Finite Group Databasebeta

← #16936573

Subgroups — #16936573

Order 4352.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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#14539727C212yesmaximal
2#14539308C211yesmaximal, characteristic, complemented
3#14529608C211yesmaximal, characteristic
4#14527349C212yesmaximal, complemented
5C17×#1281C211yesmaximal, characteristic, complemented
6#11763366C2212yes—
7C2×#10587560C2211yescharacteristic
8#11763218—22no—
9#11763218—22no—
10#11763182—22no—
11#11763156—22no—
12#11763010—24no—
13#11762962—24no—
14#11762847—22no—
15C2×#10587492—22no—
16#11762675C2211yescharacteristic
17#11762547—24no—
18C17×#431C2212yes—
19C2×C17×#97C2211yescharacteristic
20C22×D68—21no—
21C22×D68—22no—
22C22×C8×C17D412yes—
23C2×C42×C17C2311yescharacteristic
24#10587612—22no—
25C2×C4⋊Dic17—21no—
26#10587610—22no—
27#10587610—22no—
28C2×#63219—28no—
29#10587560—22no—
30#10587560—21no—
31#10587560D412yes—
32#10587556—22no—
33#10587555D412yes—
34#10587555D412yes—
35C4×Dic34—22no—
36C4×Dic34—22no—
37C17×C4⋊C8—22no—
38C17×C22⋊C8D414yes—
39C17×#97—22no—
40C4×C8×C17—22no—
41#10587502—22no—
42#10587492—28no—
43#10587492—44no—
44#10587489—24no—
45#10587488—22no—
46#10587488—22no—
47C23×D17—41no—
48C22×C4×C17C2×D412yes—
49C22×C4×C17C2×D411yescharacteristic
50C22×Dic17—41no—
51C2×D68—24no—
52C2×D68—22no—
53C2×D68—42no—
54C2×D68—42no—
55C2×D68—41no—
56C2×D68—42no—
57C2×D68—84no—
58C2×Dic34—22no—
59C2×Dic34—22no—
60C2×C8×C17—22no—
61C2×C8×C17—22no—
62C2×C8×C17—24no—
63C2×C8×C17SD1614yescomplemented
64C42×C17C2×D412yes—
65C42×C17C2×D411yescharacteristic
66C42×C17#4111yescharacteristic
67C22⋊Dic17—42no—
68C4⋊Dic17—22no—
69C4⋊Dic17—22no—
70C4⋊Dic17—22no—
71C4⋊Dic17—22no—
72C4⋊Dic17—42no—
73#63225—42no—
74#63225—42no—
75C4×Dic17—42no—
76#63219—88no—
77#10220—171nomaximal, Sylow
78C23×C17#12711yescharacteristic
79C22×D17—42no—
80C22×D17—84no—
81C22×D17—81no—
82C22×D17—81no—
83C2×C4×C17—42no—
84C2×C4×C17#12012yes—
85C2×C4×C17#12011yescharacteristic
86C2×C4×C17#12112yes—
87C2×C4×C17#12111yescharacteristic
88C2×C4×C17C2×SD1612yes—
89C2×C4×C17#13712yes—
90C2×Dic17—42no—
91C2×Dic17—42no—
92C2×Dic17—82no—
93C2×Dic17—81no—
94D68—48no—
95D68—44no—
96D68—84no—
97D68—82no—
98Dic34—44no—
99Dic34—82no—
100C8×C17—24no—
101C8×C17—44no—
102#3208—172no—
103#3162—171no—
104#1961—171no—
105#1518—172no—
106#1281D1711yescharacteristic, complemented
107C22×C17—22no—
108C22×C17#39211yescharacteristic
109C22×C17#49211yescharacteristic
110C22×C17#49711yescharacteristic
111C2×D17—84no—
112C2×D17—82no—
113C2×D17—162no—
114C2×D17—162no—
115C4×C17—22no—
116C4×C17—22no—
117C4×C17#44812yes—
118C4×C17#46112yes—
119Dic17—82no—
120Dic17—82no—
121Dic17—161no—
122#533—172no—
123C2×#127—171no—
124#501—342no—
125#501—342no—
