Finite Group Databasebeta

← #14532194

Subgroups — #14532194

Order 2176.

610 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#10587621C211yesmaximal, characteristic
2C2×#10587621C211yesmaximal, characteristic
3C2×#10587563C211yesmaximal, characteristic, complemented
4C22×C4⋊Dic17C211yesmaximal, characteristic, complemented
5#11762743C212yesmaximal, complemented
6#11762732C212yesmaximal, complemented
7#11762706C212yesmaximal, complemented
8#11762702C212yesmaximal, complemented
9#11762682C211yesmaximal, characteristic, complemented
10#11762670C211yesmaximal, characteristic, complemented
11C17×#386C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C23×Dic17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2211yescharacteristic
23#10587621—24no—
24#10587621—24no—
25C2×C4⋊Dic17—24no—
26C2×C4⋊Dic17C2212yes—
27C2×C4⋊Dic17C2212yes—
28C2×#63225C2211yescharacteristic
29C2×#63225C2212yes—
30C2×#63225C2212yes—
31C2×#63225C2212yes—
32C2×#63225C2211yescharacteristic
33C2×C4×Dic17C2212yes—
34#10587563—24no—
35C17×#95C2212yes—
36#10587502C2212yes—
37#10587502C2212yes—
38#10587502C2212yes—
39C24×C17C2311yescharacteristic
40C24×C17C2311yescharacteristic
41C2×D4×C17—22no—
42C2×D4×C17—22no—
43C2×D4×C17—22no—
44C22×C4×C17—22no—
45C22×C4×C17—24no—
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2312yes—
48C22×C4×C17C2312yes—
49C22×Dic17—22no—
50C22×Dic17—22no—
51C22×Dic17—22no—
52C22×Dic17—22no—
53C22×Dic17—22no—
54C22×Dic17D414yes—
55C22×Dic17C2312yes—
56C22×Dic17C2311yescharacteristic
57C22×Dic17C2311yescharacteristic
58C22×Dic17C2311yescharacteristic
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2312yes—
61C22⋊C4×C17—22no—
62C22⋊C4×C17—24no—
63C22⋊Dic17—22no—
64C22⋊Dic17—22no—
65C22⋊Dic17—22no—
66C22⋊Dic17—24no—
67C22⋊Dic17D414yes—
68C4⋊Dic17—24no—
69C4⋊Dic17—24no—
70#63225—24no—
71#63225—24no—
72#63225—22no—
73#63225—22no—
74#63225—24no—
75C4×Dic17—24no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—21no—
79C23×C17—21no—
80C23×C17—22no—
81C23×C17—21no—
82C23×C17—22no—
83C23×C17C2×D412yes—
84C23×C17C2×D412yes—
85C23×C17#4112yes—
86C23×C17C2411yescharacteristic
87D4×C17—44no—
88C2×C4×C17—28no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—21no—
96C2×C4×C17—22no—
97C2×C4×C17#4114yes—
98C2×Dic17—28no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—21no—
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—44no—
113C2×Dic17C2×D414yes—
114C2×Dic17C2×D412yes—
115C2×Dic17C2×D412yes—
116#2413—171nomaximal, Sylow
117C22×C17—22no—
118C22×C17—22no—
119C22×C17—24no—
120C22×C17—24no—
121C22×C17—41no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17#12114yes—
127C22×C17C22×D411yescharacteristic
128C22×C17C22×D411yescharacteristic
129C22×C17C2×#4111yescharacteristic
130C22×C17C2×#4112yes—
131C22×C172−1+412yes—
132C4×C17—24no—
133C4×C17—41no—
134C4×C17—42no—
135Dic17—22no—
136Dic17—22no—
137Dic17—24no—
138Dic17—42no—
139Dic17—41no—
140Dic17—41no—
141C2×#123—171no—
142C2×#123—171no—
143C2×#117—171no—
144C22×C4⋊C4—171no—
145#398—172no—
146#396—172no—
147#389—172no—
148#388—172no—
149#386D1711yescharacteristic
