Finite Group Databasebeta

← #16947324

Subgroups — #16947324

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#14530129C211yesmaximal, characteristic
3#14527794C212yesmaximal
4#14526791C212yesmaximal
5C17×#1325C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7C2×C4×Dic34—21no—
8#11763140—22no—
9#11762849—21no—
10#11762739—21no—
11#11762739C2212yes—
12#11762714—22no—
13#11762708—22no—
14#11762578C2211yescharacteristic
15#11762569—21no—
16#11762547—22no—
17C17×#384C2211yescharacteristic
18C17×#324C2212yes—
19C22×Dic34—21no—
20C2×C4⋊C4×C17—22no—
21C2×C4⋊C4×C17D411yescharacteristic
22C2×C4⋊C4×C17C2311yescharacteristic
23C2×C42×C17D411yescharacteristic
24#10587612—21no—
25C2×C4⋊Dic17—21no—
26C2×C4⋊Dic17—21no—
27C2×C4⋊Dic17D412yes—
28#10587610—22no—
29C2×#63225—22no—
30C2×C4×Dic17—22no—
31C2×#63223—21no—
32#10587576—41no—
33#10587574—41no—
34C4×Dic34—42no—
35C4×Dic34—42no—
36C17×C22⋊C8—22no—
37#10587502—22no—
38#10587502—21no—
39#10587502—22no—
40#10587502D412yes—
41#10587488—42no—
42#10587487—42no—
43#10587479—41no—
44#10587478—41no—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17C2×D411yescharacteristic
48C22×C4×C17C2×D412yes—
49C22×Dic17—22no—
50C22×Dic17—22no—
51C2×Dic34—24no—
52C2×Dic34—22no—
53C2×Dic34—42no—
54C2×Dic34—42no—
55C2×C8×C17—42no—
56C4⋊C4×C17—41no—
57C4⋊C4×C17—41no—
58C4⋊C4×C17—44no—
59C42×C17—21no—
60C42×C17SD1612yes—
61C22⋊Dic17—41no—
62C4⋊Dic17—21no—
63C4⋊Dic17—21no—
64C4⋊Dic17—22no—
65C4⋊Dic17—42no—
66C4⋊Dic17—42no—
67C4⋊Dic17—81no—
68#63225—42no—
69#63225—42no—
70#63225—42no—
71C4×Dic17—44no—
72C4×Dic17—41no—
73#63223—22no—
74#63223—41no—
75C2×#3622—41no—
76#12247—171nomaximal, Sylow
77C23×C17#12011yescharacteristic
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17—41no—
87C2×C4×C17C2×SD1611yescharacteristic
88C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
89C2×Dic17—24no—
90C2×Dic17—42no—
91C2×Dic17—42no—
92C2×Dic17—42no—
93C2×Dic17—42no—
94C2×Dic17—41no—
95C2×Dic17—41no—
96C2×Dic17—42no—
97Dic34—48no—
98Dic34—82no—
99C8×C17—82no—
100#3622—42no—
101#2582—171no—
102#2060—171no—
103#1611—172no—
104#1346—172no—
105#1325D1711yescharacteristic
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—
116Dic17—44no—
117Dic17—82no—
118Dic17—81no—
119Dic17—81no—
120#544—171no—
121C2×C4×Q8—341no—
122#483—342no—
123#428—341no—
124#398—172no—
125#398—341no—
126#391—342no—
127#389—342no—
128#384C2×D1711yescharacteristic
129#343—171no—
130#339—341no—
131#328—342no—
132#324C2×D1712yes—
133C2×C17—21no—
134C2×C17—21no—
135C2×C17#160411yescharacteristic
136C2×C17#219811yescharacteristic
137C2×C17#220511yescharacteristic
138C22×Q8—341no—
139C2×M16—341no—
140C4⋊Q8—681no—
141#125—681no—
142C22⋊Q8—342no—
143C4×Q8—682no—
144C4×Q8—682no—
145#117—341no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4—341no—
149C2×C4⋊C4—342no—
150C2×C4⋊C4—341no—
151C2×C4⋊C4D6811yescharacteristic
152C2×C4⋊C4C22×D1711yescharacteristic
153C2×C42—342no—
154C2×C42D6811yescharacteristic
155#107—681no—
156#106—681no—
157#106—682no—
158#103—682no—
159C22⋊C8—22no—
160#95—172no—
161#95—342no—
162#95—341no—
163#95—342no—
164C17#1224711yescharacteristic, complemented, Sylow
165C2×Q8—344no—
166C2×Q8—342no—
167C2×Q8—682no—
168C2×Q8—682no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—342no—
172C22×C4—342no—
173C22×C4C2×D6811yescharacteristic
174C22×C4D4×D1712yes—
175M16—342no—
176M16—681no—
177C2×C8—42no—
178C2×C8—681no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—44no—
182C4⋊C4—341no—
183C4⋊C4—342no—
184C4⋊C4—341no—
185C4⋊C4—682no—
186C4⋊C4—682no—
187C4⋊C4—682no—
188C4⋊C4—682no—
189C4⋊C4—682no—
190C4⋊C4—1361no—
191C22⋊C4—681no—
192C42—21no—
193C42—684no—
194C42—681no—
195C42#6321912yes—
196C23#1058756811yescharacteristic
197Q8—688no—
198Q8—1362no—
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—344no—
209C2×C4—682no—
210C2×C4—682no—
211C2×C4—682no—
212C2×C4—682no—
213C2×C4—681no—
214C2×C4—681no—
215C2×C4—682no—
216C2×C4C2×#6321911yescharacteristic
217C2×C4#1058759411yescharacteristic
218C8—82no—
219C8—682no—
220C22—21no—
221C22—21no—
222C22#1176296211yescharacteristic
223C22#1176299411yescharacteristic
224C22#1176299711yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—42no—
229C4—81no—
230C4—684no—
231C4—1362no—
232C4—1361no—
233C4—1361no—
234C2—21no—
235C2—21no—
236C2#1452775511yescharacteristic
237C2#1453078211yescharacteristic
238C2#1453081511yescharacteristic
subgroupG/H
centerC22#11762997
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1325C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#1325D17
hypocenterC17#12247
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

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
#145301292
#145277942
#145267912
C17×#13252
#1224717