Finite Group Databasebeta

← #16948006

Subgroups — #16948006

Order 4352.

298 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 69 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#14541673C211yesmaximal, characteristic, complemented
2#14531007C211yesmaximal, characteristic
3#14528141C211yesmaximal, characteristic
4#14526838C211yesmaximal, characteristic, complemented
5#14526762C211yesmaximal, characteristic, complemented
6#14526748C211yesmaximal, characteristic, complemented
7C17×#2207C211yesmaximal, characteristic
8#11763538—21no—
9#11763214C2211yescharacteristic
10#11763210—21no—
11#11763118—22no—
12#11763094—21no—
13#11763006—21no—
14C8×Dic34C2211yescharacteristic
15Q16×Dic17—21no—
16#11762610—22no—
17C4×#63214C2211yescharacteristic
18#11762562C2211yescharacteristic
19#11762553—21no—
20C2×C17×SD32—21no—
21C17×#493C2211yescharacteristic
22C4×Q16×C17C2211yescharacteristic
23C17×C4⋊C16C2211yescharacteristic
24C17×#357—21no—
25C2×Q16×C17D411yescharacteristic
26C2×D8×C17—21no—
27C2×D8×C17—21no—
28C17×#127—21no—
29C4×Q8×C17—21no—
30Q8×Dic17—22no—
31C2×#63229—41no—
32#10587576—22no—
33#10587576—42no—
34#10587555—21no—
35#10587555—41no—
36C4×Dic34—41no—
37C4×Dic34D411yescharacteristic
38C17×SD32—42no—
39C2×C16×C17—21no—
40C17×#107D411yescharacteristic
41C17×#103—21no—
42C4×C8×C17C2311yescharacteristic
43#10587506—22no—
44#10587503—41no—
45#10587489—21no—
46#10587489D412yes—
47#10587485—21no—
48C8×Dic17—21no—
49C2×#63214—21no—
50#10587479—22no—
51C4⋊#3622—21no—
52C4⋊#3622D411yescharacteristic
53C2×Q8×C17—21no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×Dic34—41no—
58C2×Dic34D811yescharacteristic
59Q16×C17—21no—
60Q16×C17SD1612yescomplemented
61D8×C17—42no—
62D8×C17—41no—
63D8×C17—41no—
64C2×C8×C17C2×D411yescharacteristic
65C2×C8×C17C2×D411yescharacteristic
66C4⋊C4×C17—21no—
67C4⋊C4×C17—41no—
68C42×C17—41no—
69C42×C17C2×D411yescharacteristic
70#63229—82no—
71C4⋊Dic17—22no—
72C4⋊Dic17—42no—
73C4⋊Dic17—42no—
74C4⋊Dic17—41no—
75C4⋊Dic17D811yescharacteristic
76#63225—21no—
77#63225—42no—
78C4×Dic17—21no—
79C4×Dic17—42no—
80C2×#3622—21no—
81C2×#3622—41no—
82C16×C17—41no—
83#63214—41no—
84#12353—171nomaximal, Sylow
85C23×C17—81no—
86Q8×C17—22no—
87Q8×C17—41no—
88D4×C17—81no—
89D4×C17—81no—
90D4×C17—81no—
91D4×C17—81no—
92D4×C17—161no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17#12011yescharacteristic
96C2×C4×C17C2×D811yescharacteristic
97C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—42no—
101Dic34—21no—
102Dic34—21no—
103Dic34—81no—
104Dic34—81no—
105C8×C17—21no—
106C8×C17C2×SD1611yescharacteristic
107C8×C17#13711yescharacteristic
108#3622—41no—
109#3622—81no—
110#3383—171no—
111#2240—171no—
112#2207D1711yescharacteristic
113#1668—171no—
114#1364—171no—
115Q8⋊C16—171no—
116#1332—171no—
117C22×C17—161no—
118C22×C17—161no—
119C22×C17—161no—
120C22×C17#44711yescharacteristic
121C4×C17—21no—
122C4×C17—42no—
123C4×C17—81no—
124C4×C17#44811yescharacteristic
125C4×C17#45411yescharacteristic
126C4×C17#50811yescharacteristic
127C4×C17#50911yescharacteristic
