Finite Group Databasebeta

← #11736662

Subgroups — #11736662

Order 896.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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
1D7×#117C211yesmaximal, characteristic, complemented
2C2×#6458C211yesmaximal, characteristic
3C7×#428C211yesmaximal, characteristic, complemented
4#89903C211yesmaximal, characteristic, complemented
5#89898C211yesmaximal, characteristic, complemented
6#89773C211yesmaximal, characteristic, complemented
7#89736C211yesmaximal, characteristic, complemented
8#89657C212yesmaximal, complemented
9#89654C212yesmaximal, complemented
10#89638C212yesmaximal, complemented
11#89635C212yesmaximal, complemented
12C22×D28—21no—
13C22×D28C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C2×C7×M16C2211yescharacteristic
16C7×#117C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4—21no—
19#6489C2211yescharacteristic
20C2×#1136C2211yescharacteristic
21#6458—21no—
22#6458—21no—
23#6458C2212yes—
24#6458C2212yes—
25#6455C2211yescharacteristic, complemented
26#6455C2211yescharacteristic, complemented
27D7×C4⋊C4C2211yescharacteristic, complemented
28D7×C4⋊C4C2211yescharacteristic, complemented
29D7×C22⋊C4—21no—
30#6440—21no—
31#6435—21no—
32C42×D7—21no—
33C7×#107C2212yescomplemented
34C7×#106C2212yescomplemented
35#6395C2212yescomplemented
36#6395C2212yescomplemented
37(C2×D7)⋊C8C2212yescomplemented
38(C2×D7)⋊C8C2212yescomplemented
39#6383C2212yescomplemented
40#6383C2212yescomplemented
41#6382C2212yescomplemented
42#6381C2212yescomplemented
43#3094—71nomaximal, Sylow
44C23×D7—21no—
45C23×D7D411yescharacteristic
46C22×C4×C7—21no—
47C22×C4×C7C2311yescharacteristic
48C22×Dic7D411yescharacteristic
49C2×D28—22no—
50C2×D28—22no—
51C2×D28—22no—
52C2×D28—22no—
53C2×D28—21no—
54C2×D28—44no—
55C2×D28C2312yes—
56C2×C4×D7—22no—
57C2×C4×D7—22no—
58C2×C4×D7—21no—
59C2×C4×D7—21no—
60C2×C4×D7D411yescharacteristic
61C2×C4×D7D411yescharacteristic
62C2×C4×D7C2311yescharacteristic
63C2×C4×D7C2311yescharacteristic
64C7×M16—22no—
65C2×C7×C8C2312yes—
66C7×C4⋊C4—21no—
67C7×C4⋊C4C2311yescharacteristic
68C7×C4⋊C4C2312yes—
69C7×C4⋊C4C2311yescharacteristic
70C7×C22⋊C4—21no—
71C42×C7—21no—
72C22⋊Dic7—21no—
73(C2×D7)⋊C4—21no—
74(C2×D7)⋊C4—21no—
75(C2×D7)⋊C4—22no—
76(C2×D7)⋊C4—22no—
77C4⋊Dic7C2311yescharacteristic
78C4⋊Dic7C2311yescharacteristic
79Dic7⋊C4—21no—
80Dic7⋊C4—21no—
81C4×Dic7—21no—
82C4×Dic7—21no—
83C4×Dic7—21no—
84#1136—22no—
85C2×C7⋊C8C2312yes—
86C2×#121—71no—
87C2×#117—71no—
88#482—72no—
89#482—72no—
90#481—72no—
91#480—72no—
92#428—71no—
93#428D711yescharacteristic, complemented
94#418—71no—
95#418—71no—
96#407—71no—
97C23×C7C2×D411yescharacteristic
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—41no—
102C22×D7—42no—
103C22×D7—42no—
104C22×D7—42no—
105C22×D7C2×D411yescharacteristic
106C22×D7C2×D411yescharacteristic
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—22no—
110C2×C4×C7—42no—
111C2×C4×C7C2×D411yescharacteristic
112C2×C4×C7#4112yes—
113C2×C4×C7#4112yes—
114C2×C4×C7C2411yescharacteristic
115C2×Dic7—22no—
116C2×Dic7—21no—
117C2×Dic7—21no—
118C2×Dic7C2×D411yescharacteristic
119C2×Dic7C2×D411yescharacteristic
120D28—22no—
121D28—22no—
122D28—44no—
123D28—44no—
124D28—42no—
125C4×D7—22no—
126C4×D7—22no—
127C4×D7—21no—
128C4×D7—21no—
129C4×D7—21no—
130C4×D7—21no—
131C4×D7—42no—
132C4×D7—42no—
133C7×C8—22no—
134C7⋊C8—22no—
135C22×D4—71no—
136C22×D4—141no—
137C23×C4—71no—
138C2×M16—71no—
139C2×M16C2×D711yescharacteristic
140#121—72no—
141#121—72no—
142#121—141no—
143#121—141no—
144#117—71no—
145#117—71no—
146#117—71no—
147#117—141no—
148#117—141no—
149#117C2×D711yescharacteristic
150C2×C4⋊C4—71no—
151C2×C4⋊C4—71no—
152C2×C4⋊C4C2×D711yescharacteristic
153C2×C22⋊C4—141no—
154C2×C22⋊C4—141no—
155C2×C42—141no—
