Finite Group Databasebeta

← #14536668

Subgroups — #14536668

Order 2176.

376 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 137 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#11762751C212yesmaximal
2#11762748C211yesmaximal, characteristic
3#11762743C212yesmaximal
4#11762729C211yesmaximal, characteristic
5#11762716C212yesmaximal
6#11762713C212yesmaximal
7#11762696C212yesmaximal
8#11762693C212yesmaximal
9C17×#391C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C42×C17C2211yescharacteristic
14C2×C4⋊Dic17C2211yescharacteristic
15C2×C4⋊Dic17C2212yes—
16C2×#63225C2212yes—
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×C4×Dic17C2212yes—
20C17×#95C2212yes—
21C17×#95C2212yes—
22#10587502C2212yes—
23#10587502C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2211yescharacteristic
31C2×Q8×C17—41no—
32C2×Q8×C17—41no—
33C2×Q8×C17Q814yescomplemented
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C2×Dic34—24no—
44C2×Dic34—41no—
45C4⋊C4×C17—41no—
46C42×C17—41no—
47C4⋊Dic17—41no—
48C4⋊Dic17—42no—
49#63225—24no—
50#63225—24no—
51#63225—42no—
52C4×Dic17—42no—
53C23×C17C2411yescharacteristic
54Q8×C17—44no—
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—22no—
62C2×C4×C17—22no—
63C2×C4×C17—21no—
64C2×C4×C17—21no—
65C2×C4×C17—22no—
66C2×C4×C17C2×Q812yes—
67C2×C4×C17C2×Q814yes—
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—22no—
77C2×Dic17—24no—
78C2×Dic17#4114yes—
79Dic34—44no—
80#2799—171nomaximal, Sylow
81C22×C17C22×Q811yescharacteristic
82C22×C17C2×#4112yes—
83C22×C17H(5,𝔽2)11yescharacteristic
84C22×C17H(5,𝔽2)11yescharacteristic
85C22×C172−1+412yes—
86C4×C17—22no—
87C4×C17—24no—
88C4×C17—42no—
89C4×C17—41no—
90C4×C17—41no—
91Dic17—24no—
92Dic17—42no—
93Dic17—42no—
94Dic17—42no—
95#400—171no—
96#400—172no—
97#398—172no—
98#395—171no—
99#391—172no—
100#391—172no—
101#391D1711yescharacteristic
102#387—172no—
103#387—172no—
104C2×C17#54212yes—
105C2×C17#54311yescharacteristic
106C2×C17#55111yescharacteristic
107C2×C17#55712yes—
108C2×C17#56311yescharacteristic
109C22×Q8—171no—
110C22×Q8C2×D1711yescharacteristic
111C2×C4⋊C4—172no—
112C2×C4⋊C4—172no—
113C2×C4⋊C4—171no—
114C2×C4⋊C4—172no—
115C2×C4⋊C4—172no—
116C2×C4⋊C4C2×D1711yescharacteristic
117C2×C42—172no—
118C2×C42C2×D1711yescharacteristic
119#95—172no—
120#95—172no—
121#95—172no—
122#95—172no—
123#95—172no—
124#95—172no—
125#95—172no—
126#95—171no—
127#95—171no—
128#95C2×D1712yes—
129#95C2×D1712yes—
130C17#279911yescharacteristic, complemented, Sylow
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—344no—
134C2×Q8—681no—
135C2×Q8Dic3414yescomplemented
136C22×C4—172no—
137C22×C4—172no—
138C22×C4—172no—
139C22×C4—172no—
140C22×C4C22×D1711yescharacteristic
141C22×C4C22×D1712yes—
142C22×C4C22×D1711yescharacteristic
143C22×C4C22×D1711yescharacteristic
144C22×C4C22×D1712yes—
145C4⋊C4—41no—
146C4⋊C4—344no—
147C4⋊C4—344no—
148C4⋊C4—682no—
149C4⋊C4—682no—
150C4⋊C4—681no—
151C42—41no—
152C42—682no—
153C23C23×D1711yescharacteristic
154Q8—44no—
155Q8—684no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—174no—
168C2×C4—342no—
169C2×C4—342no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—342no—
173C2×C4—344no—
174C2×C4—342no—
175C2×C4—342no—
176C2×C4—342no—
177C2×C4—342no—
178C2×C4C2×Dic3412yes—
179C2×C4C2×Dic3414yes—
180C22C22×Dic3411yescharacteristic
181C22#1058769111yescharacteristic
182C22#1058769111yescharacteristic
183C22#1058769412yes—
184C22#41×D1712yes—
185C4—22no—
186C4—24no—
187C4—41no—
188C4—41no—
189C4—42no—
190C4—344no—
191C4—682no—
192C4—682no—
193C4—682no—
194C2#1176342912yes—
195C2#1176343511yescharacteristic
196C2#1176348911yescharacteristic
197C2#1176353512yes—
198C2#1176355811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#391C2
hypercenter#391D17
hypocenterC17#2799
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
33. C2×Q8×C17Q8semidirect
130. C17#2799semidirect
135. C2×Q8Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627482
#117627432
#117627292
#117627162
#117627132
#117626962
#117626932
C17×#3912
#279917