Finite Group Databasebeta

← #16947341

Subgroups — #16947341

Order 4352.

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

normal subgroups only

This group has 454 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#14530134C211yesmaximal, characteristic, complemented
2#14527723C212yesmaximal, complemented
3#14526787C212yesmaximal, complemented
4#14526728C211yesmaximal, characteristic, complemented
5C17×#2819C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11762960—22no—
8#11762849—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C2×C17×#124—21no—
14C17×#394—21no—
15C17×#394C2212yes—
16C17×#393—22no—
17C17×#392—22no—
18C17×#382C2211yescharacteristic
19C22×Q8×C17—21no—
20C22×D4×C17—21no—
21C22×D4×C17—21no—
22C22×D4×C17D412yes—
23C17×#124—42no—
24C17×#124—42no—
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C22⋊C4×C17—22no—
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17—22no—
31C2×C42×C17D411yescharacteristic
32C2×#63229—42no—
33#10587612—21no—
34#10587610—22no—
35C2×#63223—21no—
36#10587576—41no—
37#10587574—41no—
38C17×#95—21no—
39C17×#95—22no—
40C17×#95D412yes—
41#10587503—42no—
42C22⋊#3622—22no—
43#10587479—41no—
44#10587478—41no—
45C24×C17—22no—
46C2×Q8×C17—42no—
47C2×Q8×C17—81no—
48C2×Q8×C17—81no—
49C2×D4×C17—42no—
50C2×D4×C17—42no—
51C2×D4×C17—82no—
52C2×D4×C17—81no—
53C2×D4×C17—81no—
54C2×D4×C17—82no—
55C2×D4×C17—81no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—22no—
59C22×C4×C17C2×D411yescharacteristic
60C22×C4×C17C2×D412yes—
61C2×Dic34—22no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—41no—
64C22⋊C4×C17—44no—
65C22⋊C4×C17—42no—
66C22⋊C4×C17—42no—
67C22⋊C4×C17—82no—
68C22⋊C4×C17—82no—
69C42×C17—21no—
70C42×C17SD1612yescomplemented
71C22⋊Dic17—41no—
72#63229—84no—
73C4⋊Dic17—21no—
74C4⋊Dic17—21no—
75C4⋊Dic17—22no—
76#63225—42no—
77C4×Dic17—41no—
78#63223—22no—
79#63223—41no—
80C2×#3622—41no—
81C2×#3622—42no—
82#12250—171nomaximal, Sylow
83C23×C17—42no—
84C23×C17—42no—
85C23×C17—42no—
86C23×C17—42no—
87C23×C17—42no—
88C23×C17—42no—
89C23×C17—42no—
90C23×C17#12011yescharacteristic
91Q8×C17—84no—
92D4×C17—44no—
93D4×C17—82no—
94D4×C17—84no—
95D4×C17—84no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—41no—
108C2×C4×C17C2×SD1611yescharacteristic
109C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
110C2×Dic17—41no—
111C2×Dic17—41no—
112Dic34—82no—
113#3622—42no—
114#3622—82no—
115#2819D1711yescharacteristic
116#2061—171no—
117#1599—172no—
118#1345—172no—
119#1324—171no—
120C22×C17—21no—
121C22×C17—21no—
122C22×C17—82no—
123C22×C17—82no—
124C22×C17—82no—
125C22×C17—82no—
126C22×C17—82no—
127C22×C17—82no—
128C22×C17—82no—
129C22×C17#45011yescharacteristic
130C22×C17#45611yescharacteristic
131C22×C17B(4,𝔽2)11yescharacteristic
132C4×C17—21no—
133C4×C17—21no—
134C4×C17—42no—
135C4×C17—42no—
136C4×C17—82no—
137C4×C17—81no—
138Dic17—81no—
139Dic17—81no—
140#544—171no—
141C2×#124—21no—
142#450—342no—
143#428—341no—
144#394—21no—
145#394C2×D1712yes—
146#393—22no—
147#392—22no—
148#382C2×D1711yescharacteristic
149#343—171no—
150#339—341no—
151#328—342no—
152#324—172no—
153C2×C17—21no—
154C2×C17—21no—
155C2×C17—162no—
156C2×C17#160411yescharacteristic
157C2×C17#219911yescharacteristic
158C2×C17#220211yescharacteristic
159C22×Q8—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C22⋊D1712yes—
163C2×SD16—682no—
164C2×M16—341no—
165C4⋊Q8—681no—
166#125—681no—
167#124—42no—
168#124—42no—
169C22⋊Q8—342no—
170#117—341no—
171C2×C4⋊C4C22⋊D1711yescharacteristic
172C2×C4⋊C4C22×D1711yescharacteristic
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42C22⋊D1711yescharacteristic
178#107—681no—
179#106—681no—
180#102—682no—
181C22⋊C8—342no—
182#95—21no—
183#95—22no—
184#95C22⋊D1712yes—
185C17#1225011yescharacteristic, complemented, Sylow
186C24—22no—
187C2×Q8—42no—
188C2×Q8—81no—
189C2×Q8—81no—
190C2×Q8—342no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—82no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—82no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4C2×C22⋊D1711yescharacteristic
202C22×C4C2×C22⋊D1712yes—
203SD16—1364no—
204M16—342no—
205M16—681no—
206C2×C8—681no—
207C2×C8—682no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—342no—
211C4⋊C4—341no—
212C4⋊C4—341no—
213C4⋊C4—682no—
214C22⋊C4—44no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—681no—
220C42—21no—
221C42—681no—
222C42#6322912yescomplemented
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23#1058763911yescharacteristic
231Q8—84no—
232Q8—1362no—
233D4—44no—
234D4—82no—
235D4—84no—
236D4—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—681no—
250C2×C4—681no—
251C2×C4#1058761811yescharacteristic
252C2×C4C2×#6322911yescharacteristic
253C8—682no—
254C8—1362no—
255C22—21no—
256C22—21no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22#1176296011yescharacteristic
265C22#1176299211yescharacteristic
266C22#1176299611yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—1361no—
274C4—1361no—
275C2—21no—
276C2—21no—
277C2—162no—
278C2#1452775311yescharacteristic
279C2#1453078511yescharacteristic
280C2#1453080111yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
Fitting, prime radicalC17×#2819C2
socleC22×C17B(4,𝔽2)
hypercenter#2819D17
hypocenterC17#12250
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2819

Complements

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

NH ≅ G/Ntype
1. #14530134C2semidirect
2. #14527723C2semidirect
3. #14526787C2semidirect
4. #14526728C2semidirect
70. C42×C17SD16semidirect
185. C17#12250semidirect
222. C42#63229semidirect

Maximal subgroups

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

subgroupindex
#145301342
#145277232
#145267872
#145267282
C17×#28192
#1225017