Finite Group Databasebeta

← #14544771

Subgroups — #14544771

Order 2176.

456 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×C4⋊#595C211yesmaximal, characteristic, complemented
2C2×#10587547C212yesmaximal, complemented
3#11763777C211yesmaximal, characteristic, complemented
4#11763682C212yesmaximal, complemented
5#121×D17C211yesmaximal, characteristic
6C23×#595—21no—
7C2×D4×D17—21no—
8C2×D4×D17C2211yescharacteristic
9C22×C4×D17C2211yescharacteristic
10C2×C4⋊#595—21no—
11C2×C4⋊#595—24no—
12C2×C4⋊#595C2212yes—
13C2×#63242C2212yes—
14C17×#121C411yescharacteristic, complemented
15#10587625—21no—
16#10587624C411yescharacteristic, complemented
17#10587616C411yescharacteristic, complemented
18#10587581C411yescharacteristic, complemented
19C4⋊C4×D17C2211yescharacteristic
20#10587570—21no—
21C22⋊C4×D17—21no—
22C22⋊#3624—21no—
23#10587547—24no—
24#10587543—21no—
25C23×D17—21no—
26C23×D17D411yescharacteristic
27C22×#595—24no—
28C22×#595—22no—
29C22×#595—42no—
30C22×#595—42no—
31C2×D4×C17—21no—
32C2×D4×C17C2×C411yescharacteristic, complemented
33C22×C4×C17C2×C411yescharacteristic
34C2×C22⋊D17—21no—
35C2×C22⋊D17—21no—
36C2×C22⋊D17—21no—
37C22×Dic17D411yescharacteristic
38D4×D17—22no—
39D4×D17—22no—
40D4×D17—42no—
41D4×D17—42no—
42C2×D68—21no—
43C2×D68C2×C411yescharacteristic, complemented
44C2×C4×D17—22no—
45C2×C4×D17—22no—
46C2×C4×D17—21no—
47C2×C4×D17C2×C411yescharacteristic
48C2×C4×D17D411yescharacteristic
49C2×C4×D17D411yescharacteristic
50C2×C4×D17C2311yescharacteristic
51C2×#3624—22no—
52C4⋊#595—24no—
53C4⋊#595—22no—
54C4⋊#595D414yescomplemented
55#63242—22no—
56#63242—22no—
57C4⋊C4×C17C2×C411yescharacteristic, complemented
58C22⋊C4×C17—21no—
59C22⋊Dic17—21no—
60#63227—21no—
61#63227—21no—
62C4⋊Dic17C2×C411yescharacteristic, complemented
63#63225—21no—
64C23×C17—21no—
65C23×C17C22⋊C411yescharacteristic
66C22×D17—21no—
67C22×D17—21no—
68C22×D17—21no—
69C22×D17—21no—
70C22×D17—21no—
71C22×D17—41no—
72C22×D17—41no—
73C22×D17—41no—
74C22×D17—41no—
75C22×D17—41no—
76C22×D17—41no—
77C22×D17C22⋊C411yescharacteristic
78C22×D17D812yes—
79C22×D17SD1612yes—
80C22×D17C2×D411yescharacteristic
81C2×#595—24no—
82C2×#595—48no—
83C2×#595—42no—
84C2×#595—42no—
85D4×C17—22no—
86D4×C17—42no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17C22⋊C411yescharacteristic
90C2×C4×C17C22×C411yescharacteristic
91C22⋊D17—42no—
92C22⋊D17—42no—
93C22⋊D17—42no—
94C2×Dic17—22no—
95C2×Dic17—21no—
96C2×Dic17C22⋊C411yescharacteristic
97C2×Dic17C2×D411yescharacteristic
98D68—22no—
99D68—42no—
100C4×D17—21no—
101C4×D17—21no—
102C4×D17—22no—
103C4×D17—22no—
104C4×D17—41no—
105C4×D17—41no—
106C4×D17C2×D412yes—
107C4×D17#4112yes—
108#3624—42no—
109#1874—171nomaximal, Sylow
110C22×C17—21no—
111C22×C17—21no—
112C22×C17—41no—
113C22×C17—41no—
114C22×C17—41no—
115C22×C17#10212yescomplemented
116C22×C17C2×C22⋊C411yescharacteristic
117C2×D17—21no—
118C2×D17—21no—
119C2×D17—22no—
120C2×D17—22no—
121C2×D17—22no—
122C2×D17—22no—
123C2×D17—41no—
124C2×D17—41no—
125C2×D17—41no—
126C2×D17—81no—
127C2×D17—81no—
128C2×D17—81no—
129C2×D17—81no—
130C2×D17#10212yescomplemented
131C2×D17#12011yescharacteristic
132C2×D17#12311yescharacteristic
133C2×D17C2×D811yescharacteristic
134C2×D17C2×SD1611yescharacteristic
135C2×D17AGL(1,ℤ/8)11yescharacteristic
136C2×D17#13711yescharacteristic
137#595—44no—
138C4×C17—21no—
139C4×C17—41no—
140C4×C17C4×D412yes—
141Dic17—21no—
142Dic17—41no—
143Dic17#12112yes—
144C2×#121—171no—
145C22×C4⋊C4—171no—
146C2×#102—172no—
147C2×C22⋊C8—171no—
148#340—172no—
149C2×C17—22no—
150C2×C17—81no—
