Finite Group Databasebeta

← #14544625

Subgroups — #14544625

Order 2176.

618 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#10587548C214yesmaximal, complemented
2#11763780C211yesmaximal, characteristic, complemented
3C4×C22⋊#595C211yesmaximal, characteristic, complemented
4C22×#63254C211yesmaximal, characteristic
5C22×D4×C17C411yescharacteristic, complemented
6C22×C22⋊D17—21no—
7C23×Dic17—21no—
8C2×#63254—28no—
9C2×#63254—21no—
10C2×#63254—21no—
11C2×#63254C412yescomplemented
12C2×#63254C2212yes—
13C22×C4×D17C2211yescharacteristic
14C22×Dic34C411yescharacteristic, complemented
15C2×C22⋊#595—21no—
16C2×C4×#595C2212yescomplemented
17C2×#63243C2212yescomplemented
18C22⋊#3624—21no—
19#10587548—28no—
20(C2×#595)⋊C4—21no—
21#10587543—21no—
22C24×C17—21no—
23C23×D17D411yescharacteristic
24C22×#595—22no—
25C2×D4×C17—24no—
26C2×D4×C17—21no—
27C2×D4×C17C2×C412yes—
28C22×C4×C17C2×C411yescharacteristic
29C2×C22⋊D17—28no—
30C2×C22⋊D17—22no—
31C2×C22⋊D17—22no—
32C22×Dic17—28no—
33C22×Dic17—21no—
34C22×Dic17—22no—
35C22×Dic17—22no—
36C22×Dic17D411yescharacteristic
37#63254—28no—
38#63254—28no—
39#63254—24no—
40#63254—24no—
41#63254—24no—
42#63254D418yes—
43C2×C4×D17—24no—
44C2×C4×D17C2×C411yescharacteristic
45C2×C4×D17D411yescharacteristic
46C2×C4×D17D411yescharacteristic
47C2×C4×D17C2311yescharacteristic
48C2×Dic34—21no—
49C2×Dic34—24no—
50C2×Dic34C2×C412yes—
51C22⋊#595—24no—
52C2×#3624—22no—
53C4×#595—24no—
54#63243—24no—
55C23×C17—24no—
56C23×C17—22no—
57C23×C17—42no—
58C23×C17—42no—
59C23×C17C22⋊C411yescharacteristic
60C22×D17—22no—
61C22×D17—22no—
62C22×D17—41no—
63C22×D17C22⋊C411yescharacteristic
64C22×D17C2×D411yescharacteristic
65C2×#595—24no—
66C2×#595—42no—
67C2×#595—42no—
68D4×C17—24no—
69D4×C17—22no—
70D4×C17C22⋊C414yescomplemented
71C2×C4×C17—22no—
72C2×C4×C17C22⋊C411yescharacteristic
73C2×C4×C17C22×C411yescharacteristic
74C22⋊D17—28no—
75C22⋊D17—28no—
76C2×Dic17—24no—
77C2×Dic17—28no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—28no—
81C2×Dic17C22⋊C411yescharacteristic
82C2×Dic17C2×D411yescharacteristic
83C4×D17—22no—
84C4×D17—22no—
85C4×D17—22no—
86C4×D17C2×D414yes—
87Dic34—24no—
88Dic34—22no—
89Dic34C22⋊C414yescomplemented
90#3624—42no—
91#1789—171nomaximal, Sylow
92C22×C17—21no—
93C22×C17—21no—
94C22×C17—24no—
95C22×C17—48no—
96C22×C17—42no—
97C22×C17—42no—
98C22×C17#10412yescomplemented
99C22×C17C2×C22⋊C411yescharacteristic
100C2×D17—21no—
101C2×D17—21no—
102C2×D17—21no—
103C2×D17—21no—
104C2×D17—21no—
105C2×D17—44no—
106C2×D17#12012yes—
107#595—44no—
108C4×C17—21no—
109C4×C17C2×C22⋊C412yes—
110Dic17—24no—
111Dic17—21no—
112Dic17#12012yes—
113C22×#41—171no—
114C2×#104—174no—
115#407—171no—
116C4×C22⋊C4—171no—
117C2×C17—22no—
118C2×C17—44no—
119C2×C17#37911yescharacteristic
120C2×C17C2×#10411yescharacteristic
121C2×C17#42111yescharacteristic
