Finite Group Databasebeta

← #11736355

Subgroups — #11736355

Order 896.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#90523C211yesmaximal, characteristic, complemented
2C2×#6512C211yesmaximal, characteristic, complemented
3C2×#6471C211yesmaximal, characteristic, complemented
4C2×#6456C211yesmaximal, characteristic, complemented
5C7×#417C211yesmaximal, characteristic, complemented
6#89890C211yesmaximal, characteristic, complemented
7#89738C211yesmaximal, characteristic, complemented
8#89591C212yesmaximal, complemented
9#89590C212yesmaximal, complemented
10#89559C212yesmaximal, complemented
11#89558C212yesmaximal, complemented
12C2×C7×#41C2211yescharacteristic
13C2×#1157C2211yescharacteristic
14C22×D28C2211yescharacteristic
15C22×C4×D7—21no—
16C2×C7×M16C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18#6512—24no—
19#6510C2211yescharacteristic, complemented
20#6507C2211yescharacteristic, complemented
21C2×#1143C2212yes—
22#6503C2211yescharacteristic, complemented
23#6502—21no—
24#6502—21no—
25#6499C2211yescharacteristic, complemented
26C2×#1141C2212yes—
27C4×C22⋊D7—21no—
28C4×C22⋊D7—21no—
29C2×(C2×D7)⋊C4—21no—
30C2×C4×Dic7C2211yescharacteristic
31C2×#1136C2211yescharacteristic
32#6471—24no—
33C2×D56C2212yes—
34C2×#1132C2212yes—
35#6456—21no—
36#6456—21no—
37#6456C2212yes—
38#6456C2212yes—
39C7×#103C2212yescomplemented
40C7×#102C2212yescomplemented
41Dic7⋊C8C2212yescomplemented
42Dic7⋊C8C2212yescomplemented
43#6384C2212yescomplemented
44#6383C2212yescomplemented
45#3036—71nomaximal, Sylow
46C23×D7—21no—
47C7×#41—22no—
48C7×#41—22no—
49C2×C7×Q8C2311yescharacteristic
50C2×C7×D4—21no—
51C2×C7×D4C2311yescharacteristic
52C22×C4×C7—21no—
53C22×C4×C7—21no—
54C22×C4×C7C2311yescharacteristic
55C2×C22⋊D7—21no—
56C2×C22⋊D7—21no—
57C2×C22⋊D7—21no—
58C22×Dic7D411yescharacteristic
59C22×Dic7D411yescharacteristic
60#1157—22no—
61#1157—22no—
62C2×D28—21no—
63C2×D28—22no—
64C2×D28D414yescomplemented
65C2×D28C2311yescharacteristic
66C2×D28C2312yes—
67C2×C4×D7—21no—
68C2×C4×D7—22no—
69C2×C4×D7—22no—
70C2×C4×D7—44no—
71C2×Dic14C2311yescharacteristic
72C7×M16—22no—
73C2×C7×C8C2312yes—
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2312yes—
76C22⋊Dic7—21no—
77C22⋊Dic7—21no—
78#1143—24no—
79#1141—24no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—22no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84Dic7⋊C4—21no—
85Dic7⋊C4—21no—
86C4×Dic7D411yescharacteristic
87C4×Dic7D411yescharacteristic
88C4×Dic7C2311yescharacteristic
89C4×Dic7C2311yescharacteristic
90#1136—22no—
91C2×C7⋊C8C2312yes—
92D56—24no—
93#1132—24no—
94C2×AGL(1,ℤ/8)—71no—
95C2×AGL(1,ℤ/8)—71no—
96#532—71no—
97C2×C4×D4—71no—
98#463—72no—
99#463—72no—
100#461—72no—
101#459—72no—
102#423—71no—
103#417—71no—
104#417D711yescharacteristic, complemented
105C23×C7—21no—
106C23×C7C2×D411yescharacteristic
107C22×D7—24no—
108C22×D7—22no—
109C22×D7—21no—
110C22×D7—42no—
111C22×D7—42no—
112C7×Q8—21no—
113C7×Q8—21no—
114C7×D4—21no—
115C7×D4—21no—
116C7×D4—42no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7C2×D411yescharacteristic
122C2×C4×C7C2×D412yes—
123C2×C4×C7#4112yes—
124C2×C4×C7C2411yescharacteristic
125C22⋊D7—42no—
126C22⋊D7—42no—
127C22⋊D7—42no—
128C2×Dic7—22no—
129C2×Dic7—21no—
130C2×Dic7—41no—
131C2×Dic7C2×D411yescharacteristic
132C2×Dic7C2×D411yescharacteristic
133C2×Dic7C2×D411yescharacteristic
134C2×Dic7C2×D411yescharacteristic
135D28—24no—
136D28—21no—
137D28—22no—
138D28—21no—
139D28C2×D414yes—
140C4×D7—44no—
141C4×D7—42no—
142C4×D7—44no—
143Dic14—21no—
144Dic14—21no—
145C7×C8—22no—
146C7⋊C8—22no—
147C2×#41—71no—
148C2×#41C2×D711yescharacteristic
149C22×D4—71no—
150C23×C4—141no—
151AGL(1,ℤ/8)—144no—
152AGL(1,ℤ/8)—144no—
153C2×SD16—72no—
154C2×SD16—72no—
155C2×D8—72no—
156C2×D8—72no—
157C2×M16—71no—
158C2×M16C2×D711yescharacteristic
159C4⋊Q8—71no—
160#127—71no—
161#124—71no—
162#124—71no—
163#121—141no—
