Finite Group Databasebeta

← #11731304

Subgroups — #11731304

Order 896.

688 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6516C211yesmaximal, characteristic
2C2×#6448C211yesmaximal, characteristic, complemented
3C7×#379C211yesmaximal, characteristic, complemented
4#90019C211yesmaximal, characteristic, complemented
5#89733C211yesmaximal, characteristic, complemented
6#89731C211yesmaximal, characteristic, complemented
7#89728C211yesmaximal, characteristic, complemented
8#89728C211yesmaximal, characteristic, complemented
9#89728C211yesmaximal, characteristic, complemented
10#89725C211yesmaximal, characteristic, complemented
11#89724C211yesmaximal, characteristic, complemented
12#89723C211yesmaximal, characteristic, complemented
13#89719C211yesmaximal, characteristic, complemented
14#89719C211yesmaximal, characteristic, complemented
15#89717C211yesmaximal, characteristic, complemented
16C25×C7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic, complemented
21C22×Dic14C2211yescharacteristic, complemented
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28#6516—24no—
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34C2×C22⋊Dic7C2211yescharacteristic
35C2×C22⋊Dic7C2211yescharacteristic
36C2×C22⋊Dic7C2211yescharacteristic
37C2×C22⋊Dic7C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
41C2×C4⋊Dic7C2211yescharacteristic, complemented
42C2×Dic7⋊C4C2211yescharacteristic, complemented
43C2×C4×Dic7C2211yescharacteristic, complemented
44#6448—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#2684—71nomaximal, Sylow
54C24×C7—22no—
55C24×C7—22no—
56C24×C7—22no—
57C24×C7—41no—
58C24×C7—41no—
59C24×C7—41no—
60C24×C7—41no—
61C24×C7C2311yescharacteristic
62C24×C7C2311yescharacteristic
63C24×C7C2311yescharacteristic
64C23×D7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—22no—
70C2×C22⋊D7—22no—
71C2×C22⋊D7—22no—
72C2×C22⋊D7—22no—
73C2×C22⋊D7—41no—
74C2×C22⋊D7—41no—
75C2×C22⋊D7—41no—
76C2×C22⋊D7—41no—
77C2×C22⋊D7D414yes—
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C2×C4×D7—41no—
86C2×C4×D7—41no—
87C2×C4×D7—41no—
88C2×Dic14—22no—
89C2×Dic14—41no—
90C2×Dic14—41no—
91C7×C22⋊C4—22no—
92C7×C22⋊C4—22no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—41no—
97C22⋊Dic7—22no—
98C22⋊Dic7—22no—
99C22⋊Dic7—22no—
100C22⋊Dic7—22no—
101C22⋊Dic7—22no—
102C22⋊Dic7—41no—
103C22⋊Dic7—41no—
104C22⋊Dic7—41no—
105C22⋊Dic7—41no—
106(C2×D7)⋊C4—22no—
107(C2×D7)⋊C4—22no—
108(C2×D7)⋊C4—41no—
109C4⋊Dic7—41no—
110Dic7⋊C4—41no—
111C4×Dic7—22no—
112C2×#124—71no—
113C2×#120—71no—
114#397—71no—
115#397—71no—
116#394—71no—
117#394—71no—
118#393—71no—
119#393—71no—
120#388—71no—
121#388—71no—
122#386—71no—
123#386—71no—
124#386—71no—
125#379—71no—
126#379D711yescharacteristic, complemented
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—44no—
146C23×C7—44no—
147C23×C7—44no—
148C23×C7—44no—
149C23×C7—44no—
150C23×C7—44no—
151C23×C7—44no—
152C23×C7C2×D412yes—
153C23×C7#4112yes—
154C23×C7#4112yes—
155C23×C7C2411yescharacteristic
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—21no—
160C22×D7—21no—
161C22×D7—21no—
162C22×D7C2×D412yes—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C22⋊D7—44no—
176C22⋊D7—44no—
177C22⋊D7—44no—
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—21no—
187C2×Dic7—21no—
188C2×Dic7—21no—
189C2×Dic7—21no—
190C2×Dic7—21no—
191C2×Dic7—21no—
192C2×Dic7—21no—
193C2×Dic7—21no—
194C2×Dic7—21no—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—21no—
198C2×Dic7C2×D412yes—
199C4×D7—44no—
200Dic14—44no—
201C25C2×D711yescharacteristic
202C22×Q8—71no—
203C22×D4—71no—
204C22×D4—71no—
205C22×D4—71no—
206C23×C4—71no—
207#124—144no—
208#120—144no—
209C2×C4⋊C4—71no—
210C2×C4⋊C4—71no—
211C2×C22⋊C4—71no—
212C2×C22⋊C4—71no—
213C2×C22⋊C4—71no—
214C2×C22⋊C4—71no—
215C2×C22⋊C4—71no—
216C2×C22⋊C4—71no—
217C2×C22⋊C4—71no—
218C2×C22⋊C4—71no—
219C2×C22⋊C4—71no—
220C2×C22⋊C4—71no—
221C2×C22⋊C4—71no—
222C2×C22⋊C4—71no—
223C2×C22⋊C4C2×D711yescharacteristic
224C2×C22⋊C4C2×D711yescharacteristic
225C2×C22⋊C4C2×D711yescharacteristic
226C2×C22⋊C4C2×D711yescharacteristic
227C2×C22⋊C4C2×D711yescharacteristic
228C2×C22⋊C4C2×D711yescharacteristic
229C2×C42—71no—
230#95—71no—
231#95—71no—
232#95—71no—
233#95—71no—
234#95—71no—
235#95—71no—
236#95—71no—
237#95—71no—
238C22×C7—22no—
239C22×C7—22no—
240C22×C7—22no—
241C22×C7—22no—
242C22×C7—22no—
243C22×C7—22no—
244C22×C7—22no—
245C22×C7—22no—
