Finite Group Databasebeta

← #14544627

Subgroups — #14544627

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×#10587550C214yesmaximal, complemented
2#11763780C211yesmaximal, characteristic, complemented
3C4×C22⋊#595C211yesmaximal, characteristic, complemented
4C22×#63256C211yesmaximal, characteristic
5C22×Q8×C17C411yescharacteristic, complemented
6C2×#63256—28no—
7C2×#63256—21no—
8C2×#63256—21no—
9C2×#63256C412yescomplemented
10C2×#63256C2212yes—
11C22×D68—21no—
12C22×D68C411yescharacteristic, complemented
13C22×C4×D17—21no—
14C22×C4×D17C2211yescharacteristic
15C2×C22⋊#595—21no—
16C2×C4×#595C2212yescomplemented
17C2×#63243C2212yescomplemented
18C22⋊#3624—21no—
19#10587550—28no—
20(C2×#595)⋊C4—21no—
21#10587543—21no—
22C23×D17—21no—
23C23×D17D411yescharacteristic
24C22×#595—22no—
25C2×Q8×C17—24no—
26C2×Q8×C17—21no—
27C2×Q8×C17C2×C412yes—
28C22×C4×C17—21no—
29C22×C4×C17C2×C411yescharacteristic
30C22×Dic17D411yescharacteristic
31#63256—28no—
32#63256—28no—
33#63256—24no—
34#63256—24no—
35#63256—24no—
36#63256D418yes—
37C2×D68—28no—
38C2×D68—22no—
39C2×D68—22no—
40C2×D68—21no—
41C2×D68—24no—
42C2×D68C2×C412yes—
43C2×C4×D17—28no—
44C2×C4×D17—22no—
45C2×C4×D17—22no—
46C2×C4×D17—24no—
47C2×C4×D17C2×C411yescharacteristic
48C2×C4×D17D411yescharacteristic
49C2×C4×D17D411yescharacteristic
50C2×C4×D17C2311yescharacteristic
51C22⋊#595—24no—
52C2×#3624—22no—
53C4×#595—24no—
54#63243—24no—
55C23×C17C22⋊C411yescharacteristic
56C22×D17—24no—
57C22×D17—22no—
58C22×D17—22no—
59C22×D17—22no—
60C22×D17—41no—
61C22×D17—42no—
62C22×D17—42no—
63C22×D17C22⋊C411yescharacteristic
64C22×D17C2×D411yescharacteristic
65C2×#595—24no—
66C2×#595—42no—
67C2×#595—42no—
68Q8×C17—24no—
69Q8×C17—22no—
70Q8×C17C22⋊C414yescomplemented
71C2×C4×C17—24no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17C22⋊C411yescharacteristic
75C2×C4×C17C22×C411yescharacteristic
76C2×Dic17—22no—
77C2×Dic17C22⋊C411yescharacteristic
78C2×Dic17C2×D411yescharacteristic
79D68—28no—
80D68—24no—
81D68—28no—
82D68—22no—
83D68C22⋊C414yescomplemented
84C4×D17—28no—
85C4×D17—22no—
86C4×D17—22no—
87C4×D17—28no—
88C4×D17—22no—
89C4×D17C2×D414yes—
90#3624—42no—
91#1789—171nomaximal, Sylow
92C22×C17—21no—
93C22×C17—21no—
94C22×C17#10412yescomplemented
95C22×C17C2×C22⋊C411yescharacteristic
96C2×D17—21no—
97C2×D17—21no—
98C2×D17—21no—
99C2×D17—24no—
100C2×D17—21no—
101C2×D17—21no—
102C2×D17—44no—
103C2×D17—48no—
104C2×D17—42no—
105C2×D17—42no—
106C2×D17#12012yes—
107#595—44no—
108C4×C17—24no—
109C4×C17—21no—
110C4×C17C2×C22⋊C412yes—
111Dic17—21no—
112Dic17#12012yes—
113C22×#41—171no—
114C2×#104—174no—
115#407—171no—
116C4×C22⋊C4—171no—
117C2×C17—22no—
118C2×C17#37911yescharacteristic
119C2×C17C2×#10411yescharacteristic
120C2×C17#42111yescharacteristic
121D17—22no—
122D17—41no—
123D17—44no—
124C2×#41—172no—
125C2×#41—172no—
126C2×#41—341no—
127C2×#41—341no—
128C2×#41—348no—
129C22×Q8#59511yescharacteristic, complemented
130C22×D4—171no—
131C22×D4—341no—
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—171no—
143C24—341no—
144#41—178no—
145#41—344no—
146#41—344no—
147#41—348no—
148#41—348no—
149#41—344no—
150C2×Q8—24no—
151C2×Q8—21no—
152C2×Q8C2×#59512yes—
153C2×D4—172no—
154C2×D4—344no—
155C2×D4—342no—
156C2×D4—342no—
157C2×D4—348no—
158C2×D4—341no—
159C22×C4—21no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4—171no—
163C22×C4—171no—
164C22×C4—171no—
165C22×C4—344no—
166C22×C4—342no—
167C22×C4—342no—
168C22×C4—348no—
169C22×C4—342no—
170C22×C4C2×#59511yescharacteristic
171M16—344no—
172C2×C8—342no—
173C22⋊C4—344no—
174C42—344no—
175C23—171no—
176C23—171no—
177C23—342no—
178C23—344no—
179C23—342no—
180C23—342no—
181C23—681no—
182C23—682no—
183C23—682no—
184C23C22⋊#59511yescharacteristic
185Q8—24no—
186Q8—22no—
187Q8C22⋊#59514yescomplemented
188D4—174no—
189D4—348no—
190D4—348no—
191D4—344no—
192D4—342no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—171no—
197C2×C4—174no—
198C2×C4—171no—
199C2×C4—342no—
200C2×C4—348no—
201C2×C4—342no—
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—172no—
214C22—341no—
215C22—341no—
216C22—344no—
217C22—341no—
218C22—341no—
219C22—341no—
220C22—684no—
221C22—688no—
222C22—682no—
223C22—682no—
224C22#1058755012yescomplemented
225C22C2×C22⋊#59511yescharacteristic
226C4—24no—
227C4—21no—
228C4—172no—
229C4—341no—
230C4—684no—
231C4C2×C22⋊#59512yes—
232C2—22no—
233C2—342no—
234C2—681no—
235C2—684no—
236C2#1176375011yescharacteristic
237C2C2×#1058755011yescharacteristic
238C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC22×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×#63256C2
cosocleC2×C4×D17C23
hypercenterC22×Q8#595
hypocenterC17#1789
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10587550C2semidirect
2. #11763780C2semidirect
3. C4×C22⋊#595C2semidirect
5. C22×Q8×C17C4semidirect
9. C2×#63256C4semidirect
12. C22×D68C4semidirect
16. C2×C4×#595C22semidirect
17. C2×#63243C22semidirect
70. Q8×C17C22⋊C4semidirect
83. D68C22⋊C4semidirect
94. C22×C17#104semidirect
129. C22×Q8#595semidirect
141. C17#1789semidirect
187. Q8C22⋊#595semidirect
224. C22#10587550semidirect

Maximal subgroups

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

subgroupindex
C2×#105875502
#117637802
C4×C22⋊#5952
C22×#632562
#178917