Finite Group Databasebeta

← #14534548

Subgroups — #14534548

Order 2176.

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

normal subgroups only

This group has 612 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×#10587633C211yesmaximal, characteristic, complemented
2C2×#10587632C211yesmaximal, characteristic, complemented
3C2×#10587581C211yesmaximal, characteristic
4C2×#10587558C211yesmaximal, characteristic, complemented
5#11762733C214yesmaximal, complemented
6#11762727C214yesmaximal, complemented
7#11762711C211yesmaximal, characteristic, complemented
8#11762678C211yesmaximal, characteristic, complemented
9C17×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×D68C2211yescharacteristic
12C22×D68C2212yes—
13C22×C4×D17C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C42×C17C2211yescharacteristic
16#10587633—24no—
17#10587632—24no—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2214yes—
21C2×#63227C2212yes—
22C2×#63227C2214yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2214yes—
27C2×C4×Dic17C2211yescharacteristic
28#10587581—24no—
29#10587558—24no—
30C17×#95C2214yes—
31C23×D17C2312yes—
32C23×D17C2311yescharacteristic
33C2×Q8×C17—22no—
34C2×Q8×C17—22no—
35C2×Q8×C17—22no—
36C22×C4×C17C2312yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2312yes—
39C22×C4×C17C2311yescharacteristic
40C22×Dic17C2312yes—
41C22×Dic17C2311yescharacteristic
42C22×Dic17C2312yes—
43C2×D68—24no—
44C2×D68—22no—
45C2×D68—22no—
46C2×D68—22no—
47C2×D68—44no—
48C2×C4×D17—22no—
49C2×C4×D17—22no—
50C2×C4×D17D414yes—
51C4⋊C4×C17—22no—
52C42×C17—22no—
53#63227—24no—
54#63227—24no—
55#63227—24no—
56#63227—28no—
57#63227—44no—
58C4⋊Dic17—22no—
59#63225—24no—
60#63225—24no—
61#63225—44no—
62C4×Dic17D414yescomplemented
63C23×C17C2411yescharacteristic
64C22×D17—21no—
65C22×D17—21no—
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—22no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—24no—
72C22×D17—24no—
73C22×D17C2×D412yes—
74Q8×C17—44no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—24no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17C2×D412yes—
84C2×C4×C17C2×D412yes—
85C2×C4×C17#4112yes—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—24no—
92C2×Dic17C2×D412yes—
93C2×Dic17C2×D414yes—
94D68—48no—
95D68—44no—
96C4×D17—28no—
97#2620—171nomaximal, Sylow
98C22×C17C22×D411yescharacteristic
99C22×C17C22×D411yescharacteristic
100C22×C17C2×#4111yescharacteristic
101C22×C17H(5,𝔽2)12yes—
102C22×C172−1+412yes—
103C2×D17—22no—
104C2×D17—22no—
105C2×D17—22no—
106C2×D17—42no—
107C2×D17—42no—
108C2×D17—42no—
109C2×D17—42no—
110C2×D17—42no—
111C2×D17—44no—
112C2×D17—44no—
113C4×C17—42no—
114C4×C17—42no—
115C4×C17—42no—
116C4×C17#12114yes—
117Dic17—22no—
118Dic17—24no—
119Dic17—42no—
120C2×#124—171no—
121C2×C22⋊Q8—171no—
122C2×#121—171no—
123C2×#117—171no—
124#396—174no—
125#394—174no—
126#391D1711yescharacteristic, complemented
127#390—171no—
128#384—171no—
129C2×C17C2×#12111yescharacteristic
130C2×C17#53511yescharacteristic
131C2×C17#53611yescharacteristic
132C2×C17#53712yes—
133C2×C17#54212yes—
134D17—42no—
135D17—82no—
136C22×Q8C2×D1711yescharacteristic
137C22×D4—171no—
138C22×D4—172no—
139C23×C4—171no—
140#124—344no—
141C22⋊Q8—344no—
142#121—344no—
143#117—344no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—171no—
146C2×C4⋊C4—172no—
147C2×C4⋊C4—174no—
148C2×C4⋊C4C2×D1711yescharacteristic
149C2×C22⋊C4—172no—
150C2×C22⋊C4—172no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4—174no—
153C2×C22⋊C4—174no—
154C2×C42—171no—
155C2×C42C2×D1711yescharacteristic
156#95C2×D1714yes—
157C17#262011yescharacteristic, complemented, Sylow
158C24—171no—
159C24—172no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×D4—344no—
164C2×D4—342no—
165C2×D4—342no—
166C2×D4—342no—
167C2×D4—684no—
168C22×C4—172no—
169C22×C4—171no—
170C22×C4—174no—
171C22×C4—172no—
172C22×C4—342no—
173C22×C4—342no—
174C22×C4C22×D1712yes—
175C22×C4C22×D1712yes—
176C22×C4C22×D1712yes—
177C22×C4C22×D1711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—344no—
180C4⋊C4—344no—
181C4⋊C4—342no—
182C4⋊C4—684no—
183C22⋊C4—344no—
184C22⋊C4—344no—
185C22⋊C4—348no—
186C22⋊C4—344no—
187C22⋊C4—684no—
188C42—22no—
189C42—174no—
190C23—172no—
191C23—341no—
192C23—341no—
193C23—342no—
194C23—342no—
195C23—342no—
196C23—342no—
197C23—342no—
198C23—344no—
199C23—344no—
200C23C23×D1711yescharacteristic
201Q8—44no—
202D4—688no—
203D4—684no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—174no—
213C2×C4—172no—
214C2×C4—341no—
215C2×C4—341no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—348no—
219C2×C4—342no—
220C2×C4—344no—
221C2×C4#6325212yes—
222C2×C4D4×D1712yes—
223C2×C4D4×D1712yes—
224C22—342no—
225C22—342no—
226C22—342no—
227C22—682no—
228C22—682no—
229C22—682no—
230C22—682no—
231C22—682no—
232C22—684no—
233C22—684no—
234C22C2×#6325211yescharacteristic
235C22C2×D4×D1711yescharacteristic
236C22C2×D4×D1711yescharacteristic
237C22#1058769412yes—
238C22#1058769612yes—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—342no—
243C4—344no—
244C4—682no—
245C4#1058757014yes—
246C2—682no—
247C2—1362no—
248C2C2×#1058757011yescharacteristic
249C2#1176337711yescharacteristic
250C2#1176338411yescharacteristic
251C2#1176338912yes—
252C2#1176343112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#391C2
prime radicalC2×#10587581C2
hypercenter#391D17
hypocenterC17#2620
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#10587633C2semidirect
2. C2×#10587632C2semidirect
4. C2×#10587558C2semidirect
5. #11762733C2semidirect
6. #11762727C2semidirect
7. #11762711C2semidirect
8. #11762678C2semidirect
9. C17×#391C2semidirect
62. C4×Dic17D4semidirect
126. #391D17semidirect
157. C17#2620semidirect

Maximal subgroups

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

subgroupindex
C2×#105876332
C2×#105876322
C2×#105875812
C2×#105875582
#117627332
#117627272
#117627112
#117626782
C17×#3912
#262017