Finite Group Databasebeta

← #16923644

Subgroups — #16923644

Order 4352.

438 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 83 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#14536730C211yesmaximal, characteristic
2#14530321C212yesmaximal, complemented
3#14530317C211yesmaximal, characteristic
4#14526609C212yesmaximal, complemented
5C17×#1270C211yesmaximal, characteristic
6C2×#10587489—22no—
7C2×#10587479—21no—
8C2×#10587487—22no—
9C2×#10587503—21no—
10#11762748—21no—
11#11762721—22no—
12#11762715C2212yes—
13#11762689—21no—
14#11762684—22no—
15#11762679C2211yescharacteristic
16#11762564C2211yescharacteristic
17C17×#390C2211yescharacteristic
18C17×#336C2212yes—
19C22×D4×C17D411yescharacteristic
20C23×Dic17—21no—
21C22×Dic34D412yes—
22C22×C8×C17—22no—
23C2×C4⋊C4×C17D411yescharacteristic
24C2×C22⋊C4×C17—22no—
25C2×C42×C17C2311yescharacteristic
26C2×C22⋊Dic17—21no—
27C2×C22⋊Dic17—22no—
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—22no—
31C2×C4⋊Dic17D412yes—
32C22×#3622—21no—
33#10587503—42no—
34#10587502—21no—
35#10587502—22no—
36#10587502—22no—
37#10587502—22no—
38#10587489—44no—
39#10587487—44no—
40#10587479—42no—
41C24×C17—21no—
42C2×D4×C17—21no—
43C2×D4×C17—41no—
44C2×D4×C17—41no—
45C2×D4×C17Q1612yes—
46C22×C4×C17—21no—
47C22×C4×C17C2×D411yescharacteristic
48C22×C4×C17C2×D412yes—
49C22×Dic17—22no—
50C22×Dic17—22no—
51C22×Dic17—41no—
52C22×Dic17—82no—
53C22×Dic17—81no—
54C2×Dic34—22no—
55C2×Dic34—22no—
56C2×Dic34—42no—
57C2×Dic34—42no—
58C2×C8×C17—24no—
59C2×C8×C17—42no—
60C2×C8×C17—42no—
61C4⋊C4×C17—41no—
62C22⋊C4×C17—42no—
63C22⋊C4×C17—42no—
64C42×C17—41no—
65C22⋊Dic17—82no—
66C22⋊Dic17—81no—
67C4⋊Dic17—42no—
68C4⋊Dic17—41no—
69C4⋊Dic17—42no—
70C4⋊Dic17—42no—
71C4⋊Dic17—41no—
72C4⋊Dic17—81no—
73C2×#3622—22no—
74C2×#3622—41no—
75C2×#3622—41no—
76#7833—171nomaximal, Sylow
77C23×C17—22no—
78C23×C17—42no—
79C23×C17—42no—
80C23×C17—41no—
81C23×C17—41no—
82C23×C17#12011yescharacteristic
83D4×C17—42no—
84D4×C17—81no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—42no—
90C2×C4×C17—41no—
91C2×C4×C17C2×Q1611yescharacteristic
92C2×C4×C17#13512yes—
93C2×C4×C17#13711yescharacteristic
94C2×C4×C17#13712yes—
95C2×Dic17—42no—
96C2×Dic17—42no—
97C2×Dic17—42no—
98C2×Dic17—42no—
99C2×Dic17—42no—
100C2×Dic17—42no—
101C2×Dic17—84no—
102C2×Dic17—82no—
103C2×Dic17—81no—
104Dic34—44no—
105Dic34—82no—
106C8×C17—44no—
107#3622—42no—
108#2804—171no—
109#2095—172no—
110#2094—171no—
111#1272—172no—
112#1270D1711yescharacteristic
113C22×C17—42no—
114C22×C17—44no—
115C22×C17—82no—
116C22×C17—81no—
117C22×C17—81no—
118C22×C17#45111yescharacteristic
119C22×C17#45212yes—
120C22×C17#45312yes—
121C22×C17#45311yescharacteristic
122C22×C17#45811yescharacteristic
123C4×C17—21no—
124C4×C17—21no—
125C4×C17—22no—
126C4×C17—22no—
127C4×C17—81no—
128Dic17—82no—
129Dic17—82no—
