Finite Group Databasebeta

← #11734166

Subgroups — #11734166

Order 896.

612 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6502C211yesmaximal, characteristic, complemented
2C2×#6446C211yesmaximal, characteristic, complemented
3C7×#392C211yesmaximal, characteristic, complemented
4#89995C211yesmaximal, characteristic, complemented
5#89995C211yesmaximal, characteristic, complemented
6#89994C211yesmaximal, characteristic, complemented
7#89993C211yesmaximal, characteristic, complemented
8#89992C211yesmaximal, characteristic, complemented
9#89728C211yesmaximal, characteristic, complemented
10#89728C211yesmaximal, characteristic, complemented
11#89724C211yesmaximal, characteristic, complemented
12#89717C211yesmaximal, characteristic, complemented
13#89714C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89431C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C23×Dic7C2211yescharacteristic
23C22×C4×D7C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34C2×C22⋊Dic7C2211yescharacteristic
35#6502—24no—
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×C4⋊Dic7C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×C4×Dic7C2211yescharacteristic
44#6446—24no—
45C7×#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2703—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C24×C7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C2×C7×D4—22no—
59C2×C7×D4—22no—
60C2×C7×D4—22no—
61C2×C7×D4—41no—
62C2×C7×D4—41no—
63C2×C7×D4—41no—
64C2×C7×D4—41no—
65C2×C7×D4—41no—
66C2×C7×D4—41no—
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C2×C22⋊D7—22no—
72C2×C22⋊D7—22no—
73C2×C22⋊D7—22no—
74C2×C22⋊D7—22no—
75C2×C22⋊D7—22no—
76C2×C22⋊D7—41no—
77C2×C22⋊D7—41no—
78C2×C22⋊D7—41no—
79C2×C22⋊D7—41no—
80C22×Dic7—22no—
81C22×Dic7—22no—
82C22×Dic7—41no—
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C22×Dic7C2311yescharacteristic
89C22×Dic7C2311yescharacteristic
90C2×C4×D7—22no—
91C2×C4×D7—41no—
92C2×C4×D7—41no—
93C7×C22⋊C4—22no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C22⋊Dic7—22no—
97C22⋊Dic7—22no—
98C22⋊Dic7—22no—
99C22⋊Dic7—41no—
100C22⋊Dic7—41no—
101C22⋊Dic7—41no—
102C22⋊Dic7—41no—
103C22⋊Dic7—41no—
104(C2×D7)⋊C4—41no—
105(C2×D7)⋊C4—41no—
106(C2×D7)⋊C4D414yescomplemented
107C4⋊Dic7—22no—
108Dic7⋊C4—22no—
109Dic7⋊C4—22no—
110Dic7⋊C4—41no—
111C4×Dic7—41no—
112C2×#123—71no—
113C2×#121—71no—
114#400—71no—
115#397—71no—
116#394—71no—
117#392—71no—
118#392—71no—
119#392D711yescharacteristic, complemented
120#390—71no—
121#388—71no—
122#388—71no—
123#386—71no—
124#386—71no—
125#386—71no—
126#380—71no—
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—21no—
146C23×C7—21no—
147C23×C7#4112yes—
148C23×C7C2411yescharacteristic
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—21no—
153C22×D7—21no—
154C22×D7—21no—
155C22×D7C2×D412yes—
156C7×D4—44no—
157C7×D4—44no—
158C7×D4—44no—
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—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7C2×D412yes—
171C22⋊D7—44no—
172C22⋊D7—44no—
173C22⋊D7—44no—
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—21no—
187C2×Dic7—21no—
188C2×Dic7—21no—
189C2×Dic7—21no—
190C2×Dic7—21no—
191C2×Dic7—21no—
192C2×Dic7—21no—
193C2×Dic7—44no—
194C2×Dic7C2×D412yes—
195C2×Dic7#4112yes—
196C4×D7—44no—
197C22×D4—71no—
198C22×D4—71no—
199C22×D4—71no—
200C22×D4C2×D711yescharacteristic
201C22×D4C2×D711yescharacteristic
202C22×D4C2×D711yescharacteristic
203C23×C4—71no—
204C23×C4—71no—
205#123—144no—
206#121—144no—
207C2×C4⋊C4—71no—
208C2×C4⋊C4—71no—
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⋊C4C2×D711yescharacteristic
223C2×C22⋊C4C2×D711yescharacteristic
224C2×C22⋊C4C2×D711yescharacteristic
225C2×C42—71no—
226#95—71no—
227#95—71no—
228#95—71no—
229#95—71no—
230#95—71no—
231#95—71no—
232#95—71no—
233#95C2×D711yescharacteristic
234C22×C7—22no—
235C22×C7—22no—
236C22×C7—22no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7—41no—
252C22×C7—41no—
253C22×C7—41no—
254C22×C7—41no—
