Finite Group Databasebeta

← #11740526

Subgroups — #11740526

Order 896.

664 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#126C211yesmaximal, characteristic, complemented
2C2×D7×C22⋊C4C211yesmaximal, characteristic
3#89766C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic, complemented
5#89762C211yesmaximal, characteristic, complemented
6#89761C211yesmaximal, characteristic, complemented
7#89761C211yesmaximal, characteristic, complemented
8#89759C211yesmaximal, characteristic, complemented
9#89727C211yesmaximal, characteristic
10#89727C211yesmaximal, characteristic
11#89724C211yesmaximal, characteristic, complemented
12#89724C211yesmaximal, characteristic, complemented
13#89723C211yesmaximal, characteristic, complemented
14#89712C211yesmaximal, characteristic, complemented
15C4×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C7×#126—24no—
20C2×C7×C4⋊C4C2211yescharacteristic, complemented
21C2×C7×C4⋊C4C2211yescharacteristic, complemented
22C2×C7×C4⋊C4C2211yescharacteristic, complemented
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C42×C7C2211yescharacteristic, complemented
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×C4⋊Dic7C2211yescharacteristic, complemented
41C2×Dic7⋊C4C2211yescharacteristic, complemented
42C2×Dic7⋊C4C2211yescharacteristic, complemented
43C2×C4×Dic7C2211yescharacteristic, complemented
44D7×C22⋊C4—24no—
45C7⋊#95C2211yescharacteristic, complemented
46C7⋊#95C2211yescharacteristic, complemented
47C7⋊#95C2211yescharacteristic, complemented
48C7⋊#95C2211yescharacteristic, complemented
49C7⋊#95C2211yescharacteristic, complemented
50C7⋊#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53#2480—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7—22no—
56C23×D7—22no—
57C23×D7—22no—
58C23×D7—22no—
59C23×D7—22no—
60C23×D7—41no—
61C23×D7—41no—
62C23×D7C2311yescharacteristic
63C23×D7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C2×C4×D7—22no—
77C2×C4×D7—22no—
78C2×C4×D7—22no—
79C2×C4×D7—22no—
80C2×C4×D7—41no—
81C2×C4×D7—41no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—22no—
84C7×C4⋊C4—22no—
85C7×C22⋊C4—22no—
86C7×C22⋊C4—22no—
87C7×C22⋊C4D414yes—
88C42×C7—22no—
89C22⋊Dic7—22no—
90C22⋊Dic7—41no—
91C22⋊Dic7—41no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4—22no—
95(C2×D7)⋊C4—22no—
96(C2×D7)⋊C4—22no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—22no—
100(C2×D7)⋊C4—41no—
101(C2×D7)⋊C4—41no—
102C4⋊Dic7—41no—
103Dic7⋊C4—41no—
104Dic7⋊C4—41no—
105C4×Dic7—41no—
106C2×#126D711yescharacteristic, complemented
107C22×C22⋊C4—71no—
108#397—71no—
109#397—71no—
110#397—71no—
111#396—71no—
112#388—71no—
113#388—71no—
114#388—71no—
115#385—71no—
116#385—71no—
117#380—71no—
118#379—71no—
119#379—71no—
120C4×C22⋊C4—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7C2×D412yes—
128C23×C7C2411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—28no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—44no—
141C22×D7—44no—
142C22×D7—44no—
143C22×D7—44no—
144C22×D7—44no—
145C22×D7—44no—
146C22×D7#4112yes—
