Finite Group Databasebeta

← #16947321

Subgroups — #16947321

Order 4352.

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

normal subgroups only

This group has 394 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#14534157C211yesmaximal, characteristic
2#14527792C212yesmaximal
3#14526789C212yesmaximal
4#14526730C211yesmaximal, characteristic
5C17×#2060C211yesmaximal, characteristic
6C2×C4×Dic34—21no—
7#11763137—22no—
8#11762739—21no—
9#11762739C2212yes—
10#11762714—22no—
11#11762708—22no—
12#11762546—22no—
13#11762535C2212yes—
14C17×#544C2211yescharacteristic
15C17×#428—21no—
16C17×#384C2211yescharacteristic
17C17×#343C2211yescharacteristic
18C17×#339—21no—
19C22×Dic34—21no—
20C2×M16×C17—21no—
21C17×C4⋊Q8—41no—
22C17×#125—41no—
23C17×C22⋊Q8—22no—
24C17×#117—21no—
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C42×C17D411yescharacteristic
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17D412yes—
32C2×#63225—22no—
33C2×C4×Dic17—22no—
34C4×Dic34—42no—
35C4×Dic34—42no—
36C17×#107—41no—
37C17×#106—41no—
38#10587506—42no—
39#10587502—22no—
40#10587502—21no—
41#10587502—22no—
42#10587502D412yes—
43C22⋊#3622—22no—
44#10587479—42no—
45C2×Q8×C17—22no—
46C22×C4×C17—21no—
47C22×C4×C17—21no—
48C22×C4×C17C2×D412yes—
49C22×C4×C17C2×D411yescharacteristic
50C22×Dic17—22no—
51C22×Dic17—22no—
52C2×Dic34—24no—
53C2×Dic34—42no—
54C2×Dic34—42no—
55M16×C17—22no—
56M16×C17—41no—
57C2×C8×C17—41no—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—21no—
60C4⋊C4×C17—21no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—42no—
64C4⋊C4×C17—44no—
65C22⋊C4×C17—41no—
66C42×C17—21no—
67C42×C17—41no—
68C42×C17SD1612yes—
69C4⋊Dic17—42no—
70C4⋊Dic17—42no—
71C4⋊Dic17—81no—
72#63225—42no—
73#63225—42no—
74C4×Dic17—44no—
75C2×#3622—42no—
76#12247—171nomaximal, Sylow
77C23×C17#12011yescharacteristic
78Q8×C17—82no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17—41no—
88C2×C4×C17—41no—
89C2×C4×C17—41no—
90C2×C4×C17C2×SD1611yescharacteristic
91C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
92C2×Dic17—24no—
93C2×Dic17—42no—
94C2×Dic17—42no—
95C2×Dic17—42no—
96C2×Dic17—42no—
97C2×Dic17—42no—
98Dic34—48no—
99C8×C17—42no—
100#3622—82no—
101#2582—171no—
102#2060D1711yescharacteristic
103#1611—172no—
104#1346—172no—
105#1325—171no—
106C22×C17—21no—
107C22×C17—21no—
108C22×C17#45011yescharacteristic
109C22×C17#45611yescharacteristic
110C22×C17B(4,𝔽2)11yescharacteristic
111C4×C17—24no—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—42no—
115C4×C17—81no—
116C4×C17—81no—
117C4×C17—81no—
118Dic17—44no—
119Dic17—82no—
120#544C2×D1711yescharacteristic
121C2×C4×Q8—341no—
122#483—342no—
123#428—21no—
124#398—172no—
125#398—341no—
126#391—342no—
127#389—342no—
128#384C2×D1711yescharacteristic
129#343C2×D1711yescharacteristic
130#339—21no—
131#328—342no—
132#324—172no—
133C2×C17—21no—
134C2×C17—21no—
135C2×C17#160411yescharacteristic
136C2×C17#219811yescharacteristic
137C2×C17#220511yescharacteristic
138C22×Q8—341no—
139C2×M16—21no—
140C4⋊Q8—41no—
141#125—41no—
142C22⋊Q8—22no—
143C4×Q8—682no—
144C4×Q8—682no—
145#117—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4—341no—
149C2×C4⋊C4—342no—
150C2×C4⋊C4—341no—
151C2×C4⋊C4C22⋊D1711yescharacteristic
152C2×C4⋊C4C22×D1711yescharacteristic
153C2×C42—342no—
154C2×C42C22⋊D1711yescharacteristic
155#107—41no—
156#106—41no—
157#106—682no—
158#103—682no—
159C22⋊C8—342no—
160#95—172no—
161#95—342no—
162#95—341no—
163#95—342no—
164C17#1224711yescharacteristic, complemented, Sylow
165C2×Q8—22no—
166C2×Q8—344no—
167C2×Q8—682no—
168C2×Q8—682no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—342no—
172C22×C4—342no—
173C22×C4D4×D1712yes—
174C22×C4C2×C22⋊D1711yescharacteristic
175M16—22no—
176M16—41no—
177C2×C8—41no—
178C2×C8—682no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—682no—
187C4⋊C4—682no—
188C4⋊C4—682no—
189C4⋊C4—682no—
190C4⋊C4—1361no—
191C22⋊C4—41no—
192C42—21no—
193C42—41no—
194C42—684no—
195C42#6323012yes—
196C23#1058762311yescharacteristic
197Q8—82no—
198Q8—688no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—344no—
211C2×C4—682no—
212C2×C4—682no—
213C2×C4—682no—
214C2×C4—682no—
215C2×C4—682no—
216C2×C4C2×#6323011yescharacteristic
217C2×C4#1058763511yescharacteristic
218C8—42no—
219C8—1362no—
220C22—21no—
221C22—21no—
222C22#1176295911yescharacteristic
223C22#1176299111yescharacteristic
224C22#1176299511yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—42no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4—684no—
233C4—1362no—
234C2—21no—
235C2—21no—
236C2#1452775211yescharacteristic
237C2#1453077911yescharacteristic
238C2#1453081311yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2060C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2060D17
hypocenterC17#12247
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2060

Complements

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

NH ≅ G/Ntype
164. C17#12247semidirect

Maximal subgroups

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

subgroupindex
#145341572
#145277922
#145267892
#145267302
C17×#20602
#1224717