Finite Group Databasebeta

← #14538317

Subgroups — #14538317

Order 2176.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587637C211yesmaximal, characteristic
2C22×Dic68C211yesmaximal, characteristic, complemented
3#11763380C211yesmaximal, characteristic
4#11763286C211yesmaximal, characteristic, complemented
5#11763030C214yesmaximal, complemented
6#11763018C214yesmaximal, complemented
7#11762835C211yesmaximal, characteristic, complemented
8#11762817C211yesmaximal, characteristic, complemented
9C17×#416C211yesmaximal, characteristic
10C2×#41×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C22×C8×C17C2211yescharacteristic
14C17×#117C2211yescharacteristic
15#10587637—24no—
16#10587630C2212yes—
17C2×#63231C2212yes—
18#10587622C2212yes—
19#10587621—22no—
20C2×#63229C2212yes—
21#10587612C2211yescharacteristic
22#10587610—22no—
23C2×#63223C2211yescharacteristic
24C2×Dic68—24no—
25C2×Dic68C2214yes—
26#10587573C2214yes—
27#10587563—22no—
28C4×Dic34—22no—
29C17×#103C2212yes—
30C17×#102C2212yes—
31#10587485C2214yes—
32#10587481C2214yes—
33#41×C17—22no—
34#41×C17—22no—
35C2×Q8×C17C2311yescharacteristic
36C2×D4×C17—21no—
37C2×D4×C17C2311yescharacteristic
38C22×C4×C17—21no—
39C22×C4×C17C2311yescharacteristic
40C22×Dic17—21no—
41C22×Dic17—21no—
42C2×Dic34—21no—
43C2×Dic34—22no—
44C2×Dic34—22no—
45C2×Dic34—22no—
46C2×Dic34—21no—
47C2×Dic34D414yes—
48C2×Dic34C2312yes—
49C2×Dic34C2312yes—
50C2×C8×C17—22no—
51C2×C8×C17C2312yes—
52C4⋊C4×C17C2312yes—
53C22⋊C4×C17—21no—
54C42×C17—21no—
55C22⋊Dic17—24no—
56C22⋊Dic17D412yes—
57#63231—24no—
58#63229—24no—
59C4⋊Dic17D412yes—
60#63225—24no—
61#63225—24no—
62C4×Dic17—24no—
63C4×Dic17C2312yes—
64#63223—22no—
65C2×#3622C2312yes—
66Dic68—28no—
67C23×C17—21no—
68C23×C17C2×D411yescharacteristic
69Q8×C17—21no—
70Q8×C17—21no—
71D4×C17—21no—
72D4×C17—21no—
73D4×C17—42no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—42no—
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2×D412yes—
79C2×C4×C17#4112yes—
80C2×C4×C17C2411yescharacteristic
81C2×Dic17—24no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—42no—
85C2×Dic17C2×D414yes—
86Dic34—24no—
87Dic34—22no—
88Dic34—22no—
89Dic34—22no—
90Dic34—42no—
91Dic34C2×D414yes—
92C8×C17—22no—
93#3622—22no—
94#3044—171nomaximal, Sylow
95C22×C17—22no—
96C22×C17—41no—
97C22×C17—42no—
98C22×C17#12112yes—
99C22×C17C22×D411yescharacteristic
100C4×C17—41no—
101C4×C17—42no—
102C4×C17#12112yes—
103C4×C17C22×D411yescharacteristic
104C4×C17C2×#4111yescharacteristic
105Dic17—24no—
106Dic17—24no—
107Dic17—42no—
108C2×#137—171no—
109C22×Q16—171no—
110#536—171no—
111#519—171no—
112#464—174no—
113#462—174no—
114#424—171no—
115#419—171no—
116#416D1711yescharacteristic
117C2×C17—22no—
118C2×C17—81no—
119C2×C17C2×#12111yescharacteristic
120C2×C17#57812yes—
121C2×#41C2×D1711yescharacteristic
122C22×Q8—171no—
123C22×Q8—171no—
124#137—344no—
125C2×Q16—172no—
126C2×Q16—174no—
127C2×Q16—344no—
128C2×SD16—172no—
129C2×M16—171no—
130C22×C8C2×D1711yescharacteristic
131C4⋊Q8—172no—
132#124—172no—
133#123—342no—
134C22⋊Q8—342no—
135C4×Q8—174no—
136C4×Q8—342no—
137#117—171no—
138#117—342no—
139#117C2×D1711yescharacteristic
140C4⋊C8—174no—
141#103—174no—
142#103C2×D1712yes—
143#102C2×D1712yes—
144C17#304411yescharacteristic, complemented, Sylow
145#41—22no—
146#41—22no—
147C2×Q8—174no—
148C2×Q8—172no—
149C2×Q8—172no—
150C2×Q8—342no—
151C2×Q8—341no—
152C2×Q8—341no—
153C2×Q8—342no—
154C2×Q8—342no—
155C2×Q8C22×D1711yescharacteristic
156C2×D4—21no—
157C2×D4C22×D1711yescharacteristic
158C22×C4—21no—
159C22×C4—341no—
160C22×C4—341no—
161C22×C4C22×D1711yescharacteristic
162Q16—348no—
163Q16—344no—
164SD16—344no—
165M16—342no—
166C2×C8—22no—
167C2×C8—172no—
168C2×C8C22×D1712yes—
169C4⋊C4—172no—
170C4⋊C4—344no—
171C4⋊C4—344no—
172C4⋊C4C22×D1712yes—
173C22⋊C4—21no—
174C22⋊C4—172no—
175C22⋊C4—344no—
176C42—21no—
177C42—172no—
178C42—344no—
179C23—21no—
180C23D4×D1711yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—174no—
184Q8—342no—
185Q8—344no—
186Q8—342no—
187Q8—342no—
188Q8—682no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—174no—
196C2×C4—342no—
197C2×C4—344no—
198C2×C4—342no—
199C2×C4—682no—
200C2×C4#6325212yes—
201C2×C4D4×D1712yes—
202C2×C4D4×D1711yescharacteristic
203C2×C4C23×D1711yescharacteristic
204C8—22no—
205C8—342no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1058757012yes—
210C22C2×D4×D1711yescharacteristic
211C4—41no—
212C4—42no—
213C4—344no—
214C4—344no—
215C4—682no—
216C4#1058757012yes—
217C4C2×#6325211yescharacteristic
218C4C2×D4×D1711yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1058757011yescharacteristic
222C2#1176362312yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#416C2
socleC22×C17C22×D4
prime radicalC2×#41×C17C22
hypercenter#416D17
hypocenterC17#3044
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic68C2semidirect
4. #11763286C2semidirect
5. #11763030C2semidirect
6. #11763018C2semidirect
7. #11762835C2semidirect
8. #11762817C2semidirect
144. C17#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#105876372
C22×Dic682
#117633802
#117632862
#117630302
#117630182
#117628352
#117628172
C17×#4162
#304417