Finite Group Databasebeta

← #14539239

Subgroups — #14539239

Order 2176.

552 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 153 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×#10587595C211yesmaximal, characteristic, complemented
2C2×#10587628C211yesmaximal, characteristic, complemented
3C2×#10587637C211yesmaximal, characteristic, complemented
4#11763355C211yesmaximal, characteristic
5C2×#10587630C211yesmaximal, characteristic, complemented
6#11763202C212yesmaximal, complemented
7#11763200C212yesmaximal, complemented
8#11763187C212yesmaximal, complemented
9#11763180C212yesmaximal, complemented
10M16×Dic17C211yesmaximal, characteristic, complemented
11C2×C17×#137C211yesmaximal, characteristic, complemented
12C2×#41×C17C2211yescharacteristic
13C22×Q8×C17C2211yescharacteristic
14C2×#63252C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C17×#137—24no—
17C2×Q16×C17C2212yes—
18C2×SD16×C17C2212yes—
19C2×M16×C17C2211yescharacteristic
20#10587637—24no—
21#10587633C2211yescharacteristic
22#10587630—22no—
23#10587630C2212yes—
24#10587630C2212yes—
25#10587630C2211yescharacteristic
26C2×#63231C2212yes—
27C2×#63231C2212yes—
28#10587628—24no—
29C2×#63230C2212yes—
30#10587626C2211yescharacteristic
31#10587625—22no—
32#10587622C2211yescharacteristic
33C2×#63229C2212yes—
34C4×C22⋊D17—22no—
35C2×#63225—22no—
36C2×C4×Dic17C2211yescharacteristic
37C2×#63223C2211yescharacteristic
38#10587595—24no—
39C2×Dic68C2212yes—
40C2×#63219C2212yes—
41#10587486C2212yes—
42C8×Dic17C2212yes—
43#41×C17—22no—
44#41×C17—22no—
45C2×Q8×C17—21no—
46C2×Q8×C17—22no—
47C2×Q8×C17—22no—
48C2×Q8×C17C2312yes—
49C2×Q8×C17C2311yescharacteristic
50C2×D4×C17—21no—
51C2×D4×C17C2311yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17C2311yescharacteristic
55C2×C22⋊D17—21no—
56C2×C22⋊D17—22no—
57C22×Dic17—21no—
58C22×Dic17D412yes—
59#63252—22no—
60#63252—22no—
61C2×D68C2311yescharacteristic
62C2×C4×D17—21no—
63C2×Dic34—22no—
64C2×Dic34—22no—
65C2×Dic34—21no—
66C2×Dic34C2312yes—
67C2×Dic34C2311yescharacteristic
68Q16×C17—24no—
69SD16×C17—24no—
70M16×C17D414yescomplemented
71C2×C8×C17C2312yes—
72C22⋊Dic17—22no—
73#63231—24no—
74#63231—24no—
75#63230—24no—
76#63229—24no—
77#63227—22no—
78#63225—24no—
79#63225—24no—
80#63225—22no—
81C4×Dic17D412yes—
82C4×Dic17C2311yescharacteristic
83C4×Dic17C2311yescharacteristic
84#63223D414yescomplemented
85C2×#3622C2312yes—
86Dic68—24no—
87#63219—24no—
88C23×C17—21no—
89C23×C17C2×D411yescharacteristic
90C22×D17—21no—
91Q8×C17—22no—
92Q8×C17—21no—
93Q8×C17—22no—
94Q8×C17—21no—
95Q8×C17—42no—
96D4×C17—21no—
97D4×C17—21no—
98D4×C17—42no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17C2×D411yescharacteristic
104C2×C4×C17C2×D412yes—
105C2×C4×C17C2×D412yes—
106C2×C4×C17C2411yescharacteristic
107C22⋊D17—42no—
108C22⋊D17—44no—
109C2×Dic17—24no—
110C2×Dic17—21no—
111C2×Dic17—22no—
112C2×Dic17—42no—
113C2×Dic17C2×D412yes—
114C2×Dic17C2×D412yes—
115D68—21no—
116D68—21no—
117C4×D17—42no—
118Dic34—22no—
119Dic34—22no—
120Dic34—21no—
121Dic34—21no—
122Dic34—42no—
123C8×C17C2×D414yes—
124#3622C2×D414yes—
125#3152—171nomaximal, Sylow
126C22×C17—22no—
127C22×C17—41no—
128C22×C17—42no—
129C22×C17#12712yes—
130C22×C17C22×D411yescharacteristic
131C2×D17—41no—
132C2×D17—42no—
133C4×C17—41no—
134C4×C17—42no—
135C4×C17#12712yes—
136C4×C17C22×D411yescharacteristic
137C4×C17C22×D411yescharacteristic
138Dic17—22no—
139Dic17—42no—
140Dic17—41no—
141Dic17#13714yescomplemented
