Finite Group Databasebeta

← #14539487

Subgroups — #14539487

Order 2176.

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

normal subgroups only

This group has 366 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
1#11763343C211yesmaximal, characteristic
2#11763215C211yesmaximal, characteristic, complemented
3#11763201C211yesmaximal, characteristic, complemented
4#11763177C211yesmaximal, characteristic, complemented
5#11763171C211yesmaximal, characteristic, complemented
6#11763048C212yesmaximal, complemented
7#11763040C212yesmaximal
8#11762914C212yesmaximal, complemented
9#11762907C212yesmaximal, complemented
10C2×#10587474C211yesmaximal, 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
27C22×#3622C2212yes—
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—
40#10587474C2211yescharacteristic
41#10587474C2212yes—
42#10587474C2211yescharacteristic
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—24no—
66#63230—24no—
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—
78C2×#3622—24no—
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—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—41no—
91C2×C4×C17C2×D411yescharacteristic
92C2×C4×C17C2×D411yescharacteristic
93C2×C4×C17C2×D411yescharacteristic
94C2×C4×C17C2×D411yescharacteristic
95C2×C4×C17C2×D411yescharacteristic
96C2×C4×C17C2411yescharacteristic
97C22⋊D17—42no—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101D68—22no—
102C4×D17—42no—
103Dic34—22no—
104#3622—22no—
105#3622#4114yes—
106#3181—171nomaximal, Sylow
107C22×C17—21no—
108C22×C17—21no—
109C22×C17#13712yes—
110C22×C17C22×D411yescharacteristic
111C2×D17—41no—
112C2×D17—41no—
113C2×D17—41no—
114C4×C17—21no—
115C4×C17—21no—
116C4×C17—21no—
117C4×C17—21no—
118C4×C17—21no—
119C4×C17—42no—
120C4×C17—42no—
121C4×C17C2×#4112yes—
122Dic17—41no—
123Dic17—41no—
124Dic17—41no—
125#533D1711yescharacteristic, complemented
126#529—171no—
127#501—171no—
128#497—171no—
129#491—171no—
130#489—171no—
131#467—172no—
132#465—172no—
133#441—172no—
134#440—172no—
135C2×#97—171no—
136C2×C17—22no—
137C2×C17#53211yescharacteristic
138C2×C17C2×#13711yescharacteristic
139C2×C17#57511yescharacteristic
140D17—81no—
141C2×Q16—172no—
142C2×SD16—172no—
143C22×C8—172no—
144C4⋊Q8C2×D1711yescharacteristic
145C4⋊Q8C2×D1711yescharacteristic
146#126—341no—
147#125—21no—
148#125—171no—
149#124—171no—
150#123—341no—
151C22⋊Q8—21no—
152C22⋊Q8—171no—
153C22⋊Q8C2×D1712yes—
154#121—171no—
155C4×Q8—21no—
156C4×Q8—171no—
157C4×Q8C2×D1712yes—
158C4×D4—171no—
159C4×D4—341no—
160#117—21no—
161#117—341no—
162C2×C42C2×D1711yescharacteristic
163#107—172no—
164#106—172no—
165#103—172no—
166#103—172no—
167#103—172no—
168#102—172no—
169#97—172no—
170#97—171no—
171#97—171no—
172C17#318111yescharacteristic, complemented, Sylow
173C2×Q8—21no—
174C2×Q8—171no—
175C2×Q8C22×D1712yes—
176C2×D4—171no—
177C2×D4—341no—
178C22×C4—341no—
179C22×C4C22⋊D1711yescharacteristic
180C22×C4C22⋊D1711yescharacteristic
181C22×C4C22×D1711yescharacteristic
182Q16—344no—
183SD16—344no—
184C2×C8—172no—
185C2×C8—172no—
186C2×C8—344no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—171no—
193C4⋊C4—171no—
194C4⋊C4—341no—
195C4⋊C4—341no—
196C4⋊C4—341no—
197C4⋊C4—341no—
198C4⋊C4C22×D1712yes—
199C22⋊C4—22no—
200C22⋊C4—341no—
201C22⋊C4—341no—
202C22⋊C4—341no—
203C22⋊C4—341no—
204C22⋊C4—341no—
205C42—22no—
206C42—341no—
207C42C22⋊D1711yescharacteristic
208C42C22⋊D1711yescharacteristic
209C42C22×D1711yescharacteristic
210C42C22×D1711yescharacteristic
211C23—341no—
212C23C2×C22⋊D1711yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—42no—
216Q8—342no—
217D4—342no—
218D4—682no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—341no—
227C2×C4—682no—
228C2×C4C2×C22⋊D1711yescharacteristic
229C2×C4C2×C22⋊D1711yescharacteristic
230C2×C4C2×C22⋊D1711yescharacteristic
231C2×C4C2×C22⋊D1711yescharacteristic
232C2×C4C2×C22⋊D1711yescharacteristic
233C2×C4C23×D1711yescharacteristic
234C8—174no—
235C8—342no—
236C22—21no—
237C22—21no—
238C22—681no—
239C22—681no—
240C22—681no—
241C22#1058762812yes—
242C22C22×C22⋊D1711yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—681no—
251C4—681no—
252C4—681no—
253C4#41×D1712yes—
254C2—22no—
255C2—1361no—
256C2#1176335511yescharacteristic
257C2C2×#1058762811yescharacteristic
258C2#1176360711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#533C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenter#533D17
hypocenterC17#3181
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. #11763215C2semidirect
3. #11763201C2semidirect
4. #11763177C2semidirect
5. #11763171C2semidirect
6. #11763048C2semidirect
8. #11762914C2semidirect
9. #11762907C2semidirect
10. C2×#10587474C2semidirect
11. C17×#533C2semidirect
125. #533D17semidirect
172. C17#3181semidirect

Maximal subgroups

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

subgroupindex
#117633432
#117632152
#117632012
#117631772
#117631712
#117630482
#117630402
#117629142
#117629072
C2×#105874742
C17×#5332
#318117