156#107—72no—
157#107C2×D712yescomplemented
158#106—72no—
159#106C2×D712yescomplemented
160#102—72no—
161#102—72no—
162#102—72no—
163#102—72no—
164C22⋊C8—72no—
165C22⋊C8—72no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7#12312yes—
169C22×C7C22×D411yescharacteristic
170C2×D7—21no—
171C2×D7—21no—
172C2×D7—21no—
173C2×D7—21no—
174C2×D7—21no—
175C2×D7—44no—
176C2×D7—42no—
177C2×D7—42no—
178C2×D7—42no—
179C2×D7—84no—
180C2×D7AGL(1,ℤ/8)12yes—
181C4×C7—41no—
182C4×C7—42no—
183C4×C7—41no—
184C4×C7#12312yes—
185C4×C7C2×#4111yescharacteristic
186C4×C7C2×#4111yescharacteristic
187Dic7—21no—
188Dic7—21no—
189Dic7—41no—
190Dic7—41no—
191C24—71no—
192C24—141no—
193C2×D4—72no—
194C2×D4—142no—
195C2×D4—142no—
196C2×D4—142no—
197C2×D4—141no—
198C2×D4—142no—
199C2×D4—284no—
200C22×C4—21no—
201C22×C4—71no—
202C22×C4—71no—
203C22×C4—71no—
204C22×C4—71no—
205C22×C4—71no—
206C22×C4—141no—
207C22×C4—141no—
208C22×C4—142no—
209C22×C4—142no—
210C22×C4C22×D711yescharacteristic
211M16—22no—
212M16—142no—
213C2×C8—72no—
214C2×C8C22×D712yes—
215C4⋊C4—21no—
216C4⋊C4—71no—
217C4⋊C4—71no—
218C4⋊C4—141no—
219C4⋊C4—141no—
220C4⋊C4C22×D711yescharacteristic
221C4⋊C4C22×D711yescharacteristic
222C4⋊C4C22×D712yes—
223C22⋊C4—21no—
224C22⋊C4—142no—
225C22⋊C4—142no—
226C22⋊C4—141no—
227C22⋊C4—141no—
228C22⋊C4—141no—
229C42—21no—
230C42—141no—
231C42—141no—
232C42—141no—
233C2×C7—22no—
234C2×C7C2×#12311yescharacteristic
235C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
236C2×C7#57511yescharacteristic
237D7—22no—
238D7—41no—
239D7—82no—
240C23—71no—
241C23—71no—
242C23—142no—
243C23—142no—
244C23—142no—
245C23—281no—
246C23—282no—
247C23—282no—
248C23—282no—
249C23D4×D711yescharacteristic
250D4—142no—
251D4—142no—
252D4—284no—
253D4—284no—
254D4—282no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—71no—
260C2×C4—71no—
261C2×C4—142no—
262C2×C4—142no—
263C2×C4—142no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—141no—
267C2×C4—141no—
268C2×C4—141no—
269C2×C4—141no—
270C2×C4—282no—
271C2×C4—282no—
272C2×C4#115712yes—
273C2×C4D4×D711yescharacteristic
274C2×C4#116112yes—
275C2×C4C23×D711yescharacteristic
276C8—22no—
277C8—142no—
278C7#309411yescharacteristic, complemented, Sylow
279C22—21no—
280C22—21no—
281C22—72no—
282C22—141no—
283C22—141no—
284C22—141no—
285C22—141no—
286C22—141no—
287C22—284no—
288C22—282no—
289C22—282no—
290C22—282no—
291C22—564no—
292C22#645712yes—
293C22C2×D4×D711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—141no—
298C4—141no—
299C4—281no—
300C4—281no—
301C4#645712yes—
302C4C2×#115711yescharacteristic
303C4C2×#116111yescharacteristic
304C2—22no—
305C2—142no—
306C2—281no—
307C2—562no—
308C2C2×#645711yescharacteristic
309C2D7×AGL(1,ℤ/8)11yescharacteristic
310C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radicalC2×#6458C2
hypercenter#428D7
hypocenterC7#3094
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. D7×#117C2semidirect
3. C7×#428C2semidirect
4. #89903C2semidirect
5. #89898C2semidirect
6. #89773C2semidirect
7. #89736C2semidirect
8. #89657C2semidirect
9. #89654C2semidirect
10. #89638C2semidirect
11. #89635C2semidirect
25. #6455C22semidirect
26. #6455C22semidirect
27. D7×C4⋊C4C22semidirect
28. D7×C4⋊C4C22semidirect
33. C7×#107C22semidirect
34. C7×#106C22semidirect
35. #6395C22semidirect
36. #6395C22semidirect
37. (C2×D7)⋊C8C22semidirect
38. (C2×D7)⋊C8C22semidirect
39. #6383C22semidirect
40. #6383C22semidirect
41. #6382C22semidirect
42. #6381C22semidirect
93. #428D7semidirect
157. #107C2×D7semidirect
159. #106C2×D7semidirect
278. C7#3094semidirect

Maximal subgroups

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

subgroupindex
D7×#1172
C2×#64582
C7×#4282
#899032
#898982
#897732
#897362
#896572
#896542
#896382
#896352
#30947