Finite Group Databasebeta

← #14538314

Subgroups — #14538314

Order 2176.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587628C211yesmaximal, characteristic
2C22×#63231C211yesmaximal, characteristic, complemented
3#11763385C211yesmaximal, characteristic
4#11763031C214yesmaximal, complemented
5#11763019C214yesmaximal, complemented
6#11762836C211yesmaximal, characteristic, complemented
7#11762817C211yesmaximal, characteristic, complemented
8#11762803C211yesmaximal, characteristic
9C17×#519C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C2×#63252C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C4×Q8×C17—22no—
14C4×Q8×C17C2214yes—
15C17×#117—22no—
16C17×#117C2211yescharacteristic
17C17×#117C2211yescharacteristic
18C2×#63231—24no—
19C2×#63231C2212yes—
20C2×#63231C2214yes—
21#10587628—24no—
22C2×#63230C2212yes—
23C2×#63223C2211yescharacteristic
24C22×#3622C2211yescharacteristic
25#10587583—22no—
26#10587572—22no—
27#10587561C2212yes—
28#10587555C2212yes—
29#10587481C2212yes—
30#10587481C2214yes—
31#10587480C2212yes—
32C4⋊#3622C2214yes—
33C2×Q8×C17—21no—
34C2×Q8×C17—22no—
35C2×Q8×C17D414yes—
36C2×Q8×C17C2312yes—
37C22×C4×C17—21no—
38C22×C4×C17C2311yescharacteristic
39C2×C22⋊D17—21no—
40C22×Dic17—21no—
41#63252—22no—
42#63252—22no—
43C2×D68C2311yescharacteristic
44C2×C4×D17—21no—
45C2×Dic34—22no—
46C2×Dic34—22no—
47C2×Dic34—21no—
48C2×Dic34C2312yes—
49C2×Dic34C2311yescharacteristic
50C4⋊C4×C17—24no—
51C4⋊C4×C17D412yes—
52C4⋊C4×C17C2312yes—
53C22⋊C4×C17—21no—
54C22⋊C4×C17D412yes—
55C42×C17—24no—
56C42×C17—21no—
57C42×C17C2312yes—
58#63231—24no—
59#63231—28no—
60#63230—24no—
61#63227—24no—
62C4⋊Dic17—24no—
63#63223—22no—
64C2×#3622—22no—
65C2×#3622C2312yes—
66C2×#3622C2312yes—
67C23×C17C2×D411yescharacteristic
68C22×D17—21no—
69Q8×C17—24no—
70Q8×C17—22no—
71Q8×C17C2×D414yes—
72C2×C4×C17—24no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17C2×D412yes—
76C2×C4×C17C2×D414yes—
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17#4112yes—
79C2×C4×C17C2411yescharacteristic
80C22⋊D17—42no—
81C2×Dic17—22no—
82C2×Dic17—21no—
83C2×Dic17—42no—
84D68—21no—
85D68—21no—
86C4×D17—42no—
87Dic34—22no—
88Dic34—21no—
89Dic34—22no—
90Dic34—21no—
91Dic34—42no—
92#3622—22no—
93#3622—22no—
94#3044—171nomaximal, Sylow
95C22×C17—22no—
96C22×C17#12112yes—
97C22×C17C22×D411yescharacteristic
98C2×D17—41no—
99C2×D17—42no—
100C4×C17—24no—
101C4×C17—24no—
102C4×C17—42no—
103C4×C17#12112yes—
104C4×C17C22×D411yescharacteristic
105C4×C17C2×#4111yescharacteristic
106Dic17—42no—
107Dic17—41no—
108C2×#137—171no—
109C22×Q16—171no—
110#536—171no—
111#519D1711yescharacteristic, complemented
112#464—174no—
113#462—174no—
114#424—171no—
115#419—171no—
116#416—171no—
117C2×C17—22no—
118C2×C17C2×#12111yescharacteristic
119C2×C17#57812yes—
120D17—81no—
121C2×#41—171no—
122C22×Q8—171no—
123C22×Q8C2×D1711yescharacteristic
124#137—344no—
125C2×Q16—174no—
126C2×Q16—172no—
127C2×Q16—344no—
128C2×SD16—172no—
129C2×M16—171no—
130C22×C8—171no—
131C4⋊Q8—172no—
132#124—172no—
133#123—342no—
134C22⋊Q8—342no—
135C4×Q8—22no—
136C4×Q8C2×D1714yes—
137#117—22no—
138#117C2×D1711yescharacteristic
139#117C2×D1711yescharacteristic
140C4⋊C8—174no—
141#103—172no—
142#103—174no—
143#102—172no—
144C17#304411yescharacteristic, complemented, Sylow
145#41—342no—
146#41—342no—
147C2×Q8—22no—
148C2×Q8—21no—
149C2×Q8—172no—
150C2×Q8—171no—
151C2×Q8—341no—
152C2×Q8—342no—
153C2×Q8—342no—
154C2×Q8D6814yes—
155C2×Q8C22×D1712yes—
156C2×D4—171no—
157C2×D4—341no—
158C22×C4—21no—
159C22×C4—341no—
160C22×C4—341no—
161C22×C4C22×D1711yescharacteristic
162Q16—348no—
163Q16—344no—
164SD16—344no—
165M16—342no—
166C2×C8—172no—
167C2×C8—172no—
168C2×C8—342no—
169C4⋊C4—24no—
170C4⋊C4—344no—
171C4⋊C4C22⋊D1712yes—
172C4⋊C4C22×D1712yes—
173C22⋊C4—21no—
174C22⋊C4—344no—
175C22⋊C4C22⋊D1712yes—
176C42—24no—
177C42—21no—
178C42C22×D1712yes—
179C23—341no—
180C23C2×C22⋊D1711yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—342no—
184Q8—341no—
185Q8—342no—
186Q8—341no—
187Q8—682no—
188Q8C2×D6814yes—
189D4—341no—
190D4—341no—
191D4—682no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—342no—
196C2×C4—341no—
197C2×C4—682no—
198C2×C4—682no—
199C2×C4C2×D6812yes—
200C2×C4#6325212yes—
201C2×C4C2×C22⋊D1714yes—
202C2×C4C2×C22⋊D1711yescharacteristic
203C2×C4C23×D1711yescharacteristic
204C8—342no—
205C8—342no—
206C22—22no—
207C22—681no—
208C22—682no—
209C22#1058761612yes—
210C22C22×C22⋊D1711yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—681no—
215C4—682no—
216C4#1058761612yes—
217C4C22×D6811yescharacteristic
218C4C2×#6325211yescharacteristic
219C2—22no—
220C2—1361no—
221C2C2×#1058761611yescharacteristic
222C2#1176362412yes—
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#519C2
socleC22×C17C22×D4
prime radicalC2×#63252C22
hypercenter#519D17
hypocenterC17#3044
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#63231C2semidirect
4. #11763031C2semidirect
5. #11763019C2semidirect
6. #11762836C2semidirect
7. #11762817C2semidirect
9. C17×#519C2semidirect
111. #519D17semidirect
144. C17#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#105876282
C22×#632312
#117633852
#117630312
#117630192
#117628362
#117628172
#117628032
C17×#5192
#304417