Finite Group Databasebeta

← #16951125

Subgroups — #16951125

Order 4352.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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#14531896C211yesmaximal, characteristic
2#14529173C211yesmaximal, characteristic
3#14529067C212yesmaximal
4#14526901C212yesmaximal
5C2×C17×#354C211yesmaximal, characteristic
6#11763284C2211yescharacteristic
7#11762818—21no—
8C2×#10587504C2211yescharacteristic
9#11762716—21no—
10#11762702—22no—
11#11762689C2212yes—
12#11762684—22no—
13C22⋊C4×Dic17—22no—
14#11762655—21no—
15#11762575—22no—
16#11762569—21no—
17C2×C17×#124C2211yescharacteristic
18C2×C17×#99C2212yes—
19C17×#354C414yescomplemented
20C22×Q8×C17D411yescharacteristic
21C22×D4×C17C2311yescharacteristic
22C23×Dic17D412yes—
23C17×#124—22no—
24C17×#124C2×C412yes—
25C17×#124Q812yes—
26C2×C22⋊C4×C17—22no—
27C2×C22⋊C4×C17—22no—
28C2×C42×C17D411yescharacteristic
29C2×C22⋊Dic17—22no—
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17D412yes—
32D4×Dic17—42no—
33#10587612—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—42no—
36C2×C4×Dic17—22no—
37C2×C4×Dic17—21no—
38C2×#63223—21no—
39C17×#99—22no—
40C17×#99C2×C414yes—
41#10587506—42no—
42#10587504—22no—
43#10587502—21no—
44#10587502—22no—
45#10587502—21no—
46#10587502—22no—
47#10587502—42no—
48C24×C17C2×D412yes—
49C2×Q8×C17—21no—
50C2×Q8×C17—41no—
51C2×Q8×C17—41no—
52C2×Q8×C17C4⋊C412yescomplemented
53C2×D4×C17—41no—
54C2×D4×C17—41no—
55C2×D4×C17C22×C411yescharacteristic
56C2×D4×C17C2×Q811yescharacteristic
57C2×D4×C17#4112yes—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17C2×D411yescharacteristic
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—22no—
66C22×Dic17—44no—
67C22×Dic17—42no—
68C22⋊C4×C17—24no—
69C22⋊C4×C17—24no—
70C22⋊C4×C17—24no—
71C22⋊C4×C17—42no—
72C42×C17—21no—
73C42×C17C4⋊C412yescomplemented
74C22⋊Dic17—44no—
75C22⋊Dic17—42no—
76C22⋊Dic17—41no—
77C22⋊Dic17—42no—
78C4⋊Dic17—22no—
79C4⋊Dic17—81no—
80C4⋊Dic17—84no—
81C4×Dic17—44no—
82C4×Dic17—41no—
83C4×Dic17—81no—
84#63223—41no—
85#63223—41no—
86C2×#3622—41no—
87#12965—171nomaximal, Sylow
88C23×C17—22no—
89C23×C17—22no—
90C23×C17—42no—
91C23×C17—42no—
92C23×C17C4×D412yes—
93C23×C17#12011yescharacteristic
94C23×C17C22⋊Q812yes—
95Q8×C17—42no—
96Q8×C17—81no—
97D4×C17—41no—
98D4×C17—41no—
99D4×C17—42no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—42no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—44no—
108C2×C4×C17—82no—
109C2×C4×C17—82no—
110C2×C4×C17C2×C4⋊C411yescharacteristic
111C2×C4×C17#12311yescharacteristic
112C2×Dic17—24no—
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—44no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—82no—
121C2×Dic17—82no—
122C2×Dic17—82no—
123#3622—81no—
124#2372—171no—
125C2×#354D1711yescharacteristic
126#1885—171no—
127#1867—172no—
128#1390—172no—
129C22×C17—22no—
130C22×C17—22no—
131C22×C17—22no—
132C22×C17—42no—
133C22×C17—42no—
134C22×C17—42no—
135C22×C17—42no—
136C22×C17—82no—
137C22×C17#38511yescharacteristic
138C22×C17B(4,𝔽2)11yescharacteristic
139C22×C17#45811yescharacteristic
140C4×C17—21no—
141C4×C17—21no—
142C4×C17—42no—
143C4×C17—42no—
144C4×C17—84no—
145Dic17—44no—
146Dic17—82no—
