Finite Group Databasebeta

← #14534482

Subgroups — #14534482

Order 2176.

476 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 145 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×#10587556C211yesmaximal, characteristic, complemented
2#11762748C211yesmaximal, characteristic, complemented
3#11762745C211yesmaximal, characteristic
4#11762739C211yesmaximal, characteristic, complemented
5#11762734C211yesmaximal, characteristic
6#11762704C212yesmaximal, complemented
7#11762703C212yesmaximal
8#11762693C212yesmaximal, complemented
9#11762682C212yesmaximal
10#11762655C211yesmaximal, characteristic, complemented
11C2×C17×#117C211yesmaximal, characteristic
12C23×C4×C17C2211yescharacteristic
13C17×#117—24no—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C42×C17C2211yescharacteristic
19C2×C42×C17C2211yescharacteristic
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2211yescharacteristic
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×C4×Dic17C2212yes—
29#10587556—24no—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C22×C4×C17—22no—
39C22×C4×C17—22no—
40C22×C4×C17—22no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C22×Dic17C2312yes—
48C22×Dic17C2312yes—
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C4⋊C4×C17—22no—
52C4⋊C4×C17—22no—
53C22⋊C4×C17—22no—
54C22⋊C4×C17—22no—
55C42×C17—22no—
56C42×C17D414yes—
57C22⋊Dic17—24no—
58C22⋊Dic17—24no—
59C4⋊Dic17—22no—
60C4⋊Dic17—22no—
61#63225—24no—
62#63225—24no—
63#63225—24no—
64#63225—24no—
65C4×Dic17—24no—
66C23×C17—21no—
67C23×C17—22no—
68C23×C17—21no—
69C23×C17—22no—
70C23×C17#4112yes—
71C23×C17C2411yescharacteristic
72C2×C4×C17—28no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17C2×D412yes—
84C2×C4×C17C2×D412yes—
85C2×C4×C17C2×D412yes—
86C2×C4×C17#4112yes—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17#4114yes—
99#2611—171nomaximal, Sylow
100C22×C17—22no—
101C22×C17—24no—
102C22×C17—41no—
103C22×C17—41no—
104C22×C17—42no—
105C22×C17C22×D411yescharacteristic
106C22×C17C2×#4111yescharacteristic
107C22×C17C2×#4111yescharacteristic
108C22×C17C2×#4112yes—
109C22×C172−1+412yes—
110C4×C17—22no—
111C4×C17—22no—
112C4×C17—22no—
113C4×C17—22no—
114C4×C17—41no—
115C4×C17—41no—
116Dic17—24no—
117Dic17—42no—
118Dic17—42no—
119Dic17—42no—
120C2×#125—171no—
121C2×#117D1711yescharacteristic
122#400—171no—
123#399—171no—
124#398—171no—
125#397—171no—
126#389—172no—
127#388—172no—
128#387—172no—
129#385—172no—
130#376—171no—
131C2×C17—42no—
132C2×C17#52511yescharacteristic
133C2×C17#53211yescharacteristic
134C2×C17#53611yescharacteristic
135C2×C17#55212yes—
136C2×C17#55212yes—
137C23×C4C2×D1711yescharacteristic
138#125—344no—
139#117—24no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4—171no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4C2×D1711yescharacteristic
147C2×C4⋊C4C2×D1711yescharacteristic
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4C2×D1711yescharacteristic
151C2×C22⋊C4C2×D1711yescharacteristic
152C2×C42—172no—
153C2×C42C2×D1711yescharacteristic
154C2×C42C2×D1711yescharacteristic
155#95—172no—
156#95—171no—
157#95—171no—
158#95—172no—
159#95—172no—
160#95—172no—
161#95—172no—
162C17#261111yescharacteristic, complemented, Sylow
163C24C22×D1711yescharacteristic
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—172no—
168C22×C4—172no—
169C22×C4—172no—
170C22×C4—172no—
171C22×C4C22×D1711yescharacteristic
172C22×C4C22×D1711yescharacteristic
173C22×C4C22×D1711yescharacteristic
174C22×C4C22×D1711yescharacteristic
175C22×C4C22×D1711yescharacteristic
176C22×C4C22×D1711yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—344no—
180C4⋊C4—344no—
181C4⋊C4—344no—
182C4⋊C4—344no—
183C4⋊C4—342no—
184C4⋊C4—342no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—344no—
188C22⋊C4—344no—
189C42—22no—
190C42—344no—
191C42C22⋊D1714yes—
192C23—21no—
193C23—22no—
194C23—22no—
195C23—21no—
196C23#6325212yes—
197C23C23×D1711yescharacteristic
198C2×C4—28no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—174no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4#6325212yes—
222C2×C4C2×C22⋊D1712yes—
223C2×C4C2×C22⋊D1712yes—
224C2×C4C2×C22⋊D1712yes—
225C22—24no—
226C22—22no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22C2×#6325211yescharacteristic
231C22C2×#6325211yescharacteristic
232C22#41×D1712yes—
233C22#1058769712yes—
234C22C22×C22⋊D1711yescharacteristic
235C4—22no—
236C4—22no—
237C4—22no—
238C4—22no—
239C4—41no—
240C4—41no—
241C4—344no—
242C4—682no—
243C4—682no—
244C4—682no—
245C2—42no—
246C2#1176332011yescharacteristic
247C2#1176335511yescharacteristic
248C2#1176338011yescharacteristic
249C2#1176349212yes—
250C2#1176349812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#117C2
prime radicalC24×C17C23
hypercenterC2×#117D17
hypocenterC17#2611
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#10587556C2semidirect
2. #11762748C2semidirect
4. #11762739C2semidirect
6. #11762704C2semidirect
8. #11762693C2semidirect
10. #11762655C2semidirect
162. C17#2611semidirect

Maximal subgroups

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

subgroupindex
C2×#105875562
#117627482
#117627452
#117627392
#117627342
#117627042
#117627032
#117626932
#117626822
#117626552
C2×C17×#1172
#261117