Finite Group Databasebeta

← #14538310

Subgroups — #14538310

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×#10587637C211yesmaximal, characteristic
2C22×#63231C211yesmaximal, characteristic, complemented
3#11763286C211yesmaximal, characteristic, complemented
4#11763028C214yesmaximal, complemented
5#11763016C214yesmaximal, complemented
6#11762836C211yesmaximal, characteristic, complemented
7#11762818C211yesmaximal, characteristic, complemented
8#11762805C211yesmaximal, characteristic
9C17×#536C211yesmaximal, characteristic
10C2×#41×C17C2211yescharacteristic
11C22×Q8×C17C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C17×C4⋊Q8C2212yes—
14C17×#124C2212yes—
15C17×#123—22no—
16C17×C22⋊Q8—22no—
17C17×#117C2211yescharacteristic
18#10587637—24no—
19C2×#63231—24no—
20C2×#63231C2212yes—
21C2×#63231C2214yes—
22C2×#63229C2212yes—
23#10587612C2211yescharacteristic
24C2×#63223C2211yescharacteristic
25C22×#3622C2211yescharacteristic
26#10587573—22no—
27#10587563—22no—
28C4×Dic34C2214yes—
29#10587506C2212yes—
30#10587506C2214yes—
31#10587503C2212yes—
32C4⋊#3622C2214yes—
33#41×C17—22no—
34#41×C17—22no—
35C2×Q8×C17—21no—
36C2×Q8×C17—22no—
37C2×Q8×C17—22no—
38C2×Q8×C17C2312yes—
39C2×Q8×C17C2311yescharacteristic
40C2×D4×C17—21no—
41C2×D4×C17C2311yescharacteristic
42C22×C4×C17—21no—
43C22×C4×C17—21no—
44C22×C4×C17C2311yescharacteristic
45C22×Dic17—21no—
46C2×Dic34—21no—
47C2×Dic34—22no—
48C2×Dic34D414yes—
49C2×Dic34C2312yes—
50C4⋊C4×C17—24no—
51C4⋊C4×C17D412yes—
52C22⋊C4×C17—24no—
53C22⋊C4×C17D412yes—
54C42×C17C2312yes—
55C22⋊Dic17—21no—
56#63231—28no—
57#63231—24no—
58#63229—24no—
59C4⋊Dic17C2312yes—
60#63225—24no—
61C4×Dic17—24no—
62C4×Dic17—21no—
63#63223—22no—
64C2×#3622—22no—
65C2×#3622C2312yes—
66C2×#3622C2312yes—
67C23×C17—21no—
68C23×C17C2×D411yescharacteristic
69Q8×C17—22no—
70Q8×C17—21no—
71Q8×C17—22no—
72Q8×C17—21no—
73Q8×C17—42no—
74D4×C17—21no—
75D4×C17—21no—
76D4×C17—42no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—42no—
80C2×C4×C17—42no—
81C2×C4×C17C2×D412yes—
82C2×C4×C17C2×D414yes—
83C2×C4×C17C2×D411yescharacteristic
84C2×C4×C17#4112yes—
85C2×C4×C17C2411yescharacteristic
86C2×Dic17—24no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89Dic34—24no—
90Dic34—22no—
91Dic34C2×D414yes—
92#3622—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—24no—
101C4×C17—41no—
102C4×C17—42no—
103C4×C17#12112yes—
104C4×C17C22×D411yescharacteristic
105C4×C17C2×#4111yescharacteristic
106Dic17—24no—
107Dic17—42no—
108C2×#137—171no—
109C22×Q16—171no—
110#536D1711yescharacteristic
111#519—171no—
112#464—174no—
113#462—174no—
114#424—171no—
115#419—171no—
116#416—171no—
117C2×C17—22no—
118C2×C17—81no—
119C2×C17C2×#12111yescharacteristic
120C2×C17#57812yes—
121C2×#41C2×D1711yescharacteristic
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⋊Q8C2×D1712yes—
132#124C2×D1712yes—
133#123—22no—
134C22⋊Q8—22no—
135C4×Q8—174no—
136C4×Q8—342no—
137#117—171no—
138#117—342no—
139#117C2×D1711yescharacteristic
140C4⋊C8—174no—
141#103—172no—
142#103—174no—
143#102—172no—
144C17#304411yescharacteristic, complemented, Sylow
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—174no—
151C2×Q8—172no—
152C2×Q8—342no—
153C2×Q8—341no—
154C2×Q8C22×D1712yes—
155C2×Q8C22×D1711yescharacteristic
156C2×D4—21no—
157C2×D4C22×D1711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
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—172no—
171C4⋊C4—344no—
172C4⋊C4C22⋊D1712yes—
173C22⋊C4—24no—
174C22⋊C4—341no—
175C22⋊C4C22⋊D1712yes—
176C42—341no—
177C42—344no—
178C42C22×D1712yes—
179C23—21no—
180C23C2×C22⋊D1711yescharacteristic
181Q8—22no—
182Q8—21no—
183Q8—22no—
184Q8—21no—
185Q8—42no—
186Q8—174no—
187Q8—342no—
188Q8—344no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—342no—
197C2×C4—344no—
198C2×C4—342no—
199C2×C4D4×D1712yes—
200C2×C4#6325412yes—
201C2×C4C2×C22⋊D1714yes—
202C2×C4C2×C22⋊D1711yescharacteristic
203C2×C4C23×D1711yescharacteristic
204C8—342no—
205C8—342no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1058762412yes—
210C22C22×C22⋊D1711yescharacteristic
211C4—24no—
212C4—41no—
213C4—42no—
214C4—344no—
215C4—682no—
216C4#1058762412yes—
217C4C2×D4×D1711yescharacteristic
218C4C2×#6325411yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1058762411yescharacteristic
222C2#1176362412yes—
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#536C2
socleC22×C17C22×D4
prime radicalC2×#41×C17C22
hypercenter#536D17
hypocenterC17#3044
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C22×#63231C2semidirect
3. #11763286C2semidirect
4. #11763028C2semidirect
5. #11763016C2semidirect
6. #11762836C2semidirect
7. #11762818C2semidirect
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×#632312
#117632862
#117630282
#117630162
#117628362
#117628182
#117628052
C17×#5362
#304417