Finite Group Databasebeta

← #14540151

Subgroups — #14540151

Order 2176.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#11763343C211yesmaximal, characteristic
2#11763116C212yesmaximal, complemented
3#11763101C211yesmaximal, characteristic, complemented
4#11763096C211yesmaximal, characteristic, complemented
5#11763065C212yesmaximal, complemented
6#11763057C212yesmaximal
7#11763031C212yesmaximal, complemented
8#11763016C211yesmaximal, characteristic, complemented
9#11763011C211yesmaximal, characteristic, complemented
10#11762833C211yesmaximal, characteristic, complemented
11C17×#533C211yesmaximal, characteristic, complemented
12C17×C4⋊Q8C2211yescharacteristic
13C17×C4⋊Q8C2211yescharacteristic
14C17×#125—21no—
15C17×C22⋊Q8—21no—
16C17×C22⋊Q8C2212yes—
17C4×Q8×C17—21no—
18C4×Q8×C17C2212yes—
19C17×#117—21no—
20C2×C42×C17C2211yescharacteristic
21C2×#63231C2212yes—
22C2×#63230C2212yes—
23#10587616C2211yescharacteristic
24#10587615—21no—
25C4×C22⋊D17—21no—
26#10587610C2211yescharacteristic
27C2×#63223C2212yes—
28#10587562—21no—
29#10587561C2211yescharacteristic
30C4×D68C2211yescharacteristic
31#10587558—21no—
32#10587556C2211yescharacteristic
33C4×Dic34C2211yescharacteristic
34#10587506C2212yes—
35#10587506C2212yes—
36#10587481C2212yes—
37#10587480C2212yes—
38#10587479C2212yes—
39#10587478C2212yes—
40C4⋊#3622C2211yescharacteristic
41C4⋊#3622C2211yescharacteristic
42C4⋊#3622C2212yes—
43C2×Q8×C17—21no—
44C2×Q8×C17C2312yes—
45C22×C4×C17D411yescharacteristic
46C22×C4×C17D411yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—21no—
49C2×D68C2311yescharacteristic
50C2×C4×D17—21no—
51C2×Dic34C2311yescharacteristic
52C4⋊C4×C17—22no—
53C4⋊C4×C17—21no—
54C4⋊C4×C17—22no—
55C4⋊C4×C17—21no—
56C4⋊C4×C17—21no—
57C4⋊C4×C17C2312yes—
58C22⋊C4×C17—22no—
59C42×C17—22no—
60C42×C17D411yescharacteristic
61C42×C17D411yescharacteristic
62C42×C17C2311yescharacteristic
63C42×C17C2311yescharacteristic
64C22⋊Dic17—21no—
65#63231—42no—
66#63230—42no—
67#63227—21no—
68#63227—21no—
69#63227—21no—
70#63227—21no—
71C4⋊Dic17C2311yescharacteristic
72C4⋊Dic17C2311yescharacteristic
73#63225—21no—
74#63225—21no—
75#63225—21no—
76#63225—21no—
77C4×Dic17—21no—
78#63223—42no—
79C2×#3622C2312yes—
80C2×#3622C2312yes—
81C23×C17C2×D411yescharacteristic
82C22×D17—21no—
83Q8×C17—22no—
84Q8×C17—22no—
85Q8×C17—42no—
86C2×C4×C17—22no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—41no—
92C2×C4×C17C2×D411yescharacteristic
93C2×C4×C17C2×D411yescharacteristic
94C2×C4×C17C2×D411yescharacteristic
95C2×C4×C17C2×D411yescharacteristic
96C2×C4×C17C2×D411yescharacteristic
97C2×C4×C17C2411yescharacteristic
98C22⋊D17—42no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—21no—
102D68—21no—
103D68—21no—
104C4×D17—42no—
105Dic34—21no—
106Dic34—21no—
107#3622—22no—
108#3622—22no—
109#3252—171nomaximal, Sylow
110C22×C17—21no—
111C22×C17—21no—
112C22×C17—21no—
113C22×C17C22×D411yescharacteristic
114C2×D17—41no—
115C2×D17—41no—
116C2×D17—41no—
117C4×C17—21no—
118C4×C17—21no—
119C4×C17—21no—
120C4×C17—21no—
121C4×C17—42no—
122C4×C17—42no—
123C4×C17#13712yes—
124C4×C17H(5,𝔽2)11yescharacteristic
125C4×C172−1+411yescharacteristic
126Dic17—41no—
127Dic17—41no—
128Dic17—41no—
129#533D1711yescharacteristic, complemented
