Finite Group Databasebeta

← #11738751

Subgroups — #11738751

Order 896.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90289C211yesmaximal, characteristic, complemented
2#90198C211yesmaximal, characteristic, complemented
3C7×#480C211yesmaximal, characteristic, complemented
4#89810C211yesmaximal, characteristic, complemented
5C2×#6384C211yesmaximal, characteristic, complemented
6#89661C211yesmaximal, characteristic, complemented
7#89658C211yesmaximal, characteristic, complemented
8#89656C211yesmaximal, characteristic, complemented
9#89647C211yesmaximal, characteristic, complemented
10#89559C211yesmaximal, characteristic, complemented
11#89553C211yesmaximal, characteristic, complemented
12#89543C211yesmaximal, characteristic, complemented
13#89537C211yesmaximal, characteristic, complemented
14#89510C211yesmaximal, characteristic, complemented
15#89502C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic
17C2×#1157—21no—
18C22×Dic14C2211yescharacteristic
19C7×#121C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21#6502—21no—
22#6499C2211yescharacteristic
23D4×Dic7—21no—
24C2×#1141C2211yescharacteristic
25#6493C2211yescharacteristic
26#6492—21no—
27C4×C22⋊D7C2211yescharacteristic
28C22×C7⋊C8C2211yescharacteristic
29C2×#1132C2211yescharacteristic
30C2×#1132C2211yescharacteristic
31C2×#1131C2211yescharacteristic
32#6460C2211yescharacteristic
33#6459—21no—
34#6458C2211yescharacteristic
35#6457—21no—
36#6456C2211yescharacteristic
37D7×C4⋊C4C2211yescharacteristic
38#6451C2211yescharacteristic
39#6450C2211yescharacteristic
40#6448—21no—
41#6441—21no—
42C7×#107C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46C7×C22⋊C8C2211yescharacteristic
47#6406C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6390C2211yescharacteristic
50#6390C2211yescharacteristic
51#6389C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6383C2211yescharacteristic
58#6381C2211yescharacteristic
59#3308—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7D411yescharacteristic
67C22×Dic7—21no—
68C22×Dic7—21no—
69#1159—22no—
70#1159—41no—
71#1157—44no—
72C2×D28—21no—
73C2×D28C2311yescharacteristic
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7C2311yescharacteristic
77C2×Dic14—22no—
78C2×Dic14—21no—
79C2×Dic14—21no—
80C2×Dic14—41no—
81C2×Dic14C2311yescharacteristic
82C2×Dic14C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C2×C7×C8C2311yescharacteristic
85C7×C4⋊C4—21no—
86C7×C4⋊C4C2311yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C7×C22⋊C4—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92C22⋊Dic7D411yescharacteristic
93#1141—41no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4D411yescharacteristic
99C4⋊Dic7—21no—
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4D411yescharacteristic
105C4×Dic7—21no—
106C4×Dic7C2311yescharacteristic
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1132—22no—
111#1132—41no—
112#1131—41no—
113#548—71no—
114#547—71no—
115#489—71no—
116#484—71no—
117#480D711yescharacteristic, complemented
118#477—71no—
119#468—71no—
120#466—71no—
121#463—71no—
122#449—71no—
123#448—71no—
124#440—71no—
125#435—71no—
126#417—71no—
127C2×#103—71no—
128C23×C7—21no—
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C22×D7C2×D411yescharacteristic
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—42no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—42no—
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2411yescharacteristic
143C22⋊D7—22no—
144C22⋊D7—42no—
145C22⋊D7—42no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154D28—21no—
155D28—21no—
156D28—42no—
157C4×D7—21no—
158C4×D7—21no—
159C4×D7—42no—
160C4×D7—42no—
161Dic14—21no—
162Dic14—21no—
163Dic14—21no—
164Dic14—42no—
165Dic14—42no—
166Dic14#4112yescomplemented
167C7×C8—21no—
168C7×C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—71no—
172C2×#41—141no—
173C22×Q8—71no—
174C2×SD16—71no—
175C2×SD16—71no—
176C2×D8—71no—
177C2×M16—71no—
178C22×C8—71no—
179C4⋊Q8—71no—
180#124—71no—
181#124—141no—
182#123—141no—
