Finite Group Databasebeta

← #14527802

Subgroups — #14527802

Order 2176.

288 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 51 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#11763367C211yesmaximal, characteristic
2#11763127C211yesmaximal, characteristic, complemented
3#11762744C211yesmaximal, characteristic, complemented
4#11762544C211yesmaximal, characteristic, complemented
5#11762543C211yesmaximal, characteristic, complemented
6#11762532C211yesmaximal, characteristic, complemented
7C17×#481C211yesmaximal, characteristic, complemented
8C2×D4×D17—21no—
9C17×#121C2211yescharacteristic
10C2×C4⋊C4×C17C2211yescharacteristic, complemented
11#10587625—21no—
12#10587623—21no—
13#10587616C2211yescharacteristic
14C2×#63227—21no—
15C2×#63227C2211yescharacteristic
16C2×#63225—21no—
17#10587580—21no—
18#10587569—21no—
19#10587568—21no—
20C17×#106—21no—
21C17×#102—21no—
22C17×C22⋊C8C2211yescharacteristic, complemented
23#10587502C2211yescharacteristic, complemented
24C22⋊#3622C2211yescharacteristic, complemented
25#10587480—21no—
26#10587479—21no—
27C23×D17—21no—
28C23×D17D411yescharacteristic
29C2×D4×C17—21no—
30C2×D4×C17D411yescharacteristic
31C22×C4×C17—21no—
32C22×C4×C17C2311yescharacteristic
33C2×C22⋊D17—21no—
34C2×C22⋊D17—21no—
35C2×C22⋊D17—21no—
36C22×Dic17—21no—
37C22×Dic17D411yescharacteristic
38D4×D17—42no—
39D4×D17—42no—
40C2×D68—21no—
41C2×D68D411yescharacteristic
42C2×C4×D17—21no—
43C2×C8×C17—21no—
44C4⋊C4×C17—21no—
45C4⋊C4×C17—21no—
46C4⋊C4×C17D411yescharacteristic
47C22⋊C4×C17—21no—
48C22⋊Dic17—21no—
49#63227—21no—
50#63227—21no—
51#63227—21no—
52#63227—21no—
53#63227—42no—
54C4⋊Dic17D411yescharacteristic
55#63225—21no—
56#63225—42no—
57C2×#3622—21no—
58C23×C17—21no—
59C23×C17C2×D411yescharacteristic
60C22×D17—21no—
61C22×D17—21no—
62C22×D17—21no—
63C22×D17—42no—
64C22×D17—41no—
65C22×D17—41no—
66C22×D17—81no—
67C22×D17—81no—
68C22×D17SD1612yes—
69D4×C17—41no—
70D4×C17—41no—
71D4×C17—41no—
72C2×C4×C17—21no—
73C2×C4×C17—41no—
74C2×C4×C17—41no—
75C2×C4×C17—41no—
76C2×C4×C17—41no—
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C22⋊D17—42no—
80C22⋊D17—41no—
81C22⋊D17—41no—
82C22⋊D17—81no—
83C2×Dic17—22no—
84C2×Dic17—21no—
85C2×Dic17—21no—
86C2×Dic17—41no—
87C2×Dic17—41no—
88C2×Dic17—41no—
89D68—41no—
90D68—42no—
91C4×D17—42no—
92C8×C17—41no—
93#3622—41no—
94#1613—171nomaximal, Sylow
95C22×C17—21no—
96C22×C17—21no—
97C22×C17—41no—
98C22×C17—41no—
99C22×C17—41no—
100C22×C17#12011yescharacteristic
101C22×C17C2×SD1611yescharacteristic
102C22×C17AGL(1,ℤ/8)11yescharacteristic
103C2×D17—22no—
104C2×D17—41no—
105C2×D17—41no—
106C2×D17—42no—
107C2×D17—41no—
108C2×D17—41no—
109C2×D17—41no—
110C2×D17—81no—
111C2×D17—84no—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—41no—
115C4×C17—81no—
116Dic17—42no—
117Dic17—41no—
118Dic17—41no—
119#534—171no—
120#481—171no—
121#481D1711yescharacteristic, complemented
122#399—171no—
123#327—171no—
124#327—171no—
125#323—171no—
126C2×C17—21no—
