Finite Group Databasebeta

← #16947171

Subgroups — #16947171

Order 4352.

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

normal subgroups only

This group has 336 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#14536772C211yesmaximal, characteristic
2#14527826C212yesmaximal
3#14526810C212yesmaximal
4#14526718C211yesmaximal, characteristic
5C17×#2028C211yesmaximal, characteristic
6#11763142—22no—
7#11762748C2212yes—
8#11762741—22no—
9#11762716—21no—
10#11762707—21no—
11#11762696—22no—
12#11762546—22no—
13#11762535C2212yes—
14C17×#544C2211yescharacteristic
15C17×#419—21no—
16C17×#391C2211yescharacteristic
17C17×#343C2211yescharacteristic
18C17×#339—21no—
19C22×Q8×C17D411yescharacteristic
20C2×M16×C17—21no—
21C17×C4⋊Q8—41no—
22C17×#125—41no—
23C17×C22⋊Q8—22no—
24C17×#117—21no—
25C2×C4⋊C4×C17C2311yescharacteristic
26C2×C42×C17D411yescharacteristic
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17D412yes—
31C2×#63225—22no—
32C2×C4×Dic17—21no—
33C17×#103—41no—
34C17×#103—41no—
35C17×#95—22no—
36#10587506—42no—
37#10587502—22no—
38#10587502—22no—
39#10587502—22no—
40#10587502D412yes—
41C22⋊#3622—22no—
42#10587478—42no—
43C2×Q8×C17—21no—
44C2×Q8×C17—22no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17Q1612yes—
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17C2×D412yes—
51C22×C4×C17C2×D411yescharacteristic
52C22×Dic17—22no—
53C22×Dic17—22no—
54M16×C17—41no—
55M16×C17—41no—
56C2×C8×C17—41no—
57C4⋊C4×C17—22no—
58C4⋊C4×C17—21no—
59C4⋊C4×C17—21no—
60C4⋊C4×C17—41no—
61C4⋊C4×C17—42no—
62C22⋊C4×C17—41no—
63C42×C17—41no—
64C42×C17—41no—
65C4⋊Dic17—42no—
66C4⋊Dic17—42no—
67C4⋊Dic17—81no—
68C4⋊Dic17—81no—
69#63225—44no—
70C4×Dic17—81no—
71C2×#3622—42no—
72#12225—171nomaximal, Sylow
73C23×C17#12011yescharacteristic
74Q8×C17—42no—
75Q8×C17—81no—
76Q8×C17—82no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—41no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
85C2×C4×C17—41no—
86C2×C4×C17—42no—
87C2×C4×C17—41no—
88C2×C4×C17—41no—
89C2×C4×C17C2×Q1611yescharacteristic
90C2×C4×C17#13711yescharacteristic
91C2×Dic17—42no—
92C2×Dic17—42no—
93C2×Dic17—42no—
94C2×Dic17—42no—
95C2×Dic17—42no—
96C2×Dic17—42no—
97C8×C17—81no—
98#3622—82no—
99#2809—171no—
100#2028D1711yescharacteristic
101#1617—172no—
102#1353—172no—
103#1319—171no—
104C22×C17—21no—
105C22×C17—21no—
106C22×C17#45111yescharacteristic
107C22×C17#45511yescharacteristic
108C22×C17B(4,𝔽2)11yescharacteristic
109C4×C17—21no—
110C4×C17—21no—
111C4×C17—42no—
112C4×C17—42no—
113C4×C17—42no—
114C4×C17—81no—
115C4×C17—81no—
116Dic17—82no—
117Dic17—82no—
118#544C2×D1711yescharacteristic
119#484—342no—
120#419—21no—
121#400—172no—
122#398—342no—
123#391—341no—
124#391C2×D1711yescharacteristic
125#389—341no—
126#387—342no—
127#343C2×D1711yescharacteristic
128#339—21no—
129#328—342no—
130#324—172no—
131C2×C17—21no—
132C2×C17—21no—
133C2×C17#160211yescharacteristic
134C2×C17#220111yescharacteristic
135C2×C17#220411yescharacteristic
136C22×Q8C22⋊D1711yescharacteristic
137C2×M16—21no—
138C4⋊Q8—41no—
139#125—41no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—342no—
144C2×C4⋊C4—341no—
145C2×C4⋊C4—341no—
146C2×C4⋊C4—341no—
147C2×C4⋊C4C22×D1711yescharacteristic
148C2×C42—341no—
149C2×C42C22⋊D1711yescharacteristic
150#107—682no—
151#103—41no—
152#103—41no—
153#103—682no—
154C22⋊C8—342no—
155#95—22no—
156#95—172no—
157#95—342no—
158#95—342no—
159#95—342no—
160C17#1222511yescharacteristic, complemented, Sylow
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#6323112yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—342no—
169C22×C4—342no—
170C22×C4D4×D1712yes—
171C22×C4C2×C22⋊D1711yescharacteristic
172M16—41no—
173M16—41no—
174C2×C8—41no—
175C2×C8—682no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—682no—
182C4⋊C4—684no—
183C4⋊C4—682no—
184C4⋊C4—1361no—
185C4⋊C4—1361no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42—1361no—
190C23#1058762311yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—682no—
207C2×C4—682no—
208C2×C4—682no—
209C2×C4—682no—
210C2×C4—682no—
211C2×C4—682no—
212C2×C4C2×#6323111yescharacteristic
213C2×C4#1058763711yescharacteristic
214C8—81no—
215C8—1362no—
216C22—21no—
217C22—21no—
218C22#1176296411yescharacteristic
219C22#1176298611yescharacteristic
220C22#1176299511yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—81no—
227C4—81no—
228C4—1362no—
229C4—1362no—
230C2—21no—
231C2—21no—
232C2#1452774011yescharacteristic
233C2#1453079611yescharacteristic
234C2#1453080911yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2028C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2028D17
hypocenterC17#12225
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

Complements

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

NH ≅ G/Ntype
160. C17#12225semidirect

Maximal subgroups

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

subgroupindex
#145367722
#145278262
#145268102
#145267182
C17×#20282
#1222517