Finite Group Databasebeta

← #14534549

Subgroups — #14534549

Order 2176.

528 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 143 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×#10587574C211yesmaximal, characteristic, complemented
2C2×#10587564C212yesmaximal
3C2×#10587563C211yesmaximal, characteristic
4#11762743C212yesmaximal, complemented
5#11762741C211yesmaximal, characteristic, complemented
6#11762740C211yesmaximal, characteristic, complemented
7#11762737C212yesmaximal
8#11762734C211yesmaximal, characteristic
9#11762713C212yesmaximal, complemented
10#11762673C211yesmaximal, characteristic, complemented
11C17×#397C211yesmaximal, characteristic
12C23×Dic17C2211yescharacteristic
13C22×Dic34C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2212yes—
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×#63225C2211yescharacteristic
24C2×#63225C2212yes—
25C2×#63225C2211yescharacteristic
26C2×C4×Dic17C2211yescharacteristic
27C2×C4×Dic17C2211yescharacteristic
28#10587574—24no—
29#10587564—28no—
30#10587563—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37#10587502C2212yes—
38C24×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×Dic17—22no—
44C22×Dic17—22no—
45C22×Dic17D414yes—
46C22×Dic17C2312yes—
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2312yes—
49C22×Dic17C2311yescharacteristic
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2311yescharacteristic
52C2×Dic34—24no—
53C2×Dic34—24no—
54C2×Dic34—24no—
55C4⋊C4×C17—22no—
56C22⋊C4×C17—24no—
57C22⋊C4×C17—22no—
58C22⋊Dic17—24no—
59C22⋊Dic17—22no—
60C4⋊Dic17—22no—
61C4⋊Dic17—24no—
62#63225—22no—
63#63225—22no—
64#63225—24no—
65#63225—24no—
66#63225—24no—
67C4×Dic17—22no—
68C4×Dic17D414yes—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17C2×D412yes—
74C23×C17C2411yescharacteristic
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17C2×D412yes—
85C2×Dic17—21no—
86C2×Dic17—21no—
87C2×Dic17—22no—
88C2×Dic17—28no—
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17C2×D412yes—
98C2×Dic17C2×D412yes—
99C2×Dic17C2×D412yes—
100C2×Dic17C2×D412yes—
101C2×Dic17#4112yes—
102Dic34—48no—
103#2621—171nomaximal, Sylow
104C22×C17—22no—
105C22×C17—24no—
106C22×C17—41no—
107C22×C17—41no—
108C22×C17—42no—
109C22×C17C22×D411yescharacteristic
110C22×C17C22×D411yescharacteristic
111C22×C17C2×#4111yescharacteristic
112C22×C172−1+412yes—
113C22×C172−1+412yes—
114C4×C17—22no—
115C4×C17—42no—
116C4×C17—41no—
117C4×C17—42no—
118Dic17—22no—
119Dic17—22no—
120Dic17—42no—
121Dic17—41no—
122Dic17—42no—
123Dic17#12114yes—
124C2×C4⋊Q8—171no—
125C2×C22⋊Q8—172no—
126C2×#117—171no—
127#398—171no—
128#398—171no—
129#398—172no—
130#397—171no—
131#397—172no—
132#397D1711yescharacteristic
133#391—172no—
134#382—171no—
135C2×C17—42no—
136C2×C17C2×#12111yescharacteristic
137C2×C17#53611yescharacteristic
138C2×C17#53611yescharacteristic
139C2×C17#53612yes—
140C2×C17#54112yes—
141C22×Q8—172no—
142C23×C4—171no—
143C4⋊Q8—344no—
144C22⋊Q8—348no—
145#117—344no—
146C2×C4⋊C4—171no—
147C2×C4⋊C4—171no—
148C2×C4⋊C4—172no—
149C2×C4⋊C4—172no—
150C2×C4⋊C4—172no—
151C2×C4⋊C4—172no—
152C2×C4⋊C4—171no—
153C2×C4⋊C4C2×D1711yescharacteristic
154C2×C22⋊C4—171no—
155C2×C22⋊C4—172no—
156C2×C22⋊C4C2×D1711yescharacteristic
157C2×C22⋊C4C2×D1712yes—
158C2×C42—171no—
159C2×C42—171no—
160#95—172no—
161#95—171no—
162#95—172no—
163#95—172no—
164#95—172no—
165#95C2×D1711yescharacteristic
166#95C2×D1712yes—
167C17#262111yescharacteristic, complemented, Sylow
168C24C22×D1711yescharacteristic
169C2×Q8—344no—
170C2×Q8—344no—
171C2×Q8—344no—
172C22×C4—171no—
173C22×C4—174no—
174C22×C4—171no—
175C22×C4—171no—
176C22×C4—171no—
177C22×C4—172no—
178C22×C4—172no—
179C22×C4—342no—
180C22×C4—342no—
181C22×C4C22×D1711yescharacteristic
182C22×C4C22×D1712yes—
183C22×C4C22×D1712yes—
184C22×C4C22×D1711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—342no—
187C4⋊C4—344no—
188C4⋊C4—344no—
189C4⋊C4—344no—
190C4⋊C4—344no—
191C4⋊C4—342no—
192C4⋊C4—342no—
193C22⋊C4—24no—
194C22⋊C4—22no—
195C22⋊C4—342no—
196C22⋊C4—344no—
197C42—174no—
198C42—342no—
199C23—21no—
200C23—21no—
201C23—22no—
202C23—22no—
203C23D4×D1712yes—
204C23C23×D1711yescharacteristic
205Q8—688no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—172no—
216C2×C4—172no—
217C2×C4—172no—
218C2×C4—172no—
219C2×C4—172no—
220C2×C4—348no—
221C2×C4—341no—
222C2×C4—341no—
223C2×C4—342no—
224C2×C4—341no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4D4×D1712yes—
233C22—22no—
234C22—24no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22C2×D4×D1711yescharacteristic
239C22C2×D4×D1711yescharacteristic
240C22#41×D1711yescharacteristic
241C22#1058769712yes—
242C22#1058769712yes—
243C4—22no—
244C4—42no—
245C4—41no—
246C4—42no—
247C4—174no—
248C4—342no—
249C4—342no—
250C4—682no—
251C4—681no—
252C4—682no—
253C2—42no—
254C2#121×D1711yescharacteristic
255C2#1176337911yescharacteristic
256C2#1176338111yescharacteristic
257C2#1176338112yes—
258C2#1176341612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC24×C17C23
hypercenter#397D17
hypocenterC17#2621
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#10587574C2semidirect
4. #11762743C2semidirect
5. #11762741C2semidirect
6. #11762740C2semidirect
9. #11762713C2semidirect
10. #11762673C2semidirect
167. C17#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105875642
C2×#105875632
#117627432
#117627412
#117627402
#117627372
#117627342
#117627132
#117626732
C17×#3972
#262117