Finite Group Databasebeta

← #14534579

Subgroups — #14534579

Order 2176.

548 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587578C211yesmaximal, characteristic, complemented
3#11762746C211yesmaximal, characteristic, complemented
4#11762745C211yesmaximal, characteristic, complemented
5#11762735C211yesmaximal, characteristic, complemented
6#11762727C212yesmaximal, complemented
7#11762725C212yesmaximal, complemented
8#11762709C212yesmaximal, complemented
9#11762706C212yesmaximal, complemented
10#11762668C211yesmaximal, characteristic, complemented
11C17×#397C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×D68C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2212yes—
27C2×#63225C2211yescharacteristic
28C2×#63225C2212yes—
29C2×#63225C2212yes—
30C2×C4×Dic17C2212yes—
31#10587578—24no—
32#10587570—24no—
33C17×#95C2212yes—
34C17×#95C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C23×D17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C2×C22⋊D17—22no—
45C2×C22⋊D17—22no—
46C2×C22⋊D17—24no—
47C2×C22⋊D17—42no—
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C2×D68—22no—
53C2×D68—42no—
54C2×C4×D17—24no—
55C2×C4×D17D414yes—
56C4⋊C4×C17—22no—
57C22⋊C4×C17—22no—
58C22⋊C4×C17—42no—
59C22⋊Dic17—42no—
60#63227—24no—
61#63227—22no—
62#63227—42no—
63#63227—42no—
64#63227—42no—
65C4⋊Dic17—22no—
66C4⋊Dic17—42no—
67#63225—22no—
68#63225—24no—
69#63225—42no—
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—22no—
79C22×D17—21no—
80C22×D17—21no—
81C22×D17—21no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—21no—
85C22×D17—21no—
86C22×D17C2×D412yes—
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—21no—
100C2×Dic17—22no—
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—
111#2624—171nomaximal, Sylow
112C22×C17—41no—
113C22×C17—41no—
114C22×C17—41no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17C22×D411yescharacteristic
118C22×C17C2×#4112yes—
119C22×C17H(5,𝔽2)11yescharacteristic
120C22×C17H(5,𝔽2)12yes—
121C22×C172−1+411yescharacteristic
122C2×D17—22no—
123C2×D17—22no—
124C2×D17—22no—
125C2×D17—42no—
126C2×D17—42no—
127C2×D17—41no—
128C2×D17—41no—
129C2×D17—41no—
130C2×D17—42no—
131C2×D17—42no—
132C4×C17—22no—
133C4×C17—41no—
134C4×C17—42no—
135C4×C17—42no—
136Dic17—42no—
137Dic17—41no—
138Dic17—42no—
139Dic17#12314yes—
140C2×#121—171no—
141C2×#117—171no—
142#399—171no—
143#399—171no—
144#397—171no—
145#397D1711yescharacteristic, complemented
146#394—172no—
147#394—172no—
148#390—172no—
149#389—172no—
150#380—171no—
151C2×C17—81no—
152C2×C17C2×#12311yescharacteristic
153C2×C17#53511yescharacteristic
154C2×C17#53711yescharacteristic
155C2×C17#54012yes—
156C2×C17#54212yes—
157D17—42no—
158D17—81no—
159C22×D4—171no—
160C22×D4—171no—
161C22×D4—171no—
162C23×C4—171no—
163#121—344no—
164#117—344no—
165C2×C4⋊C4—171no—
166C2×C4⋊C4—171no—
167C2×C4⋊C4—172no—
168C2×C4⋊C4—172no—
169C2×C4⋊C4—172no—
170C2×C4⋊C4C2×D1711yescharacteristic
171C2×C22⋊C4—172no—
172C2×C22⋊C4—171no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4C2×D1711yescharacteristic
178C2×C22⋊C4C2×D1712yes—
179C2×C42—172no—
180#95—171no—
181#95—172no—
182#95C2×D1711yescharacteristic
183#95C2×D1712yes—
184C17#262411yescharacteristic, complemented, Sylow
185C24—171no—
186C24—171no—
187C24C22×D1711yescharacteristic
188C2×D4—342no—
189C2×D4—342no—
190C2×D4—344no—
191C2×D4—342no—
192C2×D4—682no—
193C2×D4—682no—
194C22×C4—171no—
195C22×C4—174no—
196C22×C4—172no—
197C22×C4—171no—
198C22×C4—172no—
199C22×C4—344no—
200C22×C4C22×D1712yes—
201C22×C4C22×D1712yes—
202C22×C4C22×D1711yescharacteristic
203C22×C4C22×D1711yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—342no—
206C4⋊C4—344no—
207C4⋊C4—342no—
208C4⋊C4—682no—
209C4⋊C4—682no—
210C22⋊C4—22no—
211C22⋊C4—42no—
212C22⋊C4—344no—
213C22⋊C4—342no—
214C22⋊C4—682no—
215C22⋊C4—682no—
216C22⋊C4—682no—
217C22⋊C4—682no—
218C42—344no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—172no—
225C23—341no—
226C23—342no—
227C23—342no—
228C23—341no—
229C23—341no—
230C23—341no—
231C23—342no—
232C23—342no—
233C23—341no—
234C23C23×D1711yescharacteristic
235D4—684no—
236D4—684no—
237D4—684no—
238C2×C4—22no—
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—172no—
248C2×C4—174no—
249C2×C4—348no—
250C2×C4—342no—
251C2×C4—341no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4D4×D1712yes—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—42no—
264C22—342no—
265C22—342no—
266C22—342no—
267C22—682no—
268C22—682no—
269C22—681no—
270C22—681no—
271C22—681no—
272C22—682no—
273C22—682no—
274C22C2×D4×D1711yescharacteristic
275C22#1058769111yescharacteristic
276C22#41×D1712yes—
277C22#1058769612yes—
278C22#1058769711yescharacteristic
279C4—22no—
280C4—41no—
281C4—42no—
282C4—42no—
283C4—174no—
284C4—681no—
285C4—682no—
286C4—682no—
287C2—81no—
288C2—682no—
289C2—1361no—
290C2#123×D1711yescharacteristic
291C2#1176337711yescharacteristic
292C2#1176338811yescharacteristic
293C2#1176341112yes—
294C2#1176342512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC2×#10587570C2
hypercenter#397D17
hypocenterC17#2624
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#10587578C2semidirect
3. #11762746C2semidirect
4. #11762745C2semidirect
5. #11762735C2semidirect
6. #11762727C2semidirect
7. #11762725C2semidirect
8. #11762709C2semidirect
9. #11762706C2semidirect
10. #11762668C2semidirect
11. C17×#397C2semidirect
145. #397D17semidirect
184. C17#2624semidirect

Maximal subgroups

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

subgroupindex
C2×#105875702
C2×#105875782
#117627462
#117627452
#117627352
#117627272
#117627252
#117627092
#117627062
#117626682
C17×#3972
#262417