Finite Group Databasebeta

← #16947317

Subgroups — #16947317

Order 4352.

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

normal subgroups only

This group has 558 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#14530129C211yesmaximal, characteristic, complemented
2#14527723C212yesmaximal, complemented
3#14526787C212yesmaximal, complemented
4#14526730C211yesmaximal, characteristic, complemented
5C17×#2537C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11762960—22no—
8#11762849—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C2×C17×#121—22no—
14C2×C4×D4×C17—21no—
15C17×#394—21no—
16C17×#394C2212yes—
17C17×#384C2211yescharacteristic
18C17×#380—22no—
19C22×D4×C17—21no—
20C22×D4×C17—21no—
21C22×D4×C17D412yes—
22C23×C4×C17—22no—
23C17×#121—48no—
24C4×D4×C17—42no—
25C4×D4×C17—42no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17D411yescharacteristic
29C2×C4⋊C4×C17C2311yescharacteristic
30C2×C22⋊C4×C17—22no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17—22no—
33C2×C42×C17D411yescharacteristic
34C2×#63229—42no—
35#10587612—21no—
36#10587610—22no—
37C2×#63223—21no—
38#10587576—41no—
39#10587574—41no—
40C17×#95—21no—
41C17×#95D412yes—
42#10587503—42no—
43C22⋊#3622—22no—
44#10587479—41no—
45#10587478—41no—
46C24×C17—22no—
47C2×D4×C17—24no—
48C2×D4×C17—42no—
49C2×D4×C17—42no—
50C2×D4×C17—42no—
51C2×D4×C17—44no—
52C2×D4×C17—44no—
53C2×D4×C17—81no—
54C22×C4×C17—21no—
55C22×C4×C17—22no—
56C22×C4×C17—21no—
57C22×C4×C17—44no—
58C22×C4×C17—44no—
59C22×C4×C17—44no—
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D412yes—
62C2×Dic34—22no—
63C4⋊C4×C17—41no—
64C4⋊C4×C17—41no—
65C4⋊C4×C17—42no—
66C4⋊C4×C17—44no—
67C22⋊C4×C17—44no—
68C22⋊C4×C17—44no—
69C22⋊C4×C17—42no—
70C22⋊C4×C17—42no—
71C42×C17—21no—
72C42×C17SD1612yescomplemented
73C22⋊Dic17—41no—
74#63229—84no—
75C4⋊Dic17—21no—
76C4⋊Dic17—21no—
77C4⋊Dic17—22no—
78#63225—42no—
79C4×Dic17—41no—
80#63223—22no—
81#63223—41no—
82C2×#3622—41no—
83C2×#3622—42no—
84#12246—171nomaximal, Sylow
85C23×C17—24no—
86C23×C17—42no—
87C23×C17—42no—
88C23×C17—42no—
89C23×C17—42no—
90C23×C17—42no—
91C23×C17—42no—
92C23×C17#12011yescharacteristic
93D4×C17—48no—
94D4×C17—44no—
95D4×C17—82no—
96D4×C17—84no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—41no—
109C2×C4×C17—88no—
110C2×C4×C17C2×SD1611yescharacteristic
111C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
112C2×Dic17—41no—
113C2×Dic17—41no—
114Dic34—82no—
115#3622—42no—
116#3622—82no—
117#2537D1711yescharacteristic
118#2060—171no—
119#1599—172no—
120#1345—172no—
121#1325—171no—
122C22×C17—21no—
123C22×C17—21no—
124C22×C17—44no—
125C22×C17—44no—
126C22×C17—44no—
127C22×C17—82no—
128C22×C17—82no—
129C22×C17—82no—
130C22×C17—82no—
131C22×C17#45011yescharacteristic
132C22×C17#45611yescharacteristic
133C22×C17B(4,𝔽2)11yescharacteristic
134C4×C17—24no—
135C4×C17—21no—
136C4×C17—21no—
137C4×C17—42no—
138C4×C17—82no—
139C4×C17—81no—
140Dic17—81no—
141Dic17—81no—
142#544—171no—
143C2×#121—22no—
144C2×C4×D4—21no—
145#450—342no—
146#428—341no—
147#394—21no—
148#394C2×D1712yes—
149#384C2×D1711yescharacteristic
150#380—22no—
151#343—171no—
152#339—341no—
153#328—342no—
154#324—172no—
155C2×C17—21no—
156C2×C17—21no—
157C2×C17—84no—
158C2×C17#160411yescharacteristic
159C2×C17#219611yescharacteristic
160C2×C17#220411yescharacteristic
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22⋊D1712yes—
164C23×C4—22no—
165C2×SD16—682no—
166C2×M16—341no—
167C4⋊Q8—681no—
168#125—681no—
169C22⋊Q8—342no—
170#121—48no—
171C4×D4—42no—
172C4×D4—42no—
173#117—341no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊D1711yescharacteristic
177C2×C4⋊C4C22×D1711yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C42C22⋊D1711yescharacteristic
182#107—681no—
183#106—681no—
184#102—682no—
185C22⋊C8—342no—
186#95—21no—
187#95C22⋊D1712yes—
188C17#1224611yescharacteristic, complemented, Sylow
189C24—22no—
190C2×Q8—342no—
191C2×D4—24no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4C2×C22⋊D1711yescharacteristic
205C22×C4C2×C22⋊D1712yes—
206SD16—1364no—
207M16—342no—
208M16—681no—
209C2×C8—681no—
210C2×C8—682no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—342no—
216C4⋊C4—341no—
217C4⋊C4—341no—
218C4⋊C4—682no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—681no—
224C42—21no—
225C42—681no—
226C42#6322912yescomplemented
227C23—24no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23#1058763911yescharacteristic
235Q8—1362no—
236D4—48no—
237D4—44no—
238D4—82no—
239D4—84no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—88no—
253C2×C4—681no—
254C2×C4—681no—
255C2×C4#1058761811yescharacteristic
256C2×C4C2×#6322911yescharacteristic
257C8—682no—
258C8—1362no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#1176296011yescharacteristic
269C22#1176299211yescharacteristic
270C22#1176299611yescharacteristic
271C4—24no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—82no—
276C4—81no—
277C4—1361no—
278C4—1361no—
279C2—21no—
280C2—21no—
281C2—84no—
282C2#1452775311yescharacteristic
283C2#1453076811yescharacteristic
284C2#1453081011yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
Fitting, prime radicalC17×#2537C2
socleC22×C17B(4,𝔽2)
hypercenter#2537D17
hypocenterC17#12246
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. #14530129C2semidirect
2. #14527723C2semidirect
3. #14526787C2semidirect
4. #14526730C2semidirect
72. C42×C17SD16semidirect
188. C17#12246semidirect
226. C42#63229semidirect

Maximal subgroups

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

subgroupindex
#145301292
#145277232
#145267872
#145267302
C17×#25372
#1224617