Finite Group Databasebeta

← #14535754

Subgroups — #14535754

Order 2176.

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

normal subgroups only

This group has 512 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×#10587582C212yesmaximal, complemented
2#11762750C212yesmaximal, complemented
3#11762738C212yesmaximal, complemented
4#11762726C211yesmaximal, characteristic
5#11762714C211yesmaximal, characteristic, complemented
6#11762699C212yesmaximal, complemented
7#11762687C212yesmaximal
8#11762680C212yesmaximal, complemented
9C17×#398C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×C4×Dic17C2211yescharacteristic
24#10587582—28no—
25C17×#95C2211yescharacteristic
26C17×#95C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32C23×D17C2312yes—
33C22×C4×C17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C2×D68—24no—
42C2×D68—41no—
43C2×C4×D17—24no—
44C2×C4×D17—42no—
45C2×C4×D17—42no—
46C2×Dic34—41no—
47C2×Dic34—41no—
48C2×Dic34D414yescomplemented
49C4⋊C4×C17—24no—
50C4⋊C4×C17—41no—
51C4⋊C4×C17—41no—
52#63227—24no—
53#63227—24no—
54#63227—24no—
55#63227—24no—
56#63227—42no—
57#63225—24no—
58#63225—24no—
59C4×Dic17—41no—
60C23×C17C2411yescharacteristic
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17—22no—
67C22×D17#4114yes—
68C2×C4×C17—21no—
69C2×C4×C17—21no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17C2×D412yes—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17C2×D414yes—
90D68—44no—
91C4×D17—48no—
92Dic34—44no—
93#2728—171nomaximal, Sylow
94C22×C17C22×D411yescharacteristic
95C22×C17C2×#4112yes—
96C22×C17H(5,𝔽2)11yescharacteristic
97C22×C17H(5,𝔽2)11yescharacteristic
98C22×C172−1+412yes—
99C2×D17—24no—
100C2×D17—24no—
101C2×D17—24no—
102C2×D17—42no—
103C2×D17—42no—
104C2×D17—42no—
105C2×D17—42no—
106C4×C17—22no—
107C4×C17—42no—
108C4×C17—42no—
109C4×C17—41no—
110C4×C17—41no—
111Dic17—24no—
112Dic17—42no—
113Dic17—42no—
114C2×C22⋊Q8—172no—
115#400—172no—
116#398D1711yescharacteristic, complemented
117#397—172no—
118#394—171no—
119#391—171no—
120#388—172no—
121#386—172no—
122#385—172no—
123C2×C17#53411yescharacteristic
124C2×C17#53912yes—
125C2×C17#54812yes—
126C2×C17#55111yescharacteristic
127C2×C17#56311yescharacteristic
128D17—44no—
129C22×Q8—171no—
130C22×D4—171no—
131C23×C4—172no—
132C22⋊Q8—348no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4—172no—
135C2×C4⋊C4C2×D1712yes—
136C2×C4⋊C4C2×D1711yescharacteristic
137C2×C4⋊C4C2×D1711yescharacteristic
138C2×C22⋊C4—172no—
139C2×C22⋊C4—172no—
140C2×C22⋊C4—172no—
141C2×C22⋊C4—172no—
142C2×C22⋊C4—172no—
143C2×C42—171no—
144#95—172no—
145#95—172no—
146#95—172no—
147#95—172no—
148#95—171no—
149#95C2×D1712yes—
150#95C2×D1711yescharacteristic
151C17#272811yescharacteristic, complemented, Sylow
152C24—172no—
153C2×Q8—174no—
154C2×Q8—681no—
155C2×Q8—681no—
156C2×D4—344no—
157C2×D4—681no—
158C22×C4—172no—
159C22×C4—172no—
160C22×C4—172no—
161C22×C4—344no—
162C22×C4—682no—
163C22×C4—682no—
164C22×C4C22×D1711yescharacteristic
165C22×C4C22×D1711yescharacteristic
166C22×C4C22×D1712yes—
167C22×C4C22×D1712yes—
168C22×C4C22×D1711yescharacteristic
169C4⋊C4—24no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—344no—
173C4⋊C4—344no—
174C22⋊C4—344no—
175C22⋊C4—344no—
176C22⋊C4—344no—
177C22⋊C4—344no—
178C22⋊C4—682no—
179C42—681no—
180C23—174no—
181C23—342no—
182C23—342no—
183C23—342no—
184C23—342no—
185C23—342no—
186C23—342no—
187C23C23×D1711yescharacteristic
188Q8—684no—
189D4—684no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—174no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—688no—
212C2×C4D4×D1712yes—
213C22—344no—
214C22—344no—
215C22—344no—
216C22—682no—
217C22—682no—
218C22—682no—
219C22—682no—
220C22C2×D4×D1711yescharacteristic
221C22#1058769111yescharacteristic
222C22#1058769412yes—
223C22#41×D1712yes—
224C22#1058769611yescharacteristic
225C4—22no—
226C4—42no—
227C4—41no—
228C4—41no—
229C4—42no—
230C4—344no—
231C4—682no—
232C4—682no—
233C2—684no—
234C2#1176336711yescharacteristic
235C2#1176340712yes—
236C2#1176346912yes—
237C2#1176348711yescharacteristic
238C2#1176355811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC22×D68C22
hypercenter#398D17
hypocenterC17#2728
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#10587582C2semidirect
2. #11762750C2semidirect
3. #11762738C2semidirect
5. #11762714C2semidirect
6. #11762699C2semidirect
8. #11762680C2semidirect
9. C17×#398C2semidirect
48. C2×Dic34D4semidirect
116. #398D17semidirect
151. C17#2728semidirect

Maximal subgroups

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

subgroupindex
C2×#105875822
#117627502
#117627382
#117627262
#117627142
#117626992
#117626872
#117626802
C17×#3982
#272817