128Dic17—41no—
129Dic17—41no—
130Dic17—82no—
131#557—341no—
132C2×SD32—21no—
133#500—171no—
134#499—341no—
135#493C2×D1711yescharacteristic
136#479—342no—
137#475—341no—
138#460—341no—
139C8×Q8—171no—
140C4×Q16—341no—
141C4×Q16C2×D1711yescharacteristic
142C4⋊C16C2×D1711yescharacteristic
143#357—21no—
144#357—342no—
145C4×C16—171no—
146#335—171no—
147#331—341no—
148C2×C17—321no—
149C2×C17#162711yescharacteristic
150C2×C17#218611yescharacteristic
151C2×C17#219311yescharacteristic
152C2×Q16C22⋊D1711yescharacteristic
153C2×SD16—681no—
154C2×D8—21no—
155C2×D8—21no—
156C4⋊Q8—341no—
157C4⋊Q8—681no—
158#127—21no—
159#125—342no—
160#125—682no—
161C4×Q8—21no—
162C4×Q8—171no—
163C4×Q8—342no—
164C4×Q8—681no—
165SD32—42no—
166C2×C16—21no—
167C2×C16—341no—
168#107—172no—
169#107—341no—
170#107C22⋊D1711yescharacteristic
171#106—342no—
172C4⋊C8—171no—
173C4⋊C8—341no—
174C4⋊C8—341no—
175#103—21no—
176#103—342no—
177#102—681no—
178C4×C8—341no—
179C4×C8C22×D1711yescharacteristic
180C17#1235311yescharacteristic, complemented, Sylow
181C2×Q8—21no—
182C2×Q8—171no—
183C2×Q8—681no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187Q16—21no—
188Q16#6322912yescomplemented
189SD16—1362no—
190D8—42no—
191D8—41no—
192D8—41no—
193C2×C8—341no—
194C2×C8—681no—
195C2×C8D4×D1711yescharacteristic
196C2×C8C2×C22⋊D1711yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—171no—
200C4⋊C4—341no—
201C4⋊C4—342no—
202C4⋊C4—682no—
203C4⋊C4—682no—
204C4⋊C4—682no—
205C4⋊C4—681no—
206C42—41no—
207C42—341no—
208C42—682no—
209C42D4×D1711yescharacteristic
210C16—41no—
211C16—681no—
212C23—81no—
213Q8—22no—
214Q8—41no—
215Q8—341no—
216Q8—341no—
217Q8—1361no—
218Q8—1361no—
219D4—81no—
220D4—81no—
221D4—81no—
222D4—81no—
223D4—161no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—341no—
227C2×C4—341no—
228C2×C4—682no—
229C2×C4D8×D1711yescharacteristic
230C2×C4#1058759711yescharacteristic
231C2×C4#1058762311yescharacteristic
232C8—21no—
233C8—681no—
234C8—1361no—
235C8C2×#6322911yescharacteristic
236C8#1058763711yescharacteristic
237C22—161no—
238C22—161no—
239C22—161no—
240C22#1176294111yescharacteristic
241C4—21no—
242C4—42no—
243C4—81no—
244C4—681no—
245C4—681no—
246C4—1362no—
247C4#1176294711yescharacteristic
248C4#1176298011yescharacteristic
249C4#1176324011yescharacteristic
250C4#1176324611yescharacteristic
251C2—321no—
252C2#1452788611yescharacteristic
253C2#1453070611yescharacteristic
254C2#1453074511yescharacteristic
subgroupG/H
centerC22#11762941
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#2207C2
socleC22×C17#447
prime radicalC17×#493C22
hypercenter#2207D17
hypocenterC17#12353
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #14541673C2semidirect
4. #14526838C2semidirect
5. #14526762C2semidirect
6. #14526748C2semidirect
60. Q16×C17SD16semidirect
180. C17#12353semidirect
188. Q16#63229semidirect

Maximal subgroups

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

subgroupindex
#145416732
#145310072
#145281412
#145268382
#145267622
#145267482
C17×#22072
#1235317