126#492—342no—
127#486—342no—
128#460—344no—
129#450—344no—
130#431C2×D1712yes—
131#427—342no—
132C2×#102—342no—
133C2×#97C2×D1711yescharacteristic
134#382—171no—
135#328—344no—
136C2×C17—21no—
137C2×C17—21no—
138C2×C17#200812yes—
139C2×C17#201311yescharacteristic
140D17—162no—
141C22×D4—341no—
142C22×D4—342no—
143C2×SD16—348no—
144C22×C8D6812yes—
145C4⋊Q8—172no—
146C4⋊Q8—172no—
147#127—172no—
148#127—342no—
149#127—341no—
150#125—342no—
151C22⋊Q8—342no—
152C22⋊Q8—342no—
153C4×Q8—342no—
154C4×Q8—342no—
155#117—342no—
156C2×C4⋊C4—341no—
157C2×C42C22×D1711yescharacteristic
158#107—344no—
159#106—342no—
160#106—342no—
161C4⋊C8—22no—
162#102—348no—
163#102—684no—
164C22⋊C8D6814yes—
165#97—22no—
166C4×C8—22no—
167#95—342no—
168C17#1022011yescharacteristic, complemented, Sylow
169C24—681no—
170C2×Q8—342no—
171C2×Q8—342no—
172C2×D4—344no—
173C2×D4—342no—
174C2×D4—681no—
175C2×D4—682no—
176C2×D4—682no—
177C2×D4—682no—
178C2×D4—1364no—
179C22×C4—681no—
180C22×C4C2×D6812yes—
181C22×C4C2×D6811yescharacteristic
182SD16—1368no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—24no—
186C2×C8#6321914yescomplemented
187C4⋊C4—342no—
188C4⋊C4—342no—
189C4⋊C4—342no—
190C4⋊C4—342no—
191C4⋊C4—682no—
192C4⋊C4—682no—
193C4⋊C4—682no—
194C22⋊C4—682no—
195C42—682no—
196C42D4×D1712yes—
197C42D4×D1711yescharacteristic
198C42#6325611yescharacteristic
199C23—682no—
200C23—1364no—
201C23—1361no—
202C23—1361no—
203C23#1058756011yescharacteristic
204Q8—684no—
205Q8—1362no—
206D4—688no—
207D4—684no—
208D4—1364no—
209D4—1362no—
210C2×C4—42no—
211C2×C4—682no—
212C2×C4—682no—
213C2×C4—1362no—
214C2×C4—1361no—
215C2×C4#1058756812yes—
216C2×C4#1058756811yescharacteristic
217C2×C4#1058758112yes—
218C2×C4#1058758111yescharacteristic
219C2×C4C2×#6321912yes—
220C2×C4#1058759512yes—
221C8—24no—
222C8—44no—
223C22—22no—
224C22—2722no—
225C22—2722no—
226C22—1364no—
227C22—1362no—
228C22#1176271911yescharacteristic
229C22#1176318211yescharacteristic
230C22#1176320311yescharacteristic
231C4—22no—
232C4—22no—
233C4—2721no—
234C4—1362no—
235C4—1362no—
236C4#1176295012yes—
237C4#1176301512yes—
238C2—21no—
239C2—21no—
240C2—2722no—
241C2#1452984012yes—
242C2#1452986911yescharacteristic
subgroupG/H
centerC22#11762719
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1281C2
socleC22×C17#392
prime radicalC2×#10587560C22
hypercenter#1281D17
hypocenterC17#10220
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
2. #14539308C2semidirect
4. #14527349C2semidirect
5. C17×#1281C2semidirect
63. C2×C8×C17SD16semidirect
106. #1281D17semidirect
168. C17#10220semidirect
186. C2×C8#63219semidirect

Maximal subgroups

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

subgroupindex
#145397272
#145393082
#145296082
#145273492
C17×#12812
#1022017