151C2×C17#38611yescharacteristic
152C2×C17C2×#10211yescharacteristic
153C2×C17#41711yescharacteristic
154D17—22no—
155D17—81no—
156D17#44711yescharacteristic
157D17#44811yescharacteristic
158D17#48011yescharacteristic
159D17#48111yescharacteristic
160C22×D4—171no—
161C22×D4—341no—
162C23×C4—171no—
163C23×C4—341no—
164C22×C8—172no—
165#121—171no—
166#121—171no—
167#121—171no—
168#121—341no—
169#121—341no—
170#121#59511yescharacteristic, complemented
171C2×C4⋊C4—172no—
172C2×C4⋊C4—171no—
173C2×C4⋊C4—344no—
174C2×C4⋊C4—341no—
175C2×C22⋊C4—341no—
176#102—344no—
177C22⋊C8—341no—
178C22⋊C8—341no—
179C17#187411yescharacteristic, complemented, Sylow
180C24—171no—
181C24—341no—
182C2×D4—21no—
183C2×D4—171no—
184C2×D4—341no—
185C2×D4—342no—
186C2×D4—342no—
187C2×D4—341no—
188C2×D4—341no—
189C2×D4—341no—
190C2×D4—682no—
191C2×D4—682no—
192C2×D4C2×#59511yescharacteristic, complemented
193C22×C4—171no—
194C22×C4—171no—
195C22×C4—171no—
196C22×C4—171no—
197C22×C4—171no—
198C22×C4—342no—
199C22×C4—342no—
200C22×C4—341no—
201C22×C4—342no—
202C22×C4—344no—
203C22×C4—682no—
204C22×C4—682no—
205C22×C4C2×#59511yescharacteristic
206C2×C8—342no—
207C2×C8—342no—
208C2×C8—342no—
209C4⋊C4—174no—
210C4⋊C4—171no—
211C4⋊C4—341no—
212C4⋊C4—342no—
213C4⋊C4—344no—
214C4⋊C4C2×#59511yescharacteristic, complemented
215C22⋊C4—21no—
216C22⋊C4—341no—
217C22⋊C4—341no—
218C22⋊C4—341no—
219C23—21no—
220C23—171no—
221C23—171no—
222C23—172no—
223C23—172no—
224C23—341no—
225C23—341no—
226C23—341no—
227C23—341no—
228C23—341no—
229C23—681no—
230C23—681no—
231C23—681no—
232C23—681no—
233C23—681no—
234C23—681no—
235C23C22⋊#59511yescharacteristic
236D4—22no—
237D4—42no—
238D4—342no—
239D4—682no—
240D4—682no—
241D4—682no—
242D4—682no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—171no—
246C2×C4—172no—
247C2×C4—172no—
248C2×C4—171no—
249C2×C4—342no—
250C2×C4—341no—
251C2×C4—341no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—344no—
255C2×C4—341no—
256C2×C4—681no—
257C2×C4—681no—
258C2×C4—682no—
259C2×C4—682no—
260C2×C4—688no—
261C2×C4C22⋊#59511yescharacteristic
262C2×C4C22×#59511yescharacteristic
263C8—682no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—171no—
270C22—171no—
271C22—171no—
272C22—171no—
273C22—171no—
274C22—171no—
275C22—172no—
276C22—342no—
277C22—342no—
278C22—342no—
279C22—342no—
280C22—341no—
281C22—341no—
282C22—681no—
283C22—681no—
284C22—681no—
285C22—1361no—
286C22—1361no—
287C22—1361no—
288C22—1361no—
289C22#1058754712yescomplemented
290C22C2×C22⋊#59511yescharacteristic
291C4—21no—
292C4—41no—
293C4—172no—
294C4—341no—
295C4—684no—
296C4—681no—
297C4D4×#59512yes—
298C2—22no—
299C2—81no—
300C2—171no—
301C2—171no—
302C2—171no—
303C2—171no—
304C2—342no—
305C2—1361no—
306C2#1176375711yescharacteristic
307C2C2×#1058754711yescharacteristic
308C2#1176380011yescharacteristic
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#1874
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊#595C2semidirect
2. C2×#10587547C2semidirect
3. #11763777C2semidirect
4. #11763682C2semidirect
14. C17×#121C4semidirect
16. #10587624C4semidirect
17. #10587616C4semidirect
18. #10587581C4semidirect
32. C2×D4×C17C2×C4semidirect
43. C2×D68C2×C4semidirect
54. C4⋊#595D4semidirect
57. C4⋊C4×C17C2×C4semidirect
62. C4⋊Dic17C2×C4semidirect
115. C22×C17#102semidirect
130. C2×D17#102semidirect
170. #121#595semidirect
179. C17#1874semidirect
192. C2×D4C2×#595semidirect
214. C4⋊C4C2×#595semidirect
289. C22#10587547semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#5952
C2×#105875472
#117637772
#117636822
#121×D172
#187417