Finite Group Databasebeta

← #11738182

Subgroups — #11738182

Order 896.

408 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90307C211yesmaximal, characteristic, complemented
2#90290C211yesmaximal, characteristic, complemented
3C7×#466C211yesmaximal, characteristic, complemented
4#89930C211yesmaximal, characteristic, complemented
5#89922C211yesmaximal, characteristic, complemented
6#89885C211yesmaximal, characteristic, complemented
7C8×C22⋊D7C211yesmaximal, characteristic, complemented
8#89809C211yesmaximal, characteristic, complemented
9#89807C211yesmaximal, characteristic, complemented
10#89657C211yesmaximal, characteristic, complemented
11#89647C211yesmaximal, characteristic, complemented
12#89547C211yesmaximal, characteristic, complemented
13#89544C211yesmaximal, characteristic, complemented
14#89532C211yesmaximal, characteristic, complemented
15#89530C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C2×C7×D8C2211yescharacteristic
19C22×C7×C8C2211yescharacteristic
20C7×#121C2211yescharacteristic
21C7×#121C2211yescharacteristic
22#6503—21no—
23#6503C2211yescharacteristic
24#6502—21no—
25#6502—21no—
26#6501C2211yescharacteristic
27#6499—21no—
28D4×Dic7C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31C4×C22⋊D7C2211yescharacteristic
32#6489C2211yescharacteristic
33C2×C8×D7C2211yescharacteristic
34#6458C2211yescharacteristic
35#6455C2211yescharacteristic
36#6451C2211yescharacteristic
37#6450C2211yescharacteristic
38#6448—21no—
39#6447—21no—
40(C22⋊D7)⋊C4—21no—
41#6440—21no—
42C7×#107C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×#102C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47C22⋊(C7⋊C8)C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54C8×Dic7C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3305—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7D411yescharacteristic
71C22×Dic7—21no—
72C22×Dic7—21no—
73#1159—22no—
74#1159—41no—
75#1157—22no—
76#1157—41no—
77C2×D28—21no—
78C2×D28C2311yescharacteristic
79C2×C4×D7—21no—
80C2×C4×D7C2311yescharacteristic
81C2×Dic14—21no—
82C2×Dic14C2311yescharacteristic
83C7×D8—41no—
84C2×C7×C8—41no—
85C2×C7×C8C2311yescharacteristic
86C2×C7×C8C2311yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C7×C22⋊C4—21no—
90C7×C22⋊C4—21no—
91C22⋊Dic7—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)⋊C4D411yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4D411yescharacteristic
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C8×D7—41no—
112#555—71no—
113#550—71no—
114#486—71no—
115#482—71no—
116#482—71no—
117#477—71no—
118#466—71no—
119#466D711yescharacteristic, complemented
120#459—71no—
121#449—71no—
122#449—71no—
123C4×SD16—71no—
124C8×D4—71no—
125#416—71no—
126#416—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C22×D7C2×D411yescharacteristic
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—81no—
137C7×D4—81no—
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—
147C22⋊D7—42no—
148C22⋊D7—81no—
149C22⋊D7—81no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155C2×Dic7C2×D411yescharacteristic
156C2×Dic7C2×D411yescharacteristic
157C2×Dic7C2×D411yescharacteristic
158D28—21no—
159D28—21no—
160D28—81no—
161C4×D7—21no—
162C4×D7—21no—
163C4×D7—42no—
164Dic14—21no—
165Dic14—81no—
166Dic14#4112yes—
167C7×C8—21no—
168C7×C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—71no—
172C2×#41—71no—
173C2×SD16—71no—
174C2×D8—71no—
175C2×D8C2×D711yescharacteristic
176C22×C8—71no—
177C22×C8C2×D711yescharacteristic
178C4⋊Q8—71no—
179#127—71no—
180#127—141no—
181#124—141no—
182#124—141no—
183#121—71no—
184#121—71no—
185#121—141no—
186#121—141no—
187#121—141no—
188#121C2×D711yescharacteristic
189#121C2×D711yescharacteristic
