Finite Group Databasebeta

← #14532076

Subgroups — #14532076

Order 2176.

504 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 163 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×#10587573C211yesmaximal, characteristic
2C2×#10587573C211yesmaximal, characteristic
3#11762743C212yesmaximal
4#11762740C211yesmaximal, characteristic
5#11762730C211yesmaximal, characteristic
6#11762729C212yesmaximal
7#11762729C211yesmaximal, characteristic
8C4⋊C4×Dic17C212yesmaximal
9#11762660C212yesmaximal
10#11762654C211yesmaximal, characteristic
11C17×#398C211yesmaximal, characteristic
12C22×Dic34C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2212yes—
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊Dic17C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2211yescharacteristic
22C2×#63225C2211yescharacteristic
23C2×C4×Dic17C2211yescharacteristic
24C2×C4×Dic17C2212yes—
25C2×C4×Dic17C2211yescharacteristic
26C2×C4×Dic17C2211yescharacteristic
27C2×C4×Dic17C2212yes—
28#10587573—24no—
29#10587573—24no—
30C17×#95C2211yescharacteristic
31C17×#95C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×Dic17C2311yescharacteristic
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2312yes—
47C22×Dic17C2311yescharacteristic
48C2×Dic34—22no—
49C2×Dic34—22no—
50C2×Dic34—22no—
51C2×Dic34—22no—
52C2×Dic34—24no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17—22no—
55C4⋊C4×C17—24no—
56C4⋊Dic17—24no—
57#63225—22no—
58#63225—24no—
59#63225—22no—
60#63225—24no—
61#63225—24no—
62C4×Dic17—22no—
63C4×Dic17—24no—
64C4×Dic17—22no—
65C4×Dic17D414yes—
66C4×Dic17Q818yescomplemented
67C23×C17C2411yescharacteristic
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—21no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17C2×D412yes—
79C2×C4×C17C2×Q814yes—
80C2×C4×C17#4112yes—
81C2×Dic17—22no—
82C2×Dic17—21no—
83C2×Dic17—21no—
84C2×Dic17—21no—
85C2×Dic17—21no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—21no—
90C2×Dic17—22no—
91C2×Dic17C2×D412yes—
92C2×Dic17C2×D412yes—
93C2×Dic17C2×Q814yes—
94C2×Dic17C2×Q814yes—
95C2×Dic17#4112yes—
96C2×Dic17#4114yes—
97Dic34—28no—
98Dic34—28no—
99#2399—171nomaximal, Sylow
100C22×C17C22×D411yescharacteristic
101C22×C17C22×Q812yes—
102C22×C17C2×#4111yescharacteristic
103C22×C17C2×#4112yes—
104C22×C17C2×#4111yescharacteristic
105C4×C17—22no—
106C4×C17—22no—
107C4×C17—24no—
108C4×C17—41no—
109C4×C17—42no—
110Dic17—22no—
111Dic17—24no—
112Dic17—22no—
113Dic17—24no—
114Dic17—41no—
115Dic17C4⋊Q814yes—
116C2×C4×Q8—171no—
117C2×C4×Q8—171no—
118#398—171no—
119#398—172no—
120#398D1711yescharacteristic
121#395—172no—
122#395—171no—
123#395—171no—
124C4×C4⋊C4—172no—
125C4×C4⋊C4—172no—
126#376—171no—
127C2×C17#52511yescharacteristic
128C2×C17#52511yescharacteristic
129C2×C17C2×C4⋊Q811yescharacteristic
130C2×C17#53312yes—
131C2×C17#53312yes—
132C22×Q8—171no—
133C22×Q8—171no—
134C4×Q8—344no—
135C4×Q8—344no—
136C2×C4⋊C4—172no—
137C2×C4⋊C4—171no—
138C2×C4⋊C4—171no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4C2×D1711yescharacteristic
143C2×C4⋊C4C2×D1711yescharacteristic
144C2×C4⋊C4C2×D1712yes—
145C2×C42—171no—
146C2×C42—172no—
147C2×C42—172no—
148C2×C42—171no—
149C2×C42—171no—
150#95—171no—
151#95—172no—
152#95—172no—
153#95—172no—
154#95—172no—
155#95C2×D1711yescharacteristic
156#95C2×D1712yes—
157C17#239911yescharacteristic, complemented, Sylow
158C2×Q8—342no—
159C2×Q8—344no—
160C2×Q8—342no—
161C2×Q8—342no—
162C2×Q8—342no—
163C22×C4—171no—
164C22×C4—172no—
165C22×C4—171no—
166C22×C4—171no—
167C22×C4—172no—
168C22×C4—171no—
169C22×C4C22×D1712yes—
170C22×C4C22×D1711yescharacteristic
171C22×C4C22×D1711yescharacteristic
172C22×C4C22×D1711yescharacteristic
173C22×C4C22×D1712yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—344no—
178C4⋊C4—342no—
179C4⋊C4—344no—
180C4⋊C4—344no—
181C4⋊C4—344no—
182C4⋊C4—342no—
183C42—174no—
184C42—178no—
185C42—342no—
186C42—342no—
187C42—344no—
188C23C23×D1711yescharacteristic
189Q8—348no—
190Q8—348no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—172no—
202C2×C4—174no—
203C2×C4—172no—
204C2×C4—174no—
205C2×C4—174no—
206C2×C4—172no—
207C2×C4—342no—
208C2×C4—341no—
209C2×C4—341no—
210C2×C4—341no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—341no—
215C2×C4—341no—
216C2×C4—342no—
217C2×C4D4×D1712yes—
218C2×C4#6325412yes—
219C2×C4Q8×D1714yes—
220C22C2×D4×D1711yescharacteristic
221C22C2×#6325411yescharacteristic
222C22C2×Q8×D1712yes—
223C22#41×D1712yes—
224C22#41×D1711yescharacteristic
225C4—22no—
226C4—22no—
227C4—24no—
228C4—41no—
229C4—42no—
230C4—174no—
231C4—344no—
232C4—342no—
233C4—342no—
234C4—344no—
235C4—681no—
236C2#1176331711yescharacteristic
237C2#1176331911yescharacteristic
238C2C4⋊Q8×D1711yescharacteristic
239C2#1176336112yes—
240C2#1176336512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#398C2
hypercenter#398D17
hypocenterC17#2399
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
66. C4×Dic17Q8semidirect
157. C17#2399semidirect

Maximal subgroups

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

subgroupindex
C2×#105875732
#117627432
#117627402
#117627302
#117627292
C4⋊C4×Dic172
#117626602
#117626542
C17×#3982
#239917