Finite Group Databasebeta

← #14533271

Subgroups — #14533271

Order 2176.

596 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587570C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
3#11762736C211yesmaximal, characteristic, complemented
4#11762735C211yesmaximal, characteristic, complemented
5#11762734C211yesmaximal, characteristic, complemented
6#11762709C212yesmaximal, complemented
7#11762687C212yesmaximal, complemented
8#11762685C212yesmaximal, complemented
9#11762673C212yesmaximal, complemented
10#11762656C211yesmaximal, characteristic, complemented
11C17×#397C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×D68C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C22×C4×D17C2212yes—
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×#63225C2212yes—
27C2×#63225C2211yescharacteristic
28C2×C4×Dic17C2212yes—
29C4⋊C4×D17—24no—
30#10587570—24no—
31C17×#95C2211yescharacteristic
32C17×#95C2212yes—
33#10587502C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C23×D17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C2×C22⋊D17—22no—
45C2×C22⋊D17—24no—
46C2×C22⋊D17—22no—
47C2×C22⋊D17—24no—
48C22×Dic17C2312yes—
49C22×Dic17C2311yescharacteristic
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C2×D68—22no—
53C2×D68—24no—
54C2×C4×D17—24no—
55C2×C4×D17—24no—
56C2×C4×D17—42no—
57C2×C4×D17—42no—
58C2×C4×D17D414yes—
59C4⋊C4×C17—22no—
60C22⋊C4×C17—22no—
61C22⋊C4×C17—42no—
62C22⋊Dic17—42no—
63#63227—24no—
64#63227—22no—
65#63227—42no—
66#63227—42no—
67C4⋊Dic17—22no—
68#63225—22no—
69#63225—24no—
70C4×Dic17—24no—
71C23×C17—21no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—21no—
75C23×C17—21no—
76C23×C17C2411yescharacteristic
77C22×D17—22no—
78C22×D17—21no—
79C22×D17—21no—
80C22×D17—21no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—21no—
84C22×D17—21no—
85C22×D17C2×D412yes—
86C22×D17#4114yes—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17C2×D412yes—
97C22⋊D17—44no—
98C22⋊D17—44no—
99C2×Dic17—22no—
100C2×Dic17—21no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17C2×D412yes—
108C2×Dic17#4114yes—
109D68—44no—
110C4×D17—28no—
111C4×D17—48no—
112#2504—171nomaximal, Sylow
113C22×C17—41no—
114C22×C17—41no—
115C22×C17—41no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17C22×D411yescharacteristic
119C22×C17C2×#4112yes—
120C22×C17C2×#4112yes—
121C22×C17H(5,𝔽2)11yescharacteristic
122C22×C17H(5,𝔽2)11yescharacteristic
123C2×D17—22no—
124C2×D17—22no—
125C2×D17—24no—
126C2×D17—24no—
127C2×D17—41no—
128C2×D17—41no—
129C2×D17—41no—
130C2×D17—42no—
131C2×D17—42no—
132C2×D17#12414yes—
133C4×C17—22no—
134C4×C17—41no—
135C4×C17—42no—
136C4×C17—42no—
137Dic17—41no—
138Dic17—42no—
139Dic17—42no—
140Dic17#12314yes—
141C2×#121—171no—
142C22×C4⋊C4—171no—
143#397—171no—
144#397—171no—
145#397—171no—
146#397D1711yescharacteristic, complemented
147#390—172no—
148#386—172no—
149#386—172no—
150#382—172no—
151C4×C22⋊C4—171no—
152C2×C17—81no—
153C2×C17C2×#12311yescharacteristic
154C2×C17C2×#12411yescharacteristic
155C2×C17#53511yescharacteristic
156C2×C17#55312yes—
157C2×C17#55312yes—
158D17—24no—
159D17—81no—
160C22×D4—171no—
161C22×D4—171no—
162C22×D4—171no—
163C23×C4—171no—
164C23×C4—172no—
165#121—344no—
166C2×C4⋊C4—172no—
167C2×C4⋊C4—171no—
168C2×C4⋊C4—171no—
169C2×C4⋊C4—344no—
170C2×C4⋊C4C2×D1711yescharacteristic
171C2×C22⋊C4—171no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4C2×D1711yescharacteristic
177C2×C22⋊C4C2×D1712yes—
178C2×C42—172no—
179#95—171no—
180#95—172no—
181#95—172no—
182#95—172no—
183#95C2×D1711yescharacteristic
184#95C2×D1712yes—
185C17#250411yescharacteristic, complemented, Sylow
186C24—171no—
187C24—171no—
188C24C22×D1711yescharacteristic
189C2×D4—342no—
190C2×D4—344no—
191C2×D4—342no—
192C2×D4—344no—
193C2×D4—342no—
194C2×D4—344no—
195C22×C4—171no—
196C22×C4—174no—
197C22×C4—171no—
198C22×C4—172no—
199C22×C4—172no—
200C22×C4—344no—
201C22×C4—344no—
202C22×C4—682no—
203C22×C4—682no—
204C22×C4C22×D1711yescharacteristic
205C22×C4C22×D1712yes—
206C22×C4C22×D1712yes—
207C22×C4C22×D1711yescharacteristic
208C4⋊C4—22no—
209C4⋊C4—344no—
210C4⋊C4—342no—
211C4⋊C4—342no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—344no—
215C22⋊C4—342no—
216C22⋊C4—682no—
217C22⋊C4—682no—
218C22⋊C4—682no—
219C42—344no—
220C23—21no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—172no—
226C23—174no—
227C23—341no—
228C23—342no—
229C23—341no—
230C23—341no—
231C23—341no—
232C23—342no—
233C23—342no—
234C23—341no—
235C23C23×D1711yescharacteristic
236D4—684no—
237D4—684no—
238D4—684no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—172no—
249C2×C4—174no—
250C2×C4—348no—
251C2×C4—341no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—688no—
260C2×C4D4×D1712yes—
261C22—42no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—174no—
267C22—344no—
268C22—344no—
269C22—342no—
270C22—342no—
271C22—682no—
272C22—682no—
273C22—681no—
274C22—681no—
275C22—681no—
276C22C2×D4×D1711yescharacteristic
277C22#1058769111yescharacteristic
278C22#41×D1712yes—
279C22#41×D1712yes—
280C22#1058769611yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—42no—
285C4—174no—
286C4—682no—
287C4—681no—
288C4—682no—
289C2—81no—
290C2—344no—
291C2—1361no—
292C2#123×D1711yescharacteristic
293C2#124×D1711yescharacteristic
294C2#1176337511yescharacteristic
295C2#1176350412yes—
296C2#1176351012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC2×#10587570C2
hypercenter#397D17
hypocenterC17#2504
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D17C2semidirect
3. #11762736C2semidirect
4. #11762735C2semidirect
5. #11762734C2semidirect
6. #11762709C2semidirect
7. #11762687C2semidirect
8. #11762685C2semidirect
9. #11762673C2semidirect
10. #11762656C2semidirect
11. C17×#397C2semidirect
146. #397D17semidirect
185. C17#2504semidirect

Maximal subgroups

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

subgroupindex
C2×#105875702
C2×C4⋊C4×D172
#117627362
#117627352
#117627342
#117627092
#117626872
#117626852
#117626732
#117626562
C17×#3972
#250417