Finite Group Databasebeta

← #14532055

Subgroups — #14532055

Order 2176.

528 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 159 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×Q8×Dic17C211yesmaximal, characteristic, complemented
2C2×#10587578C211yesmaximal, characteristic
3#11762748C211yesmaximal, characteristic, complemented
4#11762746C211yesmaximal, characteristic
5#11762741C212yesmaximal, complemented
6#11762731C212yesmaximal
7#11762720C211yesmaximal, characteristic
8C4⋊C4×Dic17C212yesmaximal, complemented
9#11762660C211yesmaximal, characteristic, complemented
10#11762656C212yesmaximal
11C17×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16Q8×Dic17—24no—
17C2×#63227C2212yes—
18C2×#63227C2211yescharacteristic
19C2×#63227C2211yescharacteristic
20C2×#63227C2212yes—
21C2×C4⋊Dic17C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2212yes—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×#63225C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27C2×C4×Dic17C2212yes—
28C2×C4×Dic17C2212yes—
29C2×C4×Dic17C2211yescharacteristic
30#10587578—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37C23×D17C2311yescharacteristic
38C2×Q8×C17—22no—
39C2×Q8×C17—24no—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2312yes—
44C22×C4×C17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×C4×D17—22no—
51C2×C4×D17—22no—
52C2×C4×D17Q814yescomplemented
53C4⋊C4×C17—22no—
54C4⋊C4×C17—24no—
55#63227—22no—
56#63227—24no—
57#63227—24no—
58#63227—22no—
59C4⋊Dic17—22no—
60C4⋊Dic17—24no—
61C4⋊Dic17—22no—
62C4⋊Dic17—24no—
63#63225—24no—
64C4×Dic17—22no—
65C4×Dic17—24no—
66C4×Dic17—24no—
67C4×Dic17D414yes—
68C23×C17C2411yescharacteristic
69C22×D17—21no—
70C22×D17—21no—
71C22×D17—22no—
72C22×D17—22no—
73C22×D17C2×Q812yes—
74Q8×C17—28no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17C2×D412yes—
85C2×C4×C17C2×Q812yes—
86C2×C4×C17#4114yes—
87C2×C4×C17#4114yes—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17C2×D412yes—
98C2×Dic17C2×D412yes—
99C2×Dic17C2×Q812yes—
100C2×Dic17#4112yes—
101C4×D17—28no—
102#2396—171nomaximal, Sylow
103C22×C17C22×D411yescharacteristic
104C22×C17C22×Q811yescharacteristic
105C22×C17C2×#4112yes—
106C22×C17C2×#4112yes—
107C22×C17C2×#4111yescharacteristic
108C2×D17—22no—
109C2×D17—24no—
110C2×D17—41no—
111C2×D17—42no—
112C2×D17—41no—
113C4×C17—22no—
114C4×C17—22no—
115C4×C17—24no—
116C4×C17—24no—
117C4×C17—41no—
118Dic17—22no—
119Dic17—42no—
120Dic17—42no—
121Dic17—41no—
122Dic17C22⋊Q814yes—
123C2×C4×Q8—171no—
124C2×#117—171no—
125#400—171no—
126#399—171no—
127#398—172no—
128#396—172no—
129#395D1711yescharacteristic, complemented
130#393—171no—
131C4×C4⋊C4—172no—
132C4×C4⋊C4—171no—
133C4×C22⋊C4—172no—
134C2×C17C2×C22⋊Q811yescharacteristic
135C2×C17#52511yescharacteristic
136C2×C17#52912yes—
137C2×C17#53211yescharacteristic
138C2×C17#53312yes—
139D17—42no—
140C22×Q8C2×D1711yescharacteristic
141C23×C4—171no—
142C4×Q8—344no—
143#117—344no—
144C2×C4⋊C4—171no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4—171no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4—172no—
149C2×C4⋊C4C2×D1711yescharacteristic
150C2×C4⋊C4C2×D1712yes—
151C2×C22⋊C4—171no—
152C2×C22⋊C4—171no—
153C2×C22⋊C4—172no—
154C2×C22⋊C4—172no—
155C2×C42—171no—
156C2×C42—171no—
157C2×C42—172no—
158C2×C42—172no—
159#95—172no—
160#95—172no—
161#95—171no—
162#95—172no—
163#95C2×D1712yes—
164#95C2×D1711yescharacteristic
165C17#239611yescharacteristic, complemented, Sylow
166C24—171no—
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—171no—
170C22×C4—171no—
171C22×C4—174no—
172C22×C4—172no—
173C22×C4—171no—
174C22×C4—172no—
175C22×C4—342no—
176C22×C4—342no—
177C22×C4C22×D1711yescharacteristic
178C22×C4C22×D1711yescharacteristic
179C22×C4C22×D1712yes—
180C22×C4C22×D1711yescharacteristic
181C22×C4C22×D1712yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—342no—
185C4⋊C4—344no—
186C4⋊C4—342no—
187C4⋊C4—344no—
188C4⋊C4—344no—
189C22⋊C4—342no—
190C22⋊C4—342no—
191C22⋊C4—344no—
192C22⋊C4—344no—
193C42—174no—
194C42—342no—
195C42—344no—
196C42—344no—
197C23—172no—
198C23—341no—
199C23—342no—
200C23—342no—
201C23—341no—
202C23C23×D1711yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—172no—
214C2×C4—172no—
215C2×C4—172no—
216C2×C4—172no—
217C2×C4—348no—
218C2×C4—342no—
219C2×C4—341no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4D4×D1712yes—
228C2×C4#6325414yes—
229C2×C4Q8×D1712yes—
230C2×C4#6325614yes—
231C22—342no—
232C22—344no—
233C22—681no—
234C22—682no—
235C22—681no—
236C22C2×D4×D1711yescharacteristic
237C22C2×#6325412yes—
238C22C2×Q8×D1711yescharacteristic
239C22C2×#6325612yes—
240C22#41×D1711yescharacteristic
241C4—22no—
242C4—22no—
243C4—24no—
244C4—24no—
245C4—41no—
246C4—174no—
247C4—342no—
248C4—682no—
249C4—681no—
250C4—682no—
251C2—682no—
252C2C22⋊Q8×D1711yescharacteristic
253C2#1176331511yescharacteristic
254C2#1176334512yes—
255C2#1176336011yescharacteristic
256C2#1176336412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#395C2
prime radicalC23×D17C23
hypercenter#395D17
hypocenterC17#2396
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic17C2semidirect
3. #11762748C2semidirect
5. #11762741C2semidirect
8. C4⋊C4×Dic17C2semidirect
9. #11762660C2semidirect
11. C17×#395C2semidirect
52. C2×C4×D17Q8semidirect
129. #395D17semidirect
165. C17#2396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic172
C2×#105875782
#117627482
#117627462
#117627412
#117627312
#117627202
C4⋊C4×Dic172
#117626602
#117626562
C17×#3952
#239617