Finite Group Databasebeta

← #14544190

Subgroups — #14544190

Order 2176.

326 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 58 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#11763780C211yesmaximal, characteristic, complemented
2#11763752C211yesmaximal, characteristic, complemented
3#11763661C212yesmaximal, complemented
4#11763660C212yesmaximal, complemented
5#121×D17C211yesmaximal, characteristic
6C2×D4×D17—21no—
7C2×D4×D17C411yescharacteristic, complemented
8C22×C4×D17C2211yescharacteristic
9C2×C22⋊#595—21no—
10C2×#63243—21no—
11C17×#121C411yescharacteristic, complemented
12#10587625—21no—
13#10587624C2211yescharacteristic
14#10587616C411yescharacteristic, complemented
15#10587581C2211yescharacteristic
16C4⋊C4×D17C411yescharacteristic, complemented
17#10587570—21no—
18C22⋊C4×D17—21no—
19C22⋊#3624—21no—
20(C2×#595)⋊C4—21no—
21(C2×#595)⋊C4C2212yescomplemented
22#10587543C2212yescomplemented
23C23×D17—21no—
24C23×D17D411yescharacteristic
25C22×#595—22no—
26C2×D4×C17—21no—
27C2×D4×C17C2×C411yescharacteristic
28C22×C4×C17C2×C411yescharacteristic
29C2×C22⋊D17—21no—
30C2×C22⋊D17—21no—
31C2×C22⋊D17—21no—
32C22×Dic17D411yescharacteristic
33D4×D17—42no—
34D4×D17—44no—
35C2×D68—21no—
36C2×D68C2×C411yescharacteristic
37C2×C4×D17—21no—
38C2×C4×D17—42no—
39C2×C4×D17C2×C411yescharacteristic
40C2×C4×D17D411yescharacteristic
41C2×C4×D17D411yescharacteristic
42C2×C4×D17C2311yescharacteristic
43C22⋊#595—42no—
44C2×#3624—22no—
45#63243—42no—
46C4⋊C4×C17C2×C411yescharacteristic
47C22⋊C4×C17—21no—
48C22⋊Dic17—21no—
49#63227—21no—
50#63227—21no—
51C4⋊Dic17C2×C411yescharacteristic
52#63225—21no—
53C23×C17—21no—
54C23×C17C22⋊C411yescharacteristic
55C22×D17—22no—
56C22×D17—22no—
57C22×D17—21no—
58C22×D17—41no—
59C22×D17—42no—
60C22×D17—42no—
61C22×D17—42no—
62C22×D17C22⋊C411yescharacteristic
63C22×D17C2×D411yescharacteristic
64C2×#595—42no—
65C2×#595—42no—
66C2×#595—42no—
67D4×C17—41no—
68D4×C17—42no—
69C2×C4×C17—21no—
70C2×C4×C17—41no—
71C2×C4×C17C22⋊C411yescharacteristic
72C2×C4×C17C22×C411yescharacteristic
73C22⋊D17—42no—
74C22⋊D17—42no—
75C22⋊D17—42no—
76C2×Dic17—21no—
77C2×Dic17—41no—
78C2×Dic17C22⋊C411yescharacteristic
79C2×Dic17C2×D411yescharacteristic
80D68—41no—
81D68—42no—
82C4×D17—22no—
83C4×D17—22no—
84C4×D17—41no—
85C4×D17—41no—
86C4×D17—42no—
87#3624—42no—
88#1510—171nomaximal, Sylow
89C22×C17—21no—
90C22×C17—21no—
91C22×C17—21no—
92C22×C17—41no—
93C22×C17—41no—
94C22×C17—41no—
95C22×C17C2×C22⋊C411yescharacteristic
96C2×D17—21no—
97C2×D17—21no—
98C2×D17—21no—
99C2×D17—21no—
100C2×D17—44no—
101C2×D17—41no—
102C2×D17—41no—
103C2×D17—41no—
104C2×D17—84no—
105C2×D17#9912yes—
106C2×D17AGL(1,ℤ/8)12yes—
107#595—82no—
108C4×C17—21no—
109C4×C17—21no—
110C4×C17—41no—
111Dic17—21no—
112Dic17—21no—
113Dic17—41no—
114C2×#121—171no—
115#407—171no—
116#381—171no—
117#327—172no—