130#529—171no—
131#479—172no—
132#477—171no—
133#476—171no—
134#470—172no—
135#468—172no—
136#464—172no—
137#462—171no—
138#461—171no—
139#423—171no—
140C2×C17—41no—
141C2×C17#53711yescharacteristic
142C2×C17C2×#13711yescharacteristic
143C2×C17#57511yescharacteristic
144D17—81no—
145C2×Q16—172no—
146C2×SD16—172no—
147C2×M16—172no—
148C4⋊Q8C2×D1711yescharacteristic
149C4⋊Q8C2×D1711yescharacteristic
150#126—341no—
151#125—21no—
152#125—171no—
153#124—171no—
154#123—341no—
155C22⋊Q8—21no—
156C22⋊Q8—171no—
157C22⋊Q8C2×D1712yes—
158#121—171no—
159C4×Q8—21no—
160C4×Q8—171no—
161C4×Q8C2×D1712yes—
162C4×D4—171no—
163C4×D4—341no—
164#117—21no—
165#117—341no—
166C2×C42C2×D1711yescharacteristic
167#107—172no—
168#106—172no—
169C4⋊C8—171no—
170C4⋊C8—171no—
171C4⋊C8—172no—
172#103—172no—
173#103—172no—
174#103—172no—
175#102—172no—
176C17#325211yescharacteristic, complemented, Sylow
177C2×Q8—21no—
178C2×Q8—171no—
179C2×Q8C22×D1712yes—
180C2×D4—171no—
181C2×D4—341no—
182C22×C4—341no—
183C22×C4C22⋊D1711yescharacteristic
184C22×C4C22⋊D1711yescharacteristic
185C22×C4C22×D1711yescharacteristic
186Q16—682no—
187SD16—682no—
188M16—682no—
189C2×C8—172no—
190C2×C8—172no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—171no—
197C4⋊C4—171no—
198C4⋊C4—341no—
199C4⋊C4—341no—
200C4⋊C4—341no—
201C4⋊C4—341no—
202C4⋊C4C22×D1712yes—
203C22⋊C4—22no—
204C22⋊C4—341no—
205C22⋊C4—341no—
206C22⋊C4—341no—
207C22⋊C4—341no—
208C22⋊C4—341no—
209C42—22no—
210C42—341no—
211C42C22⋊D1711yescharacteristic
212C42C22⋊D1711yescharacteristic
213C42C22×D1711yescharacteristic
214C42C22×D1711yescharacteristic
215C23—341no—
216C23C2×C22⋊D1711yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—42no—
220Q8—341no—
221Q8—341no—
222D4—341no—
223D4—341no—
224D4—682no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—341no—
232C2×C4—341no—
233C2×C4—341no—
234C2×C4—682no—
235C2×C4C2×C22⋊D1711yescharacteristic
236C2×C4C2×C22⋊D1711yescharacteristic
237C2×C4C2×C22⋊D1711yescharacteristic
238C2×C4C2×C22⋊D1711yescharacteristic
239C2×C4C2×C22⋊D1711yescharacteristic
240C2×C4C23×D1711yescharacteristic
241C8—342no—
242C8—342no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—681no—
247C22—681no—
248C22—681no—
249C22C22×C22⋊D1711yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—681no—
257C4—681no—
258C4—681no—
259C4#1058762812yes—
260C4#1058769611yescharacteristic
261C4#1058769711yescharacteristic
262C2—41no—
263C2—1361no—
264C2#1176338711yescharacteristic
265C2C2×#1058762811yescharacteristic
266C2#1176360711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#533C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenter#533D17
hypocenterC17#3252
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #11763116C2semidirect
3. #11763101C2semidirect
4. #11763096C2semidirect
5. #11763065C2semidirect
7. #11763031C2semidirect
8. #11763016C2semidirect
9. #11763011C2semidirect
10. #11762833C2semidirect
11. C17×#533C2semidirect
129. #533D17semidirect
176. C17#3252semidirect

Maximal subgroups

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

subgroupindex
#117633432
#117631162
#117631012
#117630962
#117630652
#117630572
#117630312
#117630162
#117630112
#117628332
C17×#5332
#325217