Finite Group Databasebeta

← #14538311

Subgroups — #14538311

Order 2176.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
2#11763383C211yesmaximal, characteristic
3#11763287C211yesmaximal, characteristic, complemented
4#11763033C214yesmaximal, complemented
5#11763017C214yesmaximal, complemented
6#11762835C211yesmaximal, characteristic, complemented
7#11762818C211yesmaximal, characteristic, complemented
8#11762806C211yesmaximal, characteristic
9C22×Q16×C17C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×Q8×C17C2211yescharacteristic
12C2×#63252C2211yescharacteristic
13C2×Q16×C17—24no—
14C2×Q16×C17C2214yes—
15C22×C8×C17C2211yescharacteristic
16#10587633C2212yes—
17#10587632—22no—
18Q8×Dic17—22no—
19Q8×Dic17C2214yes—
20#10587630C2212yes—
21C2×#63231C2212yes—
22#10587628—24no—
23C2×#63230C2212yes—
24#10587615—22no—
25#10587612—22no—
26#10587612C2211yescharacteristic
27#10587612C2211yescharacteristic
28C2×#63223C2211yescharacteristic
29#10587506C2214yes—
30#10587492C2212yes—
31#10587487C2212yes—
32#10587485C2214yes—
33C2×Q8×C17—21no—
34C2×Q8×C17—21no—
35C2×Q8×C17—22no—
36C2×Q8×C17—22no—
37C2×Q8×C17—22no—
38C2×Q8×C17D414yes—
39C2×Q8×C17C2312yes—
40C2×Q8×C17C2312yes—
41C22×C4×C17—21no—
42C22×C4×C17—21no—
43C22×C4×C17C2311yescharacteristic
44C2×C22⋊D17—21no—
45#63252—22no—
46#63252—22no—
47C2×D68C2311yescharacteristic
48C2×C4×D17—21no—
49C2×Dic34C2311yescharacteristic
50Q16×C17—28no—
51C2×C8×C17—22no—
52C2×C8×C17C2312yes—
53C22⋊Dic17—21no—
54C22⋊Dic17D412yes—
55#63231—24no—
56#63230—24no—
57#63227—24no—
58C4⋊Dic17—24no—
59C4⋊Dic17D412yes—
60C4⋊Dic17C2312yes—
61#63225—24no—
62C4×Dic17—21no—
63C4×Dic17—24no—
64C4×Dic17C2312yes—
65#63223—22no—
66C2×#3622C2312yes—
67C23×C17C2×D411yescharacteristic
68C22×D17—21no—
69Q8×C17—24no—
70Q8×C17—22no—
71Q8×C17—22no—
72Q8×C17—22no—
73Q8×C17—42no—
74Q8×C17C2×D414yes—
75C2×C4×C17—24no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—42no—
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17C2×D412yes—
81C2×C4×C17#4112yes—
82C2×C4×C17C2411yescharacteristic
83C22⋊D17—42no—
84C2×Dic17—22no—
85C2×Dic17—21no—
86C2×Dic17C2×D414yes—
87D68—21no—
88D68—21no—
89C4×D17—42no—
90Dic34—21no—
91Dic34—21no—
92C8×C17—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—42no—
102C4×C17#12112yes—
103C4×C17C22×D411yescharacteristic
104C4×C17C2×#4111yescharacteristic
105Dic17—24no—
106Dic17—41no—
107Dic17—42no—
108C2×#137—171no—
109C22×Q16D1711yescharacteristic, complemented
110#536—171no—
111#519—171no—
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×Q8C2×D1711yescharacteristic
123C22×Q8C2×D1711yescharacteristic
124#137—344no—
125C2×Q16—24no—
126C2×Q16—172no—
127C2×Q16C2×D1714yes—
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—171no—
139#117—342no—
140C4⋊C8—174no—
141#103—174no—
142#103—172no—
143#102—172no—
144C17#304411yescharacteristic, complemented, Sylow
145#41—342no—
146#41—342no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—171no—
153C2×Q8C22⋊D1714yes—
154C2×Q8C22×D1712yes—
155C2×Q8C22×D1712yes—
156C2×D4—171no—
157C2×D4—341no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—341no—
161C22×C4C22×D1711yescharacteristic
162Q16—28no—
163Q16—344no—
164SD16—344no—
165M16—342no—
166C2×C8—22no—
167C2×C8—172no—
168C2×C8C22×D1712yes—
169C4⋊C4—172no—
170C4⋊C4—172no—
171C4⋊C4—344no—
172C4⋊C4—344no—
173C22⋊C4—172no—
174C22⋊C4—344no—
175C22⋊C4—341no—
176C42—172no—
177C42—344no—
178C42—341no—
179C23—341no—
180C23D4×D1711yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—22no—
184Q8—22no—
185Q8—42no—
186Q8—341no—
187Q8—341no—
188Q8C2×C22⋊D1714yes—
189D4—341no—
190D4—341no—
191D4—682no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—174no—
197C2×C4—342no—
198C2×C4—341no—
199C2×C4—682no—
200C2×C4D4×D1711yescharacteristic
201C2×C4#6325412yes—
202C2×C4C2×C22⋊D1712yes—
203C2×C4C23×D1711yescharacteristic
204C8—22no—
205C8—342no—
206C22—22no—
207C22—681no—
208C22—682no—
209C22#1058762512yes—
210C22C2×D4×D1711yescharacteristic
211C4—24no—
212C4—42no—
213C4—344no—
214C4—681no—
215C4—682no—
216C4#1058762512yes—
217C4C2×#6325411yescharacteristic
218C4C22×C22⋊D1711yescharacteristic
219C2—22no—
220C2—1361no—
221C2C2×#1058762511yescharacteristic
222C2#1176362212yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC22×Q16×C17C2
socleC22×C17C22×D4
prime radicalC2×#63252C22
hypercenterC22×Q16D17
hypocenterC17#3044
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #11763287C2semidirect
4. #11763033C2semidirect
5. #11763017C2semidirect
6. #11762835C2semidirect
7. #11762818C2semidirect
9. C22×Q16×C17C2semidirect
109. C22×Q16D17semidirect
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
#117633832
#117632872
#117630332
#117630172
#117628352
#117628182
#117628062
C22×Q16×C172
#304417