Finite Group Databasebeta

← #14536770

Subgroups — #14536770

Order 2176.

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

normal subgroups only

This group has 368 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#11762743C212yesmaximal
3#11762742C212yesmaximal
4#11762716C211yesmaximal, characteristic
5#11762716C211yesmaximal, characteristic
6#11762706C212yesmaximal
7#11762693C212yesmaximal
8#11762692C212yesmaximal
9C17×#391C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×C42×C17C2211yescharacteristic
13C2×C4⋊Dic17C2212yes—
14C2×C4⋊Dic17C2211yescharacteristic
15C2×C4⋊Dic17C2211yescharacteristic
16C2×#63225C2212yes—
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×C4×Dic17C2211yescharacteristic
21C2×C4×Dic17C2211yescharacteristic
22C17×#95C2212yes—
23C17×#95C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C2×Q8×C17—41no—
32C2×Q8×C17—41no—
33C2×Q8×C17Q814yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C4⋊C4×C17—41no—
44C42×C17—41no—
45C4⋊Dic17—24no—
46C4⋊Dic17—41no—
47C4⋊Dic17—41no—
48#63225—24no—
49#63225—24no—
50#63225—42no—
51#63225—42no—
52C4×Dic17—41no—
53C4×Dic17—41no—
54C23×C17C2411yescharacteristic
55Q8×C17—44no—
56C2×C4×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17—21no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17C2×Q812yes—
69C2×C4×C17C2×Q812yes—
70C2×C4×C17C2×Q812yes—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—24no—
81C2×Dic17#4114yes—
82#2809—171nomaximal, Sylow
83C22×C17C22×Q811yescharacteristic
84C22×C17C2×#4112yes—
85C22×C17H(5,𝔽2)11yescharacteristic
86C22×C17H(5,𝔽2)11yescharacteristic
87C22×C172−1+412yes—
88C4×C17—22no—
89C4×C17—22no—
90C4×C17—22no—
91C4×C17—42no—
92C4×C17—42no—
93Dic17—24no—
94Dic17—42no—
95Dic17—42no—
96Dic17—42no—
97#400—172no—
98#398—172no—
99#398—172no—
100#391—171no—
101#391—171no—
102#391D1711yescharacteristic
103#389—172no—
104#387—172no—
105#387—172no—
106C2×C17#53912yes—
107C2×C17#54311yescharacteristic
108C2×C17#55511yescharacteristic
109C2×C17#55712yes—
110C2×C17#56311yescharacteristic
111C22×Q8C2×D1711yescharacteristic
112C2×C4⋊C4—172no—
113C2×C4⋊C4—172no—
114C2×C4⋊C4—172no—
115C2×C4⋊C4—172no—
116C2×C4⋊C4—171no—
117C2×C4⋊C4—171no—
118C2×C4⋊C4—172no—
119C2×C4⋊C4C2×D1711yescharacteristic
120C2×C42—171no—
121C2×C42—171no—
122C2×C42C2×D1711yescharacteristic
123#95—172no—
124#95—172no—
125#95—172no—
126#95—172no—
127#95—172no—
128#95—172no—
129#95—172no—
130#95C2×D1712yes—
131#95C2×D1712yes—
132C17#280911yescharacteristic, complemented, Sylow
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8Dic3414yes—
136C22×C4—172no—
137C22×C4—172no—
138C22×C4—172no—
139C22×C4—172no—
140C22×C4C22×D1712yes—
141C22×C4C22×D1712yes—
142C22×C4C22×D1711yescharacteristic
143C22×C4C22×D1711yescharacteristic
144C22×C4C22×D1711yescharacteristic
145C4⋊C4—41no—
146C4⋊C4—344no—
147C4⋊C4—344no—
148C4⋊C4—344no—
149C4⋊C4—682no—
150C4⋊C4—682no—
151C4⋊C4—681no—
152C4⋊C4—681no—
153C42—41no—
154C42—681no—
155C42—681no—
156C23C23×D1711yescharacteristic
157Q8—44no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—174no—
171C2×C4—342no—
172C2×C4—342no—
173C2×C4—342no—
174C2×C4—342no—
175C2×C4—342no—
176C2×C4—344no—
177C2×C4—342no—
178C2×C4—342no—
179C2×C4—342no—
180C2×C4—342no—
181C2×C4C2×Dic3412yes—
182C2×C4C2×Dic3412yes—
183C2×C4C2×Dic3412yes—
184C22C22×Dic3411yescharacteristic
185C22#1058769111yescharacteristic
186C22#1058769111yescharacteristic
187C22#1058769412yes—
188C22#41×D1712yes—
189C4—22no—
190C4—22no—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—344no—
195C4—682no—
196C4—682no—
197C4—682no—
198C2#1176340512yes—
199C2#1176343511yescharacteristic
200C2#1176352111yescharacteristic
201C2#1176353512yes—
202C2#1176355811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#391C2
hypercenter#391D17
hypocenterC17#2809
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
132. C17#2809semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627432
#117627422
#117627162
#117627062
#117626932
#117626922
C17×#3912
#280917