Finite Group Databasebeta

← #11736656

Subgroups — #11736656

Order 896.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#6471C211yesmaximal, characteristic, complemented
2C2×D7×M16C211yesmaximal, characteristic, complemented
3#90215C211yesmaximal, characteristic, complemented
4C2×#6458C211yesmaximal, characteristic, complemented
5C7×#428C211yesmaximal, characteristic, complemented
6#89773C211yesmaximal, characteristic, complemented
7#89738C211yesmaximal, characteristic, complemented
8#89658C212yesmaximal, complemented
9#89655C212yesmaximal, complemented
10#89637C212yesmaximal, complemented
11#89636C212yesmaximal, complemented
12C2×#1157C2211yescharacteristic
13C22×D28—21no—
14C22×D28C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C2×C7×M16C2211yescharacteristic
17C7×#117C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4—21no—
20C2×#1136C2211yescharacteristic
21#6471—24no—
22D7×M16—24no—
23C2×D56C2212yescomplemented
24C2×#1132C2212yescomplemented
25C2×#1131C2212yescomplemented
26C2×C8×D7C2212yescomplemented
27#6460C2211yescharacteristic, complemented
28#6460C2211yescharacteristic, complemented
29#6458—21no—
30#6458—21no—
31#6458C2212yes—
32#6458C2212yes—
33#6458C2211yescharacteristic, complemented
34#6458C2211yescharacteristic, complemented
35#6449—21no—
36#6445—21no—
37C4×D28—21no—
38C4×D28—21no—
39C7×#107C2212yescomplemented
40C7×#106C2212yescomplemented
41#6384C2212yescomplemented
42#6383C2212yescomplemented
43#6383C2212yescomplemented
44#6383C2212yescomplemented
45#3055—71nomaximal, Sylow
46C23×D7—21no—
47C23×D7D411yescharacteristic
48C22×C4×C7—21no—
49C22×C4×C7C2311yescharacteristic
50C2×C22⋊D7—21no—
51C22×Dic7D411yescharacteristic
52#1157—22no—
53#1157—22no—
54C2×D28—22no—
55C2×D28—22no—
56C2×D28—21no—
57C2×D28—21no—
58C2×D28—22no—
59C2×D28—22no—
60C2×D28—21no—
61C2×D28—44no—
62C2×D28C2312yes—
63C2×D28C2311yescharacteristic
64C2×C4×D7—22no—
65C2×C4×D7—22no—
66C2×C4×D7—21no—
67C2×C4×D7D411yescharacteristic
68C2×C4×D7D411yescharacteristic
69C2×C4×D7C2311yescharacteristic
70C2×C4×D7C2311yescharacteristic
71C2×Dic14C2311yescharacteristic
72C7×M16D414yescomplemented
73C2×C7×C8C2312yes—
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2312yes—
77C7×C4⋊C4C2311yescharacteristic
78C7×C22⋊C4—21no—
79C42×C7—21no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—22no—
85(C2×D7)⋊C4—22no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88#1136—22no—
89C2×C7⋊C8C2312yes—
90D56—24no—
91#1132—24no—
92#1131—24no—
93C8×D7—24no—
94C2×AGL(1,ℤ/8)—71no—
95C22×M16—71no—
96#525—71no—
97C2×#121—71no—
98#469—72no—
99#468—72no—
100#466—72no—
101#465—72no—
102#428D711yescharacteristic, complemented
103#418—71no—
104#417—71no—
105C23×C7C2×D411yescharacteristic
106C22×D7—22no—
107C22×D7—22no—
108C22×D7—21no—
109C22×D7—22no—
110C22×D7—41no—
111C22×D7—42no—
112C22×D7—42no—
113C22×D7—42no—
114C22×D7C2×D411yescharacteristic
115C22×D7C2×D411yescharacteristic
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—42no—
120C2×C4×C7C2×D412yes—
121C2×C4×C7C2×D411yescharacteristic
122C2×C4×C7#4112yes—
123C2×C4×C7C2411yescharacteristic
124C22⋊D7—42no—
125C2×Dic7—22no—
126C2×Dic7—21no—
127C2×Dic7C2×D411yescharacteristic
128C2×Dic7C2×D411yescharacteristic
129D28—22no—
130D28—21no—
131D28—22no—
132D28—21no—
133D28—42no—
134D28—44no—
135D28—42no—
136D28—44no—
137D28—42no—
138C4×D7—22no—
139C4×D7—22no—
140C4×D7—21no—
141C4×D7—21no—
142C4×D7—21no—
143C4×D7—21no—
144C4×D7—42no—
145Dic14—21no—
146Dic14—21no—
147C7×C8C2×D414yescomplemented
148C7⋊C8—22no—
149C2×#41—71no—
150C22×D4—71no—
151C22×D4—141no—
152C23×C4—71no—
153AGL(1,ℤ/8)—144no—
154C2×SD16—72no—
155C2×D8—72no—
156C2×M16—71no—
157C2×M16—72no—
158C2×M16—144no—
159C2×M16C2×D711yescharacteristic
160C22×C8—72no—
161#123—141no—
162C22⋊Q8—71no—
163C22⋊Q8—71no—
164#121—72no—
165#121—72no—
166#121—71no—
167#121—71no—
168#121—141no—
169#121—141no—
170#120—141no—
171C4×D4—141no—
172C4×D4—141no—
173#117C2×D711yescharacteristic
174C2×C4⋊C4C2×D711yescharacteristic
175C2×C22⋊C4—141no—
176#107C2×D712yescomplemented