118#327—172no—
119C2×C17—41no—
120C2×C17—81no—
121C2×C17C2×#9911yescharacteristic
122C2×C17#41811yescharacteristic
123C2×C17#42111yescharacteristic
124D17—22no—
125D17—22no—
126D17—81no—
127C22×D4—171no—
128C22×D4—341no—
129C23×C4—171no—
130C2×M16—341no—
131#121—171no—
132#121—171no—
133#121—171no—
134#121—341no—
135#121—341no—
136#121#59511yescharacteristic, complemented
137C2×C4⋊C4—171no—
138C2×C22⋊C4—341no—
139C2×C22⋊C4—341no—
140C22⋊C8—172no—
141C22⋊C8—341no—
142#95—172no—
143#95—341no—
144C17#151011yescharacteristic, complemented, Sylow
145C24—171no—
146C24—341no—
147C2×D4—21no—
148C2×D4—171no—
149C2×D4—341no—
150C2×D4—341no—
151C2×D4—341no—
152C2×D4—341no—
153C2×D4—682no—
154C2×D4—684no—
155C2×D4C2×#59511yescharacteristic
156C22×C4—171no—
157C22×C4—171no—
158C22×C4—171no—
159C22×C4—171no—
160C22×C4—171no—
161C22×C4—341no—
162C22×C4—342no—
163C22×C4—682no—
164C22×C4C2×#59511yescharacteristic
165M16—682no—
166C2×C8—342no—
167C4⋊C4—171no—
168C4⋊C4—341no—
169C4⋊C4C2×#59511yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—341no—
172C22⋊C4—341no—
173C22⋊C4—341no—
174C22⋊C4—682no—
175C23—21no—
176C23—171no—
177C23—171no—
178C23—342no—
179C23—342no—
180C23—341no—
181C23—681no—
182C23—682no—
183C23—682no—
184C23—682no—
185C23C22⋊#59511yescharacteristic
186D4—41no—
187D4—42no—
188D4—681no—
189D4—682no—
190D4—682no—
191D4—682no—
192D4—682no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—171no—
196C2×C4—171no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—341no—
200C2×C4—681no—
201C2×C4—681no—
202C2×C4—682no—
203C2×C4—681no—
204C2×C4—682no—
205C2×C4—682no—
206C2×C4—682no—
207C2×C4C22⋊#59511yescharacteristic
208C2×C4C22×#59511yescharacteristic
209C8—682no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—172no—
217C22—172no—
218C22—341no—
219C22—341no—
220C22—341no—
221C22—341no—
222C22—684no—
223C22—681no—
224C22—681no—
225C22—681no—
226C22—1364no—
227C22C2×C22⋊#59511yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—341no—
232C4—341no—
233C4—681no—
234C4—1362no—
235C2—41no—
236C2—81no—
237C2—342no—
238C2—342no—
239C2—1361no—
240C2D17⋊#9911yescharacteristic
241C2#1176380111yescharacteristic
242C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#121C4
socleC22×C17C2×C22⋊C4
prime radical#121×D17C2
cosocleC2×C4×D17C23
hypercenter#121#595
hypocenterC17#1510
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #11763780C2semidirect
2. #11763752C2semidirect
3. #11763661C2semidirect
4. #11763660C2semidirect
7. C2×D4×D17C4semidirect
11. C17×#121C4semidirect
14. #10587616C4semidirect
16. C4⋊C4×D17C4semidirect
21. (C2×#595)⋊C4C22semidirect
22. #10587543C22semidirect
136. #121#595semidirect
144. C17#1510semidirect

Maximal subgroups

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

subgroupindex
#117637802
#117637522
#117636612
#117636602
#121×D172
#151017