Finite Group Databasebeta

← #11741498

Subgroups — #11741498

Order 896.

1232 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 205 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
1C23×C7×D4C211yesmaximal, characteristic, complemented
2C2×#6516C212yesmaximal, complemented
3C2×#6501C212yesmaximal, complemented
4C2×C4×C22⋊D7C211yesmaximal, characteristic, complemented
5#90019C212yesmaximal, complemented
6#89995C212yesmaximal, complemented
7#89994C212yesmaximal, complemented
8#89993C212yesmaximal, complemented
9#89984C211yesmaximal, characteristic, complemented
10C25×C7C2212yes—
11C22×C7×D4—28no—
12C22×C7×D4—24no—
13C22×C7×D4C2212yes—
14C22×C7×D4C2212yes—
15C23×C4×C7C2211yescharacteristic
16C22×C22⋊D7C2211yescharacteristic
17C22×C22⋊D7C2214yescomplemented
18C22×C4×D7C2211yescharacteristic, complemented
19#6516—28no—
20C2×C22⋊Dic7C2211yescharacteristic
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2214yescomplemented
23C2×C22⋊Dic7C2214yescomplemented
24C2×C22⋊Dic7C2214yescomplemented
25#6501—28no—
26C4×C22⋊D7—24no—
27C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
28C2×C4⋊Dic7C2212yescomplemented
29C2×Dic7⋊C4C2211yescharacteristic, complemented
30C2×C4×Dic7C2211yescharacteristic, complemented
31C7⋊#95C2212yescomplemented
32#2408—71nomaximal, Sylow
33C24×C7—28no—
34C24×C7—24no—
35C24×C7—24no—
36C24×C7—24no—
37C24×C7—44no—
38C24×C7C2314yes—
39C24×C7C2311yescharacteristic
40C23×D7C2311yescharacteristic
41C2×C7×D4—232no—
42C2×C7×D4—28no—
43C2×C7×D4—24no—
44C2×C7×D4—24no—
45C2×C7×D4—24no—
46C2×C7×D4D418yescomplemented
47C22×C4×C7—24no—
48C22×C4×C7D414yescomplemented
49C22×C4×C7C2311yescharacteristic
50C22×C4×C7C2311yescharacteristic
51C2×C22⋊D7—22no—
52C2×C22⋊D7—22no—
53C2×C22⋊D7—28no—
54C2×C22⋊D7—28no—
55C2×C22⋊D7—44no—
56C2×C22⋊D7D414yescomplemented
57C22×Dic7C2314yes—
58C22×Dic7C2311yescharacteristic
59C22×Dic7C2311yescharacteristic
60C22×Dic7C2311yescharacteristic
61C2×C4×D7—22no—
62C2×C4×D7—24no—
63C22⋊Dic7—22no—
64C22⋊Dic7—28no—
65C22⋊Dic7—24no—
66C22⋊Dic7—28no—
67C22⋊Dic7—44no—
68(C2×D7)⋊C4—22no—
69C4⋊Dic7—24no—
70Dic7⋊C4—22no—
71C4×Dic7—22no—
72C23×D4D711yescharacteristic, complemented
73C2×#121—72no—
74C2×#120—72no—
75C2×C4×D4—71no—
76#394—72no—
77#392—72no—
78#390—72no—
79#380—71no—
80#379—72no—
81C23×C7—22no—
82C23×C7—22no—
83C23×C7—216no—
84C23×C7—24no—
85C23×C7—24no—
86C23×C7—24no—
87C23×C7—24no—
88C23×C7—24no—
89C23×C7—24no—
90C23×C7—416no—
91C23×C7—416no—
92C23×C7—48no—
93C23×C7—48no—
94C23×C7C2×D412yes—
95C23×C7C2×D418yes—
96C23×C7C2×D412yes—
97C23×C7#4112yes—
98C23×C7C2411yescharacteristic
99C22×D7—21no—
100C22×D7—22no—
101C22×D7—21no—
102C22×D7—22no—
103C22×D7C2×D412yescomplemented
104C7×D4—216no—
105C7×D4—216no—
106C2×C4×C7—22no—
107C2×C4×C7—28no—
108C2×C4×C7C2×D412yes—
109C2×C4×C7C2×D412yes—
110C2×C4×C7C2×D414yes—
111C22⋊D7—28no—
112C22⋊D7—416no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—24no—
118C2×Dic7—22no—
119C2×Dic7—24no—
120C2×Dic7—22no—
121C2×Dic7—24no—
122C2×Dic7C2×D412yescomplemented
123C4×D7—44no—
124C25C2×D712yes—
125C22×D4—28no—
126C22×D4—24no—
127C22×D4—71no—
128C22×D4—74no—
129C22×D4C2×D712yes—
130C22×D4C2×D712yes—
131C23×C4—71no—
132C23×C4C2×D711yescharacteristic
133#121—148no—
134#120—148no—
135C4×D4—144no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4—72no—
138C2×C22⋊C4—71no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—74no—
141C2×C22⋊C4—74no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—74no—
144C2×C42—71no—
145#95—72no—
146C22×C7—22no—
147C22×C7—22no—
148C22×C7—24no—
149C22×C7—24no—
150C22×C7—28no—
151C22×C7—28no—
152C22×C7—28no—
153C22×C7—432no—
154C22×C7—44no—
155C22×C7—44no—
156C22×C7—44no—
157C22×C7—44no—
158C22×C7#12114yescomplemented
