Finite Group Databasebeta

← #16923136

Subgroups — #16923136

Order 4352.

536 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 87 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#14535156C211yesmaximal, characteristic
2#14530118C212yesmaximal, complemented
3#14529866C211yesmaximal, characteristic
4#14526608C212yesmaximal, complemented
5C17×#1270C211yesmaximal, characteristic
6C22×#63229—21no—
7C2×#10587555C2211yescharacteristic
8C2×#10587564—22no—
9C2×#10587488—22no—
10C2×#10587481—21no—
11#11762732—22no—
12#11762729—21no—
13#11762707C2212yes—
14#11762689—21no—
15#11762572—22no—
16#11762564C2211yescharacteristic
17C17×#390C2211yescharacteristic
18C17×#336C2212yes—
19C22×D4×C17D411yescharacteristic
20C23×Dic17—21no—
21C22×Dic34—21no—
22C22×Dic34—21no—
23C22×C8×C17—22no—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C22⋊C4×C17—22no—
26C2×C42×C17C2311yescharacteristic
27C2×C22⋊Dic17—22no—
28C2×C22⋊Dic17—21no—
29C2×#63229—42no—
30C2×#63229—42no—
31C2×#63229—42no—
32C2×C4⋊Dic17—22no—
33C2×C4⋊Dic17D412yes—
34C2×C4⋊Dic17D412yes—
35C2×#63225—22no—
36C2×#63225—22no—
37C22×#3622—21no—
38#10587564—44no—
39#10587564—44no—
40#10587555—42no—
41#10587502—21no—
42#10587502—22no—
43#10587488—44no—
44#10587481—42no—
45C24×C17—21no—
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17SD1612yes—
50C22×C4×C17—21no—
51C22×C4×C17C2×D411yescharacteristic
52C22×C4×C17C2×D412yes—
53C22×Dic17—22no—
54C22×Dic17—22no—
55C22×Dic17—44no—
56C22×Dic17—41no—
57C22×Dic17—81no—
58C2×Dic34—22no—
59C2×Dic34—41no—
60C2×Dic34—41no—
61C2×Dic34—41no—
62C2×Dic34—41no—
63C2×Dic34—44no—
64C2×Dic34—41no—
65C2×Dic34—81no—
66C2×C8×C17—24no—
67C2×C8×C17—42no—
68C2×C8×C17—42no—
69C4⋊C4×C17—41no—
70C22⋊C4×C17—42no—
71C22⋊C4×C17—42no—
72C42×C17—41no—
73C22⋊Dic17—44no—
74C22⋊Dic17—81no—
75#63229—48no—
76C4⋊Dic17—22no—
77C4⋊Dic17—24no—
78C4⋊Dic17—42no—
79C4⋊Dic17—42no—
80C4⋊Dic17SD1614yescomplemented
81#63225—44no—
82#63225—42no—
83#63225—42no—
84C2×#3622—22no—
85C2×#3622—41no—
86C2×#3622—41no—
87#7754—171nomaximal, Sylow
88C23×C17—22no—
89C23×C17—42no—
90C23×C17—41no—
91C23×C17—42no—
92C23×C17—41no—
93C23×C17#12011yescharacteristic
94D4×C17—42no—
95D4×C17—81no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—21no—
100C2×C4×C17—42no—
101C2×C4×C17—41no—
102C2×C4×C17C2×SD1611yescharacteristic
103C2×C4×C17C2×SD1612yes—
104C2×C4×C17AGL(1,ℤ/8)12yes—
105C2×C4×C17#13711yescharacteristic
106C2×Dic17—24no—
107C2×Dic17—42no—
108C2×Dic17—42no—
109C2×Dic17—42no—
110C2×Dic17—42no—
111C2×Dic17—42no—
112C2×Dic17—42no—
113C2×Dic17—84no—
114C2×Dic17—81no—
115C2×Dic17—81no—
116Dic34—42no—
117Dic34—42no—
118Dic34—84no—
119Dic34—81no—
120Dic34—81no—
121C8×C17—44no—
122#3622—42no—
123#2683—171no—
124#2058—172no—
125#2013—171no—
126#1271—172no—
127#1270D1711yescharacteristic
128C22×C17—42no—
129C22×C17—44no—
130C22×C17—82no—
131C22×C17—81no—
132C22×C17—81no—
133C22×C17#44811yescharacteristic
134C22×C17#45012yes—
135C22×C17#45311yescharacteristic
136C22×C17#45612yes—
137C22×C17B(4,𝔽2)11yescharacteristic
138C4×C17—21no—
139C4×C17—21no—
140C4×C17—22no—
141C4×C17—22no—
142C4×C17—81no—
143Dic17—44no—
144Dic17—82no—
145Dic17—82no—
