Finite Group Databasebeta

← #16937398

Subgroups — #16937398

Order 4352.

400 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 89 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
2#14529757C212yesmaximal
3#14529603C211yesmaximal, characteristic
4#14526666C212yesmaximal
5C17×#1295C211yesmaximal, characteristic
6C2×#10587555C2211yescharacteristic
7C2×#10587481—21no—
8C2×#10587481—21no—
9C2×#10587474C2211yescharacteristic
10#11762751—22no—
11#11762748—21no—
12#11762730—22no—
13#11762729—21no—
14#11762675C2212yes—
15#11762572—22no—
16#11762572—22no—
17C2×C17×#97C2212yes—
18C17×#382C2211yescharacteristic
19C22×Dic34—21no—
20C22×Dic34—21no—
21C22×C8×C17—22no—
22C2×C4⋊C4×C17D411yescharacteristic
23C2×C4⋊C4×C17D411yescharacteristic
24C2×C42×C17C2311yescharacteristic
25C2×C4⋊Dic17—21no—
26C2×C4⋊Dic17—22no—
27C2×C4⋊Dic17D412yes—
28C2×C4⋊Dic17D412yes—
29C2×#63225—22no—
30C22×#3622—21no—
31#10587555—21no—
32#10587555—22no—
33#10587555Q812yescomplemented
34C17×#97—22no—
35C17×#97—22no—
36C17×#95—22no—
37#10587502—21no—
38#10587502—22no—
39#10587502—21no—
40#10587502—22no—
41#10587502—22no—
42#10587481—42no—
43#10587481—42no—
44#10587474—22no—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17C2×D411yescharacteristic
48C22×C4×C17C2×D412yes—
49C22×Dic17—22no—
50C22×Dic17—22no—
51C22×Dic17—41no—
52C2×Dic34—22no—
53C2×Dic34—41no—
54C2×Dic34—41no—
55C2×Dic34—41no—
56C2×Dic34—41no—
57C2×Dic34—41no—
58C2×Dic34—81no—
59C2×Dic34—82no—
60C2×C8×C17—24no—
61C2×C8×C17—42no—
62C2×C8×C17—42no—
63C4⋊C4×C17—41no—
64C4⋊C4×C17—41no—
65C42×C17C2×Q811yescharacteristic
66C42×C17#4112yes—
67C42×C17#4111yescharacteristic
68C4⋊Dic17—24no—
69C4⋊Dic17—42no—
70C4⋊Dic17—42no—
71C4⋊Dic17—42no—
72C4⋊Dic17—81no—
73#63225—82no—
74C2×#3622—22no—
75C2×#3622—41no—
76C2×#3622—41no—
77#10359—171nomaximal, Sylow
78C23×C17#12011yescharacteristic
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17C22⋊Q812yes—
88C2×C4×C17C22⋊Q811yescharacteristic
89C2×C4×C17#12411yescharacteristic
90C2×C4×C17#12412yes—
91C2×Dic17—42no—
92C2×Dic17—42no—
93C2×Dic17—42no—
94C2×Dic17—42no—
95C2×Dic17—42no—
96C2×Dic17—42no—
97C2×Dic17—42no—
98C2×Dic17—81no—
99C2×Dic17—81no—
100Dic34—42no—
101Dic34—42no—
102Dic34—81no—
103Dic34—81no—
104Dic34—84no—
105C8×C17—44no—
106#3622—42no—
107#2844—171no—
108#1993—172no—
109#1960—171no—
110#1295—172no—
111#1295D1711yescharacteristic
112C22×C17#39311yescharacteristic
113C22×C17#45311yescharacteristic
114C22×C17#45411yescharacteristic
115C22×C17#45512yes—
116C22×C17#45612yes—
117C4×C17—21no—
118C4×C17—21no—
119C4×C17—22no—
120C4×C17—22no—
121C4×C17—81no—
122C4×C17—81no—
123Dic17—82no—
124Dic17—82no—
125Dic17—82no—
126C2×C4⋊Q8—171no—
127C2×#103—341no—
128C2×#103—341no—
129C2×#97—171no—
130C2×#97C2×D1712yes—
131#400—342no—
132#400—341no—
133#395—342no—
134#395—341no—
135#382—172no—
136#382C2×D1711yescharacteristic
137#340—342no—
138#340—342no—
139C2×C17#165911yescharacteristic
140C2×C17#166011yescharacteristic
141C2×C17#166412yes—
142C2×C17#202711yescharacteristic
143C2×C17#202812yes—
144C22×Q8—341no—
145C22×Q8—341no—
146C22×C8—22no—
147C22×C8—341no—
148C4⋊Q8—172no—
149C4⋊Q8—342no—
150C4⋊Q8—341no—
151C2×C4⋊C4—172no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4—342no—
154C2×C4⋊C4—342no—
155C2×C4⋊C4—341no—
156C2×C4⋊C4D6811yescharacteristic
157C2×C4⋊C4D6811yescharacteristic
158C2×C42C22×D1711yescharacteristic
159#103—682no—
160#103—682no—
161#97—22no—
162#97—22no—
163#97—342no—
164#95—22no—
165#95—342no—
166#95—341no—
167#95—341no—
168#95—342no—
169#95—342no—
170C17#1035911yescharacteristic, complemented, Sylow
171C2×Q8—342no—
172C2×Q8—681no—
173C2×Q8—681no—
174C2×Q8—681no—
175C2×Q8—681no—
176C2×Q8—681no—
177C2×Q8—1361no—
178C2×Q8—1362no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—342no—
182C22×C4—342no—
183C22×C4—681no—
184C22×C4C2×D6811yescharacteristic
185C22×C4D4×D1712yes—
186C2×C8—24no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—342no—
190C2×C8—681no—
191C2×C8—681no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—344no—
195C4⋊C4—682no—
196C4⋊C4—682no—
197C4⋊C4—682no—
198C4⋊C4—1362no—
199C4⋊C4—1361no—
200C42#6325212yes—
201C42#6325411yescharacteristic
202C42Q8×D1711yescharacteristic
203C23#1058756811yescharacteristic
204Q8—682no—
205Q8—682no—
206Q8—1361no—
207Q8—1361no—
208Q8—1364no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—682no—
218C2×C4—682no—
219C2×C4—682no—
220C2×C4—682no—
221C2×C4—682no—
222C2×C4—682no—
223C2×C4—682no—
224C2×C4—1361no—
225C2×C4—1361no—
226C2×C4#1058756111yescharacteristic
227C2×C4#1058757112yes—
228C2×C4#1058758212yes—
229C2×C4#1058758311yescharacteristic
230C8—44no—
231C8—682no—
232C22#1176272211yescharacteristic
233C22#1176297911yescharacteristic
234C22#1176298411yescharacteristic
235C22#1176298812yes—
236C22#1176299312yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—81no—
243C4—1362no—
244C4—1362no—
245C4—1362no—
246C2#1452809111yescharacteristic
247C2#1452809611yescharacteristic
248C2#1452811912yes—
249C2#1452994111yescharacteristic
250C2#1452994412yes—
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1295C2
socle, prime radicalC23×C17#120
hypercenter#1295D17
hypocenterC17#10359
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
33. #10587555Q8semidirect
170. C17#10359semidirect

Maximal subgroups

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

subgroupindex
#145370562
#145297572
#145296032
#145266662
C17×#12952
#1035917