2×Dic17—22no—
68C2×Dic17—22no—
69C2×Dic17—22no—
70C2×Dic17—22no—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17#4114yes—
77C2×Dic17#4114yes—
78Dic34—44no—
79#2612—171nomaximal, Sylow
80C22×C17C22×Q811yescharacteristic
81C22×C17C2×#4112yes—
82C22×C17C2×#4112yes—
83C22×C17H(5,𝔽2)11yescharacteristic
84C22×C172−1+411yescharacteristic
85C4×C17—24no—
86C4×C17—22no—
87C4×C17—41no—
88C4×C17—41no—
89C4×C17—42no—
90Dic17—24no—
91Dic17—24no—
92Dic17—42no—
93Dic17—42no—
94#400—171no—
95#398—172no—
96#395—171no—
97#391—172no—
98#389—172no—
99#387—172no—
100#387—172no—
101#382D1711yescharacteristic
102#376—172no—
103C2×C17#53312yes—
104C2×C17#54411yescharacteristic
105C2×C17#55111yescharacteristic
106C2×C17#55212yes—
107C2×C17#55511yescharacteristic
108C22×Q8—171no—
109C2×C4⋊C4—172no—
110C2×C4⋊C4—172no—
111C2×C4⋊C4—172no—
112C2×C4⋊C4—171no—
113C2×C4⋊C4—172no—
114C2×C4⋊C4C2×D1711yescharacteristic
115C2×C4⋊C4C2×D1711yescharacteristic
116C2×C42—172no—
117C2×C42—172no—
118C2×C42C2×D1711yescharacteristic
119#95—172no—
120#95—172no—
121#95—172no—
122#95—172no—
123#95—171no—
124#95—171no—
125#95—172no—
126#95—172no—
127#95C2×D1712yes—
128#95C2×D1712yes—
129C17#261211yescharacteristic, complemented, Sylow
130C2×Q8—344no—
131C2×Q8—681no—
132C22×C4—172no—
133C22×C4—172no—
134C22×C4—172no—
135C22×C4—172no—
136C22×C4C22×D1711yescharacteristic
137C22×C4C22×D1711yescharacteristic
138C22×C4C22×D1712yes—
139C22×C4C22×D1711yescharacteristic
140C22×C4C22×D1712yes—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—344no—
144C4⋊C4—344no—
145C4⋊C4—344no—
146C4⋊C4—682no—
147C4⋊C4—681no—
148C42—344no—
149C42—344no—
150C42Dic3414yescomplemented
151C23C23×D1711yescharacteristic
152Q8—684no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—174no—
165C2×C4—174no—
166C2×C4—342no—
167C2×C4—342no—
168C2×C4—342no—
169C2×C4—342no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—342no—
173C2×C4—342no—
174C2×C4—342no—
175C2×C4—342no—
176C2×C4C2×Dic3414yes—
177C2×C4C2×Dic3412yes—
178C22C22×Dic3411yescharacteristic
179C22#1058769111yescharacteristic
180C22#1058769411yescharacteristic
181C22#41×D1712yes—
182C22#41×D1712yes—
183C4—24no—
184C4—22no—
185C4—41no—
186C4—41no—
187C4—42no—
188C4—344no—
189C4—344no—
190C4—682no—
191C4—682no—
192C2#1176336212yes—
193C2#1176344011yescharacteristic
194C2#1176348911yescharacteristic
195C2#1176349112yes—
196C2#1176352111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#382C2
hypercenter#382D17
hypocenterC17#2612
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
44. C42×C17Q8semidirect
129. C17#2612semidirect
150. C42Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627482
#117627422
#117627292
#117627132
#117627062
#117626962
#117626922
#117626552
C17×#3822
#261217