164#121—141no—
165C4×D4—72no—
166C4×D4—72no—
167C4×D4—141no—
168C4×D4—141no—
169C4×D4—141no—
170C4×D4—141no—
171C2×C4⋊C4C2×D711yescharacteristic
172C2×C22⋊C4—141no—
173C2×C42—71no—
174C4⋊C8—72no—
175C4⋊C8—72no—
176#103—72no—
177#103C2×D712yescomplemented
178#102—72no—
179#102C2×D712yescomplemented
180C22×C7—21no—
181C22×C7—21no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7#12112yes—
186C22×C7C22×D411yescharacteristic
187C2×D7—24no—
188C2×D7—48no—
189C2×D7—42no—
190C2×D7—42no—
191C2×D7—41no—
192C2×D7—41no—
193C2×D7—41no—
194C4×C7—41no—
195C4×C7—42no—
196C4×C7#12112yes—
197C4×C7C22×D411yescharacteristic
198C4×C7C2×#4111yescharacteristic
199Dic7—21no—
200Dic7—21no—
201Dic7—21no—
202Dic7—41no—
203Dic7AGL(1,ℤ/8)12yescomplemented
204C24—141no—
205#41—22no—
206#41—22no—
207#41—142no—
208#41—142no—
209C2×Q8—71no—
210C2×Q8C22×D711yescharacteristic
211C2×D4—21no—
212C2×D4—74no—
213C2×D4—72no—
214C2×D4—71no—
215C2×D4—141no—
216C2×D4—141no—
217C2×D4—141no—
218C2×D4—142no—
219C2×D4—141no—
220C2×D4C22×D711yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—71no—
224C22×C4—71no—
225C22×C4—142no—
226C22×C4—142no—
227C22×C4—141no—
228C22×C4—284no—
229C22×C4C22×D711yescharacteristic
230SD16—144no—
231SD16—144no—
232D8—144no—
233D8—144no—
234M16—22no—
235M16—142no—
236C2×C8—72no—
237C2×C8C22×D712yes—
238C4⋊C4—21no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4C22×D712yes—
242C22⋊C4—142no—
243C22⋊C4—142no—
244C22⋊C4—141no—
245C22⋊C4—141no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C42—71no—
249C42—71no—
250C42—71no—
251C42—71no—
252C2×C7—22no—
253C2×C7—81no—
254C2×C7C2×#12111yescharacteristic
255C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
256C2×C7#57511yescharacteristic
257D7—44no—
258D7—81no—
259C23—21no—
260C23—144no—
261C23—142no—
262C23—141no—
263C23—282no—
264C23—282no—
265C23D4×D711yescharacteristic
266Q8—21no—
267Q8—21no—
268Q8—141no—
269Q8—141no—
270D4—21no—
271D4—21no—
272D4—42no—
273D4—74no—
274D4—144no—
275D4—142no—
276D4—141no—
277D4—141no—
278D4—282no—
279D4—282no—
280D4—282no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—71no—
286C2×C4—71no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—142no—
290C2×C4—141no—
291C2×C4—284no—
292C2×C4—284no—
293C2×C4—281no—
294C2×C4—282no—
295C2×C4#115712yes—
296C2×C4D4×D711yescharacteristic
297C2×C4D4×D712yes—
298C2×C4C23×D711yescharacteristic
299C8—22no—
300C8—142no—
301C7#303611yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—144no—
308C22—288no—
309C22—282no—
310C22—282no—
311C22—281no—
312C22—281no—
313C22—281no—
314C22#644712yes—
315C22C2×D4×D711yescharacteristic
316C4—41no—
317C4—42no—
318C4—72no—
319C4—141no—
320C4—141no—
321C4—141no—
322C4—281no—
323C4#644712yes—
324C4C2×#115711yescharacteristic
325C4C2×D4×D711yescharacteristic
326C2—22no—
327C2—81no—
328C2—284no—
329C2—561no—
330C2C2×#644711yescharacteristic
331C2D7×AGL(1,ℤ/8)11yescharacteristic
332C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
hypercenter#417D7
hypocenterC7#3036
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #90523C2semidirect
2. C2×#6512C2semidirect
3. C2×#6471C2semidirect
4. C2×#6456C2semidirect
5. C7×#417C2semidirect
6. #89890C2semidirect
7. #89738C2semidirect
8. #89591C2semidirect
9. #89590C2semidirect
10. #89559C2semidirect
11. #89558C2semidirect
19. #6510C22semidirect
20. #6507C22semidirect
22. #6503C22semidirect
25. #6499C22semidirect
39. C7×#103C22semidirect
40. C7×#102C22semidirect
41. Dic7⋊C8C22semidirect
42. Dic7⋊C8C22semidirect
43. #6384C22semidirect
44. #6383C22semidirect
64. C2×D28D4semidirect
104. #417D7semidirect
177. #103C2×D7semidirect
179. #102C2×D7semidirect
203. Dic7AGL(1,ℤ/8)semidirect
301. C7#3036semidirect

Maximal subgroups

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

subgroupindex
#905232
C2×#65122
C2×#64712
C2×#64562
C7×#4172
#898902
#897382
#895912
#895902
#895592
#895582
#30367