147C22×D7#4112yes—
148C22×D7#4112yes—
149C22×D7#4112yes—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7C2×D412yes—
167C2×C4×C7C2×D412yes—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178C2×Dic7—21no—
179C2×Dic7—21no—
180C2×Dic7—21no—
181C2×Dic7—21no—
182C2×Dic7—21no—
183C2×Dic7—21no—
184C2×Dic7—21no—
185C2×Dic7—21no—
186C4×D7—44no—
187C4×D7—44no—
188C25—71no—
189C23×C4—71no—
190C23×C4—71no—
191#126—24no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4C2×D711yescharacteristic, complemented
196C2×C4⋊C4C2×D711yescharacteristic, complemented
197C2×C4⋊C4C2×D711yescharacteristic, complemented
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C22⋊C4—71no—
201C2×C22⋊C4—71no—
202C2×C22⋊C4—71no—
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4—71no—
206C2×C22⋊C4—71no—
207C2×C22⋊C4—71no—
208C2×C22⋊C4—71no—
209C2×C22⋊C4—71no—
210C2×C22⋊C4—71no—
211C2×C22⋊C4—144no—
212C2×C22⋊C4C2×D711yescharacteristic
213C2×C22⋊C4C2×D711yescharacteristic
214C2×C22⋊C4C2×D711yescharacteristic
215C2×C42—71no—
216C2×C42C2×D711yescharacteristic, complemented
217#95—71no—
218#95—71no—
219#95—71no—
220#95—71no—
221#95—71no—
222#95—71no—
223#95—71no—
224#95—71no—
225C22×C7—22no—
226C22×C7—22no—
227C22×C7—22no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7C22×D411yescharacteristic
233C22×C7C2×#4111yescharacteristic
234C22×C7C2×#4111yescharacteristic
235C22×C7C2×#4111yescharacteristic
236C22×C7C2×#4111yescharacteristic
237C22×C7H(5,𝔽2)11yescharacteristic
238C22×C7H(5,𝔽2)11yescharacteristic
239C2×D7—22no—
240C2×D7—22no—
241C2×D7—22no—
242C2×D7—22no—
243C2×D7—22no—
244C2×D7—22no—
245C2×D7—22no—
246C2×D7—22no—
247C2×D7—22no—
248C2×D7—416no—
249C2×D7—41no—
250C2×D7—41no—
251C2×D7—41no—
252C2×D7—41no—
253C2×D7#12614yescomplemented
254C4×C7—22no—
255C4×C7—22no—
256C4×C7—41no—
257C4×C7—41no—
258C4×C7—41no—
259C4×C7—41no—
260Dic7—41no—
261Dic7—41no—
262Dic7—41no—
263Dic7—41no—
264Dic7—41no—
265Dic7—41no—
266C24—71no—
267C24—71no—
268C24—142no—
269C24—142no—
270C24—142no—
271C24—142no—
272C24—142no—
273C24—281no—
274C24—281no—
275C24C22×D711yescharacteristic
276C22×C4—71no—
277C22×C4—71no—
278C22×C4—71no—
279C22×C4—71no—
280C22×C4—71no—
281C22×C4—71no—
282C22×C4—142no—
283C22×C4—142no—
284C22×C4—142no—
285C22×C4—142no—
286C22×C4—281no—
287C22×C4—281no—
288C22×C4C22×D711yescharacteristic
289C22×C4C22×D711yescharacteristic
290C22×C4C22×D711yescharacteristic
291C22×C4C22×D711yescharacteristic
292C22×C4C22×D711yescharacteristic
293C22×C4C22×D711yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—281no—
298C4⋊C4—281no—
299C4⋊C4—281no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—142no—
303C22⋊C4—142no—
304C22⋊C4—142no—
305C22⋊C4—142no—
306C22⋊C4—142no—
307C22⋊C4—142no—
308C22⋊C4—142no—
309C22⋊C4—142no—
310C22⋊C4—142no—
311C22⋊C4—281no—
312C22⋊C4—281no—
313C22⋊C4—281no—
314C22⋊C4—281no—
315C22⋊C4C22⋊D714yes—
316C42—22no—
317C42—281no—
318C2×C7—42no—
319C2×C7#52511yescharacteristic
320C2×C7C2×#12611yescharacteristic
321C2×C7#53811yescharacteristic
322C2×C7#54611yescharacteristic