183#123—141no—
184C22⋊Q8—71no—
185C22⋊Q8—141no—
186C22⋊Q8—141no—
187#121—71no—
188#121—71no—
189#121—141no—
190#121C2×D711yescharacteristic
191C4×Q8—71no—
192C4×D4—71no—
193C4×D4—71no—
194C4×D4—141no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4C2×D711yescharacteristic
197#107—71no—
198#107C2×D711yescharacteristic
199#107C2×D711yescharacteristic
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#103—71no—
205#103—71no—
206#103—71no—
207#102—71no—
208#102—71no—
209#102C2×D711yescharacteristic
210#102C2×D711yescharacteristic
211C22⋊C8—71no—
212C22⋊C8C2×D711yescharacteristic
213#97—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7AGL(1,ℤ/8)12yes—
220C22×C7C22×D411yescharacteristic
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—21no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C4×C7—21no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7C2×#4111yescharacteristic
232C4×C7H(5,𝔽2)11yescharacteristic
233Dic7—22no—
234Dic7—21no—
235Dic7—21no—
236Dic7—21no—
237Dic7—41no—
238Dic7—41no—
239#41—142no—
240#41—284no—
241#41—281no—
242C2×Q8—71no—
243C2×Q8—71no—
244C2×Q8—141no—
245C2×Q8—142no—
246C2×Q8—141no—
247C2×Q8—281no—
248C2×D4—21no—
249C2×D4—71no—
250C2×D4—71no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4C22×D711yescharacteristic
255C22×C4—21no—
256C22×C4—71no—
257C22×C4—141no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4C22×D711yescharacteristic
262SD16—142no—
263SD16—281no—
264D8—281no—
265M16—281no—
266C2×C8—71no—
267C2×C8—71no—
268C2×C8—281no—
269C2×C8C22×D711yescharacteristic
270C2×C8C22×D711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—71no—
273C4⋊C4—71no—
274C4⋊C4—141no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4C22×D711yescharacteristic
279C4⋊C4C22×D711yescharacteristic
280C4⋊C4C22×D711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—71no—
283C22⋊C4—71no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C42—71no—
291C42—141no—
292C2×C7—22no—
293C2×C7—81no—
294C2×C7#54611yescharacteristic
295C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
296C2×C7#57811yescharacteristic
297D7—41no—
298D7—81no—
299C23—21no—
300C23—71no—
301C23—141no—
302C23D4×D711yescharacteristic
303Q8—72no—
304Q8—141no—
305Q8—141no—
306Q8—141no—
307Q8—282no—
308Q8—282no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—142no—
313D4—141no—
314D4—141no—
315D4—282no—
316D4—282no—
317D4—282no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—71no—
325C2×C4—71no—
326C2×C4—71no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—282no—
333C2×C4—282no—
334C2×C4—282no—
335C2×C4—282no—
336C2×C4D4×D711yescharacteristic
337C2×C4C23×D711yescharacteristic
338C8—21no—
339C8—21no—
340C8—141no—
341C8—141no—
342C7#330811yescharacteristic, complemented, Sylow
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—141no—
349C22—141no—
350C22—141no—
351C22—281no—
352C22—281no—
353C22—281no—
354C22#647412yes—
355C22C2×D4×D711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—141no—
361C4—141no—
362C4—141no—
363C4—142no—
364C4—281no—
365C4—281no—
366C4D7×#4111yescharacteristic
367C4#655311yescharacteristic
368C2—22no—
369C2—81no—
370C2—281no—
371C2—561no—
372C2#9034511yescharacteristic
373C2C2×#647411yescharacteristic
374C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#480C2
socleC22×C7C22×D4
hypercenter#480D7
hypocenterC7#3308
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. #90289C2semidirect
2. #90198C2semidirect
3. C7×#480C2semidirect
4. #89810C2semidirect
5. C2×#6384C2semidirect
6. #89661C2semidirect
7. #89658C2semidirect
8. #89656C2semidirect
9. #89647C2semidirect
10. #89559C2semidirect
11. #89553C2semidirect
12. #89543C2semidirect
13. #89537C2semidirect
14. #89510C2semidirect
15. #89502C2semidirect
117. #480D7semidirect
166. Dic14#41semidirect
342. C7#3308semidirect

Maximal subgroups

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

subgroupindex
#902892
#901982
C7×#4802
#898102
C2×#63842
#896612
#896582
#896562
#896472
#895592
#895532
#895432
#895372
#895102
#895022
#33087