Finite Group Databasebeta

← #11731898

Subgroups — #11731898

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
1C22×(C2×D7)⋊C4C211yesmaximal, characteristic
2C2×#6461C211yesmaximal, characteristic, complemented
3C7×#385C211yesmaximal, characteristic, complemented
4#89984C211yesmaximal, characteristic, complemented
5#89766C211yesmaximal, characteristic, complemented
6#89765C211yesmaximal, characteristic, complemented
7#89762C211yesmaximal, characteristic, complemented
8#89727C211yesmaximal, characteristic
9#89727C211yesmaximal, characteristic
10#89718C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
12#89447C211yesmaximal, characteristic, complemented
13#89446C211yesmaximal, characteristic, complemented
14#89442C211yesmaximal, characteristic, complemented
15#89441C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic, complemented
20C2×C7×C4⋊C4C2211yescharacteristic, complemented
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×(C2×D7)⋊C4—24no—
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
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×C4⋊Dic7C2211yescharacteristic, complemented
39C2×C4⋊Dic7C2211yescharacteristic, complemented
40C2×Dic7⋊C4C2211yescharacteristic, complemented
41C2×Dic7⋊C4C2211yescharacteristic, complemented
42C2×C4×Dic7C2211yescharacteristic, complemented
43C2×C4×Dic7C2211yescharacteristic, complemented
44#6461—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×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—41no—
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C22×Dic7—22no—
74C22×Dic7—22no—
75C22×Dic7—41no—
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C7×C4⋊C4—22no—
83C7×C4⋊C4—41no—
84C7×C22⋊C4—22no—
85C7×C22⋊C4—22no—
86C22⋊Dic7—22no—
87C22⋊Dic7—22no—
88(C2×D7)⋊C4—22no—
89(C2×D7)⋊C4—22no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4—22no—
95(C2×D7)⋊C4—41no—
96(C2×D7)⋊C4—41no—
97(C2×D7)⋊C4—41no—
98(C2×D7)⋊C4—41no—
99(C2×D7)⋊C4D414yes—
100C4⋊Dic7—22no—
101C4⋊Dic7—41no—
102Dic7⋊C4—22no—
103Dic7⋊C4—41no—
104C4×Dic7—22no—
105C4×Dic7—41no—
106C2×#126—71no—
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#385D711yescharacteristic, complemented
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#4112yes—
126C23×C7#4112yes—
127C23×C7#4112yes—
128C23×C7C2411yescharacteristic
129C22×D7—28no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—44no—
143C22×D7—44no—
144C22×D7—44no—
145C22×D7—44no—
146C22×D7—44no—
147C22×D7—44no—
148C22×D7C2×D412yes—
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×C7—21no—
167C2×C4×C7—44no—
168C2×C4×C7C2×D412yes—
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—
186C2×Dic7—44no—
187C2×Dic7C2×D412yes—
188C25—71no—
189C23×C4—71no—
190C23×C4C2×D711yescharacteristic
191#126—144no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
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—71no—
212C2×C22⋊C4—144no—
213C2×C22⋊C4C2×D711yescharacteristic
214C2×C22⋊C4C2×D711yescharacteristic
215C2×C42—71no—
216C2×C42—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95—71no—
221#95—71no—
222#95—71no—
223#95C2×D711yescharacteristic, complemented
224#95C2×D711yescharacteristic, complemented
225C22×C7—22no—
226C22×C7—22no—
227C22×C7—22no—
228C22×C7—22no—
229C22×C7—22no—
230C22×C7—22no—
231C22×C7#12614yescomplemented
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—416no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—41no—
250C2×D7—41no—
251C2×D7—41no—
252C2×D7—41no—
253C2×D7—41no—
254C4×C7—22no—
255C4×C7—41no—
