Finite Group Databasebeta

← #14544170

Subgroups — #14544170

Order 2176.

334 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 62 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#11763781C211yesmaximal, characteristic, complemented
2C2×(C2×#595)⋊C4C211yesmaximal, characteristic, complemented
3#11763663C212yesmaximal, complemented
4#11763660C212yesmaximal, complemented
5#11763316C211yesmaximal, characteristic
6C23×#595—21no—
7C2×#41×C17C411yescharacteristic, complemented
8C22×C4×D17C2211yescharacteristic
9C2×#63247—21no—
10#10587632C411yescharacteristic, complemented
11#10587632C2211yescharacteristic
12#10587624C411yescharacteristic, complemented
13#10587624C2211yescharacteristic
14#10587623—21no—
15#10587621—21no—
16C4×C22⋊D17—21no—
17C4×C22⋊D17—21no—
18#10587612C411yescharacteristic, complemented
19(C2×#595)⋊C4—21no—
20(C2×#595)⋊C4C2212yes—
21#10587543—21no—
22#10587543C2212yes—
23C23×D17D411yescharacteristic
24C22×#595—24no—
25C22×#595—22no—
26C22×#595—42no—
27C22×#595—42no—
28#41×C17—42no—
29C2×Q8×C17C2×C411yescharacteristic, complemented
30C2×D4×C17—21no—
31C2×D4×C17C2×C411yescharacteristic, complemented
32C22×C4×C17—21no—
33C22×C4×C17C2×C411yescharacteristic
34C2×C22⋊D17—21no—
35C2×C22⋊D17—21no—
36C22×Dic17D411yescharacteristic
37C2×C4×D17—42no—
38C2×C4×D17C2×C411yescharacteristic
39C2×C4×D17D411yescharacteristic
40C2×C4×D17D411yescharacteristic
41C2×C4×D17C2311yescharacteristic
42#63247—42no—
43C2×#3624—22no—
44C22⋊Dic17—21no—
45C22⋊Dic17—21no—
46C22⋊Dic17—21no—
47#63227—21no—
48#63227—21no—
49C4⋊Dic17C2×C411yescharacteristic, complemented
50C4⋊Dic17C2×C411yescharacteristic, complemented
51#63225—21no—
52#63225—21no—
53C4×Dic17—21no—
54C23×C17—21no—
55C23×C17C22⋊C411yescharacteristic
56C22×D17—22no—
57C22×D17—22no—
58C22×D17—21no—
59C22×D17—21no—
60C22×D17C22⋊C411yescharacteristic
61C22×D17C2×D411yescharacteristic
62C2×#595—24no—
63C2×#595—48no—
64C2×#595—42no—
65C2×#595—42no—
66Q8×C17—41no—
67D4×C17—41no—
68D4×C17—42no—
69C2×C4×C17—22no—
70C2×C4×C17—21no—
71C2×C4×C17—42no—
72C2×C4×C17C22⋊C411yescharacteristic
73C2×C4×C17C22×C411yescharacteristic
74C22⋊D17—42no—
75C22⋊D17—41no—
76C22⋊D17—41no—
77C2×Dic17—22no—
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17C22⋊C411yescharacteristic
81C2×Dic17C2×D411yescharacteristic
82C4×D17—41no—
83C4×D17—41no—
84C4×D17—41no—
85C4×D17—41no—
86#3624—42no—
87#1504—171nomaximal, Sylow
88C22×C17—21no—
89C22×C17—21no—
90C22×C17—41no—
91C22×C17—41no—
92C22×C17—41no—
93C22×C17#9912yes—
94C22×C17C2×C22⋊C411yescharacteristic
95C2×D17—24no—
96C2×D17—24no—
97C2×D17—21no—
98C2×D17—21no—
99C2×D17—21no—
100C2×D17—21no—
101C2×D17#10412yes—
102C2×D17AGL(1,ℤ/8)11yescharacteristic
103C2×D17#13711yescharacteristic
104#595—44no—
105C4×C17—21no—
106C4×C17—21no—
107C4×C17—41no—
108Dic17—21no—
109Dic17—21no—
110Dic17—41no—
111Dic17—41no—
112#525—171no—
113#407—171no—
114C2×#95—171no—
115#328—172no—
116#327—172no—
117C2×C17—22no—
118C2×C17—81no—