190C4×Q8—71no—
191C4×D4—71no—
192C4×D4—71no—
193C4×D4—141no—
194#117—71no—
195#117—71no—
196#117—141no—
197#107—71no—
198#107C2×D711yescharacteristic
199#106—71no—
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208#102—71no—
209#102C2×D711yescharacteristic
210#102C2×D711yescharacteristic
211C22⋊C8—71no—
212C22⋊C8—71no—
213C4×C8—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7C22×D411yescharacteristic
224C2×D7—21no—
225C2×D7—21no—
226C2×D7—21no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C4×C7—21no—
231C4×C7—41no—
232C4×C7—41no—
233C4×C7C2×#4111yescharacteristic
234C4×C7H(5,𝔽2)11yescharacteristic
235Dic7—22no—
236Dic7—21no—
237Dic7—21no—
238Dic7—41no—
239Dic7—41no—
240Dic7#13512yes—
241#41—142no—
242#41—142no—
243#41—281no—
244#41—281no—
245C2×Q8—71no—
246C2×Q8—141no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—71no—
250C2×D4—71no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4—141no—
256C2×D4—141no—
257C2×D4C22×D711yescharacteristic
258C2×D4C22×D711yescharacteristic
259C22×C4—71no—
260C22×C4—141no—
261C22×C4—141no—
262C22×C4—141no—
263C22×C4C22×D711yescharacteristic
264SD16—142no—
265D8—41no—
266D8—281no—
267C2×C8—41no—
268C2×C8—71no—
269C2×C8—71no—
270C2×C8—281no—
271C2×C8C22×D711yescharacteristic
272C2×C8C22×D711yescharacteristic
273C4⋊C4—71no—
274C4⋊C4—71no—
275C4⋊C4—71no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4C22×D711yescharacteristic
279C4⋊C4C22×D711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—71no—
283C22⋊C4—71no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C42—71no—
290C42—141no—
291C42—141no—
292C42—141no—
293C2×C7—41no—
294C2×C7—81no—
295C2×C7—81no—
296C2×C7#54611yescharacteristic
297C2×C7C2×#13511yescharacteristic
298C2×C7#57611yescharacteristic
299D7—41no—
300D7—81no—
301C23—21no—
302C23—21no—
303C23—71no—
304C23—141no—
305C23D4×D711yescharacteristic
306Q8—72no—
307Q8—141no—
308Q8—561no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—81no—
314D4—81no—
315D4—142no—
316D4—141no—
317D4—141no—
318D4—282no—
319D4—282no—
320D4—282no—
321D4—561no—
322D4—561no—
323D4—561no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—71no—
329C2×C4—71no—
330C2×C4—71no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—282no—
337C2×C4—282no—
338C2×C4—282no—
339C2×C4D4×D711yescharacteristic
340C2×C4C23×D711yescharacteristic
341C8—21no—
342C8—21no—
343C8—141no—
344C8—141no—
345C7#330511yescharacteristic, complemented, Sylow
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—141no—
356C22—141no—
357C22—141no—
358C22—281no—
359C22—281no—
360C22—281no—
361C22C2×D4×D711yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—72no—
366C4—141no—
367C4—141no—
368C4—142no—
369C4—281no—
370C4—281no—
371C4#654811yescharacteristic
372C4D7×#4111yescharacteristic
373C2—41no—
374C2—81no—
375C2—81no—
376C2—281no—
377C2—561no—
378C2#9030111yescharacteristic
379C2#9044811yescharacteristic
380C2D7×#13511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#466C2
socleC22×C7C22×D4
hypercenter#466D7
hypocenterC7#3305
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #90307C2semidirect
2. #90290C2semidirect
3. C7×#466C2semidirect
4. #89930C2semidirect
5. #89922C2semidirect
6. #89885C2semidirect
7. C8×C22⋊D7C2semidirect
8. #89809C2semidirect
9. #89807C2semidirect
10. #89657C2semidirect
11. #89647C2semidirect
12. #89547C2semidirect
13. #89544C2semidirect
14. #89532C2semidirect
15. #89530C2semidirect
119. #466D7semidirect
345. C7#3305semidirect

Maximal subgroups

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

subgroupindex
#903072
#902902
C7×#4662
#899302
#899222
#898852
C8×C22⋊D72
#898092
#898072
#896572
#896472
#895472
#895442
#895322
#895302
#33057