Finite Group Databasebeta

← #14540016

Subgroups — #14540016

Order 2176.

362 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 125 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×#10587555C211yesmaximal, characteristic, complemented
2#11763343C211yesmaximal, characteristic
3#11763218C212yesmaximal, complemented
4#11763169C212yesmaximal, complemented
5#11763145C212yesmaximal, complemented
6#11763128C212yesmaximal
7#11763095C212yesmaximal, complemented
8#11763090C212yesmaximal, complemented
9C17×#432C211yesmaximal, characteristic, complemented
10C22×Dic34—21no—
11C2×C42×C17C2211yescharacteristic
12#10587616C2211yescharacteristic
13#10587615—21no—
14C4×C22⋊D17—21no—
15C2×C4⋊Dic17—21no—
16C2×C4⋊Dic17C2212yes—
17#10587610C2211yescharacteristic
18#10587562—21no—
19#10587561C2211yescharacteristic
20C4×D68C2211yescharacteristic
21#10587558—21no—
22#10587556C2211yescharacteristic
23#10587555—21no—
24#10587555—21no—
25#10587555C2212yes—
26#10587555C2212yes—
27C4×Dic34C2211yescharacteristic
28C17×C4⋊C8C2212yes—
29C17×C22⋊C8C2212yes—
30C17×#97C2212yes—
31#10587492C2214yes—
32#10587489C2214yes—
33#10587488C2214yes—
34#10587487C2214yes—
35C22×C4×C17D411yescharacteristic
36C22×C4×C17D411yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C2×C22⋊D17—21no—
39C22×Dic17—22no—
40C2×D68C2311yescharacteristic
41C2×C4×D17—21no—
42C2×Dic34—22no—
43C2×Dic34—22no—
44C2×Dic34—44no—
45C2×Dic34C2311yescharacteristic
46C2×C8×C17C2314yes—
47C42×C17D411yescharacteristic
48C42×C17D411yescharacteristic
49C42×C17C2311yescharacteristic
50C42×C17C2311yescharacteristic
51C22⋊Dic17—21no—
52#63227—21no—
53#63227—21no—
54#63227—21no—
55#63227—21no—
56C4⋊Dic17—22no—
57C4⋊Dic17—22no—
58C4⋊Dic17—22no—
59C4⋊Dic17C2314yes—
60C4⋊Dic17C2311yescharacteristic
61C4⋊Dic17C2311yescharacteristic
62#63225—21no—
63#63225—21no—
64#63225—21no—
65#63225—21no—
66C4×Dic17—21no—
67C23×C17C2×D411yescharacteristic
68C22×D17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2×D411yescharacteristic
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2×D411yescharacteristic
76C2×C4×C17C2×D411yescharacteristic
77C2×C4×C17C2411yescharacteristic
78C22⋊D17—42no—
79C2×Dic17—21no—
80C2×Dic17—21no—
81C2×Dic17—24no—
82C2×Dic17—21no—
83C2×Dic17—44no—
84D68—22no—
85C4×D17—42no—
86Dic34—22no—
87Dic34—44no—
88Dic34—44no—
89C8×C17—24no—
90#3239—171nomaximal, Sylow
91C22×C17—21no—
92C22×C17—21no—
93C22×C17#13712yes—
94C22×C17C22×D411yescharacteristic
95C2×D17—41no—
96C2×D17—41no—
97C2×D17—41no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—21no—
102C4×C17AGL(1,ℤ/8)12yes—
103C4×C172−1+412yes—
104Dic17—41no—
105Dic17—44no—
106Dic17—41no—
107Dic17—41no—
108C2×C4⋊Q8—171no—
109#529—171no—
110#501—172no—
111#489—172no—
112#484—172no—
113#481—172no—
114#475—172no—
115#474—172no—
116#432D1711yescharacteristic, complemented
117C2×C17—22no—
118C2×C17#53611yescharacteristic
119C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C17C2×#13711yescharacteristic
121D17—81no—
122C22×Q8—341no—
123C4⋊Q8—172no—
124C4⋊Q8—172no—
125C4⋊Q8—341no—
126C4⋊Q8—341no—
127#126—341no—
128#125—171no—
129#124—171no—
130#123—341no—
131C22⋊Q8—171no—
132#121—171no—
133C4×Q8—171no—
134C4×D4—171no—
135C4×D4—341no—
136#117—341no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—341no—
139C2×C42C2×D1711yescharacteristic
140#107—174no—
141#106—174no—
142C4⋊C8C2×D1712yes—
143#103—174no—
144#102—174no—
145C22⋊C8C2×D1712yes—
146#97C2×D1712yes—
147C17#323911yescharacteristic, complemented, Sylow
148C2×Q8—171no—
149C2×Q8—342no—
150C2×Q8—342no—
151C2×Q8—684no—
152C2×D4—171no—
153C2×D4—341no—
154C22×C4—342no—
155C22×C4—341no—
156C22×C4D6811yescharacteristic
157C22×C4D6811yescharacteristic
158C22×C4C22×D1711yescharacteristic
159C2×C8C22×D1714yes—
160C4⋊C4—174no—
161C4⋊C4—171no—
162C4⋊C4—171no—
163C4⋊C4—342no—
164C4⋊C4—342no—
165C4⋊C4—342no—
166C4⋊C4—341no—
167C4⋊C4—341no—
168C4⋊C4—341no—
169C4⋊C4—341no—
170C22⋊C4—341no—
171C22⋊C4—341no—
172C22⋊C4—341no—
173C22⋊C4—341no—
174C22⋊C4—341no—
175C42—341no—
176C42D6811yescharacteristic
177C42D6811yescharacteristic
178C42C22×D1711yescharacteristic
179C42C22×D1711yescharacteristic
180C23—341no—
181C23C2×D6811yescharacteristic
182Q8—342no—
183Q8—684no—
184Q8—684no—
185D4—342no—
186D4—682no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—341no—
191C2×C4—341no—
192C2×C4—341no—
193C2×C4—344no—
194C2×C4—684no—
195C2×C4—682no—
196C2×C4C2×D6811yescharacteristic
197C2×C4C2×D6811yescharacteristic
198C2×C4C2×D6811yescharacteristic
199C2×C4C2×D6811yescharacteristic
200C2×C4C2×D6811yescharacteristic
201C2×C4C23×D1711yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—681no—
206C22—681no—
207C22—681no—
208C22#1058759512yes—
209C22C22×D6811yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—684no—
215C4—681no—
216C4—681no—
217C4—681no—
218C4#1058759412yes—
219C4#1058769712yes—
220C2—22no—
221C2—1361no—
222C2#1176338511yescharacteristic
223C2C2×#1058759411yescharacteristic
224C2C2×#1058759511yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#432C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenter#432D17
hypocenterC17#3239
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. C2×#10587555C2semidirect
3. #11763218C2semidirect
4. #11763169C2semidirect
5. #11763145C2semidirect
7. #11763095C2semidirect
8. #11763090C2semidirect
9. C17×#432C2semidirect
116. #432D17semidirect
147. C17#3239semidirect

Maximal subgroups

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

subgroupindex
C2×#105875552
#117633432
#117632182
#117631692
#117631452
#117631282
#117630952
#117630902
C17×#4322
#323917