Finite Group Databasebeta

← #14532402

Subgroups — #14532402

Order 2176.

952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#63254C211yesmaximal, characteristic, complemented
2C2×#10587626C211yesmaximal, characteristic, complemented
3C2×#10587561C211yesmaximal, characteristic, complemented
4C2×#10587581C211yesmaximal, characteristic, complemented
5#11762718C214yesmaximal, complemented
6#11762714C211yesmaximal, characteristic, complemented
7#11762686C214yesmaximal, complemented
8C4×#63227C211yesmaximal, characteristic, complemented
9C17×#390C211yesmaximal, characteristic, complemented
10C22×D4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×C22⋊D17C2214yes—
13C23×Dic17C2212yes—
14C2×#63254—24no—
15C2×#63254—24no—
16C2×#63254—24no—
17C22×D68C2211yescharacteristic
18C22×D68C2212yes—
19C22×C4×D17C2211yescharacteristic
20C22×Dic34C2211yescharacteristic
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2214yes—
23C2×C42×C17C2211yescharacteristic
24#10587626—24no—
25C2×#63227C2212yes—
26C2×#63227C2214yes—
27C2×#63227C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29#10587581—24no—
30#10587561—24no—
31#10587502C2214yes—
32#10587502C2212yes—
33C24×C17C2312yes—
34C23×D17C2312yes—
35C23×D17C2311yescharacteristic
36C2×D4×C17—22no—
37C2×D4×C17—22no—
38C2×D4×C17D414yescomplemented
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C2×C22⋊D17—24no—
43C2×C22⋊D17—24no—
44C2×C22⋊D17—28no—
45C2×C22⋊D17—24no—
46C2×C22⋊D17—28no—
47C2×C22⋊D17—44no—
48C22×Dic17—24no—
49C22×Dic17—24no—
50C22×Dic17—24no—
51C22×Dic17C2312yes—
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2312yes—
54#63254—232no—
55C2×D68—24no—
56C2×D68—22no—
57C2×D68—22no—
58C2×D68—22no—
59C2×D68—44no—
60C2×C4×D17—22no—
61C2×C4×D17—22no—
62C2×C4×D17D414yescomplemented
63C2×Dic34—22no—
64C2×Dic34—22no—
65C2×Dic34D414yescomplemented
66C4⋊C4×C17—22no—
67C22⋊C4×C17—44no—
68C42×C17—22no—
69#63227—28no—
70#63227—24no—
71#63227—24no—
72C4×Dic17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—24no—
76C23×C17—24no—
77C23×C17C2×D414yes—
78C23×C17C2411yescharacteristic
79C22×D17—21no—
80C22×D17—21no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—24no—
87C22×D17—24no—
88C22×D17C2×D412yes—
89D4×C17—28no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—24no—
95C2×C4×C17C2×D412yes—
96C2×C4×C17C2×D412yes—
97C2×C4×C17C2×D412yes—
98C22⋊D17—216no—
99C22⋊D17—416no—
100C2×Dic17—216no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—21no—
104C2×Dic17—21no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17C2×D414yes—
108C2×Dic17C2×D412yes—
109C2×Dic17#4114yescomplemented
110D68—48no—
111D68—44no—
112C4×D17—28no—
113Dic34—28no—
114#2431—171nomaximal, Sylow
115C22×C17—24no—
116C22×C17—28no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17—44no—
120C22×C17C22×D411yescharacteristic
121C22×C17C22×D411yescharacteristic
122C22×C17C22×D411yescharacteristic
123C22×C17C2×#4112yes—
124C22×C17H(5,𝔽2)12yes—
125C2×D17—22no—
126C2×D17—22no—
127C2×D17—22no—
128C2×D17—42no—
129C2×D17—42no—
130C2×D17—42no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—44no—
134C2×D17—44no—
135C4×C17—42no—
136C4×C17—42no—
137C4×C17#12014yes—
138Dic17—24no—
139Dic17—22no—
140Dic17—24no—
141C22×#41—171no—
142C2×#127—171no—
143C2×#124—171no—
144C2×#121—171no—
145#392—174no—
146#391—171no—
147#390D1711yescharacteristic, complemented
148#386—174no—
149C4×C22⋊C4—171no—
150C2×C17—44no—