122D17—22no—
123D17—41no—
124C2×#41—172no—
125C2×#41—172no—
126C2×#41—341no—
127C2×#41—341no—
128C2×#41—348no—
129C22×Q8—171no—
130C22×D4—341no—
131C22×D4#59511yescharacteristic, complemented
132C23×C4—171no—
133C23×C4—341no—
134C2×M16—172no—
135C2×C22⋊C4—341no—
136C2×C42—172no—
137#104—348no—
138C22⋊C8—341no—
139C22⋊C8—341no—
140#95—341no—
141C17#178911yescharacteristic, complemented, Sylow
142C24—21no—
143C24—171no—
144#41—178no—
145#41—344no—
146#41—344no—
147#41—348no—
148#41—348no—
149#41—344no—
150C2×Q8—172no—
151C2×Q8—344no—
152C2×Q8—341no—
153C2×D4—24no—
154C2×D4—21no—
155C2×D4—342no—
156C2×D4—342no—
157C2×D4—348no—
158C2×D4C2×#59512yes—
159C22×C4—171no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4—171no—
163C22×C4—171no—
164C22×C4—344no—
165C22×C4—342no—
166C22×C4—342no—
167C22×C4—348no—
168C22×C4—341no—
169C22×C4—342no—
170C22×C4C2×#59511yescharacteristic
171M16—344no—
172C2×C8—342no—
173C22⋊C4—344no—
174C42—344no—
175C23—24no—
176C23—22no—
177C23—42no—
178C23—42no—
179C23—171no—
180C23—171no—
181C23—342no—
182C23—342no—
183C23—681no—
184C23C22⋊#59511yescharacteristic
185Q8—174no—
186Q8—344no—
187Q8—342no—
188D4—24no—
189D4—22no—
190D4—348no—
191D4—348no—
192D4C22⋊#59514yescomplemented
193C2×C4—22no—
194C2×C4—171no—
195C2×C4—174no—
196C2×C4—171no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—348no—
201C2×C4—344no—
202C2×C4—348no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—344no—
206C2×C4—682no—
207C2×C4—682no—
208C2×C4C22⋊#59511yescharacteristic
209C2×C4C22×#59511yescharacteristic
210C8—682no—
211C22—21no—
212C22—21no—
213C22—24no—
214C22—48no—
215C22—42no—
216C22—42no—
217C22—172no—
218C22—341no—
219C22—341no—
220C22—341no—
221C22—341no—
222C22—341no—
223C22—684no—
224C22#1058754812yescomplemented
225C22C2×C22⋊#59511yescharacteristic
226C4—21no—
227C4—172no—
228C4—341no—
229C4—344no—
230C4—684no—
231C4C2×C22⋊#59512yes—
232C2—22no—
233C2—44no—
234C2—342no—
235C2—681no—
236C2#1176375011yescharacteristic
237C2C2×#1058754811yescharacteristic
238C2#1176380711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC22×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×#63254C2
cosocleC2×C4×D17C23
hypercenterC22×D4#595
hypocenterC17#1789
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#10587548C2semidirect
2. #11763780C2semidirect
3. C4×C22⋊#595C2semidirect
5. C22×D4×C17C4semidirect
11. C2×#63254C4semidirect
14. C22×Dic34C4semidirect
16. C2×C4×#595C22semidirect
17. C2×#63243C22semidirect
70. D4×C17C22⋊C4semidirect
89. Dic34C22⋊C4semidirect
98. C22×C17#104semidirect
131. C22×D4#595semidirect
141. C17#1789semidirect
192. D4C22⋊#595semidirect
224. C22#10587548semidirect

Maximal subgroups

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

subgroupindex
C2×#105875482
#117637802
C4×C22⋊#5952
C22×#632542
#178917