323C2×C7#54611yescharacteristic
324C2×C7#55111yescharacteristic
325C2×C7#55111yescharacteristic
326D7—24no—
327D7—42no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—72no—
335C23—72no—
336C23—72no—
337C23—72no—
338C23—148no—
339C23—141no—
340C23—141no—
341C23—141no—
342C23—141no—
343C23—141no—
344C23—141no—
345C23—141no—
346C23—141no—
347C23—141no—
348C23—141no—
349C23—284no—
350C23—284no—
351C23—284no—
352C23—284no—
353C23—284no—
354C23—284no—
355C23C2×C22⋊D712yes—
356C23C23×D711yescharacteristic
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—141no—
376C2×C4—141no—
377C2×C4—141no—
378C2×C4—141no—
379C2×C4—141no—
380C2×C4—141no—
381C2×C4—141no—
382C2×C4—141no—
383C2×C4—141no—
384C2×C4—141no—
385C2×C4—141no—
386C2×C4—141no—
387C2×C4—141no—
388C2×C4—141no—
389C2×C4—141no—
390C2×C4—141no—
391C2×C4—284no—
392C2×C4—284no—
393C2×C4C2×C22⋊D712yes—
394C2×C4C2×C22⋊D712yes—
395C7#248011yescharacteristic, complemented, Sylow
396C22—22no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—74no—
404C22—142no—
405C22—142no—
406C22—142no—
407C22—142no—
408C22—142no—
409C22—142no—
410C22—142no—
411C22—142no—
412C22—142no—
413C22—2816no—
414C22—281no—
415C22—281no—
416C22—281no—
417C22—281no—
418C22D7×#4111yescharacteristic
419C22D7×#4111yescharacteristic
420C22D7×#4111yescharacteristic
421C22D7×#4111yescharacteristic
422C22#655311yescharacteristic
423C22#655311yescharacteristic
424C22C22×C22⋊D711yescharacteristic
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—281no—
432C4—281no—
433C4—281no—
434C4—281no—
435C4—281no—
436C4—281no—
437C2—42no—
438C2—144no—
439C2—282no—
440C2D7×#12611yescharacteristic
441C2#9039511yescharacteristic
442C2#9039611yescharacteristic
443C2#9039611yescharacteristic
444C2#9051911yescharacteristic
445C2#9052011yescharacteristic
446C2#9052011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#126C2
prime radicalC24×D7C22
hypercenterC2×#126D7
hypocenterC7#2480
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C7×#126C2semidirect
3. #89766C2semidirect
4. #89765C2semidirect
5. #89762C2semidirect
6. #89761C2semidirect
7. #89761C2semidirect
8. #89759C2semidirect
11. #89724C2semidirect
12. #89724C2semidirect
13. #89723C2semidirect
14. #89712C2semidirect
15. C4×(C2×D7)⋊C4C2semidirect
20. C2×C7×C4⋊C4C22semidirect
21. C2×C7×C4⋊C4C22semidirect
22. C2×C7×C4⋊C4C22semidirect
26. C2×C42×C7C22semidirect
40. C2×C4⋊Dic7C22semidirect
41. C2×Dic7⋊C4C22semidirect
42. C2×Dic7⋊C4C22semidirect
43. C2×C4×Dic7C22semidirect
45. C7⋊#95C22semidirect
46. C7⋊#95C22semidirect
47. C7⋊#95C22semidirect
48. C7⋊#95C22semidirect
49. C7⋊#95C22semidirect
50. C7⋊#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
106. C2×#126D7semidirect
195. C2×C4⋊C4C2×D7semidirect
196. C2×C4⋊C4C2×D7semidirect
197. C2×C4⋊C4C2×D7semidirect
216. C2×C42C2×D7semidirect
253. C2×D7#126semidirect
395. C7#2480semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1262
C2×D7×C22⋊C42
#897662
#897652
#897622
#897612
#897592
#897272
#897242
#897232
#897122
C4×(C2×D7)⋊C42
#24807