Finite Group Databasebeta

← #14527751

Subgroups — #14527751

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#11762948C211yesmaximal, characteristic, complemented
3#11762728C211yesmaximal, characteristic, complemented
4#11762544C211yesmaximal, characteristic, complemented
5#11762543C211yesmaximal, characteristic, complemented
6#11762534C211yesmaximal, characteristic, complemented
7C17×#448C211yesmaximal, characteristic, complemented
8C22×Q8×C17C2211yescharacteristic
9C2×#63254—21no—
10C2×SD16×C17—21no—
11C17×#121C2211yescharacteristic
12C2×#63230—21no—
13#10587625—21no—
14#10587625—21no—
15#10587616C2211yescharacteristic
16C2×#63225—21no—
17C2×#63225C2211yescharacteristic
18#10587582—21no—
19#10587570—21no—
20#10587564—21no—
21C17×#103—21no—
22C17×C22⋊C8C2211yescharacteristic
23#10587506—21no—
24#10587502—21no—
25#10587502C2211yescharacteristic
26C22⋊#3622C2211yescharacteristic
27C2×Q8×C17—21no—
28C2×Q8×C17—21no—
29C2×Q8×C17—42no—
30C2×D4×C17—21no—
31C2×D4×C17D411yescharacteristic
32C22×C4×C17—21no—
33C22×C4×C17C2311yescharacteristic
34C2×C22⋊D17—21no—
35C2×C22⋊D17—21no—
36C2×C22⋊D17—21no—
37C22×Dic17—21no—
38C22×Dic17—21no—
39C22×Dic17D411yescharacteristic
40C22×Dic17D411yescharacteristic
41#63254—42no—
42#63254—42no—
43C2×D68D411yescharacteristic
44C2×C4×D17—21no—
45C2×Dic34—21no—
46SD16×C17—42no—
47C2×C8×C17—21no—
48C4⋊C4×C17D411yescharacteristic
49C22⋊C4×C17—21no—
50C22⋊Dic17—21no—
51#63230—42no—
52#63227—21no—
53#63227—21no—
54C4⋊Dic17D411yescharacteristic
55#63225—21no—
56#63225—21no—
57#63225—21no—
58#63225—42no—
59C2×#3622—21no—
60C23×C17—21no—
61C23×C17C2×D411yescharacteristic
62C22×D17—21no—
63Q8×C17—22no—
64Q8×C17—22no—
65Q8×C17—41no—
66Q8×C17—41no—
67D4×C17—41no—
68D4×C17—41no—
69D4×C17—41no—
70C2×C4×C17—22no—
71C2×C4×C17—21no—
72C2×C4×C17—41no—
73C2×C4×C17—41no—
74C2×C4×C17—41no—
75C2×C4×C17C2×D411yescharacteristic
76C2×C4×C17C2×D411yescharacteristic
77C22⋊D17—42no—
78C22⋊D17—41no—
79C22⋊D17—41no—
80C22⋊D17—81no—
81C2×Dic17—21no—
82C2×Dic17—21no—
83C2×Dic17—22no—
84C2×Dic17—21no—
85C2×Dic17—21no—
86C2×Dic17—42no—
87C2×Dic17—41no—
88C2×Dic17—41no—
89C2×Dic17—41no—
90C2×Dic17SD1612yescomplemented
91D68—41no—
92C4×D17—42no—
93Dic34—42no—
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—41no—
111C4×C17—42no—
112Dic17—22no—
113Dic17—42no—
114Dic17—41no—
115Dic17—41no—
116#537—171no—
117#448—171no—
118#448D1711yescharacteristic, complemented
119#395—171no—
120#327—171no—
121#327—171no—
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×Q8C2×D1711yescharacteristic
132C2×SD16—21no—
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—21no—
144#103—341no—
145C22⋊C8—171no—
146C22⋊C8C2×D1711yescharacteristic
147#95—171no—
148#95—341no—
149C17#160411yescharacteristic, complemented, Sylow
150#41—682no—
151#41—682no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—42no—
155C2×Q8—341no—
156C2×D4—21no—
157C2×D4—171no—
158C2×D4—341no—
159C2×D4—341no—
160C2×D4—341no—
161C2×D4C22⋊D1711yescharacteristic
162C22×C4—21no—
163C22×C4—171no—
164C22×C4—171no—
165C22×C4—341no—
166C22×C4—341no—
167C22×C4—341no—
168C22×C4C22×D1711yescharacteristic
169SD16—42no—
170SD16—682no—
171C2×C8—21no—
172C2×C8—341no—
173C4⋊C4—171no—
174C4⋊C4—341no—
175C4⋊C4—341no—
176C4⋊C4—341no—
177C4⋊C4—682no—
178C4⋊C4C22⋊D1711yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—341no—
181C22⋊C4—341no—
182C22⋊C4—341no—
183C23—21no—
184C23—341no—
185C23D4×D1711yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
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×C4D4×D1711yescharacteristic
216C2×C4C2×C22⋊D1711yescharacteristic
217C8—41no—
218C8—681no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—681no—
225C22—681no—
226C22—681no—
227C22SD16×D1711yescharacteristic
228C22#1058760011yescharacteristic
229C22#1058762311yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—42no—
234C4—342no—
235C4—682no—
236C4—681no—
237C4—681no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—1361no—
242C2#1176295811yescharacteristic
243C2#1176299011yescharacteristic
244C2#1176299511yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#448C2
socleC22×C17#120
prime radical#11763386C2
hypercenter#448D17
hypocenterC17#1604
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #11762948C2semidirect
3. #11762728C2semidirect
4. #11762544C2semidirect
5. #11762543C2semidirect
6. #11762534C2semidirect
7. C17×#448C2semidirect
90. C2×Dic17SD16semidirect
118. #448D17semidirect
149. C17#1604semidirect

Maximal subgroups

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

subgroupindex
#117633862
#117629482
#117627282
#117625442
#117625432
#117625342
C17×#4482
#160417