256C4×C7—41no—
257C4×C7—41no—
258C4×C7—41no—
259C4×C7—41no—
260Dic7—22no—
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—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—71no—
280C22×C4—71no—
281C22×C4—71no—
282C22×C4—71no—
283C22×C4—71no—
284C22×C4—71no—
285C22×C4—142no—
286C22×C4—142no—
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—41no—
296C4⋊C4—142no—
297C4⋊C4—142no—
298C4⋊C4—281no—
299C4⋊C4—281no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—74no—
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—142no—
312C22⋊C4—281no—
313C22⋊C4—281no—
314C22⋊C4—281no—
315C22⋊C4—281no—
316C42—142no—
317C42—281no—
318C2×C7—24no—
319C2×C7#52511yescharacteristic
320C2×C7C2×#12611yescharacteristic
321C2×C7#53811yescharacteristic
322C2×C7#54611yescharacteristic
323C2×C7#54611yescharacteristic
324C2×C7#55111yescharacteristic
325C2×C7#55111yescharacteristic
326D7—42no—
327D7—42no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—72no—
333C23—72no—
334C23—148no—
335C23—141no—
336C23—141no—
337C23—141no—
338C23—141no—
339C23—141no—
340C23—141no—
341C23—141no—
342C23—141no—
343C23—141no—
344C23—141no—
345C23—141no—
346C23—141no—
347C23—284no—
348C23—284no—
349C23—284no—
350C23—284no—
351C23—284no—
352C23—284no—
353C23#115712yes—
354C23#115912yes—
355C23#116112yes—
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—21no—
374C2×C4—44no—
375C2×C4—72no—
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—141no—
392C2×C4—141no—
393C2×C4—284no—
394C2×C4D4×D712yes—
395C7#248011yescharacteristic, complemented, Sylow
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—142no—
403C22—142no—
404C22—142no—
405C22—142no—
406C22—142no—
407C22—142no—
408C22—2816no—
409C22—281no—
410C22—281no—
411C22—281no—
412C22—281no—
413C22—281no—
414C22—281no—
415C22—281no—
416C22—281no—
417C22#646114yescomplemented
418C22C2×#115711yescharacteristic
419C22C2×D4×D711yescharacteristic
420C22C2×#115911yescharacteristic
421C22#654811yescharacteristic
422C22C2×#116111yescharacteristic
423C22D7×#4111yescharacteristic
424C22#655311yescharacteristic
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—142no—
432C4—281no—
433C4—281no—
434C4—281no—
435C4—281no—
436C4—281no—
437C2—24no—
438C2—282no—
439C2—282no—
440C2C2×#646111yescharacteristic
441C2#9024111yescharacteristic
442C2#9024711yescharacteristic
443C2#9028211yescharacteristic
444C2#9031811yescharacteristic
445C2#9033511yescharacteristic
446C2#9034911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC24×D7C22
hypercenter#385D7
hypocenterC7#2480
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#6461C2semidirect
3. C7×#385C2semidirect
4. #89984C2semidirect
5. #89766C2semidirect
6. #89765C2semidirect
7. #89762C2semidirect
10. #89718C2semidirect
11. C22⋊C4×Dic7C2semidirect
12. #89447C2semidirect
13. #89446C2semidirect
14. #89442C2semidirect
15. #89441C2semidirect
19. C2×C7×C4⋊C4C22semidirect
20. C2×C7×C4⋊C4C22semidirect
38. C2×C4⋊Dic7C22semidirect
39. C2×C4⋊Dic7C22semidirect
40. C2×Dic7⋊C4C22semidirect
41. C2×Dic7⋊C4C22semidirect
42. C2×C4×Dic7C22semidirect
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
116. #385D7semidirect
196. C2×C4⋊C4C2×D7semidirect
197. C2×C4⋊C4C2×D7semidirect
223. #95C2×D7semidirect
224. #95C2×D7semidirect
231. C22×C7#126semidirect
395. C7#2480semidirect
417. C22#6461semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D7)⋊C42
C2×#64612
C7×#3852
#899842
#897662
#897652
#897622
#897272
#897182
C22⋊C4×Dic72
#894472
#894462
#894422
#894412
#24807