150#385—171no—
151#381—171no—
152C2×C17—24no—
153C2×C17—81no—
154C2×C17C2×#12111yescharacteristic
155C2×C17#52511yescharacteristic
156C2×C17#53611yescharacteristic
157C2×C17#54712yes—
158C2×C17#55212yes—
159C22×D4C2×D1711yescharacteristic
160C23×C4—171no—
161C23×C4—171no—
162C23×C4C2×D1711yescharacteristic
163#123—344no—
164#123—344no—
165#117—344no—
166C2×C4⋊C4—172no—
167C2×C4⋊C4—172no—
168C2×C4⋊C4—172no—
169C2×C4⋊C4—171no—
170C2×C4⋊C4—172no—
171C2×C4⋊C4—171no—
172C2×C4⋊C4—172no—
173C2×C4⋊C4—344no—
174C2×C22⋊C4—171no—
175C2×C22⋊C4—171no—
176C2×C22⋊C4—171no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—171no—
179C2×C22⋊C4C2×D1711yescharacteristic
180C2×C22⋊C4C2×D1712yes—
181C2×C42—172no—
182#95—172no—
183#95—172no—
184#95—172no—
185#95C2×D1712yes—
186C17#241311yescharacteristic, complemented, Sylow
187C24C22×D1711yescharacteristic
188C24C22×D1711yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—174no—
195C22×C4—172no—
196C22×C4—171no—
197C22×C4—171no—
198C22×C4—171no—
199C22×C4—171no—
200C22×C4—172no—
201C22×C4—342no—
202C22×C4—342no—
203C22×C4—342no—
204C22×C4—342no—
205C22×C4—342no—
206C22×C4C22×D1712yes—
207C22×C4C22×D1711yescharacteristic
208C22×C4C22×D1712yes—
209C4⋊C4—344no—
210C4⋊C4—344no—
211C4⋊C4—344no—
212C4⋊C4—344no—
213C4⋊C4—342no—
214C4⋊C4—342no—
215C4⋊C4—344no—
216C22⋊C4—22no—
217C22⋊C4—24no—
218C22⋊C4—174no—
219C22⋊C4—344no—
220C22⋊C4—342no—
221C22⋊C4—342no—
222C22⋊C4—342no—
223C42—344no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23#6325212yes—
232C23D4×D1712yes—
233C23D4×D1712yes—
234C23C23×D1711yescharacteristic
235D4—44no—
236C2×C4—28no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—174no—
246C2×C4—172no—
247C2×C4—172no—
248C2×C4—341no—
249C2×C4—341no—
250C2×C4—341no—
251C2×C4—341no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—348no—
255C2×C4—342no—
256C2×C4—341no—
257C2×C4—341no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—684no—
263C2×C4#6325414yes—
264C22—24no—
265C22—24no—
266C22—22no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22#1058757014yes—
274C22C2×#6325211yescharacteristic
275C22C2×D4×D1711yescharacteristic
276C22C2×D4×D1711yescharacteristic
277C22C2×#6325412yes—
278C22#1058769712yes—
279C4—24no—
280C4—42no—
281C4—41no—
282C4—344no—
283C4—342no—
284C4—342no—
285C4—682no—
286C4—681no—
287C4—681no—
288C2—24no—
289C2—81no—
290C2C2×#1058757011yescharacteristic
291C2#1176331511yescharacteristic
292C2#1176338111yescharacteristic
293C2#1176345912yes—
294C2#1176349912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC22×D4×C17C22
hypercenter#386D17
hypocenterC17#2413
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. C2×#10587563C2semidirect
4. C22×C4⋊Dic17C2semidirect
5. #11762743C2semidirect
6. #11762732C2semidirect
7. #11762706C2semidirect
8. #11762702C2semidirect
9. #11762682C2semidirect
10. #11762670C2semidirect
186. C17#2413semidirect

Maximal subgroups

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

subgroupindex
C2×#105876212
C2×#105875632
C22×C4⋊Dic172
#117627432
#117627322
#117627062
#117627022
#117626822
#117626702
C17×#3862
#241317