127C2×C17—21no—
128C2×C17—81no—
129C2×C17#45011yescharacteristic
130C2×C17#45311yescharacteristic
131C2×C17#45811yescharacteristic
132D17—42no—
133D17—81no—
134C22×D4—341no—
135#123—341no—
136#123—341no—
137#121—171no—
138#121—341no—
139#121C2×D1711yescharacteristic
140#120—341no—
141#120—341no—
142C2×C4⋊C4—341no—
143C2×C4⋊C4C2×D1711yescharacteristic, complemented
144C2×C22⋊C4—171no—
145C2×C22⋊C4—341no—
146#106—21no—
147#106—341no—
148#102—21no—
149#102—341no—
150C22⋊C8—171no—
151C22⋊C8C2×D1711yescharacteristic, complemented
152#95—171no—
153C17#161311yescharacteristic, complemented, Sylow
154C24—171no—
155C24—341no—
156C2×D4—21no—
157C2×D4—171no—
158C2×D4—341no—
159C2×D4—341no—
160C2×D4—341no—
161C2×D4—341no—
162C2×D4—682no—
163C2×D4—682no—
164C2×D4C22⋊D1711yescharacteristic
165C22×C4—21no—
166C22×C4—171no—
167C22×C4—341no—
168C22×C4—341no—
169C22×C4C22×D1711yescharacteristic
170C2×C8—21no—
171C2×C8—341no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—171no—
175C4⋊C4—341no—
176C4⋊C4—682no—
177C4⋊C4C22⋊D1711yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—341no—
180C22⋊C4—341no—
181C22⋊C4—341no—
182C22⋊C4—341no—
183C22⋊C4—341no—
184C22⋊C4—682no—
185C23—21no—
186C23—172no—
187C23—341no—
188C23—341no—
189C23—341no—
190C23—681no—
191C23—681no—
192C23—682no—
193C23—1361no—
194C23—1361no—
195C23D4×D1711yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—681no—
200D4—682no—
201D4—682no—
202D4—681no—
203D4—681no—
204D4—1361no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—342no—
211C2×C4—341no—
212C2×C4—341no—
213C2×C4—682no—
214C2×C4—681no—
215C2×C4—681no—
216C2×C4—681no—
217C2×C4D4×D1711yescharacteristic
218C2×C4C2×C22⋊D1711yescharacteristic
219C8—41no—
220C8—681no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—342no—
227C22—682no—
228C22—681no—
229C22—681no—
230C22—681no—
231C22—681no—
232C22—681no—
233C22—1361no—
234C22—1364no—
235C22SD16×D1711yescharacteristic
236C22#1058760011yescharacteristic
237C22#1058762311yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—81no—
242C4—682no—
243C4—681no—
244C4—681no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—682no—
249C2—1361no—
250C2#1176295811yescharacteristic
251C2#1176297511yescharacteristic
252C2#1176299811yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#481C2
socleC22×C17#120
prime radical#11763367C2
hypercenter#481D17
hypocenterC17#1613
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #11763127C2semidirect
3. #11762744C2semidirect
4. #11762544C2semidirect
5. #11762543C2semidirect
6. #11762532C2semidirect
7. C17×#481C2semidirect
10. C2×C4⋊C4×C17C22semidirect
22. C17×C22⋊C8C22semidirect
23. #10587502C22semidirect
24. C22⋊#3622C22semidirect
121. #481D17semidirect
143. C2×C4⋊C4C2×D17semidirect
151. C22⋊C8C2×D17semidirect
153. C17#1613semidirect

Maximal subgroups

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

subgroupindex
#117633672
#117631272
#117627442
#117625442
#117625432
#117625322
C17×#4812
#161317