Finite Group Databasebeta

← #11738744

Subgroups — #11738744

Order 896.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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×#121C211yesmaximal, characteristic, complemented
2C2×#6458C211yesmaximal, characteristic, complemented
3C7×#480C211yesmaximal, characteristic, complemented
4#89808C211yesmaximal, characteristic, complemented
5#89739C211yesmaximal, characteristic, complemented
6#89655C212yesmaximal, complemented
7#89654C212yesmaximal, complemented
8#89547C212yesmaximal, complemented
9#89545C212yesmaximal, complemented
10#89504C211yesmaximal, characteristic, complemented
11D7×C22⋊C8C211yesmaximal, characteristic, complemented
12C2×D4×D7—21no—
13C2×D4×D7C2211yescharacteristic, complemented
14C22×D28—21no—
15C22×D28C2211yescharacteristic, complemented
16C22×C4×D7C2211yescharacteristic
17C7×#121C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic, complemented
19#6502—21no—
20#6501C2211yescharacteristic, complemented
21C2×#1141C2212yescomplemented
22#6493C2211yescharacteristic
23C2×(C2×D7)⋊C4—21no—
24C2×D56C2212yescomplemented
25C2×C8×D7C2212yescomplemented
26#6458—21no—
27#6458—21no—
28#6458C2212yes—
29#6458C2211yescharacteristic, complemented
30#6458C2212yescomplemented
31D7×C4⋊C4C2211yescharacteristic, complemented
32#6447—21no—
33D7×C22⋊C4—21no—
34C7×#107C2212yescomplemented
35C7×#102C2212yescomplemented
36C7×C22⋊C8C2211yescharacteristic, complemented
37C22⋊(C7⋊C8)C2211yescharacteristic, complemented
38#6395C2212yescomplemented
39(C2×D7)⋊C8C2212yescomplemented
40#6383C2212yescomplemented
41#6383C2212yescomplemented
42#6381C2212yescomplemented
43#3186—71nomaximal, Sylow
44C23×D7—21no—
45C23×D7—21no—
46C23×D7D411yescharacteristic
47C2×C7×D4—21no—
48C2×C7×D4C2311yescharacteristic
49C22×C4×C7—21no—
50C22×C4×C7C2311yescharacteristic
51C2×C22⋊D7—21no—
52C2×C22⋊D7—21no—
53C2×C22⋊D7—21no—
54C22×Dic7D411yescharacteristic
55D4×D7—42no—
56D4×D7—44no—
57C2×D28—21no—
58C2×D28—22no—
59C2×D28—22no—
60C2×D28—21no—
61C2×D28—42no—
62C2×D28—44no—
63C2×D28C2312yes—
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×C4×D7—42no—
67C2×C4×D7D411yescharacteristic
68C2×C4×D7D411yescharacteristic
69C2×C4×D7C2311yescharacteristic
70C2×C4×D7C2311yescharacteristic
71C2×C7×C8C2312yes—
72C7×C4⋊C4—21no—
73C7×C4⋊C4C2312yes—
74C7×C4⋊C4C2311yescharacteristic
75C7×C22⋊C4—21no—
76C22⋊Dic7—21no—
77#1141—42no—
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—22no—
82C4⋊Dic7C2311yescharacteristic
83Dic7⋊C4—21no—
84C2×C7⋊C8C2312yes—
85D56—42no—
86C8×D7—42no—
87C2×#121—71no—
88C2×#121—71no—
89#480—71no—
90#480—72no—
91#480D711yescharacteristic, complemented
92#466—72no—
93#466—72no—
94#447—71no—
95#447—72no—
96#447—71no—
97C2×C22⋊C8—71no—
98C23×C7—21no—
99C23×C7C2×D411yescharacteristic
100C22×D7—22no—
101C22×D7—21no—
102C22×D7—41no—
103C22×D7—41no—
104C22×D7—42no—
105C22×D7—42no—
106C22×D7—42no—
107C22×D7—42no—
108C22×D7—42no—
109C22×D7—42no—
110C22×D7D814yescomplemented
111C22×D7C2×D411yescharacteristic
112C22×D7C2×D411yescharacteristic
113C7×D4—22no—
114C7×D4—42no—
115C2×C4×C7—22no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—42no—
119C2×C4×C7C2×D411yescharacteristic
120C2×C4×C7C2411yescharacteristic
121C22⋊D7—42no—
122C22⋊D7—42no—
123C22⋊D7—42no—
124C2×Dic7—21no—
125C2×Dic7—41no—
126C2×Dic7C2×D411yescharacteristic
127C2×Dic7C2×D411yescharacteristic
128D28—22no—
129D28—22no—
130D28—22no—
131D28—42no—
132D28—44no—
133D28—44no—
134D28—42no—
135C4×D7—22no—
136C4×D7—22no—
137C4×D7—41no—
138C4×D7—41no—
139C4×D7—42no—
140C7×C8—22no—
141C7⋊C8—22no—
142C22×D4—71no—
143C22×D4—71no—
144C22×D4—141no—
145C22×D4—141no—
146C23×C4—71no—
147C2×D8—72no—
148C2×D8—72no—
149C22×C8—72no—
150#121—71no—
151#121—71no—
152#121—72no—
153#121—71no—
154#121—72no—
155#121—141no—
156#121—141no—
157#121—141no—
158#121—141no—
159#121C2×D711yescharacteristic
160C2×C4⋊C4—71no—
161C2×C4⋊C4C2×D711yescharacteristic, complemented
162C2×C22⋊C4—141no—
163C2×C22⋊C4—141no—
164#107—72no—
165#107C2×D712yescomplemented
166#102—72no—
167#102—72no—
