Finite Group Databasebeta

← #14534455

Subgroups — #14534455

Order 2176.

540 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 137 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×#10587563C212yesmaximal, complemented
2#11762739C211yesmaximal, characteristic, complemented
3#11762732C212yesmaximal, complemented
4#11762702C212yesmaximal, complemented
5#11762691C212yesmaximal, complemented
6#11762689C212yesmaximal
7#11762681C212yesmaximal, complemented
8#11762674C211yesmaximal, characteristic, complemented
9C17×#394C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×Dic17C2212yes—
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C22⋊C4×C17C2212yes—
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2211yescharacteristic
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×C4×Dic17C2212yes—
24C2×C4×Dic17C2211yescharacteristic
25#10587563—28no—
26C17×#95C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32C24×C17C2312yes—
33C2×D4×C17—24no—
34C2×D4×C17—41no—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×Dic17—24no—
40C22×Dic17—24no—
41C22×Dic17—24no—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C4⋊C4×C17—41no—
47C22⋊C4×C17—24no—
48C22⋊C4×C17—24no—
49C22⋊Dic17—24no—
50C22⋊Dic17—24no—
51C22⋊Dic17—42no—
52C4⋊Dic17—41no—
53C4⋊Dic17—41no—
54#63225—24no—
55#63225—24no—
56#63225—24no—
57C4×Dic17—24no—
58C4×Dic17D414yescomplemented
59C23×C17—22no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17—22no—
64C23×C17—22no—
65C23×C17#4114yes—
66C23×C17C2411yescharacteristic
67D4×C17—44no—
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—21no—
77C2×C4×C17C2×D412yes—
78C2×Dic17—216no—
79C2×Dic17—22no—
80C2×Dic17—22no—
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—
90C2×Dic17#4114yes—
91#2609—171nomaximal, Sylow
92C22×C17—24no—
93C22×C17—24no—
94C22×C17—24no—
95C22×C17—42no—
96C22×C17—42no—
97C22×C17—42no—
98C22×C17—42no—
99C22×C17C22×D411yescharacteristic
100C22×C17C2×#4112yes—
101C22×C17C2×#4112yes—
102C22×C17H(5,𝔽2)11yescharacteristic
103C22×C172−1+411yescharacteristic
104C4×C17—22no—
105C4×C17—42no—
106C4×C17—41no—
107C4×C17—41no—
108Dic17—24no—
109Dic17—24no—
110Dic17—42no—
111Dic17—42no—
112C2×#117—172no—
113#398—171no—
114#396—172no—
115#394D1711yescharacteristic
116#388—172no—
117#387—172no—
118#386—172no—
119#385—172no—
120#382—171no—
121C2×C17—44no—
122C2×C17#52512yes—
123C2×C17#53711yescharacteristic
124C2×C17#55111yescharacteristic
125C2×C17#55212yes—
126C2×C17#55511yescharacteristic
127C22×D4C2×D1711yescharacteristic
128C23×C4—172no—
129#117—348no—
130C2×C4⋊C4—172no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—172no—
133C2×C4⋊C4—171no—
134C2×C4⋊C4—171no—
135C2×C4⋊C4C2×D1711yescharacteristic
136C2×C22⋊C4—172no—
137C2×C22⋊C4—172no—
138C2×C22⋊C4—172no—
139C2×C22⋊C4C2×D1712yes—
140C2×C22⋊C4C2×D1712yes—
141C2×C42—171no—
142C2×C42—172no—
143#95—172no—
144#95—172no—
145#95—171no—
146#95—172no—
147#95—172no—
148#95C2×D1711yescharacteristic
149C17#260911yescharacteristic, complemented, Sylow
150C24C22×D1712yes—
151C2×D4—24no—
152C2×D4—41no—
153C22×C4—172no—
154C22×C4—172no—
155C22×C4—172no—
156C22×C4—172no—
157C22×C4—344no—
158C22×C4—344no—
159C22×C4—344no—
160C22×C4C22×D1711yescharacteristic
161C22×C4C22×D1711yescharacteristic
162C22×C4C22×D1712yes—
163C22×C4C22×D1711yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—344no—
166C4⋊C4—344no—
167C4⋊C4—344no—
168C4⋊C4—681no—
169C4⋊C4—681no—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—344no—
173C22⋊C4—344no—
174C22⋊C4—682no—
175C42—174no—
176C42—344no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23#6325414yes—
184C23C23×D1711yescharacteristic
185D4—44no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—174no—
196C2×C4—174no—
197C2×C4—3416no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4D4×D1712yes—
209C22—24no—
210C22—24no—
211C22—24no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22C2×D4×D1711yescharacteristic
217C22C2×#6325412yes—
218C22#1058769111yescharacteristic
219C22#41×D1712yes—
220C22#1058769711yescharacteristic
221C4—22no—
222C4—41no—
223C4—41no—
224C4—42no—
225C4—344no—
226C4—344no—
227C4—682no—
228C4—682no—
229C2—44no—
230C2#1176331912yes—
231C2#1176338611yescharacteristic
232C2#1176349011yescharacteristic
233C2#1176349612yes—
234C2#1176352111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×C17C22
hypercenter#394D17
hypocenterC17#2609
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587563C2semidirect
2. #11762739C2semidirect
3. #11762732C2semidirect
4. #11762702C2semidirect
5. #11762691C2semidirect
7. #11762681C2semidirect
8. #11762674C2semidirect
58. C4×Dic17D4semidirect
149. C17#2609semidirect

Maximal subgroups

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

subgroupindex
C2×#105875632
#117627392
#117627322
#117627022
#117626912
#117626892
#117626812
#117626742
C17×#3942
#260917