Finite Group Databasebeta

← #16936570

Subgroups — #16936570

Order 4352.

542 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 103 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#14539725C212yesmaximal
2#14539308C211yesmaximal, characteristic, complemented
3#14527347C212yesmaximal, complemented
4#14526629C211yesmaximal, characteristic, complemented
5C17×#1961C211yesmaximal, characteristic
6#11763366C2212yes—
7#11763218—22no—
8#11763218—22no—
9#11763181—22no—
10#11763155—22no—
11#11763006—24no—
12#11762960—24no—
13#11762862C2212yes—
14#11762847—22no—
15#11762545—24no—
16C2×C17×#127C2211yescharacteristic
17C2×C17×#102—22no—
18C2×C17×#97C2211yescharacteristic
19C17×#382C2211yescharacteristic
20C22×D4×C17—21no—
21C22×D4×C17—22no—
22C22×C8×C17D412yes—
23C17×#127—22no—
24C17×#127—21no—
25C17×#127D412yes—
26C2×C4⋊C4×C17—21no—
27C2×C42×C17C2311yescharacteristic
28C2×#63229—28no—
29#10587612—22no—
30#10587610—22no—
31#10587610—22no—
32#10587556—22no—
33#10587555D412yes—
34#10587555D412yes—
35C4×Dic34—22no—
36C4×Dic34—22no—
37C17×#102—44no—
38C17×#97—22no—
39C17×#95—22no—
40#10587503—28no—
41C22⋊#3622D414yes—
42#10587489—24no—
43#10587488—22no—
44#10587488—22no—
45C4⋊#3622—22no—
46C4×#3622—22no—
47C24×C17—41no—
48C2×D4×C17—24no—
49C2×D4×C17—22no—
50C2×D4×C17—41no—
51C2×D4×C17—42no—
52C2×D4×C17—42no—
53C2×D4×C17—42no—
54C2×D4×C17—84no—
55C22×C4×C17—41no—
56C22×C4×C17C2×D412yes—
57C22×C4×C17C2×D411yescharacteristic
58C2×Dic34—22no—
59C2×Dic34—22no—
60C2×C8×C17—22no—
61C2×C8×C17—22no—
62C2×C8×C17SD1614yescomplemented
63C4⋊C4×C17—42no—
64C42×C17C2×D411yescharacteristic
65C42×C17C2×D412yes—
66C42×C17#4111yescharacteristic
67C22⋊Dic17—42no—
68#63229—88no—
69C4⋊Dic17—22no—
70C4⋊Dic17—22no—
71C4⋊Dic17—22no—
72C4⋊Dic17—22no—
73#63225—42no—
74#63225—42no—
75C4×Dic17—42no—
76C2×#3622—24no—
77#10220—171nomaximal, Sylow
78C23×C17—42no—
79C23×C17—84no—
80C23×C17—81no—
81C23×C17—81no—
82C23×C17#12711yescharacteristic
83D4×C17—48no—
84D4×C17—44no—
85D4×C17—84no—
86D4×C17—82no—
87C2×C4×C17—42no—
88C2×C4×C17—82no—
89C2×C4×C17—81no—
90C2×C4×C17#12012yes—
91C2×C4×C17#12011yescharacteristic
92C2×C4×C17#12112yes—
93C2×C4×C17#12111yescharacteristic
94C2×C4×C17C2×SD1612yes—
95C2×C4×C17#13712yes—
96C2×Dic17—42no—
97C2×Dic17—42no—
98Dic34—44no—
99Dic34—82no—
100C8×C17—24no—
101#3622—44no—
102#3208—172no—
103#3162—171no—
104#1961D1711yescharacteristic
105#1518—172no—
106#1281—171no—
107C22×C17—22no—
108C22×C17—84no—
109C22×C17—82no—
110C22×C17—162no—
111C22×C17—162no—
112C22×C17#39211yescharacteristic
113C22×C17#49211yescharacteristic
114C22×C17#49711yescharacteristic
115C4×C17—22no—
116C4×C17—22no—
117C4×C17—161no—
118C4×C17#44812yes—
119C4×C17#46112yes—
120Dic17—82no—
121Dic17—82no—
122#533—172no—