177#106C2×D712yescomplemented
178#103—72no—
179#102—72no—
180#102—72no—
181#102—72no—
182C22×C7—21no—
183C22×C7—21no—
184C22×C7#12112yes—
185C22×C7C22×D411yescharacteristic
186C2×D7—21no—
187C2×D7—21no—
188C2×D7—21no—
189C2×D7—21no—
190C2×D7—21no—
191C2×D7—44no—
192C2×D7—41no—
193C2×D7—42no—
194C2×D7—42no—
195C2×D7—41no—
196C2×D7—41no—
197C2×D7—42no—
198C2×D7—84no—
199C2×D7AGL(1,ℤ/8)12yescomplemented
200C4×C7—42no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7#12112yes—
204C4×C7C22×D411yescharacteristic
205C4×C7C2×#4111yescharacteristic
206Dic7—21no—
207Dic7—21no—
208Dic7—41no—
209C24—71no—
210C24—141no—
211#41—142no—
212#41—142no—
213C2×Q8—71no—
214C2×D4—71no—
215C2×D4—72no—
216C2×D4—141no—
217C2×D4—141no—
218C2×D4—142no—
219C2×D4—142no—
220C2×D4—141no—
221C2×D4—142no—
222C2×D4—141no—
223C2×D4—142no—
224C2×D4—284no—
225C22×C4—21no—
226C22×C4—71no—
227C22×C4—71no—
228C22×C4—71no—
229C22×C4—71no—
230C22×C4—71no—
231C22×C4—142no—
232C22×C4—142no—
233C22×C4—141no—
234C22×C4C22×D711yescharacteristic
235SD16—144no—
236D8—144no—
237M16—142no—
238M16—144no—
239M16D2814yescomplemented
240C2×C8—72no—
241C2×C8—144no—
242C2×C8C22×D712yes—
243C4⋊C4—21no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4C22×D711yescharacteristic
247C4⋊C4C22×D711yescharacteristic
248C4⋊C4C22×D712yes—
249C22⋊C4—21no—
250C22⋊C4—142no—
251C22⋊C4—142no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C42—21no—
257C2×C7—22no—
258C2×C7C2×#12111yescharacteristic
259C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
260C2×C7#57511yescharacteristic
261D7—22no—
262D7—41no—
263D7—81no—
264D7—82no—
265C23—71no—
266C23—71no—
267C23—142no—
268C23—142no—
269C23—141no—
270C23—142no—
271C23—281no—
272C23—282no—
273C23—282no—
274C23—282no—
275C23D4×D711yescharacteristic
276Q8—141no—
277Q8—141no—
278D4—141no—
279D4—142no—
280D4—142no—
281D4—141no—
282D4—282no—
283D4—284no—
284D4—282no—
285D4—284no—
286D4—282no—
287D4—282no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—42no—
292C2×C4—71no—
293C2×C4—71no—
294C2×C4—142no—
295C2×C4—142no—
296C2×C4—142no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—282no—
303C2×C4C2×D2812yes—
304C2×C4D4×D711yescharacteristic
305C2×C4#116112yes—
306C2×C4C23×D711yescharacteristic
307C8—142no—
308C8C2×D2814yescomplemented
309C7#305511yescharacteristic, complemented, Sylow
310C22—21no—
311C22—21no—
312C22—72no—
313C22—141no—
314C22—141no—
315C22—141no—
316C22—141no—
317C22—141no—
318C22—284no—
319C22—281no—
320C22—282no—
321C22—282no—
322C22—281no—
323C22—281no—
324C22—282no—
325C22—564no—
326C22#645812yes—
327C22C2×D4×D711yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—141no—
332C4—141no—
333C4—281no—
334C4#645812yes—
335C4C22×D2811yescharacteristic
336C4C2×#116111yescharacteristic
337C2—22no—
338C2—142no—
339C2—281no—
340C2—562no—
341C2—561no—
342C2C2×#645811yescharacteristic
343C2D7×AGL(1,ℤ/8)11yescharacteristic
344C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
hypercenter#428D7
hypocenterC7#3055
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#6471C2semidirect
2. C2×D7×M16C2semidirect
3. #90215C2semidirect
4. C2×#6458C2semidirect
5. C7×#428C2semidirect
6. #89773C2semidirect
7. #89738C2semidirect
8. #89658C2semidirect
9. #89655C2semidirect
10. #89637C2semidirect
11. #89636C2semidirect
23. C2×D56C22semidirect
24. C2×#1132C22semidirect
25. C2×#1131C22semidirect
26. C2×C8×D7C22semidirect
27. #6460C22semidirect
28. #6460C22semidirect
33. #6458C22semidirect
34. #6458C22semidirect
39. C7×#107C22semidirect
40. C7×#106C22semidirect
41. #6384C22semidirect
42. #6383C22semidirect
43. #6383C22semidirect
44. #6383C22semidirect
72. C7×M16D4semidirect
102. #428D7semidirect
147. C7×C8C2×D4semidirect
176. #107C2×D7semidirect
177. #106C2×D7semidirect
199. C2×D7AGL(1,ℤ/8)semidirect
239. M16D28semidirect
308. C8C2×D28semidirect
309. C7#3055semidirect

Maximal subgroups

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

subgroupindex
C2×#64712
C2×D7×M162
#902152
C2×#64582
C7×#4282
#897732
#897382
#896582
#896552
#896372
#896362
#30557