Finite Group Databasebeta

← #14527754

Subgroups — #14527754

Order 2176.

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

normal subgroups only

This group has 282 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#11763386C211yesmaximal, characteristic
2#11762950C211yesmaximal, characteristic, complemented
3#11762947C211yesmaximal, characteristic, complemented
4#11762730C211yesmaximal, characteristic, complemented
5#11762542C211yesmaximal, characteristic, complemented
6#11762534C211yesmaximal, characteristic, complemented
7C17×#327C211yesmaximal, characteristic, complemented
8C2×#63254—21no—
9C22×Dic34C2211yescharacteristic
10C17×#121C2211yescharacteristic
11#10587625—21no—
12#10587625—21no—
13C2×#63229—21no—
14#10587616C2211yescharacteristic
15C2×C4⋊Dic17—21no—
16C2×#63225C2211yescharacteristic
17C2×#63219—21no—
18#10587582—21no—
19#10587570—21no—
20#10587564—21no—
21C17×C22⋊C8C2211yescharacteristic
22C17×#95C2211yescharacteristic
23#10587502—21no—
24C22⋊#3622C2211yescharacteristic
25#10587487—21no—
26#10587481—21no—
27C2×D4×C17—21no—
28C2×D4×C17D411yescharacteristic
29C22×C4×C17—21no—
30C22×C4×C17C2311yescharacteristic
31C2×C22⋊D17—21no—
32C2×C22⋊D17—21no—
33C2×C22⋊D17—21no—
34C22×Dic17—21no—
35C22×Dic17—21no—
36C22×Dic17D411yescharacteristic
37C22×Dic17D411yescharacteristic
38#63254—42no—
39#63254—42no—
40C2×D68D411yescharacteristic
41C2×C4×D17—21no—
42C2×Dic34—21no—
43C2×Dic34—21no—
44C2×Dic34—21no—
45C2×Dic34—42no—
46C2×C8×C17—21no—
47C4⋊C4×C17D411yescharacteristic
48C22⋊C4×C17—21no—
49C22⋊Dic17—21no—
50#63229—42no—
51#63227—21no—
52#63227—21no—
53C4⋊Dic17—42no—
54C4⋊Dic17D411yescharacteristic
55#63225—21no—
56#63225—21no—
57#63225—21no—
58C2×#3622—21no—
59#63219—42no—
60C23×C17—21no—
61C23×C17C2×D411yescharacteristic
62C22×D17—21no—
63D4×C17—41no—
64D4×C17—41no—
65D4×C17—41no—
66C2×C4×C17—22no—
67C2×C4×C17—21no—
68C2×C4×C17—41no—
69C2×C4×C17—41no—
70C2×C4×C17—41no—
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2×D411yescharacteristic
73C22⋊D17—42no—
74C22⋊D17—41no—
75C22⋊D17—41no—
76C22⋊D17—81no—
77C2×Dic17—21no—
78C2×Dic17—21no—
79C2×Dic17—22no—
80C2×Dic17—21no—
81C2×Dic17—21no—
82C2×Dic17—42no—
83C2×Dic17—41no—
84C2×Dic17—41no—
85C2×Dic17—41no—
86C2×Dic17SD1612yescomplemented
87D68—41no—
88C4×D17—42no—
89Dic34—22no—
90Dic34—22no—
91Dic34—42no—
92Dic34—41no—
93Dic34—41no—
94C8×C17—41no—
95#3622—41no—
96#1604—171nomaximal, Sylow
97C22×C17—21no—
98C22×C17—21no—
99C22×C17—41no—
100C22×C17—41no—
101C22×C17—41no—
102C22×C17#12011yescharacteristic
103C22×C17C2×SD1611yescharacteristic
104C22×C17AGL(1,ℤ/8)11yescharacteristic
105C2×D17—41no—
106C2×D17—41no—
107C2×D17—41no—
108C4×C17—21no—
109C4×C17—21no—
110C4×C17—42no—
111C4×C17—41no—
112Dic17—22no—
113Dic17—41no—
114Dic17—42no—
115Dic17—41no—
116#537—171no—
117#448—171no—
118#448—171no—
119#395—171no—
120#327—171no—
121#327D1711yescharacteristic, complemented
122#324—171no—
123C2×C17—21no—
124C2×C17—21no—
125C2×C17—81no—
126C2×C17#45011yescharacteristic
127C2×C17#45611yescharacteristic
128C2×C17B(4,𝔽2)11yescharacteristic
129D17—81no—
130C2×#41—341no—
131C22×Q8—171no—
132C2×SD16—341no—
133C2×SD16—341no—
134C22⋊Q8—341no—
135C22⋊Q8—341no—
136#121—171no—
137#121—341no—
138#121—341no—
139#121—341no—
140#121C2×D1711yescharacteristic
141C2×C4⋊C4—171no—
142C2×C4⋊C4—341no—
143#103—341no—
144#103—341no—
145C22⋊C8—171no—
146C22⋊C8C2×D1711yescharacteristic
147#95—341no—
148#95C2×D1711yescharacteristic
149C17#160411yescharacteristic, complemented, Sylow
150#41—682no—
151#41—682no—
152C2×Q8—341no—
153C2×Q8—341no—
154C2×Q8—341no—
155C2×Q8—682no—
156C2×D4—21no—
157C2×D4—171no—
158C2×D4—341no—
159C2×D4—341no—
160C2×D4—341no—
161C2×D4D6811yescharacteristic
162C22×C4—21no—
163C22×C4—171no—
164C22×C4—171no—
165C22×C4—341no—
166C22×C4—341no—
167C22×C4—341no—
168C22×C4C22×D1711yescharacteristic
169SD16—682no—
170SD16—682no—
171C2×C8—21no—
172C2×C8—341no—
173C4⋊C4—171no—
174C4⋊C4—341no—
175C4⋊C4—341no—
176C4⋊C4—341no—
177C4⋊C4—682no—
178C4⋊C4D6811yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—341no—
181C22⋊C4—341no—
182C22⋊C4—341no—
183C23—21no—
184C23—341no—
185C23D4×D1711yescharacteristic
186Q8—342no—
187Q8—342no—
188Q8—681no—
189Q8—681no—
190Q8—682no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—681no—
195D4—682no—
196D4—681no—
197D4—681no—
198D4—1361no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—172no—
205C2×C4—341no—
206C2×C4—341no—
207C2×C4—342no—
208C2×C4—341no—
209C2×C4—341no—
210C2×C4—682no—
211C2×C4—682no—
212C2×C4—681no—
213C2×C4—681no—
214C2×C4—681no—
215C2×C4C2×D6811yescharacteristic
216C2×C4D4×D1711yescharacteristic
217C8—41no—
218C8—681no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—681no—
225C22—681no—
226C22—681no—
227C22#1058756811yescharacteristic
228C22#1058759711yescharacteristic
229C22SD16×D1711yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—342no—
235C4—681no—
236C4—682no—
237C4—681no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—1361no—
242C2#1176296111yescharacteristic
243C2#1176299311yescharacteristic
244C2#1176299711yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#327C2
socleC22×C17#120
prime radical#11763386C2
hypercenter#327D17
hypocenterC17#1604
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #11762950C2semidirect
3. #11762947C2semidirect
4. #11762730C2semidirect
5. #11762542C2semidirect
6. #11762534C2semidirect
7. C17×#327C2semidirect
86. C2×Dic17SD16semidirect
121. #327D17semidirect
149. C17#1604semidirect

Maximal subgroups

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

subgroupindex
#117633862
#117629502
#117629472
#117627302
#117625422
#117625342
C17×#3272
#160417