Finite Group Databasebeta

← #14537050

Subgroups — #14537050

Order 2176.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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#11762750C212yesmaximal
2#11762748C211yesmaximal, characteristic
3#11762743C212yesmaximal
4#11762742C212yesmaximal
5#11762740C212yesmaximal
6#11762728C212yesmaximal
7#11762678C211yesmaximal, characteristic
8#11762676C212yesmaximal
9C17×#395C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2212yes—
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊Dic17C2211yescharacteristic
14C2×C4⋊Dic17C2211yescharacteristic
15C2×#63225C2212yes—
16C2×#63225C2212yes—
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×C4×Dic17C2211yescharacteristic
21C17×#95C2212yes—
22C17×#95C2211yescharacteristic
23#10587502C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C2×Q8×C17—41no—
32C2×Q8×C17—42no—
33C22×C4×C17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C4⋊C4×C17—41no—
43C4⋊C4×C17—42no—
44C4⋊Dic17—41no—
45C4⋊Dic17—41no—
46#63225—42no—
47#63225—42no—
48#63225—42no—
49#63225—42no—
50#63225—42no—
51C4×Dic17Q814yescomplemented
52C23×C17C2411yescharacteristic
53Q8×C17—44no—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17—21no—
57C2×C4×C17—22no—
58C2×C4×C17—22no—
59C2×C4×C17—22no—
60C2×C4×C17—21no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17C2×Q812yes—
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—24no—
76C2×Dic17—22no—
77C2×Dic17C2×Q814yes—
78#2843—171nomaximal, Sylow
79C22×C17C22×Q811yescharacteristic
80C22×C17H(5,𝔽2)11yescharacteristic
81C22×C17H(5,𝔽2)12yes—
82C22×C172−1+412yes—
83C22×C172−1+411yescharacteristic
84C4×C17—22no—
85C4×C17—41no—
86C4×C17—41no—
87C4×C17—42no—
88C4×C17—42no—
89Dic17—24no—
90Dic17—42no—
91Dic17—42no—
92Dic17—42no—
93#400—171no—
94#400—172no—
95#398—172no—
96#398—172no—
97#398—172no—
98#395—172no—
99#395D1711yescharacteristic
100#384—171no—
101#383—172no—
102C2×C17#54412yes—
103C2×C17#54411yescharacteristic
104C2×C17#56011yescharacteristic
105C2×C17#56311yescharacteristic
106C2×C17#56312yes—
107C22×Q8C2×D1711yescharacteristic
108C2×C4⋊C4—172no—
109C2×C4⋊C4—172no—
110C2×C4⋊C4—171no—
111C2×C4⋊C4—172no—
112C2×C4⋊C4—172no—
113C2×C4⋊C4—172no—
114C2×C4⋊C4—171no—
115C2×C4⋊C4C2×D1711yescharacteristic
116C2×C4⋊C4C2×D1712yes—
117C2×C42—171no—
118#95—172no—
119#95—172no—
120#95—172no—
121#95—172no—
122#95—172no—
123#95—172no—
124#95—171no—
125#95—172no—
126#95C2×D1712yes—
127#95C2×D1711yescharacteristic
128C17#284311yescharacteristic, complemented, Sylow
129C2×Q8—41no—
130C2×Q8—42no—
131C22×C4—172no—
132C22×C4—172no—
133C22×C4—172no—
134C22×C4—172no—
135C22×C4C22×D1712yes—
136C22×C4C22×D1712yes—
137C22×C4C22×D1711yescharacteristic
138C22×C4C22×D1711yescharacteristic
139C22×C4C22×D1711yescharacteristic
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—682no—
143C4⋊C4—682no—
144C4⋊C4—682no—
145C4⋊C4—682no—
146C4⋊C4—682no—
147C4⋊C4—681no—
148C4⋊C4—681no—
149C42—174no—
150C23C23×D1711yescharacteristic
151Q8—44no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—174no—
165C2×C4—342no—
166C2×C4—342no—
167C2×C4—342no—
168C2×C4—342no—
169C2×C4—342no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—344no—
173C2×C4—342no—
174C2×C4—342no—
175C2×C4Q8×D1712yes—
176C22#1058769111yescharacteristic
177C22#1058769112yes—
178C22C2×Q8×D1711yescharacteristic
179C22#1058769412yes—
180C22#1058769711yescharacteristic
181C4—22no—
182C4—41no—
183C4—42no—
184C4—41no—
185C4—42no—
186C4—344no—
187C4—682no—
188C4—682no—
189C4—682no—
190C2#1176343612yes—
191C2#1176343811yescharacteristic
192C2#1176354811yescharacteristic
193C2#1176355811yescharacteristic
194C2#1176356012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#395C2
hypercenter#395D17
hypocenterC17#2843
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
51. C4×Dic17Q8semidirect
128. C17#2843semidirect

Maximal subgroups

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

subgroupindex
#117627502
#117627482
#117627432
#117627422
#117627402
#117627282
#117626782
#117626762
C17×#3952
#284317