255C22×C7C22×D411yescharacteristic
256C22×C7C2×#4111yescharacteristic
257C22×C7C2×#4111yescharacteristic
258C22×C7H(5,𝔽2)11yescharacteristic
259C22×C7H(5,𝔽2)11yescharacteristic
260C22×C7H(5,𝔽2)11yescharacteristic
261C22×C7H(5,𝔽2)11yescharacteristic
262C2×D7—22no—
263C2×D7—22no—
264C2×D7—22no—
265C2×D7—41no—
266C2×D7—41no—
267C2×D7—41no—
268C2×D7—41no—
269C4×C7—22no—
270C4×C7—41no—
271C4×C7—41no—
272C4×C7—41no—
273Dic7—22no—
274Dic7—22no—
275Dic7—41no—
276Dic7—41no—
277Dic7—41no—
278Dic7—41no—
279Dic7—41no—
280C24—71no—
281C24C22×D711yescharacteristic
282C24C22×D711yescharacteristic
283C24C22×D711yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—142no—
294C2×D4—142no—
295C2×D4—142no—
296C2×D4—142no—
297C2×D4—142no—
298C2×D4—281no—
299C2×D4—281no—
300C2×D4—281no—
301C2×D4—281no—
302C22×C4—71no—
303C22×C4—71no—
304C22×C4—71no—
305C22×C4—71no—
306C22×C4—71no—
307C22×C4—71no—
308C22×C4—71no—
309C22×C4—142no—
310C22×C4—142no—
311C22×C4—142no—
312C22×C4—281no—
313C22×C4—281no—
314C22×C4—281no—
315C22×C4C22×D711yescharacteristic
316C22×C4C22×D711yescharacteristic
317C22×C4C22×D711yescharacteristic
318C22×C4C22×D711yescharacteristic
319C4⋊C4—142no—
320C4⋊C4—142no—
321C4⋊C4—142no—
322C4⋊C4—281no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—74no—
327C22⋊C4—142no—
328C22⋊C4—142no—
329C22⋊C4—142no—
330C22⋊C4—281no—
331C22⋊C4—281no—
332C22⋊C4—281no—
333C22⋊C4—281no—
334C22⋊C4—281no—
335C22⋊C4—281no—
336C22⋊C4—281no—
337C42—281no—
338C2×C7—42no—
339C2×C7—81no—
340C2×C7—81no—
341C2×C7#53511yescharacteristic
342C2×C7#53811yescharacteristic
343C2×C7#54011yescharacteristic
344C2×C7#54611yescharacteristic
345C2×C7#54611yescharacteristic
346C2×C7#55311yescharacteristic
347C2×C7#56011yescharacteristic
348D7—42no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
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—72no—
370C23—141no—
371C23—141no—
372C23—141no—
373C23—141no—
374C23—141no—
375C23—141no—
376C23#115912yes—
377C23C23×D711yescharacteristic
378D4—44no—
379D4—44no—
380D4—44no—
381D4—284no—
382D4—284no—
383D4—284no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—72no—
396C2×C4—72no—
397C2×C4—141no—
398C2×C4—141no—
399C2×C4—141no—
400C2×C4—141no—
401C2×C4—141no—
402C2×C4—141no—
403C2×C4—141no—
404C2×C4—141no—
405C2×C4—141no—
406C2×C4—141no—
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—284no—
417C2×C4—284no—
418C2×C4D4×D712yes—
419C7#270311yescharacteristic, complemented, Sylow
420C22—22no—
421C22—22no—
422C22—22no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—142no—
442C22—142no—
443C22—142no—
444C22—281no—
445C22—281no—
446C22—281no—
447C22—281no—
448C22C2×D4×D711yescharacteristic
449C22C2×#115911yescharacteristic
450C22#654811yescharacteristic
451C22#654811yescharacteristic
452C22#654811yescharacteristic
453C22#654811yescharacteristic
454C22D7×#4111yescharacteristic
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—142no—
460C4—142no—
461C4—281no—
462C4—281no—
463C4—281no—
464C4—281no—
465C4—281no—
466C2—42no—
467C2—81no—
468C2—81no—
469C2—282no—
470C2#9028211yescharacteristic
471C2#9028411yescharacteristic
472C2#9028611yescharacteristic
473C2#9029211yescharacteristic
474C2#9030311yescharacteristic
475C2#9030611yescharacteristic
476C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2703
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#6502C2semidirect
2. C2×#6446C2semidirect
3. C7×#392C2semidirect
4. #89995C2semidirect
5. #89995C2semidirect
6. #89994C2semidirect
7. #89993C2semidirect
8. #89992C2semidirect
9. #89728C2semidirect
10. #89728C2semidirect
11. #89724C2semidirect
12. #89717C2semidirect
13. #89714C2semidirect
14. #89441C2semidirect
15. #89431C2semidirect
106. (C2×D7)⋊C4D4semidirect
119. #392D7semidirect
419. C7#2703semidirect

Maximal subgroups

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

subgroupindex
C2×#65022
C2×#64462
C7×#3922
#899952
#899942
#899932
#899922
#897282
#897242
#897172
#897142
#894412
#894312
#27037