Finite Group Databasebeta

← #14534935

Subgroups — #14534935

Order 2176.

498 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 137 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×#10587615C211yesmaximal, characteristic
2#11762747C211yesmaximal, characteristic, complemented
3#11762744C211yesmaximal, characteristic, complemented
4#11762737C212yesmaximal, complemented
5#11762728C211yesmaximal, characteristic, complemented
6#11762724C212yesmaximal
7#11762720C211yesmaximal, characteristic, complemented
8#11762704C212yesmaximal, complemented
9#11762699C212yesmaximal, complemented
10C4×C22⋊Dic17C211yesmaximal, characteristic, complemented
11C2×C17×C22⋊Q8C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C17×C22⋊Q8—24no—
16C2×C4⋊C4×C17C2212yes—
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic
21#10587615—24no—
22C2×#63227C2211yescharacteristic
23C2×#63227C2212yes—
24C2×#63227C2211yescharacteristic
25C2×#63227C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2212yes—
28C2×#63225C2211yescharacteristic
29C2×#63225C2212yes—
30C2×C4×Dic17C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C2×Q8×C17—22no—
40C2×Q8×C17—42no—
41C22×C4×C17—42no—
42C22×C4×C17D414yes—
43C22×C4×C17C2312yes—
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C2×C22⋊D17—22no—
48C2×C22⋊D17—42no—
49C22×Dic17C2311yescharacteristic
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2312yes—
54C4⋊C4×C17—22no—
55C4⋊C4×C17—24no—
56C22⋊C4×C17—24no—
57C22⋊Dic17—22no—
58C22⋊Dic17—24no—
59#63227—22no—
60#63227—22no—
61#63227—24no—
62#63227—42no—
63#63225—22no—
64#63225—22no—
65#63225—24no—
66#63225—42no—
67C4×Dic17—42no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17—21no—
71C23×C17—21no—
72C23×C17C2×D412yes—
73C23×C17C2411yescharacteristic
74C22×D17—21no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—21no—
78C22×D17—21no—
79Q8×C17—44no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—44no—
89C2×C4×C17C2×D412yes—
90C2×C4×C17C2×D412yes—
91C22⋊D17—44no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—21no—
95C2×Dic17—21no—
96C2×Dic17—21no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17#4114yes—
105#2663—171nomaximal, Sylow
106C22×C17—22no—
107C22×C17—22no—
108C22×C17—22no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17C22×D411yescharacteristic
112C22×C17C2×#4112yes—
113C22×C17H(5,𝔽2)12yes—
114C22×C172−1+411yescharacteristic
115C22×C172−1+411yescharacteristic
116C2×D17—41no—
117C2×D17—41no—
118C2×D17—41no—
119C2×D17—42no—
120C2×D17—42no—
121C4×C17—22no—
122C4×C17—22no—
123C4×C17—42no—
124C4×C17—42no—
125Dic17—24no—
126Dic17—41no—
127Dic17—41no—
128Dic17—41no—
129Dic17—42no—
130C2×#123—171no—
131C2×C22⋊Q8D1711yescharacteristic, complemented
132#400—171no—
133#399—171no—
134#397—172no—
135#395—171no—
136#394—172no—
137#393—171no—
138#389—172no—
139#388—172no—
140C4×C22⋊C4—171no—
141C2×C17—42no—
142C2×C17#53211yescharacteristic
143C2×C17#53411yescharacteristic
144C2×C17#53611yescharacteristic
145C2×C17#53912yes—
146C2×C17#54212yes—
147D17—81no—
148C22×Q8C2×D1711yescharacteristic
149C22×D4—171no—
150C23×C4C2×D1711yescharacteristic
151#123—344no—
152C22⋊Q8—24no—
153C2×C4⋊C4—171no—
154C2×C4⋊C4—171no—
155C2×C4⋊C4—172no—
156C2×C4⋊C4—172no—
157C2×C4⋊C4C2×D1712yes—
158C2×C4⋊C4C2×D1711yescharacteristic
159C2×C22⋊C4—171no—
160C2×C22⋊C4—172no—
161C2×C22⋊C4—171no—
162C2×C22⋊C4—171no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4C2×D1712yes—
166C2×C42—172no—
167#95—171no—
168#95—171no—
169#95—172no—
170#95—172no—
171#95—172no—
172#95—172no—
173C17#266311yescharacteristic, complemented, Sylow
174C24—171no—
175C24C22×D1711yescharacteristic
176C2×Q8—22no—
177C2×Q8—42no—
178C2×D4—342no—
179C2×D4—682no—
180C22×C4—42no—
181C22×C4—172no—
182C22×C4—172no—
183C22×C4—171no—
184C22×C4—171no—
185C22×C4—171no—
186C22×C4C22⋊D1714yes—
187C22×C4C22×D1712yes—
188C22×C4C22×D1712yes—
189C22×C4C22×D1711yescharacteristic
190C22×C4C22×D1711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—344no—
194C4⋊C4—342no—
195C4⋊C4—342no—
196C4⋊C4—682no—
197C22⋊C4—24no—
198C22⋊C4—344no—
199C22⋊C4—344no—
200C22⋊C4—342no—
201C22⋊C4—342no—
202C22⋊C4—342no—
203C22⋊C4—682no—
204C42—682no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—342no—
210C23—341no—
211C23—341no—
212C23—342no—
213C23—341no—
214C23C2×C22⋊D1712yes—
215C23C23×D1711yescharacteristic
216Q8—44no—
217D4—684no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—174no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4—341no—
233C2×C4—341no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—341no—
238C2×C4—341no—
239C2×C4—341no—
240C2×C4C2×C22⋊D1712yes—
241C2×C4C2×C22⋊D1712yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—681no—
248C22—681no—
249C22—681no—
250C22—682no—
251C22—682no—
252C22#1058769112yes—
253C22#1058769411yescharacteristic
254C22#1058769411yescharacteristic
255C22#41×D1712yes—
256C22C22×C22⋊D1711yescharacteristic
257C4—22no—
258C4—22no—
259C4—42no—
260C4—42no—
261C4—344no—
262C4—681no—
263C4—681no—
264C4—681no—
265C4—682no—
266C2—42no—
267C2—1361no—
268C2#1176335511yescharacteristic
269C2#1176336811yescharacteristic
270C2#1176338311yescharacteristic
271C2#1176340512yes—
272C2#1176342912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×C22⋊Q8C2
prime radicalC22×C22⋊D17C22
hypercenterC2×C22⋊Q8D17
hypocenterC17#2663
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11762747C2semidirect
3. #11762744C2semidirect
4. #11762737C2semidirect
5. #11762728C2semidirect
7. #11762720C2semidirect
8. #11762704C2semidirect
9. #11762699C2semidirect
10. C4×C22⋊Dic17C2semidirect
11. C2×C17×C22⋊Q8C2semidirect
131. C2×C22⋊Q8D17semidirect
173. C17#2663semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
#117627472
#117627442
#117627372
#117627282
#117627242
#117627202
#117627042
#117626992
C4×C22⋊Dic172
C2×C17×C22⋊Q82
#266317