146C22×SD16—341no—
147C2×C4⋊Q8—171no—
148C2×C22⋊Q8—342no—
149C2×#106—342no—
150C2×#103—341no—
151#396—342no—
152#395—341no—
153#390C2×D1711yescharacteristic
154#389—172no—
155#386—341no—
156#340—342no—
157#336—171no—
158#336C2×D1712yes—
159C2×C17—82no—
160C2×C17#159611yescharacteristic
161C2×C17#160412yes—
162C2×C17#162911yescharacteristic
163C2×C17#165112yes—
164C2×C17#166011yescharacteristic
165C22×Q8—341no—
166C22×Q8—341no—
167C22×D4D6811yescharacteristic
168C23×C4—341no—
169C2×SD16—682no—
170C2×SD16—682no—
171C2×SD16—682no—
172C22×C8—22no—
173C22×C8—341no—
174C4⋊Q8—682no—
175C22⋊Q8—684no—
176C22⋊Q8—684no—
177C2×C4⋊C4—172no—
178C2×C4⋊C4—172no—
179C2×C4⋊C4—342no—
180C2×C4⋊C4—342no—
181C2×C4⋊C4—342no—
182C2×C4⋊C4D6811yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—342no—
185C2×C22⋊C4—341no—
186C2×C42C22×D1711yescharacteristic
187#106—684no—
188#103—682no—
189#95—341no—
190#95—342no—
191C17#775411yescharacteristic, complemented, Sylow
192C24—21no—
193C2×Q8—342no—
194C2×Q8—681no—
195C2×Q8—681no—
196C2×Q8—681no—
197C2×Q8—684no—
198C2×Q8—681no—
199C2×Q8—681no—
200C2×Q8—1361no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4#6321912yes—
205C22×C4—21no—
206C22×C4—342no—
207C22×C4—342no—
208C22×C4—684no—
209C22×C4—681no—
210C22×C4—1361no—
211C22×C4C2×D6811yescharacteristic
212C22×C4D4×D1712yes—
213SD16—688no—
214C2×C8—24no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—342no—
218C2×C8—681no—
219C2×C8—681no—
220C4⋊C4—41no—
221C4⋊C4—174no—
222C4⋊C4—344no—
223C4⋊C4—342no—
224C4⋊C4—682no—
225C4⋊C4—682no—
226C4⋊C4—684no—
227C4⋊C4—682no—
228C4⋊C4—682no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—684no—
232C22⋊C4—1361no—
233C42—41no—
234C23—22no—
235C23—42no—
236C23—41no—
237C23—42no—
238C23—41no—
239C23#1058756811yescharacteristic
240Q8—682no—
241Q8—682no—
242Q8—1361no—
243Q8—1361no—
244Q8—1364no—
245D4—42no—
246D4—81no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—344no—
254C2×C4—682no—
255C2×C4—682no—
256C2×C4—682no—
257C2×C4—682no—
258C2×C4—682no—
259C2×C4—682no—
260C2×C4—1364no—
261C2×C4—1361no—
262C2×C4—1361no—
263C2×C4C2×#6321911yescharacteristic
264C2×C4#1058759511yescharacteristic
265C2×C4#1058759712yes—
266C2×C4SD16×D1712yes—
267C8—44no—
268C8—682no—
269C22—44no—
270C22—42no—
271C22—82no—
272C22—81no—
273C22—81no—
274C22#1176295011yescharacteristic
275C22#1176296112yes—
276C22#1176297911yescharacteristic
277C22#1176299312yes—
278C22#1176299711yescharacteristic
279C4—22no—
280C4—22no—
281C4—21no—
282C4—21no—
283C4—81no—
284C4—684no—
285C4—1362no—
286C4—1362no—
287C2—82no—
288C2#1452770911yescharacteristic
289C2#1452775412yes—
290C2#1452790111yescharacteristic
291C2#1452804912yes—
292C2#1452809611yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1270C2
socleC23×C17#120
prime radicalC22×D4×C17D4
hypercenter#1270D17
hypocenterC17#7754
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #14530118C2semidirect
4. #14526608C2semidirect
80. C4⋊Dic17SD16semidirect
191. C17#7754semidirect

Maximal subgroups

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

subgroupindex
#145351562
#145301182
#145298662
#145266082
C17×#12702
#775417