123C2×#127C2×D1711yescharacteristic
124#501—342no—
125#501—342no—
126#492—342no—
127#486—342no—
128#460—344no—
129#450—344no—
130#431—172no—
131#427—342no—
132C2×#102—22no—
133C2×#97C2×D1711yescharacteristic
134#382C2×D1711yescharacteristic
135#328—344no—
136C2×C17—21no—
137C2×C17—21no—
138C2×C17—162no—
139C2×C17#200812yes—
140C2×C17#201311yescharacteristic
141C22×D4—22no—
142C22×D4—21no—
143C2×SD16—348no—
144C22×C8C22⋊D1712yes—
145C4⋊Q8—172no—
146C4⋊Q8—172no—
147#127—21no—
148#127—22no—
149#127D6812yes—
150#125—342no—
151C22⋊Q8—342no—
152C22⋊Q8—342no—
153C4×Q8—342no—
154C4×Q8—342no—
155#117—342no—
156C2×C4⋊C4—21no—
157C2×C42C22×D1711yescharacteristic
158#107—344no—
159#106—342no—
160#106—342no—
161C4⋊C8—342no—
162#102—44no—
163#102—348no—
164C22⋊C8—174no—
165#97—22no—
166C4×C8—342no—
167#95—22no—
168C17#1022011yescharacteristic, complemented, Sylow
169C24—41no—
170C2×Q8—342no—
171C2×Q8—342no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—41no—
177C2×D4—42no—
178C2×D4—84no—
179C22×C4—41no—
180C22×C4D4×D1712yes—
181C22×C4C2×C22⋊D1711yescharacteristic
182SD16—1368no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—344no—
186C2×C8#6322914yescomplemented
187C4⋊C4—42no—
188C4⋊C4—342no—
189C4⋊C4—342no—
190C4⋊C4—342no—
191C4⋊C4—342no—
192C4⋊C4—682no—
193C4⋊C4—682no—
194C22⋊C4—682no—
195C42—682no—
196C42C2×D6811yescharacteristic
197C42#6325211yescharacteristic
198C42D4×D1712yes—
199C23—42no—
200C23—84no—
201C23—81no—
202C23—81no—
203C23#1058762611yescharacteristic
204Q8—684no—
205Q8—1362no—
206D4—48no—
207D4—44no—
208D4—84no—
209D4—82no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—81no—
213C2×C4—682no—
214C2×C4—682no—
215C2×C4#1058756812yes—
216C2×C4#1058757012yes—
217C2×C4#1058761611yescharacteristic
218C2×C4C2×#6322912yes—
219C2×C4#1058762311yescharacteristic
220C2×C4#1058763712yes—
221C8—24no—
222C8—684no—
223C22—22no—
224C22—84no—
225C22—82no—
226C22—162no—
227C22—162no—
228C22#1176271811yescharacteristic
229C22#1176318111yescharacteristic
230C22#1176320411yescharacteristic
231C4—22no—
232C4—22no—
233C4—161no—
234C4—1362no—
235C4—1362no—
236C4#1176294712yes—
237C4#1176301412yes—
238C2—21no—
239C2—21no—
240C2—162no—
241C2#1452983912yes—
242C2#1452986611yescharacteristic
subgroupG/H
centerC22#11762718
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1961C2
socleC22×C17#392
prime radicalC2×C17×#127C22
hypercenter#1961D17
hypocenterC17#10220
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. #14539308C2semidirect
3. #14527347C2semidirect
4. #14526629C2semidirect
62. C2×C8×C17SD16semidirect
168. C17#10220semidirect
186. C2×C8#63229semidirect

Maximal subgroups

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

subgroupindex
#145397252
#145393082
#145273472
#145266292
C17×#19612
#1022017