Finite Group Databasebeta

← #11738611

Subgroups — #11738611

Order 896.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90379C211yesmaximal, characteristic, complemented
2C4×D4×D7C211yesmaximal, characteristic
3C7×#478C211yesmaximal, characteristic, complemented
4#89838C211yesmaximal, characteristic, complemented
5#89781C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89653C211yesmaximal, characteristic, complemented
8#89642C211yesmaximal, characteristic, complemented
9#89633C211yesmaximal, characteristic, complemented
10#89619C211yesmaximal, characteristic, complemented
11#89609C211yesmaximal, characteristic, complemented
12#89542C211yesmaximal, characteristic
13#89542C211yesmaximal, characteristic
14#89533C211yesmaximal, characteristic, complemented
15#89531C211yesmaximal, characteristic, complemented
16C2×D4×D7C2211yescharacteristic
17C22×C4×D7—21no—
18C7×#125C2211yescharacteristic, complemented
19C4×C7×D4C2211yescharacteristic
20D4×Dic7C2211yescharacteristic
21C4×C22⋊D7—21no—
22C2×#1131C2211yescharacteristic
23C2×#1131C2211yescharacteristic
24#6459—21no—
25#6459—21no—
26#6456C2211yescharacteristic
27#6455C2211yescharacteristic
28#6455C2211yescharacteristic
29D7×C4⋊C4C2211yescharacteristic
30#6451C2211yescharacteristic, complemented
31#6451C2211yescharacteristic, complemented
32#6450—21no—
33#6450—21no—
34(C22⋊D7)⋊C4—21no—
35D7×C22⋊C4—21no—
36C4×D28C2211yescharacteristic
37C42×D7C2211yescharacteristic
38#6433C2211yescharacteristic, complemented
39C7×#107C2211yescharacteristic, complemented
40C7×#106C2211yescharacteristic, complemented
41C7×C4⋊C8C2211yescharacteristic, complemented
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6392C2211yescharacteristic, complemented
49#6391C2211yescharacteristic, complemented
50Dic7⋊C8C2211yescharacteristic, complemented
51Dic7⋊C8C2211yescharacteristic, complemented
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic, complemented
55#6382C2211yescharacteristic, complemented
56#6381C2211yescharacteristic, complemented
57#6381C2211yescharacteristic, complemented
58C4⋊(C7⋊C8)C2211yescharacteristic, complemented
59#3077—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C22×Dic7—21no—
65D4×D7—22no—
66D4×D7Q814yescomplemented
67C2×D28C2311yescharacteristic
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7—44no—
72C2×C4×D7D411yescharacteristic
73C2×C4×D7D411yescharacteristic
74C2×C4×D7C2311yescharacteristic
75C2×Dic14—21no—
76C2×Dic14—21no—
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C42×C7C2311yescharacteristic
86C22⋊Dic7—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7D411yescharacteristic
102C4×Dic7D411yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#1131—22no—
107#1131—22no—
108#533—71no—
109C2×C4×D4—71no—
110#478—71no—
111#478—71no—
112#478—71no—
113#478D711yescharacteristic, complemented
114#476—71no—
115#476—71no—
116#474—71no—
117#474—71no—
118#428—71no—
119#428—71no—
120#423—71no—
121#418—71no—
122#418—71no—
123C23×C7—21no—
124C22×D7—21no—
125C22×D7—24no—
126C22×D7—42no—
127C22×D7—42no—
128C22×D7C2×D411yescharacteristic
129C7×D4—21no—
130C7×D4C2×Q812yes—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—42no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C22⋊D7—24no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—42no—
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146D28—21no—
147D28C2×Q812yes—
148C4×D7—22no—
149C4×D7—41no—
150C4×D7—42no—
151C4×D7—42no—
152C4×D7—42no—
153C4×D7C2×Q812yes—
154C4×D7#4112yes—
155Dic14—42no—
156Dic14—42no—
157C7×C8—21no—
158C7×C8—21no—
159C7⋊C8—21no—
160C7⋊C8—21no—
161C22×D4—71no—
162C23×C4—141no—
163C2×M16—71no—
164C2×M16—71no—
165C4⋊Q8—71no—
166C4⋊Q8—71no—
167#125—71no—
168#125C2×D711yescharacteristic, complemented
169C22⋊Q8—141no—
170C22⋊Q8—141no—
171C4×Q8—141no—
172C4×Q8—141no—
173C4×D4—71no—
174C4×D4—71no—
175C4×D4—71no—
176C4×D4—141no—
177C4×D4—141no—
178C4×D4C2×D711yescharacteristic
179#117—71no—
180#117—71no—
181C2×C4⋊C4—71no—
182C2×C22⋊C4—141no—
183C2×C42—71no—
