Finite Group Databasebeta

← #11738265

Subgroups — #11738265

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#90324C211yesmaximal, characteristic, complemented
2#90302C211yesmaximal, characteristic, complemented
3C7×#468C211yesmaximal, characteristic, complemented
4#89942C211yesmaximal, characteristic, complemented
5#89937C211yesmaximal, characteristic, complemented
6#89905C211yesmaximal, characteristic, complemented
7#89899C211yesmaximal, characteristic, complemented
8#89816C211yesmaximal, characteristic, complemented
9#89809C211yesmaximal, characteristic, complemented
10#89654C211yesmaximal, characteristic, complemented
11#89647C211yesmaximal, characteristic, complemented
12D7×#103C211yesmaximal, characteristic, complemented
13#89567C211yesmaximal, characteristic, complemented
14#89547C211yesmaximal, characteristic, complemented
15#89529C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×#1159—21no—
18C2×#1157C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C2×C7×M16C2211yescharacteristic
21C7×C22⋊Q8C2211yescharacteristic
22C7×#121C2211yescharacteristic
23#6510C2211yescharacteristic
24#6501C2211yescharacteristic
25#6498—21no—
26D4×Dic7C2211yescharacteristic
27C2×#1142C2211yescharacteristic
28C2×#1141C2211yescharacteristic
29C4×C22⋊D7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×C8×D7C2211yescharacteristic
32#6459—21no—
33#6458C2211yescharacteristic
34#6456—21no—
35D7×C4⋊C4C2211yescharacteristic
36#6451C2211yescharacteristic
37#6450C2211yescharacteristic
38#6448—21no—
39#6447—21no—
40#6446—21no—
41#6441—21no—
42C7×#107C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#102C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47C22⋊(C7⋊C8)C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3308—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7D411yescharacteristic
67C22×Dic7—21no—
68C22×Dic7—21no—
69Q8×D7—22no—
70Q8×D7—41no—
71#1159—44no—
72#1157—22no—
73#1157—41no—
74C2×D28—21no—
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C4×D7C2311yescharacteristic
80C2×Dic14—21no—
81C2×Dic14—21no—
82C2×Dic14C2311yescharacteristic
83C7×SD16—41no—
84C7×M16—41no—
85C2×C7×C8C2311yescharacteristic
86C2×C7×C8C2311yescharacteristic
87C7×C4⋊C4—21no—
88C7×C4⋊C4C2311yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C22⋊C4—21no—
91C7×C22⋊C4—21no—
92C22⋊Dic7—21no—
93C22⋊Dic7—21no—
94C22⋊Dic7D411yescharacteristic
95#1142—22no—
96#1141—41no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4D411yescharacteristic
101C4⋊Dic7—21no—
102C4⋊Dic7C2311yescharacteristic
103C4⋊Dic7C2311yescharacteristic
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107Dic7⋊C4D411yescharacteristic
108C4×Dic7—21no—
109C4×Dic7C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112C8×D7—41no—
113#548—71no—
114#547—71no—
115#489—71no—
116#484—71no—
117#480—71no—
118#477—71no—
119#468D711yescharacteristic, complemented
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×Q8—21no—
133C7×Q8—21no—
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—42no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2411yescharacteristic
144C22⋊D7—22no—
145C22⋊D7—42no—
146C22⋊D7—42no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—42no—
151C2×Dic7—42no—
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155D28—21no—
156D28—21no—
157D28—42no—
158C4×D7—21no—
159C4×D7—21no—
160C4×D7—42no—
161C4×D7—42no—
162C4×D7—42no—
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×SD16C2×D711yescharacteristic
176C2×D8—71no—
177C2×M16C2×D711yescharacteristic
178C22×C8—71no—
179C4⋊Q8—71no—
180#124—71no—
181#124—141no—
182#123—141no—
183#123—141no—
184C22⋊Q8—141no—
185C22⋊Q8—141no—
186C22⋊Q8C2×D711yescharacteristic
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⋊C4—71no—
197#107—71no—
198#107—71no—
199#107C2×D711yescharacteristic
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#103—71no—
205#103—71no—
206#103C2×D711yescharacteristic
207#102—71no—
208#102—71no—
209#102—71no—
210#102C2×D711yescharacteristic
211C22⋊C8—71no—
212C22⋊C8—71no—
213#97—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7C22×D411yescharacteristic
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7AGL(1,ℤ/8)12yes—
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—281no—
241#41—284no—
242C2×Q8—71no—
243C2×Q8—141no—
244C2×Q8—141no—
245C2×Q8—142no—
246C2×Q8—281no—
247C2×Q8C22×D711yescharacteristic
248C2×D4—21no—
249C2×D4—71no—
250C2×D4—71no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4C22×D711yescharacteristic
255C22×C4—71no—
256C22×C4—141no—
257C22×C4—141no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4C22×D711yescharacteristic
262SD16—41no—
263SD16—142no—
264D8—281no—
265M16—41no—
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—71no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4C22×D711yescharacteristic
280C4⋊C4C22×D711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—71no—
284C22⋊C4—71no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C42—71no—
291C42—141no—
292C2×C7—41no—
293C2×C7—81no—
294C2×C7#54611yescharacteristic
295C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
296C2×C7#57811yescharacteristic
297D7—22no—
298D7—81no—
299C23—21no—
300C23—71no—
301C23—141no—
302C23D4×D711yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—72no—
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—71no—
324C2×C4—71no—
325C2×C4—71no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—282no—
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—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—72no—
350C22—141no—
351C22—141no—
352C22—281no—
353C22—281no—
354C22—281no—
355C22C2×D4×D711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—141no—
361C4—141no—
362C4—141no—
363C4—142no—
364C4—281no—
365C4—281no—
366C4#654811yescharacteristic
367C4D7×#4111yescharacteristic
368C2—41no—
369C2—81no—
370C2—142no—
371C2—561no—
372C2#9030111yescharacteristic
373C2D7×AGL(1,ℤ/8)11yescharacteristic
374C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#468C2
socleC22×C7C22×D4
hypercenter#468D7
hypocenterC7#3308
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #90324C2semidirect
2. #90302C2semidirect
3. C7×#468C2semidirect
4. #89942C2semidirect
5. #89937C2semidirect
6. #89905C2semidirect
7. #89899C2semidirect
8. #89816C2semidirect
9. #89809C2semidirect
10. #89654C2semidirect
11. #89647C2semidirect
12. D7×#103C2semidirect
13. #89567C2semidirect
14. #89547C2semidirect
15. #89529C2semidirect
119. #468D7semidirect
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
#903242
#903022
C7×#4682
#899422
#899372
#899052
#898992
#898162
#898092
#896542
#896472
D7×#1032
#895672
#895472
#895292
#33087