147Dic17—82no—
148C2×#124C2×D1711yescharacteristic
149#518—171no—
150#419—341no—
151C2×#100—171no—
152C2×#99C2×D1712yes—
153#391—341no—
154#388—342no—
155#386—172no—
156#385—342no—
157C4×C22⋊C4—342no—
158#376—341no—
159#354Dic1714yescomplemented
160#342—342no—
161#339—341no—
162C2×C17—21no—
163C2×C17—21no—
164C2×C17—42no—
165C2×C17—42no—
166C2×C17#186711yescharacteristic
167C2×C17#219712yes—
168C22×Q8D6811yescharacteristic
169C22×D4C22×D1711yescharacteristic
170C23×C4—172no—
171C2×M16—341no—
172#124—22no—
173#124Dic3412yes—
174#124C2×Dic1712yes—
175C4×D4—682no—
176#117—341no—
177C2×C4⋊C4—341no—
178C2×C4⋊C4—682no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—172no—
182C2×C22⋊C4—342no—
183C2×C22⋊C4—342no—
184C2×C42—342no—
185C2×C42—341no—
186C2×C42D6811yescharacteristic
187#103—682no—
188#100—342no—
189#99—22no—
190#99C2×Dic1714yes—
191#95—341no—
192#95—341no—
193#95—342no—
194#95—342no—
195#95—682no—
196C17#1296511yescharacteristic, complemented, Sylow
197C24D4×D1712yes—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8C4⋊Dic1712yescomplemented
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×Dic3411yescharacteristic
205C2×D4#6325412yes—
206C2×D4C22×Dic1711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—342no—
211C22×C4—342no—
212C22×C4—342no—
213C22×C4—342no—
214C22×C4—684no—
215C22×C4—682no—
216C22×C4C2×D6811yescharacteristic
217M16—681no—
218M16—681no—
219C2×C8—681no—
220C4⋊C4—342no—
221C4⋊C4—1361no—
222C4⋊C4—1364no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—42no—
227C22⋊C4—684no—
228C22⋊C4—682no—
229C22⋊C4—681no—
230C22⋊C4—682no—
231C42—21no—
232C42—681no—
233C42—684no—
234C42—1361no—
235C42C4⋊Dic1712yescomplemented
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23#1058756412yes—
241C23#1058756811yescharacteristic
242C23D4×Dic1712yes—
243Q8—42no—
244Q8—81no—
245D4—41no—
246D4—42no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—44no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—344no—
259C2×C4—682no—
260C2×C4—682no—
261C2×C4—684no—
262C2×C4—682no—
263C2×C4—682no—
264C2×C4—682no—
265C2×C4—682no—
266C2×C4—1362no—
267C2×C4—1362no—
268C2×C4—1362no—
269C2×C4#1058757211yescharacteristic
270C2×C4C2×C4⋊Dic1711yescharacteristic
271C8—1361no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—82no—
280C22#1176268411yescharacteristic
281C22#1176299711yescharacteristic
282C22#1176300011yescharacteristic
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—84no—
288C4—684no—
289C4—1362no—
290C4—1362no—
291C2—21no—
292C2—21no—
293C2—42no—
294C2—42no—
295C2#1452906711yescharacteristic
296C2#1453077712yes—
subgroupG/H
centerC22#11762997
commutatorC2×D4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC2×C17×#354C2
socleC22×C17B(4,𝔽2)
prime radical, cosocleC22×D4×C17C23
hypercenterC2×#354D17
hypocenterC17#12965
LCSG ▷ C2×D4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
19. C17×#354C4semidirect
52. C2×Q8×C17C4⋊C4semidirect
73. C42×C17C4⋊C4semidirect
159. #354Dic17semidirect
196. C17#12965semidirect
201. C2×Q8C4⋊Dic17semidirect
235. C42C4⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
#145318962
#145291732
#145290672
#145269012
C2×C17×#3542
#1296517