Finite Group Databasebeta

← #16946918

Subgroups — #16946918

Order 4352.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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#14537056C211yesmaximal, characteristic, complemented
2#14527753C212yesmaximal
3#14526813C212yesmaximal, complemented
4#14526728C211yesmaximal, characteristic, complemented
5C17×#2040C211yesmaximal, characteristic
6#11762947—22no—
7#11762751—22no—
8#11762748—21no—
9#11762730—22no—
10#11762729C2212yes—
11#11762675—21no—
12#11762543—22no—
13#11762535C2212yes—
14C17×#537C2211yescharacteristic
15C2×C17×#104—21no—
16C17×#417—21no—
17C17×#382C2211yescharacteristic
18C17×#343C2211yescharacteristic
19C2×#41×C17—21no—
20C22×Dic34D412yes—
21C2×M16×C17—21no—
22C17×C22⋊Q8—41no—
23C17×#121—22no—
24C17×#121—41no—
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17D411yescharacteristic
28C2×#63229—42no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×#63225—22no—
33C17×#104—42no—
34C17×#103—41no—
35C17×#102—41no—
36C17×#95—22no—
37#10587502—22no—
38#10587502—22no—
39#10587502—21no—
40#10587502—22no—
41#10587502D412yes—
42C22⋊#3622—22no—
43#10587481—42no—
44#41×C17—41no—
45#41×C17—41no—
46#41×C17—41no—
47#41×C17—41no—
48C2×Q8×C17—21no—
49C2×D4×C17—21no—
50C2×D4×C17—22no—
51C2×D4×C17—41no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—41no—
55C22×C4×C17C2×D412yes—
56C22×C4×C17C2×D411yescharacteristic
57C22×Dic17—22no—
58C22×Dic17—22no—
59C2×Dic34—42no—
60C2×Dic34—82no—
61M16×C17—41no—
62M16×C17—41no—
63C2×C8×C17—41no—
64C4⋊C4×C17—22no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—41no—
67C22⋊C4×C17—42no—
68C42×C17—21no—
69C42×C17SD1612yescomplemented
70#63229—84no—
71C4⋊Dic17—81no—
72C4⋊Dic17—81no—
73C4⋊Dic17—81no—
74#63225—82no—
75C2×#3622—42no—
76#12184—171nomaximal, Sylow
77C23×C17—41no—
78C23×C17#12011yescharacteristic
79Q8×C17—41no—
80Q8×C17—41no—
81D4×C17—41no—
82D4×C17—41no—
83D4×C17—82no—
84D4×C17—81no—
85D4×C17—81no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—42no—
93C2×C4×C17—42no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17—81no—
97C2×C4×C17—81no—
98C2×C4×C17C2×SD1611yescharacteristic
99C2×C4×C17#13711yescharacteristic
100C2×Dic17—42no—
101C2×Dic17—42no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17—42no—
106Dic34—44no—
107Dic34—82no—
108C8×C17—81no—
109#3622—82no—
110#2844—171no—
111#2040D1711yescharacteristic
112#1604—172no—
113#1354—172no—
114#1324—171no—
115C22×C17—21no—
116C22×C17—21no—
117C22×C17—81no—
118C22×C17—81no—
119C22×C17—81no—
120C22×C17#44811yescharacteristic
121C22×C17#45411yescharacteristic
122C22×C17B(4,𝔽2)11yescharacteristic
123C4×C17—21no—
124C4×C17—21no—
125C4×C17—42no—
126C4×C17—42no—
127C4×C17—81no—
128C4×C17—81no—
129Dic17—82no—
130Dic17—82no—
131#537C2×D1711yescharacteristic
132#448—342no—
133C2×#104—21no—
134#417—21no—
135#400—342no—
136#400—341no—
137#395—172no—
138#395—342no—
139#382—341no—
140#382C2×D1711yescharacteristic
141#343C2×D1711yescharacteristic
142#327—342no—
143#324—172no—
144C2×C17—21no—
145C2×C17—21no—
146C2×C17—161no—
147C2×C17#159911yescharacteristic
148C2×C17#219911yescharacteristic
149C2×C17#220311yescharacteristic
150C2×#41—21no—
151C22×Q8—172no—
152C2×SD16—682no—
153C2×M16—21no—
154C22⋊Q8—41no—
155#121—22no—
156#121—41no—
157C2×C4⋊C4—341no—
158C2×C4⋊C4—341no—
159C2×C4⋊C4—341no—
160C2×C4⋊C4—342no—
161C2×C4⋊C4C22⋊D1711yescharacteristic
162C2×C4⋊C4C22×D1711yescharacteristic
163C2×C42C22⋊D1711yescharacteristic
164#104—42no—
165#103—41no—
166#103—682no—
167#102—41no—
168C22⋊C8—342no—
169#95—22no—
170#95—172no—
171#95—342no—
172#95—342no—
173#95—341no—
174#95—342no—
175C17#1218411yescharacteristic, complemented, Sylow
176#41—41no—
177#41—41no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—682no—
182C2×Q8—1362no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—41no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—342no—
190C22×C4—342no—
191C22×C4D4×D1712yes—
192C22×C4C2×C22⋊D1711yescharacteristic
193SD16—1364no—
194M16—41no—
195M16—41no—
196C2×C8—41no—
197C2×C8—682no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—1362no—
202C4⋊C4—1361no—
203C4⋊C4—1361no—
204C4⋊C4—1361no—
205C22⋊C4—42no—
206C42—21no—
207C42#6322912yescomplemented
208C23—41no—
209C23#1058762311yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—684no—
213Q8—1362no—
214D4—41no—
215D4—41no—
216D4—82no—
217D4—81no—
218D4—81no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—682no—
232C2×C4—682no—
233C2×C4—682no—
234C2×C4—682no—
235C2×C4—682no—
236C2×C4—682no—
237C2×C4C2×#6322911yescharacteristic
238C2×C4#1058763711yescharacteristic
239C8—81no—
240C8—1362no—
241C22—21no—
242C22—21no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22#1176294711yescharacteristic
247C22#1176298111yescharacteristic
248C22#1176299511yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—1362no—
256C4—1362no—
257C2—21no—
258C2—21no—
259C2—161no—
260C2#1452772311yescharacteristic
261C2#1453078411yescharacteristic
262C2#1453080411yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2040C2
socleC22×C17B(4,𝔽2)
prime radicalC17×#537C22
hypercenter#2040D17
hypocenterC17#12184
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #14537056C2semidirect
3. #14526813C2semidirect
4. #14526728C2semidirect
69. C42×C17SD16semidirect
175. C17#12184semidirect
207. C42#63229semidirect

Maximal subgroups

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

subgroupindex
#145370562
#145277532
#145268132
#145267282
C17×#20402
#1218417