130Dic17—161no—
131C2×#107—342no—
132C2×#106—341no—
133C2×#103—342no—
134C2×#102—341no—
135#400—341no—
136#393—342no—
137#391—172no—
138#390C2×D1711yescharacteristic
139#386—341no—
140#385—342no—
141#384—171no—
142#336—171no—
143#336C2×D1712yes—
144C2×C17—82no—
145C2×C17#161411yescharacteristic
146C2×C17#161812yes—
147C2×C17#163011yescharacteristic
148C2×C17#163212yes—
149C2×C17#165511yescharacteristic
150C22×Q8—172no—
151C22×D4D6811yescharacteristic
152C23×C4—341no—
153C22×C8—22no—
154C22×C8—341no—
155C2×C4⋊C4—172no—
156C2×C4⋊C4—341no—
157C2×C4⋊C4—342no—
158C2×C4⋊C4—341no—
159C2×C4⋊C4D6811yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—342no—
162C2×C22⋊C4—341no—
163C2×C42C22×D1711yescharacteristic
164#107—684no—
165#106—682no—
166#103—684no—
167#102—682no—
168#95—342no—
169#95—341no—
170#95—342no—
171#95—342no—
172C17#783311yescharacteristic, complemented, Sylow
173C24—21no—
174C2×Q8—342no—
175C2×Q8—342no—
176C2×Q8—682no—
177C2×Q8—682no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4Dic6812yes—
182C22×C4—21no—
183C22×C4—342no—
184C22×C4—342no—
185C22×C4—681no—
186C22×C4—1362no—
187C22×C4—1361no—
188C22×C4C2×D6811yescharacteristic
189C22×C4D4×D1712yes—
190C2×C8—24no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—342no—
194C2×C8—681no—
195C2×C8—681no—
196C4⋊C4—41no—
197C4⋊C4—682no—
198C4⋊C4—681no—
199C4⋊C4—682no—
200C4⋊C4—681no—
201C4⋊C4—682no—
202C4⋊C4—1361no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—1362no—
206C22⋊C4—1361no—
207C42—41no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23#1058756811yescharacteristic
214Q8—684no—
215Q8—1362no—
216D4—42no—
217D4—81no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—682no—
225C2×C4—682no—
226C2×C4—682no—
227C2×C4—682no—
228C2×C4—682no—
229C2×C4—682no—
230C2×C4—1362no—
231C2×C4—1361no—
232C2×C4—1364no—
233C2×C4C2×Dic6811yescharacteristic
234C2×C4#1058759511yescharacteristic
235C2×C4#1058759812yes—
236C2×C4#1058760112yes—
237C8—44no—
238C8—682no—
239C22—42no—
240C22—44no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22#1176296711yescharacteristic
245C22#1176297212yes—
246C22#1176297812yes—
247C22#1176297911yescharacteristic
248C22#1176300011yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—81no—
254C4—2721no—
255C4—1362no—
256C4—1362no—
257C2—82no—
258C2#1452781111yescharacteristic
259C2#1452783312yes—
260C2#1452790611yescharacteristic
261C2#1452792012yes—
262C2#1452807411yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1270C2
socleC23×C17#120
prime radicalC22×D4×C17D4
hypercenter#1270D17
hypocenterC17#7833
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #14530321C2semidirect
4. #14526609C2semidirect
172. C17#7833semidirect

Maximal subgroups

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

subgroupindex
#145367302
#145303212
#145303172
#145266092
C17×#12702
#783317