151C2×C17C2×#12011yescharacteristic
152C2×C17#53211yescharacteristic
153C2×C17#53511yescharacteristic
154C2×C17#54612yes—
155C2×C17#55012yes—
156D17—42no—
157D17—82no—
158C2×#41—344no—
159C2×#41—344no—
160C2×#41—344no—
161C22×Q8—171no—
162C22×D4—172no—
163C22×D4—174no—
164C22×D4—171no—
165C22×D4—172no—
166C22×D4C2×D1711yescharacteristic
167C23×C4—171no—
168C23×C4—172no—
169#127—344no—
170#124—344no—
171#121—344no—
172C2×C4⋊C4C2×D1711yescharacteristic
173C2×C22⋊C4—172no—
174C2×C22⋊C4—174no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4C2×D1714yes—
177C2×C42—171no—
178C2×C42C2×D1711yescharacteristic
179#95—172no—
180#95—174no—
181C17#243111yescharacteristic, complemented, Sylow
182C24—171no—
183C24—172no—
184C24C22×D1712yes—
185#41—3432no—
186C2×Q8—174no—
187C2×Q8—342no—
188C2×Q8—342no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—344no—
192C2×D4—344no—
193C2×D4—348no—
194C2×D4—344no—
195C2×D4—344no—
196C2×D4—348no—
197C2×D4—342no—
198C2×D4—342no—
199C2×D4—342no—
200C2×D4—684no—
201C2×D4—684no—
202C2×D4D6814yescomplemented
203C22×C4—172no—
204C22×C4—171no—
205C22×C4—172no—
206C22×C4—174no—
207C22×C4—344no—
208C22×C4—344no—
209C22×C4—344no—
210C22×C4—342no—
211C22×C4—342no—
212C22×C4C22×D1712yes—
213C22×C4C22×D1712yes—
214C22×C4C22×D1711yescharacteristic
215C4⋊C4—22no—
216C22⋊C4—44no—
217C22⋊C4—344no—
218C22⋊C4—348no—
219C22⋊C4—344no—
220C42—22no—
221C42—342no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—24no—
226C23—172no—
227C23—341no—
228C23—341no—
229C23—342no—
230C23—342no—
231C23—342no—
232C23—342no—
233C23—342no—
234C23—344no—
235C23—344no—
236C23C2×D6814yes—
237C23C23×D1711yescharacteristic
238Q8—348no—
239D4—28no—
240D4—3416no—
241D4—688no—
242D4—6816no—
243D4—684no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—172no—
250C2×C4—174no—
251C2×C4—174no—
252C2×C4—3416no—
253C2×C4—342no—
254C2×C4—341no—
255C2×C4—342no—
256C2×C4—341no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—348no—
260C2×C4C2×D6812yes—
261C2×C4D4×D1712yes—
262C2×C4D4×D1712yes—
263C22—24no—
264C22—28no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—342no—
269C22—342no—
270C22—342no—
271C22—682no—
272C22—682no—
273C22—684no—
274C22—682no—
275C22—682no—
276C22—682no—
277C22—684no—
278C22C22×D6811yescharacteristic
279C22C2×D4×D1711yescharacteristic
280C22C2×D4×D1711yescharacteristic
281C22#1058769112yes—
282C22#41×D1712yes—
283C4—42no—
284C4—42no—
285C4—342no—
286C4—344no—
287C4—344no—
288C4#1058756814yes—
289C2—44no—
290C2—682no—
291C2—1362no—
292C2C2×#1058756811yescharacteristic
293C2#1176335811yescharacteristic
294C2#1176337211yescharacteristic
295C2#1176345412yes—
296C2#1176347712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
hypercenter#390D17
hypocenterC17#2431
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×#63254C2semidirect
2. C2×#10587626C2semidirect
3. C2×#10587561C2semidirect
4. C2×#10587581C2semidirect
5. #11762718C2semidirect
6. #11762714C2semidirect
7. #11762686C2semidirect
8. C4×#63227C2semidirect
9. C17×#390C2semidirect
38. C2×D4×C17D4semidirect
62. C2×C4×D17D4semidirect
65. C2×Dic34D4semidirect
109. C2×Dic17#41semidirect
147. #390D17semidirect
181. C17#2431semidirect
202. C2×D4D68semidirect

Maximal subgroups

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

subgroupindex
C22×#632542
C2×#105876262
C2×#105875612
C2×#105875812
#117627182
#117627142
#117626862
C4×#632272
C17×#3902
#243117