Finite Group Databasebeta

← #11741376

Subgroups — #11741376

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×C7×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#90523C211yesmaximal, characteristic, complemented
3C2×D4×Dic7C211yesmaximal, characteristic, complemented
4C2×#6495C211yesmaximal, characteristic, complemented
5#90004C211yesmaximal, characteristic, complemented
6#89937C212yesmaximal, complemented
7#89934C212yesmaximal, complemented
8#89925C212yesmaximal, complemented
9#89922C212yesmaximal, complemented
10#89905C211yesmaximal, characteristic, complemented
11#89890C211yesmaximal, characteristic, complemented
12C2×C7×#41C2211yescharacteristic
13C22×C7×D4C2211yescharacteristic
14C23×Dic7—21no—
15C2×#1157C2211yescharacteristic
16C7×AGL(1,ℤ/8)—24no—
17C2×C7×SD16C2212yes—
18C2×C7×D8C2212yes—
19C2×C7×M16C2211yescharacteristic
20C2×C22⋊Dic7—21no—
21#6510C2211yescharacteristic, complemented
22#6507C2211yescharacteristic, complemented
23#6503C2211yescharacteristic, complemented
24#6502—21no—
25#6502—21no—
26#6499C2211yescharacteristic, complemented
27D4×Dic7—21no—
28D4×Dic7—21no—
29D4×Dic7C2212yes—
30D4×Dic7C2212yes—
31C2×#1142C2212yes—
32#6495—24no—
33C2×#1141C2212yes—
34C4×C22⋊D7—21no—
35C4×C22⋊D7—21no—
36C2×C4⋊Dic7C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38C2×#1136C2211yescharacteristic
39#6409C2212yescomplemented
40#6406C2212yescomplemented
41#6395C2212yescomplemented
42#6390C2212yescomplemented
43Dic7⋊C8C2212yescomplemented
44Dic7⋊C8C2212yescomplemented
45#3036—71nomaximal, Sylow
46C24×C7—21no—
47C7×#41—22no—
48C7×#41—22no—
49C2×C7×Q8C2311yescharacteristic
50C2×C7×D4—21no—
51C2×C7×D4—21no—
52C2×C7×D4—22no—
53C2×C7×D4D414yescomplemented
54C2×C7×D4C2311yescharacteristic
55C2×C7×D4C2312yes—
56C22×C4×C7—21no—
57C22×C4×C7C2311yescharacteristic
58C2×C22⋊D7—21no—
59C2×C22⋊D7—21no—
60C2×C22⋊D7—21no—
61C22×Dic7—21no—
62C22×Dic7—22no—
63C22×Dic7—22no—
64C22×Dic7—44no—
65C22×Dic7D411yescharacteristic
66C22×Dic7D411yescharacteristic
67#1157—22no—
68#1157—22no—
69C2×D28C2311yescharacteristic
70C2×C4×D7—21no—
71C2×Dic14C2311yescharacteristic
72C7×SD16—24no—
73C7×D8—24no—
74C7×M16—22no—
75C2×C7×C8C2312yes—
76C22⋊Dic7—21no—
77C22⋊Dic7—21no—
78C22⋊Dic7—22no—
79C22⋊Dic7—22no—
80#1142—24no—
81#1141—24no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7C2312yes—
86Dic7⋊C4—21no—
87Dic7⋊C4—21no—
88C4×Dic7D411yescharacteristic
89C4×Dic7D411yescharacteristic
90C4×Dic7C2311yescharacteristic
91C4×Dic7C2311yescharacteristic
92#1136—22no—
93C2×C7⋊C8C2312yes—
94C2×AGL(1,ℤ/8)—71no—
95C2×AGL(1,ℤ/8)D711yescharacteristic, complemented
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#417—71no—
105C23×C7—24no—
106C23×C7—22no—
107C23×C7—21no—
108C23×C7—42no—
109C23×C7—42no—
110C23×C7C2×D411yescharacteristic
111C22×D7—21no—
112C7×Q8—21no—
113C7×Q8—21no—
114C7×D4—24no—
115C7×D4—21no—
116C7×D4—22no—
117C7×D4—21no—
118C7×D4—42no—
119C7×D4C2×D414yes—
120C2×C4×C7—21no—
121C2×C4×C7—42no—
122C2×C4×C7C2×D411yescharacteristic
123C2×C4×C7C2×D412yes—
124C2×C4×C7#4112yes—
125C2×C4×C7C2411yescharacteristic
126C22⋊D7—42no—
127C22⋊D7—42no—
128C22⋊D7—42no—
129C2×Dic7—22no—
130C2×Dic7—21no—
131C2×Dic7—22no—
132C2×Dic7—41no—
133C2×Dic7—44no—
134C2×Dic7—42no—
135C2×Dic7—44no—
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140D28—21no—
141D28—21no—
142C4×D7—42no—
143Dic14—21no—
144Dic14—21no—
145C7×C8—22no—
146C7⋊C8—22no—
147C2×#41—71no—
148C2×#41C2×D711yescharacteristic
149C22×D4C2×D711yescharacteristic
150C23×C4—141no—
151AGL(1,ℤ/8)—24no—
152AGL(1,ℤ/8)—144no—
153C2×SD16—72no—
154C2×SD16C2×D712yes—
155C2×D8—72no—
156C2×D8C2×D712yes—
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⋊C4—71no—
172C2×C22⋊C4—141no—
173C2×C42—71no—
