Finite Group Databasebeta

← #16927895

Subgroups — #16927895

Order 4352.

498 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 131 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#14528776C211yesmaximal, characteristic
2#14527105C211yesmaximal, characteristic
3#14526687C212yesmaximal, complemented
4#14526642C212yesmaximal, complemented
5C17×#2349C211yesmaximal, characteristic
6#11762862C2212yes—
7#11762833C2212yes—
8C2×#10587504—21no—
9#11762578—21no—
10#11762533—22no—
11C2×C4×D4×C17—21no—
12C2×C4×D4×C17C2211yescharacteristic
13C17×#389—21no—
14C17×#385—22no—
15C17×#382C2211yescharacteristic
16C17×#380C412yes—
17C4×C4⋊C4×C17C2211yescharacteristic
18C2×C17×#95C412yes—
19C22×D4×C17D411yescharacteristic
20C23×C4×C17—22no—
21C23×C4×C17C2×C412yes—
22C4×D4×C17—22no—
23C4×D4×C17—22no—
24C4×D4×C17—44no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17D412yes—
31C2×C42×C17—21no—
32C2×C42×C17C2311yescharacteristic
33C2×#63223—22no—
34C17×#95—22no—
35C17×#95—21no—
36C17×#95C2×C411yescharacteristic
37C17×#95C2×C412yes—
38C17×#95C2×C411yescharacteristic
39#10587504—42no—
40C22⋊#3622—22no—
41C4⋊#3622—22no—
42C4⋊#3622—22no—
43C4⋊#3622—22no—
44C4×#3622—22no—
45C24×C17C22⋊C412yes—
46C2×D4×C17—22no—
47C2×D4×C17—22no—
48C2×D4×C17SD1612yes—
49C2×D4×C17Q1612yes—
50C22×C4×C17—21no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—21no—
55C22×C4×C17—44no—
56C22×C4×C17—44no—
57C22×C4×C17—82no—
58C22×C4×C17—82no—
59C22×C4×C17C22⋊C412yes—
60C22×C4×C17C22×C411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C4⋊C4×C17—22no—
64C4⋊C4×C17—42no—
65C4⋊C4×C17—42no—
66C4⋊C4×C17—82no—
67C22⋊C4×C17—24no—
68C22⋊C4×C17—44no—
69C42×C17—42no—
70C42×C17#4112yes—
71C42×C17#4112yes—
72#63223—42no—
73#63223—42no—
74C2×#3622—42no—
75C2×#3622—42no—
76#8765—171nomaximal, Sylow
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—42no—
80C23×C17—42no—
81C23×C17#10314yes—
82C23×C17C2×C22⋊C411yescharacteristic
83D4×C17—24no—
84D4×C17—24no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—41no—
88C2×C4×C17—41no—
89C2×C4×C17—41no—
90C2×C4×C17—41no—
91C2×C4×C17—42no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—41no—
97C2×C4×C17—42no—
98C2×C4×C17—82no—
99C2×C4×C17—82no—
100C2×C4×C17—88no—
101C2×C4×C17#11711yescharacteristic
102C2×C4×C17#11711yescharacteristic
103C2×C4×C17#12312yes—
104C2×C4×C17#12312yes—
105C2×C4×C17C2×SD1611yescharacteristic
106C2×C4×C17C2×Q1611yescharacteristic
107C2×C4×C17AGL(1,ℤ/8)12yes—
108#3622—82no—
109#3622—82no—
110#2349D1711yescharacteristic
111#1800—171no—
112#1460—171no—
113#1303—172no—
114#1286—172no—
115C22×C17—21no—
116C22×C17—21no—
117C22×C17—24no—
118C22×C17—44no—
119C22×C17—44no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17#38111yescharacteristic
123C22×C17C2×#10311yescharacteristic
124C22×C17#41811yescharacteristic
125C4×C17—22no—
126C4×C17—21no—
127C4×C17—21no—
128C4×C17—42no—
129C4×C17—42no—
130C4×C17—81no—
131C4×C17—82no—
132C4×C17#48112yes—
133C4×C17#48412yes—
134C2×C4×D4—21no—
135C2×C4×D4C2×D1711yescharacteristic
136#431—172no—
137#423—172no—
138C2×#100—341no—
139#389—21no—
140#385—22no—
141#382C2×D1711yescharacteristic
142#380Dic1712yes—
143C4×C4⋊C4C2×D1711yescharacteristic
144C2×#95Dic1712yes—
145#343—341no—
146#323—342no—
147C2×C17—21no—
148C2×C17—21no—
149C2×C17—44no—
150C2×C17#182411yescharacteristic
151C2×C17#182711yescharacteristic
152C2×C17#182911yescharacteristic
153C22×D4C22⋊D1711yescharacteristic
154C23×C4—22no—
155C23×C4C2×Dic1712yes—
156C2×M16—342no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—44no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4C22⋊D1711yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4C22⋊D1712yes—
166C2×C42—21no—
167C2×C42C22×D1711yescharacteristic
168C4⋊C8—342no—
169C4⋊C8—342no—
170C4⋊C8—342no—
171#100—682no—
172C22⋊C8—342no—
173C4×C8—342no—
174#95—22no—
175#95—21no—
176#95C2×Dic1711yescharacteristic
177#95C2×Dic1712yes—
178#95C2×Dic1711yescharacteristic
179C17#876511yescharacteristic, complemented, Sylow
180C24C22⋊Dic1712yes—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4#6323012yes—
184C2×D4#6323112yes—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4C22⋊Dic1712yes—
195C22×C4C22×Dic1711yescharacteristic
196C22×C4C2×C22⋊D1711yescharacteristic
197C22×C4C2×C22⋊D1711yescharacteristic
198M16—682no—
199M16—682no—
200C2×C8—682no—
201C2×C8—682no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—82no—
206C22⋊C4—24no—
207C22⋊C4—44no—
208C42—42no—
209C42#6325212yes—
210C42#6325212yes—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23#1058750614yes—
216C23C2×C22⋊Dic1711yescharacteristic
217D4—24no—
218D4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—88no—
235C2×C4#1058761211yescharacteristic
236C2×C4#1058761211yescharacteristic
237C2×C4#1058761512yes—
238C2×C4#1058761512yes—
239C2×C4#1058761812yes—
240C2×C4C2×#6323011yescharacteristic
241C2×C4C2×#6323111yescharacteristic
242C8—1362no—
243C8—1362no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22#1176267211yescharacteristic
252C22C2×#1058750611yescharacteristic
253C22#1176281411yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—82no—
260C4—81no—
261C4#1176312712yes—
262C4#1176314412yes—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#1452887411yescharacteristic
267C2#1452888811yescharacteristic
268C2#1452889511yescharacteristic
subgroupG/H
centerC22#11762672
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2349C2
socleC22×C17#381
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#2349D17
hypocenterC17#8765
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #14526687C2semidirect
4. #14526642C2semidirect
179. C17#8765semidirect

Maximal subgroups

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

subgroupindex
#145287762
#145271052
#145266872
#145266422
C17×#23492
#876517