246C22×C7—22no—
247C22×C7—416no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7—41no—
252C22×C7—41no—
253C22×C7—41no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7C22×D411yescharacteristic
261C22×C7C2×#4111yescharacteristic
262C22×C7C2×#4111yescharacteristic
263C22×C7H(5,𝔽2)11yescharacteristic
264C22×C7H(5,𝔽2)11yescharacteristic
265C22×C7H(5,𝔽2)11yescharacteristic
266C22×C7H(5,𝔽2)11yescharacteristic
267C2×D7—22no—
268C2×D7—22no—
269C2×D7—22no—
270C2×D7—41no—
271C2×D7—41no—
272C2×D7—41no—
273C2×D7—41no—
274C4×C7—41no—
275C4×C7—41no—
276C4×C7—41no—
277C4×C7—41no—
278Dic7—22no—
279Dic7—41no—
280Dic7—41no—
281Dic7—41no—
282Dic7—41no—
283Dic7—41no—
284Dic7—41no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—71no—
293C24C22×D711yescharacteristic
294C24C22×D711yescharacteristic
295C24C22×D711yescharacteristic
296C2×Q8—142no—
297C2×Q8—281no—
298C2×Q8—281no—
299C2×D4—74no—
300C2×D4—142no—
301C2×D4—142no—
302C2×D4—142no—
303C2×D4—142no—
304C2×D4—281no—
305C2×D4—281no—
306C2×D4—281no—
307C2×D4—281no—
308C22×C4—71no—
309C22×C4—71no—
310C22×C4—71no—
311C22×C4—71no—
312C22×C4—71no—
313C22×C4—71no—
314C22×C4—71no—
315C22×C4—281no—
316C22×C4—281no—
317C22×C4—281no—
318C22×C4C22×D711yescharacteristic
319C22×C4C22×D711yescharacteristic
320C22×C4C22×D711yescharacteristic
321C22×C4C22×D711yescharacteristic
322C4⋊C4—281no—
323C4⋊C4—281no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—142no—
331C22⋊C4—142no—
332C22⋊C4—142no—
333C22⋊C4—142no—
334C22⋊C4—142no—
335C22⋊C4—142no—
336C22⋊C4—142no—
337C22⋊C4—281no—
338C22⋊C4—281no—
339C22⋊C4—281no—
340C22⋊C4—281no—
341C22⋊C4—281no—
342C42—142no—
343C2×C7—42no—
344C2×C7—42no—
345C2×C7—42no—
346C2×C7#53411yescharacteristic
347C2×C7#53811yescharacteristic
348C2×C7#53811yescharacteristic
349C2×C7#54611yescharacteristic
350C2×C7#54611yescharacteristic
351C2×C7#55111yescharacteristic
352C2×C7H(3,𝔽4)11yescharacteristic
353D7—42no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—72no—
380C23—141no—
381C23—141no—
382C23—141no—
383C23—141no—
384C23—141no—
385C23—141no—
386C23#115712yes—
387C23D4×D712yes—
388C23#115912yes—
389C23C23×D711yescharacteristic
390Q8—284no—
391D4—284no—
392D4—284no—
393D4—284no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—72no—
407C2×C4—141no—
408C2×C4—141no—
409C2×C4—141no—
410C2×C4—141no—
411C2×C4—141no—
412C2×C4—141no—
413C2×C4—141no—
414C2×C4—141no—
415C2×C4—141no—
416C2×C4—141no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—141no—
420C2×C4—141no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4—141no—
424C2×C4—141no—
425C2×C4—141no—
426C2×C4—141no—
427C2×C4—284no—
428C7#268411yescharacteristic, complemented, Sylow
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—416no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—142no—
452C22—142no—
453C22—142no—
454C22—281no—
455C22—281no—
456C22—281no—
457C22—281no—
458C22C2×#115711yescharacteristic
459C22C2×D4×D711yescharacteristic
460C22C2×#115911yescharacteristic
461C22#654811yescharacteristic
462C22#654811yescharacteristic
463C22#654811yescharacteristic
464C22#654811yescharacteristic
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—142no—
470C4—281no—
471C4—281no—
472C4—281no—
473C4—281no—
474C4—281no—
475C4—281no—
476C2—42no—
477C2—42no—
478C2—42no—
479C2—282no—
480C2#9018111yescharacteristic
481C2#9018511yescharacteristic
482C2#9018611yescharacteristic
483C2#9028011yescharacteristic
484C2#9028211yescharacteristic
485C2#9028411yescharacteristic
486C2#9028511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#379C2
prime radicalC2×#6516C2
hypercenter#379D7
hypocenterC7#2684
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#6448C2semidirect
3. C7×#379C2semidirect
4. #90019C2semidirect
5. #89733C2semidirect
6. #89731C2semidirect
7. #89728C2semidirect
8. #89728C2semidirect
9. #89728C2semidirect
10. #89725C2semidirect
11. #89724C2semidirect
12. #89723C2semidirect
13. #89719C2semidirect
14. #89719C2semidirect
15. #89717C2semidirect
20. C22×C4×D7C22semidirect
21. C22×Dic14C22semidirect
38. C2×(C2×D7)⋊C4C22semidirect
39. C2×(C2×D7)⋊C4C22semidirect
40. C2×(C2×D7)⋊C4C22semidirect
41. C2×C4⋊Dic7C22semidirect
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
126. #379D7semidirect
428. C7#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#65162
C2×#64482
C7×#3792
#900192
#897332
#897312
#897282
#897252
#897242
#897232
#897192
#897172
#26847