168#102—72no—
169#102C2×D712yescomplemented
170C22⋊C8—71no—
171C22⋊C8—72no—
172C22⋊C8C2×D711yescharacteristic, complemented
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—41no—
176C22×C7—41no—
177C22×C7—41no—
178C22×C7C2×D812yes—
179C22×C7C22×D411yescharacteristic
180C2×D7—21no—
181C2×D7—21no—
182C2×D7—21no—
183C2×D7—21no—
184C2×D7—44no—
185C2×D7—42no—
186C2×D7—41no—
187C2×D7—42no—
188C2×D7—41no—
189C2×D7—42no—
190C2×D7—41no—
191C2×D7—84no—
192C2×D7—84no—
193C2×D7C2×D812yes—
194C2×D7C2×D812yes—
195C4×C7—21no—
196C4×C7—41no—
197C4×C7—42no—
198C4×C7H(5,𝔽2)12yes—
199Dic7—21no—
200Dic7—21no—
201Dic7—41no—
202C24—71no—
203C24—141no—
204C24—141no—
205C2×D4—21no—
206C2×D4—72no—
207C2×D4—71no—
208C2×D4—141no—
209C2×D4—142no—
210C2×D4—142no—
211C2×D4—141no—
212C2×D4—141no—
213C2×D4—141no—
214C2×D4—141no—
215C2×D4—282no—
216C2×D4—284no—
217C2×D4—282no—
218C2×D4—284no—
219C2×D4C22×D711yescharacteristic
220C22×C4—21no—
221C22×C4—71no—
222C22×C4—71no—
223C22×C4—71no—
224C22×C4—71no—
225C22×C4—71no—
226C22×C4—141no—
227C22×C4—282no—
228C22×C4C22×D711yescharacteristic
229D8—282no—
230D8—282no—
231C2×C8—72no—
232C2×C8—282no—
233C2×C8C22×D712yes—
234C4⋊C4—21no—
235C4⋊C4—71no—
236C4⋊C4—141no—
237C4⋊C4C22×D711yescharacteristic
238C4⋊C4C22×D712yes—
239C22⋊C4—21no—
240C22⋊C4—141no—
241C22⋊C4—142no—
242C22⋊C4—142no—
243C22⋊C4—141no—
244C22⋊C4—141no—
245C2×C7—22no—
246C2×C7—81no—
247C2×C7#53411yescharacteristic
248C2×C7C22×D811yescharacteristic
249C2×C7#57511yescharacteristic
250D7—22no—
251D7—22no—
252D7—82no—
253D7—81no—
254C23—21no—
255C23—71no—
256C23—71no—
257C23—74no—
258C23—142no—
259C23—141no—
260C23—281no—
261C23—281no—
262C23—282no—
263C23—282no—
264C23—282no—
265C23—282no—
266C23—282no—
267C23—282no—
268C23D4×D711yescharacteristic
269D4—22no—
270D4—42no—
271D4—142no—
272D4—142no—
273D4—142no—
274D4—282no—
275D4—284no—
276D4—284no—
277D4—282no—
278D4—282no—
279D4—282no—
280D4—282no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—71no—
286C2×C4—71no—
287C2×C4—142no—
288C2×C4—142no—
289C2×C4—141no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—282no—
293C2×C4—281no—
294C2×C4D4×D711yescharacteristic
295C2×C4C23×D711yescharacteristic
296C8—22no—
297C8—142no—
298C7#318611yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—72no—
305C22—72no—
306C22—141no—
307C22—141no—
308C22—141no—
309C22—141no—
310C22—284no—
311C22—282no—
312C22—281no—
313C22—282no—
314C22—281no—
315C22—282no—
316C22—281no—
317C22—564no—
318C22—564no—
319C22D7×D812yes—
320C22C2×D4×D711yescharacteristic
321C4—21no—
322C4—42no—
323C4—41no—
324C4—141no—
325C4—141no—
326C4—281no—
327C4#655312yes—
328C2—22no—
329C2—81no—
330C2—142no—
331C2—142no—
332C2—562no—
333C2—561no—
334C2#9034411yescharacteristic
335C2C2×D7×D811yescharacteristic
336C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#480C2
socleC22×C7C22×D4
hypercenter#480D7
hypocenterC7#3186
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. D7×#121C2semidirect
2. C2×#6458C2semidirect
3. C7×#480C2semidirect
4. #89808C2semidirect
5. #89739C2semidirect
6. #89655C2semidirect
7. #89654C2semidirect
8. #89547C2semidirect
9. #89545C2semidirect
10. #89504C2semidirect
11. D7×C22⋊C8C2semidirect
13. C2×D4×D7C22semidirect
15. C22×D28C22semidirect
18. C2×C7×C4⋊C4C22semidirect
20. #6501C22semidirect
21. C2×#1141C22semidirect
24. C2×D56C22semidirect
25. C2×C8×D7C22semidirect
29. #6458C22semidirect
30. #6458C22semidirect
31. D7×C4⋊C4C22semidirect
34. C7×#107C22semidirect
35. C7×#102C22semidirect
36. C7×C22⋊C8C22semidirect
37. C22⋊(C7⋊C8)C22semidirect
38. #6395C22semidirect
39. (C2×D7)⋊C8C22semidirect
40. #6383C22semidirect
41. #6383C22semidirect
42. #6381C22semidirect
91. #480D7semidirect
110. C22×D7D8semidirect
161. C2×C4⋊C4C2×D7semidirect
165. #107C2×D7semidirect
169. #102C2×D7semidirect
172. C22⋊C8C2×D7semidirect
298. C7#3186semidirect

Maximal subgroups

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

subgroupindex
D7×#1212
C2×#64582
C7×#4802
#898082
#897392
#896552
#896542
#895472
#895452
#895042
D7×C22⋊C82
#31867