Finite Group Databasebeta

← #14527701

Subgroups — #14527701

Order 2176.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#11763371C211yesmaximal, characteristic, complemented
2#11762943C212yesmaximal, complemented
3#11762542C212yesmaximal, complemented
4#11762533C211yesmaximal, characteristic, complemented
5C17×#392C211yesmaximal, characteristic, complemented
6C22×D4×C17—21no—
7C22×D4×C17C2211yescharacteristic, complemented
8C22×C22⋊D17—21no—
9C2×C22⋊C4×C17—21no—
10C2×C22⋊C4×C17C2211yescharacteristic
11C2×#63228—22no—
12#10587616—21no—
13#10587616C2212yes—
14#10587572—22no—
15#10587568—22no—
16C17×#95C2211yescharacteristic, complemented
17#10587503—22no—
18C22⋊#3622C2212yescomplemented
19C24×C17—21no—
20C24×C17D411yescharacteristic
21C23×D17—21no—
22C2×D4×C17—22no—
23C2×D4×C17—42no—
24C2×D4×C17—42no—
25C2×D4×C17—42no—
26C2×D4×C17—42no—
27C22×C4×C17—21no—
28C22×C4×C17D411yescharacteristic
29C22×C4×C17C2311yescharacteristic
30C2×C22⋊D17—22no—
31C2×C22⋊D17—22no—
32C2×C22⋊D17—42no—
33C2×C22⋊D17—42no—
34C22×Dic17—21no—
35C2×D68—21no—
36C2×D68D412yescomplemented
37C22⋊C4×C17—22no—
38C22⋊C4×C17—42no—
39#63228—44no—
40#63227—22no—
41#63227—22no—
42C4⋊Dic17—21no—
43C4⋊Dic17D412yescomplemented
44C2×#3622—22no—
45C23×C17—21no—
46C23×C17—21no—
47C23×C17—22no—
48C23×C17—41no—
49C23×C17—41no—
50C23×C17—41no—
51C23×C17—41no—
52C23×C17—41no—
53C23×C17—41no—
54C23×C17—41no—
55C23×C17—41no—
56C23×C17D812yescomplemented
57C23×C17C2×D411yescharacteristic
58C22×D17—22no—
59C22×D17—42no—
60C22×D17—81no—
61C22×D17—81no—
62D4×C17—24no—
63D4×C17—42no—
64D4×C17—48no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—41no—
68C2×C4×C17—41no—
69C2×C4×C17—41no—
70C2×C4×C17—41no—
71C2×C4×C17C2×D412yes—
72C22⋊D17—44no—
73C22⋊D17—42no—
74C22⋊D17—42no—
75C2×Dic17—22no—
76C2×Dic17—42no—
77D68—42no—
78D68—81no—
79#3622—42no—
80#1595—171nomaximal, Sylow
81C22×C17—21no—
82C22×C17—21no—
83C22×C17—22no—
84C22×C17—42no—
85C22×C17—42no—
86C22×C17—42no—
87C22×C17—41no—
88C22×C17—41no—
89C22×C17—42no—
90C22×C17—81no—
91C22×C17—81no—
92C22×C17—81no—
93C22×C17—81no—
94C22×C17—81no—
95C22×C17#12011yescharacteristic
96C22×C17C2×D811yescharacteristic
97C22×C17AGL(1,ℤ/8)11yescharacteristic
98C2×D17—42no—
99C2×D17—42no—
100C2×D17—42no—
101C2×D17—84no—
102C4×C17—22no—
103C4×C17—42no—
104C4×C17—41no—
105Dic17—42no—
106#534—171no—
107#447—172no—
108#392D1711yescharacteristic, complemented
109#327—172no—
110#323—171no—
111C2×C17—21no—
112C2×C17—21no—
113C2×C17—42no—
114C2×C17—82no—
115C2×C17#44711yescharacteristic
116C2×C17#45311yescharacteristic
117C2×C17B(4,𝔽2)11yescharacteristic
118D17—82no—
119C22×D4—21no—
120C22×D4—341no—
121C22×D4C2×D1711yescharacteristic, complemented
122C2×D8—342no—
123#123—342no—
124#121—172no—
125#121—341no—
126#120—342no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4C2×D1711yescharacteristic
129#102—342no—
130C22⋊C8—172no—
131#95C2×D1711yescharacteristic, complemented
132C17#159511yescharacteristic, complemented, Sylow
133C24—21no—
134C24—341no—
135C24C22⋊D1711yescharacteristic
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—172no—
142C2×D4—341no—
143C2×D4—342no—
144C2×D4—342no—
145C2×D4—682no—
146C2×D4—682no—
147C22×C4—21no—
148C22×C4—341no—
149C22×C4C22⋊D1711yescharacteristic
150C22×C4C22×D1711yescharacteristic
151D8—684no—
152C2×C8—342no—
153C4⋊C4—172no—
154C4⋊C4—341no—
155C22⋊C4—22no—
156C22⋊C4—42no—
157C22⋊C4—342no—
158C22⋊C4—342no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—342no—
171C23—682no—
172C23—1361no—
173C23—1361no—
174C23#6322812yescomplemented
175C23C2×C22⋊D1711yescharacteristic
176D4—24no—
177D4—42no—
178D4—48no—
179D4—682no—
180D4—684no—
181D4—682no—
182D4—682no—
183D4—1361no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—342no—
191C2×C4—682no—
192C2×C4D4×D1712yes—
193C8—682no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—682no—
209C22—682no—
210C22—682no—
211C22—1364no—
212C22C2×#6322811yescharacteristic
213C22#1058762311yescharacteristic
214C22#1058763511yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—682no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—1362no—
224C2#1176294211yescharacteristic
225C2#1176297611yescharacteristic
226C2#1176299511yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#392C2
socleC22×C17#120
hypercenter#392D17
hypocenterC17#1595
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #11763371C2semidirect
2. #11762943C2semidirect
3. #11762542C2semidirect
4. #11762533C2semidirect
5. C17×#392C2semidirect
7. C22×D4×C17C22semidirect
16. C17×#95C22semidirect
18. C22⋊#3622C22semidirect
36. C2×D68D4semidirect
43. C4⋊Dic17D4semidirect
56. C23×C17D8semidirect
108. #392D17semidirect
121. C22×D4C2×D17semidirect
131. #95C2×D17semidirect
132. C17#1595semidirect
174. C23#63228semidirect

Maximal subgroups

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

subgroupindex
#117633712
#117629432
#117625422
#117625332
C17×#3922
#159517