Finite Group Databasebeta

← #14534417

Subgroups — #14534417

Order 2176.

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

normal subgroups only

This group has 470 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×#10587565C211yesmaximal, characteristic
2C2×Q8×Dic17C211yesmaximal, characteristic, complemented
3#11762749C211yesmaximal, characteristic, complemented
4#11762745C211yesmaximal, characteristic
5#11762728C211yesmaximal, characteristic, complemented
6#11762706C212yesmaximal, complemented
7#11762703C212yesmaximal
8#11762702C212yesmaximal
9#11762696C212yesmaximal, complemented
10#11762654C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic
12C22×Q8×C17C2211yescharacteristic
13C2×C22⋊C4×C17C2212yes—
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊Dic17C2211yescharacteristic
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2211yescharacteristic
18Q8×Dic17—24no—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2211yescharacteristic
25C2×#63225C2211yescharacteristic
26C2×C4×Dic17C2212yes—
27C2×C4×Dic17C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29#10587565—24no—
30C17×#95C2212yes—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C2×Q8×C17—22no—
39C2×Q8×C17—24no—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2312yes—
44C22×Dic17C2311yescharacteristic
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2312yes—
50C22⋊C4×C17—22no—
51C22⋊C4×C17—24no—
52C22⋊Dic17—22no—
53C22⋊Dic17—22no—
54C22⋊Dic17—24no—
55C4⋊Dic17—22no—
56C4⋊Dic17—22no—
57C4⋊Dic17—24no—
58#63225—22no—
59#63225—24no—
60#63225—22no—
61#63225—24no—
62C4×Dic17—24no—
63C4×Dic17—24no—
64C4×Dic17D414yes—
65C23×C17—21no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17—21no—
69C23×C17—21no—
70C23×C17C2411yescharacteristic
71Q8×C17—28no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17C2×D412yes—
81C2×C4×C17#4114yes—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—21no—
85C2×Dic17—22no—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—21no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17C2×D412yes—
96C2×Dic17C2×D412yes—
97C2×Dic17#4114yes—
98#2604—171nomaximal, Sylow
99C22×C17—41no—
100C22×C17—41no—
101C22×C17—41no—
102C22×C17—42no—
103C22×C17—42no—
104C22×C17C22×D411yescharacteristic
105C22×C17C2×#4112yes—
106C22×C17C2×#4112yes—
107C22×C172−1+411yescharacteristic
108C22×C172−1+411yescharacteristic
109C4×C17—22no—
110C4×C17—24no—
111C4×C17—41no—
112C4×C17—42no—
113Dic17—22no—
114Dic17—22no—
115Dic17—24no—
116Dic17—41no—
117Dic17—42no—
118Dic17—41no—
119C2×#126—171no—
120C2×C4×Q8—171no—
121#400—171no—
122#399—171no—
123#395—171no—
124#393D1711yescharacteristic
125#389—172no—
126#388—172no—
127#388—172no—
128#387—172no—
129#376—171no—
130C2×C17—81no—
131C2×C17#52511yescharacteristic
132C2×C17#53211yescharacteristic
133C2×C17#53611yescharacteristic
134C2×C17#55212yes—
135C2×C17#55612yes—
136C22×Q8C2×D1711yescharacteristic
137#126—344no—
138C4×Q8—344no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4—171no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—172no—
146C2×C22⋊C4—171no—
147C2×C22⋊C4—171no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4C2×D1711yescharacteristic
150C2×C22⋊C4C2×D1712yes—
151C2×C42—171no—
152C2×C42—172no—
153C2×C42—172no—
154#95—172no—
155#95—171no—
156#95—172no—
157#95—172no—
158#95—172no—
159#95C2×D1711yescharacteristic
160#95C2×D1712yes—
161C17#260411yescharacteristic, complemented, Sylow
162C24C22×D1711yescharacteristic
163C2×Q8—22no—
164C2×Q8—24no—
165C22×C4—172no—
166C22×C4—171no—
167C22×C4—171no—
168C22×C4—171no—
169C22×C4—172no—
170C22×C4—171no—
171C22×C4C22×D1712yes—
172C22×C4C22×D1711yescharacteristic
173C22×C4C22×D1711yescharacteristic
174C22×C4C22×D1712yes—
175C4⋊C4—344no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—344no—
179C4⋊C4—342no—
180C4⋊C4—342no—
181C4⋊C4—344no—
182C22⋊C4—22no—
183C22⋊C4—24no—
184C22⋊C4—344no—
185C22⋊C4—342no—
186C22⋊C4—342no—
187C42—174no—
188C42—344no—
189C42—344no—
190C23—21no—
191C23—22no—
192C23—21no—
193C23—22no—
194C23—21no—
195C23C23×D1711yescharacteristic
196Q8—28no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—172no—
206C2×C4—172no—
207C2×C4—174no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—341no—
211C2×C4—342no—
212C2×C4—341no—
213C2×C4—342no—
214C2×C4—341no—
215C2×C4—341no—
216C2×C4—342no—
217C2×C4—341no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—341no—
221C2×C4D4×D1712yes—
222C2×C4#6325414yes—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22C2×D4×D1711yescharacteristic
229C22C2×#6325412yes—
230C22#1058769411yescharacteristic
231C22#41×D1712yes—
232C22#1058769711yescharacteristic
233C4—22no—
234C4—24no—
235C4—42no—
236C4—41no—
237C4—344no—
238C4—342no—
239C4—342no—
240C4—681no—
241C4—682no—
242C4—681no—
243C2—81no—
244C2#1176331511yescharacteristic
245C2#1176335811yescharacteristic
246C2#1176338211yescharacteristic
247C2#1176349312yes—
248C2#1176352812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC24×C17C23
hypercenter#393D17
hypocenterC17#2604
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic17C2semidirect
3. #11762749C2semidirect
5. #11762728C2semidirect
6. #11762706C2semidirect
9. #11762696C2semidirect
10. #11762654C2semidirect
161. C17#2604semidirect

Maximal subgroups

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

subgroupindex
C2×#105875652
C2×Q8×Dic172
#117627492
#117627452
#117627282
#117627062
#117627032
#117627022
#117626962
#117626542
C17×#3932
#260417