Finite Group Databasebeta

← #14534693

Subgroups — #14534693

Order 2176.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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×#10587573C211yesmaximal, characteristic, complemented
3#11762743C212yesmaximal, complemented
4#11762740C211yesmaximal, characteristic, complemented
5#11762732C212yesmaximal
6#11762730C211yesmaximal, characteristic, complemented
7#11762721C211yesmaximal, characteristic
8#11762703C212yesmaximal
9#11762692C212yesmaximal, complemented
10#11762673C211yesmaximal, characteristic, complemented
11C17×#397C211yesmaximal, characteristic
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2211yescharacteristic
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2211yescharacteristic
22C2×#63225C2211yescharacteristic
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×C4×Dic17C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28#10587573—24no—
29#10587565—24no—
30C17×#95C2212yes—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×Dic17C2312yes—
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2311yescharacteristic
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2312yes—
48C2×Dic34—22no—
49C2×Dic34—24no—
50C4⋊C4×C17—22no—
51C22⋊C4×C17—22no—
52C22⋊C4×C17—42no—
53C22⋊Dic17—22no—
54C22⋊Dic17—22no—
55C22⋊Dic17—24no—
56C4⋊Dic17—22no—
57C4⋊Dic17—24no—
58#63225—22no—
59#63225—22no—
60#63225—24no—
61#63225—42no—
62#63225—42no—
63C4×Dic17—24no—
64C4×Dic17D414yes—
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—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17C2×D412yes—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—21no—
84C2×Dic17—22no—
85C2×Dic17—21no—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—21no—
93C2×Dic17—22no—
94C2×Dic17C2×D412yes—
95C2×Dic17C2×D412yes—
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—41no—
111C4×C17—42no—
112C4×C17—42no—
113Dic17—24no—
114Dic17—22no—
115Dic17—22no—
116Dic17—41no—
117Dic17—41no—
118Dic17—42no—
119C2×#126—171no—
120C2×C4×Q8—171no—
121#398—171no—
122#398—172no—
123#397D1711yescharacteristic
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—344no—
138C4×Q8—344no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4C2×D1711yescharacteristic
147C2×C22⋊C4—171no—
148C2×C22⋊C4—171no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4C2×D1711yescharacteristic
151C2×C22⋊C4C2×D1712yes—
152C2×C42—171no—
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#263711yescharacteristic, complemented, Sylow
162C24C22×D1711yescharacteristic
163C2×Q8—344no—
164C2×Q8—342no—
165C22×C4—171no—
166C22×C4—172no—
167C22×C4—171no—
168C22×C4—172no—
169C22×C4—171no—
170C22×C4—171no—
171C22×C4C22×D1711yescharacteristic
172C22×C4C22×D1712yes—
173C22×C4C22×D1712yes—
174C22×C4C22×D1711yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—344no—
177C4⋊C4—342no—
178C4⋊C4—342no—
179C4⋊C4—344no—
180C4⋊C4—342no—
181C4⋊C4—682no—
182C4⋊C4—682no—
183C22⋊C4—22no—
184C22⋊C4—42no—
185C22⋊C4—344no—
186C22⋊C4—342no—
187C22⋊C4—342no—
188C42—174no—
189C42—344no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23—21no—
195C23C23×D1711yescharacteristic
196Q8—348no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—172no—
207C2×C4—172no—
208C2×C4—174no—
209C2×C4—342no—
210C2×C4—341no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—341no—
214C2×C4—341no—
215C2×C4—341no—
216C2×C4—341no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—341no—
221C2×C4—342no—
222C2×C4D4×D1712yes—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22C2×D4×D1711yescharacteristic
229C22#1058769111yescharacteristic
230C22#41×D1712yes—
231C22#1058769712yes—
232C22#1058769711yescharacteristic
233C4—22no—
234C4—41no—
235C4—42no—
236C4—42no—
237C4—344no—
238C4—342no—
239C4—342no—
240C4—681no—
241C4—681no—
242C4—682no—
243C2—81no—
244C2#1176331711yescharacteristic
245C2#1176337911yescharacteristic
246C2#1176338811yescharacteristic
247C2#1176341612yes—
248C2#1176342712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC24×C17C23
hypercenter#397D17
hypocenterC17#2637
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#10587573C2semidirect
3. #11762743C2semidirect
4. #11762740C2semidirect
6. #11762730C2semidirect
9. #11762692C2semidirect
10. #11762673C2semidirect
161. C17#2637semidirect

Maximal subgroups

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

subgroupindex
C2×#105875652
C2×#105875732
#117627432
#117627402
#117627322
#117627302
#117627212
#117627032
#117626922
#117626732
C17×#3972
#263717