Finite Group Databasebeta

← #14534164

Subgroups — #14534164

Order 2176.

544 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#10587610C211yesmaximal, characteristic
2#11762739C212yesmaximal, complemented
3#11762729C212yesmaximal, complemented
4#11762704C214yesmaximal, complemented
5#11762682C212yesmaximal
6#11762682C212yesmaximal
7C4×C22⋊Dic17C211yesmaximal, characteristic
8C22×C4⋊C4×C17C211yesmaximal, characteristic
9C23×C4×C17C2211yescharacteristic
10C23×C4×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17—24no—
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19#10587610—24no—
20C2×#63225C2212yes—
21C2×#63225C2214yes—
22C2×#63225C2214yes—
23C2×C4×Dic17C2212yes—
24#10587502C2214yes—
25#10587502C2214yes—
26#10587502C2212yes—
27C24×C17C2311yescharacteristic
28C22×C4×C17—22no—
29C22×C4×C17—24no—
30C22×C4×C17—22no—
31C22×C4×C17—42no—
32C22×C4×C17—42no—
33C22×C4×C17D414yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2314yes—
41C2×Dic34—22no—
42C2×Dic34—42no—
43C4⋊C4×C17—24no—
44C4⋊C4×C17—24no—
45C22⋊Dic17—24no—
46C22⋊Dic17Q818yescomplemented
47C4⋊Dic17—22no—
48#63225—24no—
49#63225—28no—
50#63225—44no—
51C4×Dic17—42no—
52C23×C17—22no—
53C23×C17—22no—
54C23×C17C2×D412yes—
55C23×C17C2×Q814yes—
56C23×C17C2411yescharacteristic
57C2×C4×C17—28no—
58C2×C4×C17—22no—
59C2×C4×C17—22no—
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—24no—
64C2×C4×C17—44no—
65C2×C4×C17—44no—
66C2×C4×C17C2×D412yes—
67C2×C4×C17C2×D412yes—
68C2×C4×C17#4114yes—
69C2×Dic17—24no—
70C2×Dic17—24no—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—24no—
74C2×Dic17—24no—
75C2×Dic17C2×Q818yes—
76Dic34—44no—
77#2583—171nomaximal, Sylow
78C22×C17—22no—
79C22×C17—22no—
80C22×C17—28no—
81C22×C17C4⋊Q814yes—
82C22×C17C22×D411yescharacteristic
83C22×C17C22×Q812yes—
84C22×C17C2×#4112yes—
85C22×C17H(5,𝔽2)12yes—
86C4×C17—24no—
87C4×C17—22no—
88C4×C17—22no—
89C4×C17—42no—
90Dic17—28no—
91Dic17—42no—
92Dic17—42no—
93C2×C22⋊Q8—171no—
94C22×C4⋊C4D1711yescharacteristic
95#398—172no—
96#395—172no—
97#389—174no—
98#385—172no—
99#385—172no—
100C4×C22⋊C4—171no—
101C2×C17—24no—
102C2×C17#52511yescharacteristic
103C2×C17C2×C4⋊Q811yescharacteristic
104C2×C17#53411yescharacteristic
105C2×C17#55414yes—
106C22×Q8—171no—
107C23×C4C2×D1711yescharacteristic
108C23×C4C2×D1711yescharacteristic
109C23×C4C2×D1711yescharacteristic
110C22⋊Q8—344no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—172no—
113C2×C4⋊C4—171no—
114C2×C4⋊C4—174no—
115C2×C4⋊C4—174no—
116C2×C4⋊C4C2×D1712yes—
117C2×C4⋊C4C2×D1712yes—
118C2×C22⋊C4—172no—
119C2×C22⋊C4—172no—
120C2×C42—172no—
121#95—172no—
122#95—174no—
123#95—174no—
124C17#258311yescharacteristic, complemented, Sylow
125C24C22×D1711yescharacteristic
126C2×Q8—342no—
127C2×Q8—682no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—174no—
134C22×C4—172no—
135C22×C4—172no—
136C22×C4C22⋊D1714yes—
137C22×C4C22×D1712yes—
138C22×C4C22×D1712yes—
139C22×C4C22×D1711yescharacteristic
140C22×C4C22×D1711yescharacteristic
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—348no—
144C4⋊C4—344no—
145C4⋊C4—342no—
146C4⋊C4—684no—
147C22⋊C4—178no—
148C22⋊C4—344no—
149C42—682no—
150C23—22no—
151C23—22no—
152C23Q8×D1714yes—
153C23C2×C22⋊D1712yes—
154C23C23×D1711yescharacteristic
155Q8—684no—
156C2×C4—28no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—178no—
166C2×C4—344no—
167C2×C4—344no—
168C2×C4—344no—
169C2×C4—344no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4#6325214yes—
173C2×C4C2×C22⋊D1712yes—
174C2×C4C2×C22⋊D1712yes—
175C22—28no—
176C22—22no—
177C22—22no—
178C22#1058763014yes—
179C22C2×#6325212yes—
180C22#1058769112yes—
181C22C2×Q8×D1712yes—
182C22C22×C22⋊D1711yescharacteristic
183C4—24no—
184C4—22no—
185C4—22no—
186C4—42no—
187C4—348no—
188C4—682no—
189C4—682no—
190C2—24no—
191C2#1176332011yescharacteristic
192C2C2×#1058763011yescharacteristic
193C2#1176336811yescharacteristic
194C2#1176351414yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C4⋊C4×C17C2
prime radicalC24×C17C23
hypercenterC22×C4⋊C4D17
hypocenterC17#2583
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11762739C2semidirect
3. #11762729C2semidirect
4. #11762704C2semidirect
46. C22⋊Dic17Q8semidirect
124. C17#2583semidirect

Maximal subgroups

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

subgroupindex
C2×#105876102
#117627392
#117627292
#117627042
#117626822
C4×C22⋊Dic172
C22×C4⋊C4×C172
#258317