Finite Group Databasebeta

← #14532191

Subgroups — #14532191

Order 2176.

610 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 165 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×#10587615C211yesmaximal, characteristic
2C2×#10587615C211yesmaximal, characteristic
3C2×#10587612C211yesmaximal, characteristic, complemented
4#11762742C212yesmaximal, complemented
5#11762731C212yesmaximal, complemented
6#11762704C212yesmaximal, complemented
7#11762699C212yesmaximal, complemented
8#11762688C211yesmaximal, characteristic
9#11762682C211yesmaximal, characteristic, complemented
10#11762672C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C17C211yesmaximal, characteristic, complemented
12C23×C4×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C2×C4⋊C4×C17—24no—
17C2×C4⋊C4×C17C2212yes—
18C2×C4⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2211yescharacteristic
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2211yescharacteristic
22#10587615—24no—
23#10587615—24no—
24C2×#63227C2212yes—
25C2×#63227C2211yescharacteristic
26C2×#63227C2212yes—
27C2×#63227C2211yescharacteristic
28#10587612—24no—
29C2×C4⋊Dic17C2212yes—
30C2×#63225C2211yescharacteristic
31C2×#63225C2212yes—
32C2×#63225C2212yes—
33C2×#63225C2211yescharacteristic
34C2×C4×Dic17C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37#10587502C2212yes—
38#10587502C2212yes—
39C24×C17C2311yescharacteristic
40C23×D17C2311yescharacteristic
41C22×C4×C17—22no—
42C22×C4×C17—22no—
43C22×C4×C17—22no—
44C22×C4×C17—24no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—22no—
48C22×C4×C17D414yes—
49C22×C4×C17C2312yes—
50C22×C4×C17C2311yescharacteristic
51C22×C4×C17C2311yescharacteristic
52C22×C4×C17C2312yes—
53C2×C22⋊D17—22no—
54C2×C22⋊D17—22no—
55C2×C22⋊D17—22no—
56C22×Dic17C2311yescharacteristic
57C22×Dic17C2311yescharacteristic
58C22×Dic17C2312yes—
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2312yes—
61C4⋊C4×C17—24no—
62C4⋊C4×C17—24no—
63C22⋊Dic17—22no—
64C22⋊Dic17—22no—
65C22⋊Dic17D414yes—
66#63227—22no—
67#63227—22no—
68#63227—24no—
69#63227—24no—
70C4⋊Dic17—24no—
71#63225—22no—
72#63225—22no—
73#63225—24no—
74#63225—24no—
75C4×Dic17—24no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17C2×D412yes—
79C23×C17C2×D412yes—
80C23×C17#4112yes—
81C23×C17C2411yescharacteristic
82C22×D17—21no—
83C22×D17—21no—
84C22×D17—21no—
85C22×D17—22no—
86C22×D17—22no—
87C2×C4×C17—28no—
88C2×C4×C17—28no—
89C2×C4×C17—22no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—44no—
99C2×C4×C17C2×D412yes—
100C2×C4×C17C2×D412yes—
101C2×C4×C17#4114yes—
102C22⋊D17—44no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17C2×D414yes—
116#2413—171nomaximal, Sylow
117C22×C17—22no—
118C22×C17—22no—
119C22×C17—24no—
120C22×C17—24no—
121C22×C17#12114yes—
122C22×C17C22×D411yescharacteristic
123C22×C17C22×D411yescharacteristic
124C22×C17C2×#4112yes—
125C22×C17C2×#4111yescharacteristic
126C22×C172−1+412yes—
127C2×D17—41no—
128C2×D17—41no—
129C2×D17—41no—
130C2×D17—42no—
131C2×D17—42no—
132C4×C17—22no—
133C4×C17—22no—
134C4×C17—24no—
135C4×C17—42no—
136Dic17—24no—
137Dic17—41no—
138Dic17—41no—
139Dic17—41no—
140Dic17—42no—
141C2×#123—171no—
142C2×#123—171no—
143C2×#117—171no—
144C22×C4⋊C4D1711yescharacteristic, complemented
