Finite Group Databasebeta

← #16947080

Subgroups — #16947080

Order 4352.

404 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#14529940C211yesmaximal, characteristic
2#14527791C212yesmaximal, complemented
3#14526790C212yesmaximal, complemented
4#14526716C211yesmaximal, characteristic
5C17×#2724C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11763139—22no—
8#11762814—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C2×C17×#123—21no—
14C17×#398C2212yes—
15C17×#397—22no—
16C17×#390C2211yescharacteristic
17C17×#389—21no—
18C17×#385—22no—
19C22×D4×C17D411yescharacteristic
20C23×C4×C17—21no—
21C17×#123—42no—
22C17×#123—42no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17D412yes—
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17—21no—
30C2×C22⋊C4×C17—22no—
31C2×C42×C17D411yescharacteristic
32#10587612—21no—
33#10587610—22no—
34C2×#63223—21no—
35#10587576—41no—
36#10587574—41no—
37C17×#95—22no—
38C17×#95—22no—
39C17×#95D412yes—
40#10587503—41no—
41#10587503—41no—
42C22⋊#3622—22no—
43#10587481—42no—
44#10587479—42no—
45C24×C17—21no—
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17SD1612yes—
50C22×C4×C17—21no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—42no—
54C22×C4×C17—81no—
55C22×C4×C17—81no—
56C22×C4×C17C2×D411yescharacteristic
57C22×C4×C17C2×D412yes—
58C2×Dic34—22no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—42no—
61C4⋊C4×C17—42no—
62C4⋊C4×C17—42no—
63C4⋊C4×C17—42no—
64C4⋊C4×C17—81no—
65C22⋊C4×C17—44no—
66C22⋊C4×C17—42no—
67C22⋊C4×C17—44no—
68C42×C17—41no—
69C22⋊Dic17—41no—
70C4⋊Dic17—22no—
71C4⋊Dic17—21no—
72C4⋊Dic17—21no—
73#63225—42no—
74C4×Dic17—41no—
75#63223—41no—
76#63223—41no—
77C2×#3622—42no—
78C2×#3622—41no—
79#12209—171nomaximal, Sylow
80C23×C17—22no—
81C23×C17—42no—
82C23×C17—42no—
83C23×C17—41no—
84C23×C17—41no—
85C23×C17#12011yescharacteristic
86D4×C17—42no—
87D4×C17—81no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17—84no—
102C2×C4×C17C2×SD1611yescharacteristic
103C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
104C2×Dic17—41no—
105C2×Dic17—41no—
106Dic34—82no—
107#3622—81no—
108#3622—82no—
109#2724D1711yescharacteristic
110#2027—171no—
111#1611—172no—
112#1346—172no—
113#1318—171no—
114C22×C17—21no—
115C22×C17—21no—
116C22×C17—44no—
117C22×C17—42no—
118C22×C17—82no—
119C22×C17—81no—
120C22×C17—81no—
121C22×C17#45011yescharacteristic
122C22×C17#45511yescharacteristic
123C22×C17B(4,𝔽2)11yescharacteristic
124C4×C17—21no—
125C4×C17—21no—
126C4×C17—42no—
127C4×C17—42no—
128C4×C17—82no—
129C4×C17—82no—
130Dic17—81no—
131Dic17—81no—
132#544—171no—
133C2×#123—21no—
134#483—342no—
135#418—341no—
136#398C2×D1712yes—
137#397—22no—
138#390C2×D1711yescharacteristic
139#389—21no—
140#385—22no—
141#343—171no—
142#339—341no—
143#328—342no—
144#324—172no—
145C2×C17—21no—
146C2×C17—21no—
147C2×C17—82no—
148C2×C17#160111yescharacteristic
149C2×C17#219811yescharacteristic
150C2×C17#220311yescharacteristic
151C22×D4C22⋊D1711yescharacteristic
152C23×C4—21no—
153C2×M16—341no—
154C4⋊Q8—681no—
155#125—681no—
156#123—42no—
157#123—42no—
158C22⋊Q8—342no—
159#117—341no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊D1712yes—
164C2×C4⋊C4C22×D1711yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C42C22⋊D1711yescharacteristic
169#106—682no—
170#103—682no—
171#102—681no—
172#102—681no—
173C22⋊C8—342no—
174#95—22no—
175#95—22no—
176#95C22⋊D1712yes—
177C17#1220911yescharacteristic, complemented, Sylow
178C24—21no—
179C2×Q8—342no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4#6322912yes—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×C22⋊D1711yescharacteristic
191C22×C4C2×C22⋊D1712yes—
192M16—681no—
193M16—681no—
194C2×C8—682no—
195C2×C8—681no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—341no—
203C4⋊C4—341no—
204C4⋊C4—342no—
205C4⋊C4—682no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—681no—
210C42—41no—
211C42—681no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23#1058763911yescharacteristic
218Q8—1362no—
219D4—42no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4—681no—
236C2×C4—681no—
237C2×C4#1058761811yescharacteristic
238C2×C4C2×#6322911yescharacteristic
239C8—1361no—
240C8—1362no—
241C22—21no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#1176296011yescharacteristic
249C22#1176298711yescharacteristic
250C22#1176299611yescharacteristic
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—1361no—
258C4—1361no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#1452773611yescharacteristic
263C2#1453078011yescharacteristic
264C2#1453080511yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2724C2
socleC22×C17B(4,𝔽2)
prime radicalC22×D4×C17D4
hypercenter#2724D17
hypocenterC17#12209
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2724

Complements

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

NH ≅ G/Ntype
2. #14527791C2semidirect
3. #14526790C2semidirect
177. C17#12209semidirect

Maximal subgroups

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

subgroupindex
#145299402
#145277912
#145267902
#145267162
C17×#27242
#1220917