Finite Group Databasebeta

← #16947407

Subgroups — #16947407

Order 4352.

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

normal subgroups only

This group has 330 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#14537056C211yesmaximal, characteristic
2#14527826C212yesmaximal
3#14526810C212yesmaximal
4#14526728C211yesmaximal, characteristic
5C17×#2061C211yesmaximal, characteristic
6#11763142—22no—
7#11762751—22no—
8#11762748C2212yes—
9#11762730—22no—
10#11762729—21no—
11#11762675—21no—
12#11762546—22no—
13#11762535C2212yes—
14C17×#544C2211yescharacteristic
15C17×#428—21no—
16C17×#382C2211yescharacteristic
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×C17D411yescharacteristic
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17D411yescharacteristic
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17D412yes—
31C2×#63225—22no—
32C17×#107—41no—
33C17×#106—41no—
34C17×#95—22no—
35#10587506—42no—
36#10587502—21no—
37#10587502—22no—
38#10587502—22no—
39#10587502—22no—
40#10587502D412yes—
41C22⋊#3622—22no—
42#10587478—42no—
43C2×Q8×C17—22no—
44C22×C4×C17—21no—
45C22×C4×C17—21no—
46C22×C4×C17C2×D412yes—
47C22×C4×C17C2×D411yescharacteristic
48C22×Dic17—22no—
49C22×Dic17—22no—
50C2×Dic34—82no—
51C2×Dic34—81no—
52M16×C17—22no—
53M16×C17—41no—
54C2×C8×C17—41no—
55C4⋊C4×C17—22no—
56C4⋊C4×C17—21no—
57C4⋊C4×C17—21no—
58C4⋊C4×C17—41no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—42no—
61C22⋊C4×C17—41no—
62C42×C17—21no—
63C42×C17—41no—
64C42×C17Q1612yes—
65C4⋊Dic17—42no—
66C4⋊Dic17—81no—
67C4⋊Dic17—81no—
68#63225—82no—
69C2×#3622—42no—
70#12261—171nomaximal, Sylow
71C23×C17#12011yescharacteristic
72Q8×C17—82no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—41no—
78C2×C4×C17—41no—
79C2×C4×C17—42no—
80C2×C4×C17—42no—
81C2×C4×C17—41no—
82C2×C4×C17—41no—
83C2×C4×C17—41no—
84C2×C4×C17C2×Q1611yescharacteristic
85C2×C4×C17#13711yescharacteristic
86C2×Dic17—42no—
87C2×Dic17—42no—
88C2×Dic17—42no—
89C2×Dic17—42no—
90C2×Dic17—42no—
91C2×Dic17—42no—
92Dic34—84no—
93C8×C17—42no—
94#3622—82no—
95#2844—171no—
96#2061D1711yescharacteristic
97#1617—172no—
98#1353—172no—
99#1324—171no—
100C22×C17—21no—
101C22×C17—21no—
102C22×C17#45111yescharacteristic
103C22×C17#45611yescharacteristic
104C22×C17B(4,𝔽2)11yescharacteristic
105C4×C17—21no—
106C4×C17—21no—
107C4×C17—42no—
108C4×C17—42no—
109C4×C17—81no—
110C4×C17—81no—
111C4×C17—81no—
112Dic17—82no—
113Dic17—82no—
114#544C2×D1711yescharacteristic
115#484—342no—
116#428—21no—
117#400—172no—
118#400—342no—
119#395—342no—
120#395—341no—
121#382—341no—
122#382C2×D1711yescharacteristic
123#343C2×D1711yescharacteristic
124#339—21no—
125#328—342no—
126#324—172no—
127C2×C17—21no—
128C2×C17—21no—
129C2×C17#160511yescharacteristic
130C2×C17#220111yescharacteristic
131C2×C17#220211yescharacteristic
132C22×Q8—341no—
133C2×M16—21no—
134C4⋊Q8—41no—
135#125—41no—
136C22⋊Q8—22no—
137#117—21no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4—341no—
140C2×C4⋊C4—341no—
141C2×C4⋊C4—342no—
142C2×C4⋊C4C22⋊D1711yescharacteristic
143C2×C4⋊C4C22×D1711yescharacteristic
144C2×C42C22⋊D1711yescharacteristic
145#107—41no—
146#107—682no—
147#106—41no—
148#103—682no—
149C22⋊C8—342no—
150#95—22no—
151#95—172no—
152#95—342no—
153#95—341no—
154#95—342no—
155#95—342no—
156C17#1226111yescharacteristic, complemented, Sylow
157C2×Q8—22no—
158C2×Q8—1362no—
159C2×Q8—1361no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—342no—
163C22×C4—342no—
164C22×C4D4×D1712yes—
165C22×C4C2×C22⋊D1711yescharacteristic
166M16—22no—
167M16—41no—
168C2×C8—41no—
169C2×C8—682no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—682no—
177C4⋊C4—1362no—
178C4⋊C4—1361no—
179C4⋊C4—1361no—
180C22⋊C4—41no—
181C42—21no—
182C42—41no—
183C42#6323112yes—
184C23#1058762311yescharacteristic
185Q8—82no—
186Q8—1364no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—682no—
199C2×C4—682no—
200C2×C4—682no—
201C2×C4—682no—
202C2×C4—682no—
203C2×C4—682no—
204C2×C4C2×#6323111yescharacteristic
205C2×C4#1058763711yescharacteristic
206C8—42no—
207C8—1362no—
208C22—21no—
209C22—21no—
210C22#1176296411yescharacteristic
211C22#1176299111yescharacteristic
212C22#1176299511yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4—1362no—
221C4—1362no—
222C2—21no—
223C2—21no—
224C2#1452775711yescharacteristic
225C2#1453079611yescharacteristic
226C2#1453080011yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2061C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2061D17
hypocenterC17#12261
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
156. C17#12261semidirect

Maximal subgroups

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

subgroupindex
#145370562
#145278262
#145268102
#145267282
C17×#20612
#1226117