Finite Group Databasebeta

← #14534692

Subgroups — #14534692

Order 2176.

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

normal subgroups only

This group has 450 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×C4×Dic34C211yesmaximal, characteristic, complemented
2#11762742C212yesmaximal, complemented
3#11762739C211yesmaximal, characteristic, complemented
4#11762734C211yesmaximal, characteristic
5#11762732C212yesmaximal
6#11762729C211yesmaximal, characteristic, complemented
7#11762721C211yesmaximal, characteristic
8#11762697C212yesmaximal
9#11762690C212yesmaximal, complemented
10#11762675C211yesmaximal, characteristic, complemented
11C2×C17×#126C211yesmaximal, characteristic
12C22×Dic34C2211yescharacteristic
13C17×#126—24no—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C42×C17C2211yescharacteristic
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×C4×Dic17C2212yes—
29C4×Dic34—24no—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C22×Dic17C2312yes—
48C2×Dic34—22no—
49C2×Dic34—24no—
50C4⋊C4×C17—22no—
51C4⋊C4×C17—22no—
52C4⋊C4×C17—22no—
53C22⋊C4×C17—22no—
54C22⋊C4×C17—22no—
55C22⋊C4×C17—22no—
56C42×C17D414yes—
57C22⋊Dic17—24no—
58C22⋊Dic17—42no—
59C4⋊Dic17—22no—
60C4⋊Dic17—42no—
61#63225—24no—
62#63225—24no—
63#63225—42no—
64C4×Dic17—24no—
65C23×C17—21no—
66C23×C17—22no—
67C23×C17—21no—
68C23×C17—22no—
69C23×C17—21no—
70C23×C17C2411yescharacteristic
71C2×C4×C17—22no—
72C2×C4×C17—21no—
73C2×C4×C17—22no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—21no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17C2×D412yes—
83C2×C4×C17C2×D412yes—
84C2×C4×C17C2×D412yes—
85C2×Dic17—22no—
86C2×Dic17—22no—
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#4114yes—
97Dic34—28no—
98#2637—171nomaximal, Sylow
99C22×C17—41no—
100C22×C17—41no—
101C22×C17—41no—
102C22×C17—42no—
103C22×C17—42no—
104C22×C17C22×D411yescharacteristic
105C22×C17C2×#4112yes—
106C22×C17H(5,𝔽2)11yescharacteristic
107C22×C172−1+412yes—
108C22×C172−1+411yescharacteristic
109C4×C17—22no—
110C4×C17—22no—
111C4×C17—22no—
112C4×C17—41no—
113C4×C17—41no—
114C4×C17—41no—
115Dic17—24no—
116Dic17—42no—
117Dic17—42no—
118Dic17—42no—
119C2×#126D1711yescharacteristic
120C2×C4×Q8—171no—
121#398—171no—
122#398—172no—
123#397—171no—
124#396—172no—
125#395—171no—
126#393—171no—
127#388—172no—
128#387—172no—
129#382—171no—
130C2×C17—81no—
131C2×C17#52511yescharacteristic
132C2×C17#53611yescharacteristic
133C2×C17#53711yescharacteristic
134C2×C17#54112yes—
135C2×C17#54212yes—
136C22×Q8—171no—
137#126—24no—
138C4×Q8—344no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4C2×D1711yescharacteristic
145C2×C4⋊C4C2×D1711yescharacteristic
146C2×C4⋊C4C2×D1711yescharacteristic
147C2×C22⋊C4—172no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4C2×D1711yescharacteristic
150C2×C22⋊C4C2×D1711yescharacteristic
151C2×C22⋊C4C2×D1711yescharacteristic
152C2×C42—172no—
153C2×C42C2×D1711yescharacteristic
154#95—172no—
155#95—171no—
156#95—171no—
157#95—172no—
158#95—172no—
159#95—172no—
160#95—172no—
161C17#263711yescharacteristic, complemented, Sylow
162C24C22×D1711yescharacteristic
163C2×Q8—344no—
164C2×Q8—342no—
165C22×C4—172no—
166C22×C4—172no—
167C22×C4—172no—
168C22×C4—172no—
169C22×C4C22×D1711yescharacteristic
170C22×C4C22×D1711yescharacteristic
171C22×C4C22×D1711yescharacteristic
172C22×C4C22×D1711yescharacteristic
173C22×C4C22×D1711yescharacteristic
174C22×C4C22×D1711yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—344no—
179C4⋊C4—344no—
180C4⋊C4—342no—
181C4⋊C4—682no—
182C4⋊C4—682no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—344no—
187C22⋊C4—682no—
188C42—344no—
189C42C22⋊D1714yes—
190C23—22no—
191C23—21no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23C23×D1711yescharacteristic
196Q8—348no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—174no—
209C2×C4—342no—
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×C4C2×C22⋊D1712yes—
221C2×C4C2×C22⋊D1712yes—
222C2×C4C2×C22⋊D1712yes—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22#41×D1712yes—
229C22#1058769611yescharacteristic
230C22#1058769712yes—
231C22#1058769711yescharacteristic
232C22C22×C22⋊D1711yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—344no—
240C4—682no—
241C4—682no—
242C4—682no—
243C2—81no—
244C2#1176331611yescharacteristic
245C2#1176338011yescharacteristic
246C2#1176338711yescharacteristic
247C2#1176341612yes—
248C2#1176342512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#126C2
prime radicalC24×C17C23
hypercenterC2×#126D17
hypocenterC17#2637
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic34C2semidirect
2. #11762742C2semidirect
3. #11762739C2semidirect
6. #11762729C2semidirect
9. #11762690C2semidirect
10. #11762675C2semidirect
161. C17#2637semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic342
#117627422
#117627392
#117627342
#117627322
#117627292
#117627212
#117626972
#117626902
#117626752
C2×C17×#1262
#263717