174C4⋊C8—72no—
175C4⋊C8—72no—
176#103—72no—
177#103—72no—
178#102—72no—
179#102—72no—
180C22×C7—21no—
181C22×C7—21no—
182C22×C7—24no—
183C22×C7—48no—
184C22×C7—42no—
185C22×C7—42no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—41no—
189C22×C7#12112yes—
190C22×C7C22×D411yescharacteristic
191C2×D7—41no—
192C2×D7—41no—
193C2×D7—41no—
194C4×C7—41no—
195C4×C7#12112yes—
196C4×C7C22×D411yescharacteristic
197C4×C7C2×#4111yescharacteristic
198Dic7—21no—
199Dic7—21no—
200Dic7—21no—
201Dic7—41no—
202Dic7—42no—
203Dic7AGL(1,ℤ/8)12yescomplemented
204C24—21no—
205#41—22no—
206#41—22no—
207#41—142no—
208#41—142no—
209C2×Q8—71no—
210C2×Q8C22×D711yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—71no—
215C2×D4—141no—
216C2×D4—141no—
217C2×D4—141no—
218C2×D4C22⋊D714yescomplemented
219C2×D4C22×D711yescharacteristic
220C2×D4C22×D712yes—
221C22×C4—21no—
222C22×C4—71no—
223C22×C4—71no—
224C22×C4—142no—
225C22×C4—142no—
226C22×C4—141no—
227C22×C4—141no—
228C22×C4—284no—
229C22×C4C22×D711yescharacteristic
230SD16—24no—
231SD16—144no—
232D8—24no—
233D8—144no—
234M16—22no—
235M16—142no—
236C2×C8—72no—
237C2×C8C22×D712yes—
238C4⋊C4—72no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C22⋊C4—141no—
243C22⋊C4—141no—
244C22⋊C4—142no—
245C22⋊C4—142no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C42—71no—
249C42—71no—
250C42—71no—
251C42—71no—
252C2×C7—22no—
253C2×C7—44no—
254C2×C7—81no—
255C2×C7C2×#12111yescharacteristic
256C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
257C2×C7#57511yescharacteristic
258D7—81no—
259C23—24no—
260C23—22no—
261C23—21no—
262C23—42no—
263C23—42no—
264C23—141no—
265C23D4×D711yescharacteristic
266Q8—21no—
267Q8—21no—
268Q8—141no—
269Q8—141no—
270D4—24no—
271D4—22no—
272D4—21no—
273D4—21no—
274D4—42no—
275D4—141no—
276D4—141no—
277D4—282no—
278D4—282no—
279D4—282no—
280D4C2×C22⋊D714yes—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—71no—
284C2×C4—71no—
285C2×C4—71no—
286C2×C4—71no—
287C2×C4—142no—
288C2×C4—142no—
289C2×C4—141no—
290C2×C4—284no—
291C2×C4—281no—
292C2×C4—284no—
293C2×C4—282no—
294C2×C4—282no—
295C2×C4D4×D711yescharacteristic
296C2×C4#115912yes—
297C2×C4C2×C22⋊D712yes—
298C2×C4C23×D711yescharacteristic
299C8—22no—
300C8—142no—
301C7#303611yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—281no—
312C22—281no—
313C22—281no—
314C22#650212yes—
315C22C2×D4×D711yescharacteristic
316C4—41no—
317C4—72no—
318C4—141no—
319C4—141no—
320C4—141no—
321C4—282no—
322C4—281no—
323C4#650212yes—
324C4C2×#115911yescharacteristic
325C4C22×C22⋊D711yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—561no—
330C2D7×AGL(1,ℤ/8)11yescharacteristic
331C2#9046411yescharacteristic
332C2C2×#650211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C7×AGL(1,ℤ/8)C2
socleC22×C7C22×D4
hypercenterC2×AGL(1,ℤ/8)D7
hypocenterC7#3036
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C7×AGL(1,ℤ/8)C2semidirect
2. #90523C2semidirect
3. C2×D4×Dic7C2semidirect
4. C2×#6495C2semidirect
5. #90004C2semidirect
6. #89937C2semidirect
7. #89934C2semidirect
8. #89925C2semidirect
9. #89922C2semidirect
10. #89905C2semidirect
11. #89890C2semidirect
21. #6510C22semidirect
22. #6507C22semidirect
23. #6503C22semidirect
26. #6499C22semidirect
39. #6409C22semidirect
40. #6406C22semidirect
41. #6395C22semidirect
42. #6390C22semidirect
43. Dic7⋊C8C22semidirect
44. Dic7⋊C8C22semidirect
53. C2×C7×D4D4semidirect
95. C2×AGL(1,ℤ/8)D7semidirect
203. Dic7AGL(1,ℤ/8)semidirect
218. C2×D4C22⋊D7semidirect
301. C7#3036semidirect

Maximal subgroups

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

subgroupindex
C2×C7×AGL(1,ℤ/8)2
#905232
C2×D4×Dic72
C2×#64952
#900042
#899372
#899342
#899252
#899222
#899052
#898902
#30367