145#398—172no—
146#396—172no—
147#389—172no—
148#388—172no—
149#386—171no—
150#385—171no—
151#381—171no—
152C2×C17—24no—
153C2×C17C2×#12111yescharacteristic
154C2×C17#52511yescharacteristic
155C2×C17#53611yescharacteristic
156C2×C17#54712yes—
157C2×C17#55212yes—
158D17—81no—
159C22×D4—171no—
160C23×C4C2×D1711yescharacteristic
161C23×C4C2×D1711yescharacteristic
162C23×C4C2×D1711yescharacteristic
163#123—344no—
164#123—344no—
165#117—344no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—172no—
168C2×C4⋊C4—171no—
169C2×C4⋊C4—172no—
170C2×C4⋊C4—171no—
171C2×C4⋊C4—172no—
172C2×C4⋊C4C2×D1712yes—
173C2×C4⋊C4C2×D1712yes—
174C2×C22⋊C4—171no—
175C2×C22⋊C4—171no—
176C2×C22⋊C4—171no—
177C2×C22⋊C4—171no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4—171no—
180C2×C22⋊C4—172no—
181C2×C42—172no—
182#95—172no—
183#95—172no—
184#95—172no—
185#95—172no—
186C17#241311yescharacteristic, complemented, Sylow
187C24—171no—
188C24C22×D1711yescharacteristic
189C2×D4—342no—
190C2×D4—342no—
191C2×D4—342no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—172no—
200C22×C4—171no—
201C22×C4—172no—
202C22×C4—171no—
203C22×C4—171no—
204C22×C4C22⋊D1714yes—
205C22×C4C22×D1712yes—
206C22×C4C22×D1712yes—
207C22×C4C22×D1711yescharacteristic
208C22×C4C22×D1711yescharacteristic
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—344no—
212C4⋊C4—344no—
213C4⋊C4—344no—
214C4⋊C4—342no—
215C4⋊C4—342no—
216C22⋊C4—174no—
217C22⋊C4—344no—
218C22⋊C4—342no—
219C22⋊C4—342no—
220C22⋊C4—344no—
221C22⋊C4—342no—
222C22⋊C4—342no—
223C42—344no—
224C23—22no—
225C23—22no—
226C23—342no—
227C23—341no—
228C23—341no—
229C23—342no—
230C23—341no—
231C23D4×D1712yes—
232C23#6325412yes—
233C23C2×C22⋊D1712yes—
234C23C23×D1711yescharacteristic
235D4—684no—
236C2×C4—28no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—44no—
248C2×C4—174no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—341no—
255C2×C4—341no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—341no—
259C2×C4—341no—
260C2×C4—341no—
261C2×C4#6325214yes—
262C2×C4C2×C22⋊D1712yes—
263C2×C4C2×C22⋊D1712yes—
264C22—24no—
265C22—24no—
266C22—22no—
267C22—22no—
268C22—682no—
269C22—682no—
270C22—681no—
271C22—681no—
272C22—681no—
273C22#1058762514yes—
274C22C2×#6325212yes—
275C22C2×D4×D1711yescharacteristic
276C22C2×#6325411yescharacteristic
277C22#1058769412yes—
278C22C22×C22⋊D1711yescharacteristic
279C4—22no—
280C4—22no—
281C4—24no—
282C4—42no—
283C4—344no—
284C4—682no—
285C4—681no—
286C4—681no—
287C4—681no—
288C2—24no—
289C2—1361no—
290C2C2×#1058762511yescharacteristic
291C2#1176332011yescharacteristic
292C2#1176338311yescharacteristic
293C2#1176346012yes—
294C2#1176349412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C4⋊C4×C17C2
prime radicalC22×C22⋊D17C22
hypercenterC22×C4⋊C4D17
hypocenterC17#2413
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#10587612C2semidirect
4. #11762742C2semidirect
5. #11762731C2semidirect
6. #11762704C2semidirect
7. #11762699C2semidirect
9. #11762682C2semidirect
10. #11762672C2semidirect
11. C22×C4⋊C4×C17C2semidirect
144. C22×C4⋊C4D17semidirect
186. C17#2413semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×#105876122
#117627422
#117627312
#117627042
#117626992
#117626882
#117626822
#117626722
C22×C4⋊C4×C172
#241317