Finite Group Databasebeta

← #14534260

Subgroups — #14534260

Order 2176.

596 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 139 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×#10587581C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
3#11762747C211yesmaximal, characteristic, complemented
4#11762727C211yesmaximal, characteristic, complemented
5#11762726C211yesmaximal, characteristic, complemented
6#11762701C212yesmaximal, complemented
7#11762698C212yesmaximal, complemented
8#11762687C212yesmaximal, complemented
9#11762687C212yesmaximal, complemented
10#11762656C211yesmaximal, characteristic, complemented
11C17×#400C211yesmaximal, characteristic, complemented
12C22×D68C2212yes—
13C22×D68C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C22×C4×D17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×#63225C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×C4×Dic17C2212yes—
29#10587581—24no—
30C4⋊C4×D17—24no—
31C17×#95C2212yes—
32C17×#95C2212yes—
33C17×#95C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C23×D17C2312yes—
38C23×D17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2311yescharacteristic
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C2×D68—24no—
49C2×D68—24no—
50C2×D68—24no—
51C2×D68—22no—
52C2×D68—42no—
53C2×C4×D17—22no—
54C2×C4×D17—22no—
55C2×C4×D17—22no—
56C2×C4×D17—22no—
57C2×C4×D17—42no—
58C2×C4×D17—42no—
59C2×C4×D17D414yes—
60C4⋊C4×C17—22no—
61C4⋊C4×C17—22no—
62#63227—24no—
63#63227—24no—
64#63227—24no—
65#63227—24no—
66#63227—42no—
67#63227—42no—
68C4⋊Dic17—22no—
69#63225—22no—
70#63225—22no—
71C4×Dic17—42no—
72C23×C17C2411yescharacteristic
73C22×D17—21no—
74C22×D17—21no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17C2×D412yes—
84C22×D17#4114yes—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
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—
97C2×C4×C17#4112yes—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17C2×D412yes—
107C2×Dic17#4112yes—
108D68—44no—
109D68—48no—
110C4×D17—28no—
111C4×D17—44no—
112C4×D17—44no—
113#2591—171nomaximal, Sylow
114C22×C17C22×D411yescharacteristic
115C22×C17C2×#4111yescharacteristic
116C22×C17C2×#4111yescharacteristic
117C22×C17C2×#4112yes—
118C22×C17H(5,𝔽2)12yes—
119C2×D17—22no—
120C2×D17—22no—
121C2×D17—24no—
122C2×D17—24no—
123C2×D17—42no—
124C2×D17—42no—
125C2×D17—42no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17—42no—
130C2×D17#12414yes—
131C4×C17—22no—
132C4×C17—22no—
133C4×C17—42no—
134C4×C17—41no—
135C4×C17—42no—
136Dic17—22no—
137Dic17—22no—
138Dic17—41no—
139Dic17—42no—
140C2×#121—171no—
141C22×C4⋊C4—171no—
142#400—171no—
143#400D1711yescharacteristic, complemented
144#394—171no—
145#394—171no—
146#388—172no—
147#388—172no—
148#386—172no—
149#386—172no—
150C4×C22⋊C4—171no—
151C2×C17#52511yescharacteristic
152C2×C17C2×#12411yescharacteristic
153C2×C17#53411yescharacteristic
154C2×C17#55312yes—
155C2×C17#55312yes—
156D17—24no—
157D17—82no—
158C22×D4—172no—
159C22×D4—171no—
160C23×C4—171no—
161C23×C4—171no—
162C23×C4—171no—
163#121—344no—
164C2×C4⋊C4—171no—
165C2×C4⋊C4—171no—
166C2×C4⋊C4—171no—
167C2×C4⋊C4—344no—
168C2×C4⋊C4C2×D1711yescharacteristic
169C2×C4⋊C4C2×D1711yescharacteristic
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—172no—
176C2×C42—172no—
177#95—171no—
178#95—172no—
179#95—172no—
180#95C2×D1711yescharacteristic
181#95C2×D1712yes—
182#95C2×D1712yes—
183C17#259111yescharacteristic, complemented, Sylow
184C24—171no—
185C24—172no—
186C2×D4—344no—
187C2×D4—344no—
188C2×D4—344no—
189C2×D4—342no—
190C2×D4—682no—
191C22×C4—171no—
192C22×C4—171no—
193C22×C4—174no—
194C22×C4—172no—
195C22×C4—171no—
196C22×C4—342no—
197C22×C4—342no—
198C22×C4—342no—
199C22×C4—342no—
200C22×C4—682no—
201C22×C4—682no—
202C22×C4C22×D1712yes—
203C22×C4C22×D1712yes—
204C22×C4C22×D1711yescharacteristic
205C22×C4C22×D1711yescharacteristic
206C22×C4C22×D1711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—342no—
210C4⋊C4—342no—
211C4⋊C4—342no—
212C22⋊C4—344no—
213C22⋊C4—344no—
214C22⋊C4—344no—
215C22⋊C4—344no—
216C22⋊C4—682no—
217C22⋊C4—682no—
218C42—682no—
219C23—172no—
220C23—174no—
221C23—341no—
222C23—341no—
223C23—342no—
224C23—342no—
225C23—342no—
226C23—342no—
227C23—342no—
228C23—342no—
229C23—342no—
230C23—342no—
231C23C23×D1711yescharacteristic
232D4—684no—
233D4—688no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—172no—
246C2×C4—172no—
247C2×C4—348no—
248C2×C4—341no—
249C2×C4—341no—
250C2×C4—342no—
251C2×C4—341no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—684no—
257C2×C4—684no—
258C2×C4D4×D1712yes—
259C2×C4#6325612yes—
260C22—174no—
261C22—344no—
262C22—344no—
263C22—342no—
264C22—342no—
265C22—682no—
266C22—682no—
267C22—682no—
268C22—682no—
269C22—682no—
270C22—682no—
271C22—682no—
272C22C2×D4×D1711yescharacteristic
273C22C2×#6325611yescharacteristic
274C22#41×D1711yescharacteristic
275C22#41×D1712yes—
276C22#1058769612yes—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—41no—
282C4—342no—
283C4—342no—
284C4—682no—
285C4—681no—
286C2—344no—
287C2—1362no—
288C2#1176332211yescharacteristic
289C2#124×D1711yescharacteristic
290C2#1176337011yescharacteristic
291C2#1176351012yes—
292C2#1176351212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#400C2
prime radicalC2×#10587581C2
hypercenter#400D17
hypocenterC17#2591
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D17C2semidirect
3. #11762747C2semidirect
4. #11762727C2semidirect
5. #11762726C2semidirect
6. #11762701C2semidirect
7. #11762698C2semidirect
8. #11762687C2semidirect
9. #11762687C2semidirect
10. #11762656C2semidirect
11. C17×#400C2semidirect
143. #400D17semidirect
183. C17#2591semidirect

Maximal subgroups

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

subgroupindex
C2×#105875812
C2×C4⋊C4×D172
#117627472
#117627272
#117627262
#117627012
#117626982
#117626872
#117626562
C17×#4002
#259117