119C2×C17C2×#9911yescharacteristic
120C2×C17#41711yescharacteristic
121C2×C17C2×#10411yescharacteristic
122D17—21no—
123D17—22no—
124D17—21no—
125C2×#41#59511yescharacteristic, complemented
126C23×C4—171no—
127C23×C4—341no—
128C2×M16—341no—
129#123—341no—
130C22⋊Q8—171no—
131C22⋊Q8—171no—
132#121—171no—
133#121—171no—
134#120—341no—
135C4×D4—341no—
136C4×D4—341no—
137#117—171no—
138C22⋊C8—172no—
139C22⋊C8—341no—
140#95—172no—
141#95—341no—
142C17#150411yescharacteristic, complemented, Sylow
143C24—171no—
144#41—42no—
145C2×Q8C2×#59511yescharacteristic, complemented
146C2×D4—21no—
147C2×D4—341no—
148C2×D4—341no—
149C2×D4C2×#59511yescharacteristic, complemented
150C22×C4—21no—
151C22×C4—171no—
152C22×C4—171no—
153C22×C4—171no—
154C22×C4—171no—
155C22×C4—171no—
156C22×C4—342no—
157C22×C4—344no—
158C22×C4—682no—
159C22×C4—682no—
160C22×C4—682no—
161C22×C4C2×#59511yescharacteristic
162M16—682no—
163C2×C8—342no—
164C4⋊C4—171no—
165C4⋊C4—171no—
166C4⋊C4—341no—
167C4⋊C4—341no—
168C22⋊C4—341no—
169C22⋊C4—341no—
170C22⋊C4—341no—
171C22⋊C4—341no—
172C22⋊C4—341no—
173C42—341no—
174C23—21no—
175C23—171no—
176C23—171no—
177C23—342no—
178C23—341no—
179C23—341no—
180C23—342no—
181C23C22⋊#59511yescharacteristic
182Q8—41no—
183D4—41no—
184D4—42no—
185D4—681no—
186D4—681no—
187D4—682no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—42no—
191C2×C4—171no—
192C2×C4—171no—
193C2×C4—342no—
194C2×C4—344no—
195C2×C4—341no—
196C2×C4—341no—
197C2×C4—681no—
198C2×C4—681no—
199C2×C4—681no—
200C2×C4—681no—
201C2×C4—682no—
202C2×C4—688no—
203C2×C4—682no—
204C2×C4C22⋊#59511yescharacteristic
205C2×C4C22×#59511yescharacteristic
206C8—682no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—171no—
213C22—171no—
214C22—172no—
215C22—344no—
216C22—344no—
217C22—341no—
218C22—341no—
219C22—341no—
220C22—341no—
221C22#1058755112yes—
222C22C2×C22⋊#59511yescharacteristic
223C4—21no—
224C4—21no—
225C4—41no—
226C4—341no—
227C4—341no—
228C4—684no—
229C4—681no—
230C4—681no—
231C2—22no—
232C2—81no—
233C2—341no—
234C2—341no—
235C2—342no—
236C2C2×#1058755111yescharacteristic
237C2#1176380011yescharacteristic
238C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC2×#41×C17C4
socleC22×C17C2×C22⋊C4
prime radical#11763316C2
cosocleC2×C4×D17C23
hypercenterC2×#41#595
hypocenterC17#1504
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #11763781C2semidirect
2. C2×(C2×#595)⋊C4C2semidirect
3. #11763663C2semidirect
4. #11763660C2semidirect
7. C2×#41×C17C4semidirect
10. #10587632C4semidirect
12. #10587624C4semidirect
18. #10587612C4semidirect
29. C2×Q8×C17C2×C4semidirect
31. C2×D4×C17C2×C4semidirect
49. C4⋊Dic17C2×C4semidirect
50. C4⋊Dic17C2×C4semidirect
125. C2×#41#595semidirect
142. C17#1504semidirect
145. C2×Q8C2×#595semidirect
149. C2×D4C2×#595semidirect

Maximal subgroups

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

subgroupindex
#117637812
C2×(C2×#595)⋊C42
#117636632
#117636602
#117633162
#150417