159C22×C7C22×D411yescharacteristic
160C22×C7C22×D411yescharacteristic
161C22×C7C22×D412yes—
162C22×C7C2×#4111yescharacteristic
163C22×C7H(5,𝔽2)12yes—
164C2×D7—22no—
165C2×D7—24no—
166C2×D7—41no—
167C2×D7—41no—
168C2×D7—42no—
169C4×C7—22no—
170C4×C7#12014yescomplemented
171Dic7—22no—
172Dic7—41no—
173Dic7—41no—
174Dic7—44no—
175C24—28no—
176C24—24no—
177C24—24no—
178C24—24no—
179C24—44no—
180C24—71no—
181C24C22×D714yes—
182C24C22×D711yescharacteristic
183C2×D4—232no—
184C2×D4—28no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—74no—
189C2×D4—142no—
190C2×D4—142no—
191C2×D4—148no—
192C2×D4—148no—
193C2×D4—284no—
194C2×D4C22⋊D718yescomplemented
195C22×C4—24no—
196C22×C4—71no—
197C22×C4—74no—
198C22×C4—71no—
199C22×C4—71no—
200C22×C4—142no—
201C22×C4—144no—
202C22×C4C22⋊D714yescomplemented
203C22×C4C22×D711yescharacteristic
204C22×C4C22×D711yescharacteristic
205C4⋊C4—142no—
206C4⋊C4—144no—
207C22⋊C4—148no—
208C22⋊C4—144no—
209C22⋊C4—142no—
210C22⋊C4—148no—
211C22⋊C4—142no—
212C22⋊C4—284no—
213C42—142no—
214C2×C7—24no—
215C2×C7—48no—
216C2×C7C2×#12011yescharacteristic
217C2×C7C2×#12111yescharacteristic
218C2×C7#53511yescharacteristic
219C2×C7D4212yes—
220C2×C7#54612yes—
221D7—42no—
222C23—22no—
223C23—22no—
224C23—216no—
225C23—24no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—416no—
232C23—416no—
233C23—48no—
234C23—48no—
235C23—72no—
236C23—141no—
237C23—142no—
238C23—141no—
239C23—142no—
240C23D4×D712yes—
241C23#115912yes—
242C23C2×C22⋊D718yes—
243C23C2×C22⋊D712yes—
244C23C23×D711yescharacteristic
245D4—216no—
246D4—216no—
247D4—148no—
248D4—2816no—
249C2×C4—22no—
250C2×C4—28no—
251C2×C4—72no—
252C2×C4—141no—
253C2×C4—141no—
254C2×C4—144no—
255C2×C4—142no—
256C2×C4—144no—
257C2×C4—141no—
258C2×C4—142no—
259C2×C4—141no—
260C2×C4—144no—
261C2×C4—284no—
262C2×C4C2×C22⋊D712yes—
263C2×C4C2×C22⋊D712yes—
264C2×C4C2×C22⋊D714yes—
265C7#240811yescharacteristic, complemented, Sylow
266C22—24no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—28no—
271C22—28no—
272C22—28no—
273C22—432no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—44no—
278C22—142no—
279C22—144no—
280C22—281no—
281C22—282no—
282C22—281no—
283C22#650114yescomplemented
284C22C2×D4×D711yescharacteristic
285C22C2×#115911yescharacteristic
286C22#654812yes—
287C22C22×C22⋊D711yescharacteristic
288C22C22×C22⋊D712yes—
289C4—22no—
290C4—142no—
291C4—284no—
292C4—281no—
293C4—281no—
294C4#651614yescomplemented
295C2—24no—
296C2—48no—
297C2—282no—
298C2C2×#650111yescharacteristic
299C2D4×C22⋊D712yes—
300C2#9049611yescharacteristic
301C2#9049712yes—
302C2C2×#651611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC23×C7×D4C2
hypercenterC23×D4D7
hypocenterC7#2408
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C7×D4C2semidirect
2. C2×#6516C2semidirect
3. C2×#6501C2semidirect
4. C2×C4×C22⋊D7C2semidirect
5. #90019C2semidirect
6. #89995C2semidirect
7. #89994C2semidirect
8. #89993C2semidirect
9. #89984C2semidirect
17. C22×C22⋊D7C22semidirect
18. C22×C4×D7C22semidirect
22. C2×C22⋊Dic7C22semidirect
23. C2×C22⋊Dic7C22semidirect
24. C2×C22⋊Dic7C22semidirect
27. C2×(C2×D7)⋊C4C22semidirect
28. C2×C4⋊Dic7C22semidirect
29. C2×Dic7⋊C4C22semidirect
30. C2×C4×Dic7C22semidirect
31. C7⋊#95C22semidirect
46. C2×C7×D4D4semidirect
48. C22×C4×C7D4semidirect
56. C2×C22⋊D7D4semidirect
72. C23×D4D7semidirect
103. C22×D7C2×D4semidirect
122. C2×Dic7C2×D4semidirect
158. C22×C7#121semidirect
170. C4×C7#120semidirect
194. C2×D4C22⋊D7semidirect
202. C22×C4C22⋊D7semidirect
265. C7#2408semidirect
283. C22#6501semidirect
294. C4#6516semidirect

Maximal subgroups

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

subgroupindex
C23×C7×D42
C2×#65162
C2×#65012
C2×C4×C22⋊D72
#900192
#899952
#899942
#899932
#899842
#24087