Finite Group Databasebeta

← #14534564

Subgroups — #14534564

Order 2176.

526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587615C211yesmaximal, characteristic
2#11762747C212yesmaximal, complemented
3#11762744C212yesmaximal, complemented
4#11762740C212yesmaximal, complemented
5#11762735C212yesmaximal, complemented
6#11762688C212yesmaximal
7#11762682C212yesmaximal, complemented
8#11762672C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C17C211yesmaximal, characteristic, complemented
10C23×C4×C17C2211yescharacteristic
11C23×C4×C17C2212yes—
12C22×C22⋊D17C2211yescharacteristic
13C2×C4⋊C4×C17—24no—
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2211yescharacteristic
18#10587615—24no—
19C2×#63227C2211yescharacteristic
20C2×#63227C2211yescharacteristic
21C2×#63227C2214yes—
22C2×#63225C2211yescharacteristic
23C2×#63225C2211yescharacteristic
24C2×#63225C2214yes—
25#10587502C2214yes—
26#10587502C2212yes—
27#10587502C2214yes—
28#10587502C2212yes—
29C24×C17C2311yescharacteristic
30C23×D17C2311yescharacteristic
31C22×C4×C17—24no—
32C22×C4×C17—24no—
33C22×C4×C17—44no—
34C22×C4×C17D414yes—
35C22×C4×C17C2314yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C2×C22⋊D17—22no—
39C2×C22⋊D17—24no—
40C22×Dic17C2311yescharacteristic
41C22×Dic17C2311yescharacteristic
42C22×Dic17C2314yes—
43C22×Dic17C2311yescharacteristic
44C4⋊C4×C17—24no—
45C4⋊C4×C17—24no—
46C22⋊Dic17—22no—
47C22⋊Dic17—24no—
48#63227—22no—
49#63227—22no—
50#63227—44no—
51#63225—22no—
52#63225—22no—
53#63225—44no—
54C23×C17—22no—
55C23×C17—22no—
56C23×C17C2×D412yes—
57C23×C17#4114yes—
58C23×C17C2411yescharacteristic
59C22×D17—21no—
60C22×D17—22no—
61C22×D17—22no—
62C22×D17—22no—
63C2×C4×C17—28no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—24no—
67C2×C4×C17—24no—
68C2×C4×C17—24no—
69C2×C4×C17—48no—
70C2×C4×C17C2×D412yes—
71C2×C4×C17C2×D412yes—
72C22⋊D17—44no—
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17—21no—
76C2×Dic17—24no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—24no—
80C2×Dic17—22no—
81C2×Dic17—24no—
82#2623—171nomaximal, Sylow
83C22×C17—22no—
84C22×C17—22no—
85C22×C17—28no—
86C22×C17#12314yes—
87C22×C17C22×D411yescharacteristic
88C22×C17C2×#4112yes—
89C22×C17H(5,𝔽2)12yes—
90C22×C172−1+412yes—
91C2×D17—41no—
92C2×D17—41no—
93C2×D17—41no—
94C2×D17—44no—
95C4×C17—22no—
96C4×C17—22no—
97C4×C17—44no—
98Dic17—41no—
99Dic17—41no—
100Dic17—41no—
101Dic17—44no—
102C2×#123—171no—
103C22×C4⋊C4D1711yescharacteristic, complemented
104#400—172no—
105#399—172no—
106#398—172no—
107#397—172no—
108#386—172no—
109#385—172no—
110#381—171no—
111C2×C17—24no—
112C2×C17C2×#12311yescharacteristic
113C2×C17#53511yescharacteristic
114C2×C17#53611yescharacteristic
115C2×C17#53914yes—
116D17—81no—
117C22×D4—171no—
118C23×C4C2×D1711yescharacteristic
119C23×C4C2×D1712yes—
120#123—344no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—171no—
123C2×C4⋊C4—171no—
124C2×C4⋊C4—174no—
125C2×C4⋊C4C2×D1712yes—
126C2×C4⋊C4C2×D1712yes—
127C2×C22⋊C4—171no—
128C2×C22⋊C4—172no—
129C2×C22⋊C4—171no—
130C2×C22⋊C4—171no—
131C2×C22⋊C4—174no—
132#95—172no—
133#95—172no—
134#95—174no—
135#95—174no—
136C17#262311yescharacteristic, complemented, Sylow
137C24—171no—
138C24C22×D1711yescharacteristic
139C2×D4—344no—
140C2×D4—342no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—44no—
144C22×C4—174no—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—171no—
148C22×C4C22⋊D1714yes—
149C22×C4C22×D1714yes—
150C22×C4C22×D1711yescharacteristic
151C22×C4C22×D1711yescharacteristic
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—342no—
155C4⋊C4—342no—
156C4⋊C4—684no—
157C22⋊C4—344no—
158C22⋊C4—342no—
159C22⋊C4—342no—
160C22⋊C4—342no—
161C22⋊C4—684no—
162C23—22no—
163C23—22no—
164C23—342no—
165C23—342no—
166C23—342no—
167C23—341no—
168C23#6325214yes—
169C23C2×C22⋊D1712yes—
170C23C23×D1711yescharacteristic
171D4—684no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—28no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—48no—
179C2×C4—344no—
180C2×C4—342no—
181C2×C4—344no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—344no—
185C2×C4—341no—
186C2×C4—341no—
187C2×C4—341no—
188C2×C4C2×C22⋊D1712yes—
189C2×C4C2×C22⋊D1712yes—
190C22—22no—
191C22—22no—
192C22—28no—
193C22—681no—
194C22—681no—
195C22—681no—
196C22—684no—
197C22#1058761514yes—
198C22C2×#6325212yes—
199C22#1058769112yes—
200C22#1058769412yes—
201C22C22×C22⋊D1711yescharacteristic
202C4—22no—
203C4—22no—
204C4—44no—
205C4—681no—
206C4—681no—
207C4—681no—
208C4—684no—
209C2—24no—
210C2—1361no—
211C2C2×#1058761511yescharacteristic
212C2#1176337611yescharacteristic
213C2#1176338311yescharacteristic
214C2#1176340614yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C4⋊C4×C17C2
prime radicalC22×C22⋊D17C22
hypercenterC22×C4⋊C4D17
hypocenterC17#2623
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11762747C2semidirect
3. #11762744C2semidirect
4. #11762740C2semidirect
5. #11762735C2semidirect
7. #11762682C2semidirect
8. #11762672C2semidirect
9. C22×C4⋊C4×C17C2semidirect
103. C22×C4⋊C4D17semidirect
136. C17#2623semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
#117627472
#117627442
#117627402
#117627352
#117626882
#117626822
#117626722
C22×C4⋊C4×C172
#262317