Finite Group Databasebeta

← #14535549

Subgroups — #14535549

Order 2176.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587622C212yesmaximal, complemented
2#11762734C211yesmaximal, characteristic, complemented
3#11762723C212yesmaximal, complemented
4#11762721C212yesmaximal, complemented
5#11762713C212yesmaximal, complemented
6#11762702C212yesmaximal, complemented
7#11762689C212yesmaximal, complemented
8#11762681C211yesmaximal, characteristic, complemented
9C17×#392C211yesmaximal, characteristic
10C22×D4×C17C2212yes—
11C22×D4×C17C2211yescharacteristic
12C23×Dic17C2211yescharacteristic
13C22×Dic34C2212yes—
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2211yescharacteristic
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2211yescharacteristic
22#10587622—28no—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×#63225C2212yes—
25C2×C4×Dic17C2212yes—
26C17×#95C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32C24×C17C2312yes—
33C24×C17C2311yescharacteristic
34C2×D4×C17—24no—
35C2×D4×C17—24no—
36C2×D4×C17—42no—
37C2×D4×C17—41no—
38C2×D4×C17—42no—
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×Dic17—41no—
43C22×Dic17—42no—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C22×Dic17C2312yes—
48C2×Dic34—24no—
49C2×Dic34—24no—
50C2×Dic34—42no—
51C22⋊C4×C17—24no—
52C22⋊C4×C17—41no—
53C22⋊Dic17—24no—
54C22⋊Dic17—24no—
55C22⋊Dic17—24no—
56C22⋊Dic17—42no—
57C22⋊Dic17—42no—
58C22⋊Dic17D414yescomplemented
59C4⋊Dic17—41no—
60#63225—42no—
61C4×Dic17—24no—
62C23×C17—22no—
63C23×C17—21no—
64C23×C17—22no—
65C23×C17—21no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17C2×D412yes—
74C23×C17C2411yescharacteristic
75D4×C17—48no—
76D4×C17—44no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17#4114yes—
84C2×Dic17—22no—
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—44no—
96C2×Dic17C2×D414yes—
97Dic34—48no—
98#2712—171nomaximal, Sylow
99C22×C17—22no—
100C22×C17—24no—
101C22×C17—42no—
102C22×C17—41no—
103C22×C17—41no—
104C22×C17—42no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17C22×D411yescharacteristic
112C22×C17C2×#4112yes—
113C22×C17H(5,𝔽2)11yescharacteristic
114C22×C17H(5,𝔽2)12yes—
115C22×C172−1+411yescharacteristic
116C4×C17—24no—
117C4×C17—41no—
118C4×C17—41no—
119Dic17—24no—
120Dic17—42no—
121Dic17—42no—
122Dic17—42no—
123C2×#124—172no—
124#397—171no—
125#394—172no—
126#393—172no—
127#392D1711yescharacteristic
128#391—172no—
129#388—172no—
130#386—172no—
131#385—171no—
132C2×C17—42no—
133C2×C17—82no—
134C2×C17#53711yescharacteristic
135C2×C17#54212yes—
136C2×C17#54612yes—
137C2×C17#55911yescharacteristic
138C2×C17H(3,𝔽4)11yescharacteristic
139C22×Q8—172no—
140C22×D4C2×D1712yes—
141C22×D4C2×D1711yescharacteristic
142C23×C4—171no—
143#124—348no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—171no—
146C2×C22⋊C4—171no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4—172no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4C2×D1712yes—
153C2×C22⋊C4C2×D1711yescharacteristic
154C2×C42—172no—
155#95—172no—
156#95—172no—
157#95—172no—
158#95—172no—
159#95—171no—
160#95C2×D1711yescharacteristic
161C17#271211yescharacteristic, complemented, Sylow
162C24C22×D1712yes—
163C24C22×D1711yescharacteristic
164C2×Q8—344no—
165C2×Q8—344no—
166C2×Q8—682no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—42no—
170C2×D4—41no—
171C2×D4—42no—
172C22×C4—172no—
173C22×C4—172no—
174C22×C4—172no—
175C22×C4—172no—
176C22×C4—681no—
177C22×C4—682no—
178C22×C4C22×D1711yescharacteristic
179C22×C4C22×D1711yescharacteristic
180C22×C4C22×D1712yes—
181C4⋊C4—682no—
182C4⋊C4—681no—
183C22⋊C4—24no—
184C22⋊C4—41no—
185C22⋊C4—174no—
186C22⋊C4—344no—
187C22⋊C4—344no—
188C22⋊C4—344no—
189C22⋊C4—682no—
190C22⋊C4—682no—
191C42—344no—
192C23—22no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23D4×D1712yes—
204C23C23×D1711yescharacteristic
205Q8—688no—
206D4—48no—
207D4—44no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—174no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—684no—
227C2×C4#6325414yes—
228C22—22no—
229C22—24no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22C2×D4×D1711yescharacteristic
241C22C2×#6325412yes—
242C22#1058769111yescharacteristic
243C22#1058769112yes—
244C22#1058769711yescharacteristic
245C4—24no—
246C4—41no—
247C4—41no—
248C4—344no—
249C4—682no—
250C4—682no—
251C4—682no—
252C2—42no—
253C2—82no—
254C2#1176338611yescharacteristic
255C2#1176342612yes—
256C2#1176344612yes—
257C2#1176354511yescharacteristic
258C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC17×#392C2
hypercenter#392D17
hypocenterC17#2712
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587622C2semidirect
2. #11762734C2semidirect
3. #11762723C2semidirect
4. #11762721C2semidirect
5. #11762713C2semidirect
6. #11762702C2semidirect
7. #11762689C2semidirect
8. #11762681C2semidirect
58. C22⋊Dic17D4semidirect
161. C17#2712semidirect

Maximal subgroups

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

subgroupindex
C2×#105876222
#117627342
#117627232
#117627212
#117627132
#117627022
#117626892
#117626812
C17×#3922
#271217