184#107—71no—
185#107—71no—
186#107—71no—
187#107C2×D711yescharacteristic, complemented
188#106—71no—
189#106—71no—
190#106—71no—
191#106C2×D711yescharacteristic, complemented
192C4⋊C8—71no—
193C4⋊C8—71no—
194C4⋊C8—71no—
195C4⋊C8C2×D711yescharacteristic, complemented
196#102—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102C2×D711yescharacteristic
203#102C2×D711yescharacteristic
204C22×C7—22no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7C22×D411yescharacteristic
208C2×D7—21no—
209C2×D7—21no—
210C2×D7—22no—
211C2×D7—48no—
212C2×D7—41no—
213C2×D7—41no—
214C2×D7C22⋊Q812yes—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7C22×Q811yescharacteristic
221C4×C7C2×#4111yescharacteristic
222Dic7—21no—
223Dic7—41no—
224Dic7—41no—
225Dic7—41no—
226Dic7C22⋊Q812yes—
227Dic7AGL(1,ℤ/8)12yes—
228C24—141no—
229C2×Q8—141no—
230C2×Q8—141no—
231C2×D4—74no—
232C2×D4—71no—
233C2×D4—142no—
234C2×D4—141no—
235C2×D4C22×D711yescharacteristic
236C22×C4—21no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—141no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4—141no—
244C22×C4—284no—
245M16—142no—
246M16—142no—
247C2×C8—71no—
248C2×C8—71no—
249C2×C8C22×D711yescharacteristic
250C2×C8C22×D711yescharacteristic
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4C22×D711yescharacteristic
262C4⋊C4C22×D711yescharacteristic
263C4⋊C4C22×D711yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C42—71no—
271C42—71no—
272C42—71no—
273C42—141no—
274C42—141no—
275C42C22×D711yescharacteristic
276C2×C7—42no—
277C2×C7C2×C22⋊Q811yescharacteristic
278C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
279C2×C7#57511yescharacteristic
280D7—22no—
281D7—42no—
282C23—21no—
283C23—71no—
284C23—144no—
285C23—141no—
286C23—282no—
287C23—282no—
288Q8—282no—
289Q8—282no—
290D4—21no—
291D4—72no—
292D4—144no—
293D4—141no—
294D4Q8×D712yes—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—71no—
300C2×C4—71no—
301C2×C4—72no—
302C2×C4—72no—
303C2×C4—71no—
304C2×C4—142no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—281no—
309C2×C4—282no—
310C2×C4—282no—
311C2×C4—282no—
312C2×C4—282no—
313C2×C4D4×D711yescharacteristic
314C2×C4D4×D711yescharacteristic
315C2×C4C23×D711yescharacteristic
316C8—21no—
317C8—21no—
318C8—141no—
319C8—141no—
320C7#307711yescharacteristic, complemented, Sylow
321C22—22no—
322C22—41no—
323C22—41no—
324C22—72no—
325C22—141no—
326C22—141no—
327C22—142no—
328C22—288no—
329C22—281no—
330C22—281no—
331C22C2×D4×D711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—72no—
338C4—72no—
339C4—141no—
340C4—281no—
341C4—281no—
342C4—281no—
343C4C2×Q8×D711yescharacteristic
344C4D7×#4111yescharacteristic
345C2—42no—
346C2—142no—
347C2—282no—
348C2D7×C22⋊Q811yescharacteristic
349C2#9045911yescharacteristic
350C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#478C2
socleC22×C7C22×D4
prime radicalC2×D4×D7C22
hypercenter#478D7
hypocenterC7#3077
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #90379C2semidirect
3. C7×#478C2semidirect
4. #89838C2semidirect
5. #89781C2semidirect
6. #89662C2semidirect
7. #89653C2semidirect
8. #89642C2semidirect
9. #89633C2semidirect
10. #89619C2semidirect
11. #89609C2semidirect
14. #89533C2semidirect
15. #89531C2semidirect
18. C7×#125C22semidirect
30. #6451C22semidirect
31. #6451C22semidirect
38. #6433C22semidirect
39. C7×#107C22semidirect
40. C7×#106C22semidirect
41. C7×C4⋊C8C22semidirect
48. #6392C22semidirect
49. #6391C22semidirect
50. Dic7⋊C8C22semidirect
51. Dic7⋊C8C22semidirect
54. #6382C22semidirect
55. #6382C22semidirect
56. #6381C22semidirect
57. #6381C22semidirect
58. C4⋊(C7⋊C8)C22semidirect
66. D4×D7Q8semidirect
113. #478D7semidirect
168. #125C2×D7semidirect
187. #107C2×D7semidirect
191. #106C2×D7semidirect
195. C4⋊C8C2×D7semidirect
320. C7#3077semidirect

Maximal subgroups

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

subgroupindex
#903792
C4×D4×D72
C7×#4782
#898382
#897812
#896622
#896532
#896422
#896332
#896192
#896092
#895422
#895332
#895312
#30777