Finite Group Databasebeta

← #14536813

Subgroups — #14536813

Order 2176.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#10587583C211yesmaximal, characteristic, complemented
2C2×#10587570C211yesmaximal, characteristic
3#11762744C211yesmaximal, characteristic, complemented
4#11762742C212yesmaximal, complemented
5#11762726C212yesmaximal, complemented
6#11762725C212yesmaximal, complemented
7#11762724C212yesmaximal, complemented
8#11762722C211yesmaximal, characteristic, complemented
9#11762721C211yesmaximal, characteristic, complemented
10#11762671C211yesmaximal, characteristic, complemented
11C17×#399C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×D68C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C22×Dic34C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2212yes—
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×#63227C2211yescharacteristic
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×C4⋊Dic17C2212yes—
28C2×#63225C2211yescharacteristic
29C2×#63225C2212yes—
30#10587583—24no—
31#10587570—24no—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
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—42no—
47C2×C22⋊D17—42no—
48C22×Dic17C2312yes—
49C22×Dic17C2311yescharacteristic
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C2×D68—22no—
53C2×D68—42no—
54C2×C4×D17—42no—
55C2×C4×D17D414yes—
56C2×Dic34—22no—
57C2×Dic34—42no—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—42no—
60C22⋊C4×C17—22no—
61C22⋊C4×C17—42no—
62C22⋊Dic17—42no—
63#63227—24no—
64#63227—22no—
65#63227—42no—
66#63227—42no—
67#63227—42no—
68C4⋊Dic17—24no—
69#63225—22no—
70#63225—42no—
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—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17C2×D412yes—
109D68—44no—
110C4×D17—44no—
111Dic34—44no—
112#2816—171nomaximal, Sylow
113C22×C17—41no—
114C22×C17—41no—
115C22×C17—41no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17C22×D411yescharacteristic
119C22×C17H(5,𝔽2)12yes—
120C22×C17H(5,𝔽2)12yes—
121C22×C172−1+411yescharacteristic
122C22×C172−1+411yescharacteristic
123C2×D17—22no—
124C2×D17—22no—
125C2×D17—22no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—41no—
129C2×D17—41no—
130C2×D17—41no—
131C2×D17—42no—
132C2×D17—42no—
133C4×C17—22no—
134C4×C17—41no—
135C4×C17—42no—
136C4×C17—42no—
137Dic17—22no—
138Dic17—42no—
139Dic17—41no—
140Dic17—42no—
141C2×C22⋊Q8—171no—
142C2×#121—171no—
143#399—171no—
144#399D1711yescharacteristic, complemented
145#398—172no—
146#394—172no—
147#394—172no—
148#394—172no—
149#393—171no—
150#393—171no—
151#381—171no—
152C2×C17—81no—
153C2×C17#53411yescharacteristic
154C2×C17#53511yescharacteristic
155C2×C17#53611yescharacteristic
156C2×C17#56212yes—
157C2×C17#56212yes—
158D17—42no—
159D17—81no—
160C22×Q8—171no—
161C22×D4—171no—
162C22×D4—171no—
163C22×D4—171no—
164C23×C4—171no—
165C22⋊Q8—344no—
166#121—344no—
167C2×C4⋊C4—171no—
168C2×C4⋊C4—172no—
169C2×C4⋊C4—172no—
170C2×C4⋊C4C2×D1711yescharacteristic
171C2×C4⋊C4C2×D1712yes—
172C2×C22⋊C4—171no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4C2×D1711yescharacteristic
179C2×C22⋊C4C2×D1712yes—
180#95—171no—
181#95—172no—
182#95—172no—
183#95—172no—
184#95C2×D1711yescharacteristic
185C17#281611yescharacteristic, complemented, Sylow
186C24—171no—
187C24—171no—
188C24C22×D1711yescharacteristic
189C2×Q8—342no—
190C2×Q8—682no—
191C2×D4—342no—
192C2×D4—342no—
193C2×D4—342no—
194C2×D4—682no—
195C2×D4—682no—
196C2×D4—682no—
197C22×C4—171no—
198C22×C4—174no—
199C22×C4—172no—
200C22×C4—172no—
201C22×C4—171no—
202C22×C4—682no—
203C22×C4C22×D1712yes—
204C22×C4C22×D1711yescharacteristic
205C22×C4C22×D1712yes—
206C22×C4C22×D1711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—342no—
210C4⋊C4—344no—
211C4⋊C4—682no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—344no—
215C22⋊C4—342no—
216C22⋊C4—682no—
217C22⋊C4—682no—
218C22⋊C4—682no—
219C22⋊C4—682no—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—172no—
226C23—342no—
227C23—341no—
228C23—342no—
229C23—341no—
230C23—341no—
231C23—342no—
232C23—341no—
233C23—342no—
234C23—341no—
235C23C23×D1711yescharacteristic
236Q8—684no—
237D4—684no—
238D4—684no—
239D4—684no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—172no—
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×C4—342no—
259C2×C4—684no—
260C2×C4D4×D1712yes—
261C22—42no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—342no—
267C22—342no—
268C22—342no—
269C22—682no—
270C22—682no—
271C22—682no—
272C22—682no—
273C22—681no—
274C22—681no—
275C22—681no—
276C22C2×D4×D1711yescharacteristic
277C22#1058769112yes—
278C22#1058769411yescharacteristic
279C22#1058769612yes—
280C22#1058769711yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—42no—
285C4—342no—
286C4—682no—
287C4—682no—
288C4—681no—
289C2—81no—
290C2—682no—
291C2—1361no—
292C2#1176337011yescharacteristic
293C2#1176337211yescharacteristic
294C2#1176338211yescharacteristic
295C2#1176355512yes—
296C2#1176355612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
prime radicalC2×#10587570C2
hypercenter#399D17
hypocenterC17#2816
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#10587583C2semidirect
3. #11762744C2semidirect
4. #11762742C2semidirect
5. #11762726C2semidirect
6. #11762725C2semidirect
7. #11762724C2semidirect
8. #11762722C2semidirect
9. #11762721C2semidirect
10. #11762671C2semidirect
11. C17×#399C2semidirect
144. #399D17semidirect
185. C17#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
C2×#105875702
#117627442
#117627422
#117627262
#117627252
#117627242
#117627222
#117627212
#117626712
C17×#3992
#281617