142C2×#137—171no—
143C2×#137—171no—
144C2×#137—171no—
145C2×#137D1711yescharacteristic, complemented
146#532—171no—
147C2×C4⋊Q8—171no—
148#497—172no—
149#497—172no—
150#494—172no—
151#492—172no—
152C4×M16—171no—
153C2×C17—22no—
154C2×C17—81no—
155C2×C17C2×#12711yescharacteristic
156C2×C17C2×#13712yes—
157D17—81no—
158C2×#41—171no—
159C2×#41C2×D1711yescharacteristic
160C22×Q8—171no—
161C22×Q8C2×D1711yescharacteristic
162#137—24no—
163#137—344no—
164#137—344no—
165#137—344no—
166C2×Q16—172no—
167C2×Q16—172no—
168C2×Q16—172no—
169C2×Q16C2×D1712yes—
170C2×SD16—172no—
171C2×SD16—172no—
172C2×SD16—172no—
173C2×SD16C2×D1712yes—
174C2×M16—171no—
175C2×M16C2×D1711yescharacteristic
176C4⋊Q8—172no—
177C4⋊Q8—172no—
178C4⋊Q8—171no—
179C4⋊Q8—342no—
180#127—171no—
181#124—171no—
182#124—171no—
183#121—342no—
184C4×D4—342no—
185C2×C4⋊C4—342no—
186C2×C42—171no—
187#97—172no—
188C4×C8—172no—
189C17#315211yescharacteristic, complemented, Sylow
190#41—22no—
191#41—22no—
192#41—342no—
193#41—342no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—171no—
198C2×Q8—172no—
199C2×Q8—341no—
200C2×Q8—342no—
201C2×Q8—342no—
202C2×Q8C22×D1711yescharacteristic
203C2×Q8C22×D1712yes—
204C2×D4—21no—
205C2×D4—171no—
206C2×D4—342no—
207C2×D4—341no—
208C2×D4C22×D1711yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—172no—
212C22×C4—341no—
213C22×C4—341no—
214C22×C4C22×D1711yescharacteristic
215Q16—24no—
216Q16—344no—
217Q16—344no—
218Q16—344no—
219SD16—24no—
220SD16—344no—
221SD16—344no—
222SD16—344no—
223M16—174no—
224M16C22⋊D1714yescomplemented
225C2×C8—172no—
226C2×C8C22×D1712yes—
227C4⋊C4—344no—
228C4⋊C4—344no—
229C4⋊C4—342no—
230C22⋊C4—342no—
231C22⋊C4—342no—
232C42—172no—
233C42—171no—
234C42—171no—
235C23—21no—
236C23—341no—
237C23D4×D1711yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—342no—
244Q8—341no—
245Q8—342no—
246Q8—341no—
247Q8—682no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—341no—
252D4—341no—
253D4—684no—
254D4—682no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—172no—
260C2×C4—172no—
261C2×C4—344no—
262C2×C4—341no—
263C2×C4—342no—
264C2×C4—682no—
265C2×C4—682no—
266C2×C4D4×D1712yes—
267C2×C4D4×D1711yescharacteristic
268C2×C4C2×C22⋊D1712yes—
269C2×C4C23×D1711yescharacteristic
270C8—174no—
271C8C2×C22⋊D1714yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—682no—
276C22—681no—
277C22#1058762612yes—
278C22C2×D4×D1711yescharacteristic
279C4—41no—
280C4—42no—
281C4—174no—
282C4—342no—
283C4—681no—
284C4—682no—
285C4#1058762612yes—
286C4C2×D4×D1711yescharacteristic
287C4C22×C22⋊D1711yescharacteristic
288C2—22no—
289C2—81no—
290C2—1361no—
291C2C2×#1058762611yescharacteristic
292C2#137×D1712yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#137C2
socleC22×C17C22×D4
prime radical#11763355C2
hypercenterC2×#137D17
hypocenterC17#3152
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#10587595C2semidirect
2. C2×#10587628C2semidirect
3. C2×#10587637C2semidirect
5. C2×#10587630C2semidirect
6. #11763202C2semidirect
7. #11763200C2semidirect
8. #11763187C2semidirect
9. #11763180C2semidirect
10. M16×Dic17C2semidirect
11. C2×C17×#137C2semidirect
70. M16×C17D4semidirect
84. #63223D4semidirect
141. Dic17#137semidirect
145. C2×#137D17semidirect
189. C17#3152semidirect
224. M16C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875952
C2×#105876282
C2×#105876372
#117633552
C2×#105876302
#117632022
#117632002
#117631872
#117631802
M16×Dic172
C2×C17×#1372
#315217