Finite Group Databasebeta

← #14534603

Subgroups — #14534603

Order 2176.

508 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587564C211yesmaximal, characteristic
2C22×#63225C211yesmaximal, characteristic
3#11762750C212yesmaximal, complemented
4#11762743C212yesmaximal, complemented
5#11762740C212yesmaximal, complemented
6#11762729C212yesmaximal, complemented
7#11762682C211yesmaximal, characteristic
8#11762670C211yesmaximal, characteristic
9#11762670C211yesmaximal, characteristic
10#11762667C211yesmaximal, characteristic
11C17×#385C211yesmaximal, characteristic
12C23×C4×C17C2211yescharacteristic
13C23×Dic17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C2×C4⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊Dic17C2211yescharacteristic
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C4⋊Dic17C2211yescharacteristic
22C2×#63225—24no—
23C2×#63225C2212yes—
24C2×#63225C2211yescharacteristic
25C2×#63225C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2212yes—
28C2×#63225C2212yes—
29#10587564—24no—
30C17×#95C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C22×C4×C17—22no—
39C22×C4×C17—42no—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×Dic17—22no—
45C22×Dic17—22no—
46C22×Dic17—42no—
47C22×Dic17D414yes—
48C22×Dic17Q814yescomplemented
49C22×Dic17C2311yescharacteristic
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2312yes—
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2312yes—
55C2×Dic34—22no—
56C2×Dic34—42no—
57C4⋊C4×C17—42no—
58C22⋊C4×C17—22no—
59C22⋊C4×C17—22no—
60C22⋊Dic17—22no—
61C22⋊Dic17—22no—
62C4⋊Dic17—22no—
63#63225—22no—
64#63225—22no—
65#63225—24no—
66#63225—24no—
67#63225—42no—
68#63225—42no—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17C2×D412yes—
72C23×C17C2×Q812yes—
73C23×C17#4112yes—
74C23×C17C2411yescharacteristic
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—44no—
86C2×Dic17—28no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—44no—
99C2×Dic17C2×D412yes—
100C2×Dic17C2×D412yes—
101C2×Dic17C2×Q814yes—
102Dic34—44no—
103#2627—171nomaximal, Sylow
104C22×C17—22no—
105C22×C17—22no—
106C22×C17—24no—
107C22×C17—24no—
108C22×C17C22⋊Q814yes—
109C22×C17C22×D411yescharacteristic
110C22×C17C22×Q811yescharacteristic
111C22×C17C2×#4111yescharacteristic
112C22×C17H(5,𝔽2)12yes—
113C22×C172−1+412yes—
114C4×C17—41no—
115C4×C17—42no—
116C4×C17—41no—
117C4×C17—42no—
118Dic17—24no—
119Dic17—22no—
120Dic17—22no—
121Dic17—41no—
122Dic17—41no—
123Dic17—42no—
124C2×C22⋊Q8—171no—
125C22×C4⋊C4—171no—
126#400—172no—
127#398—172no—
128#398—172no—
129#395—172no—
130#385—171no—
131#385D1711yescharacteristic
132#381—171no—
133#381—171no—
134#380—171no—
135C2×C17—24no—
136C2×C17C2×C22⋊Q811yescharacteristic
137C2×C17#53411yescharacteristic
138C2×C17#53611yescharacteristic
139C2×C17#53912yes—
140C2×C17#54412yes—
141C22×Q8—171no—
142C23×C4—171no—
143C23×C4—171no—
144C23×C4C2×D1711yescharacteristic
145C22⋊Q8—344no—
146C2×C4⋊C4—172no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4—171no—
149C2×C4⋊C4—171no—
150C2×C4⋊C4—171no—
151C2×C4⋊C4—172no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4—344no—
154C2×C4⋊C4C2×D1712yes—
155C2×C22⋊C4—171no—
156C2×C22⋊C4—171no—
157C2×C22⋊C4C2×D1711yescharacteristic
158C2×C22⋊C4C2×D1711yescharacteristic
159#95—172no—
160#95—172no—
161#95—172no—
162#95—172no—
163#95—172no—
164#95—172no—
165#95C2×D1712yes—
166C17#262711yescharacteristic, complemented, Sylow
167C24C22×D1711yescharacteristic
168C2×Q8—342no—
169C2×Q8—682no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—174no—
173C22×C4—172no—
174C22×C4—171no—
175C22×C4—171no—
176C22×C4—174no—
177C22×C4—172no—
178C22×C4—171no—
179C22×C4—171no—
180C22×C4—342no—
181C22×C4—342no—
182C22×C4—682no—
183C22×C4C22×D1712yes—
184C22×C4C22×D1711yescharacteristic
185C22×C4C22×D1711yescharacteristic
186C22×C4C22×D1712yes—
187C4⋊C4—42no—
188C4⋊C4—344no—
189C4⋊C4—344no—
190C4⋊C4—342no—
191C4⋊C4—342no—
192C4⋊C4—342no—
193C4⋊C4—682no—
194C4⋊C4—682no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—342no—
198C22⋊C4—342no—
199C23—22no—
200C23—22no—
201C23#6325212yes—
202C23D4×D1712yes—
203C23Q8×D1712yes—
204C23C23×D1711yescharacteristic
205Q8—684no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—44no—
217C2×C4—172no—
218C2×C4—172no—
219C2×C4—174no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—348no—
225C2×C4—342no—
226C2×C4—341no—
227C2×C4—341no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4—684no—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22#1058758214yes—
238C22C2×#6325211yescharacteristic
239C22C2×D4×D1711yescharacteristic
240C22#1058769112yes—
241C22C2×Q8×D1711yescharacteristic
242C22#1058769712yes—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—344no—
248C4—342no—
249C4—342no—
250C4—681no—
251C4—681no—
252C4—682no—
253C2—24no—
254C2C2×#1058758211yescharacteristic
255C2#1176336911yescharacteristic
256C2#1176338111yescharacteristic
257C2#1176340112yes—
258C2#1176343812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC24×C17C23
hypercenter#385D17
hypocenterC17#2627
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #11762750C2semidirect
4. #11762743C2semidirect
5. #11762740C2semidirect
6. #11762729C2semidirect
48. C22×Dic17Q8semidirect
166. C17#2627semidirect

Maximal subgroups

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

subgroupindex
C2×#105875642
C22×#632252
#117627502
#117627432
#117627402
#117627292
#117626822
#117626702
#117626672
C17×#3852
#262717