Finite Group Databasebeta

← #11732244

Subgroups — #11732244

Order 896.

626 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6502C211yesmaximal, characteristic, complemented
2C2×#6501C211yesmaximal, characteristic, complemented
3C7×#386C211yesmaximal, characteristic, complemented
4#89985C211yesmaximal, characteristic, complemented
5#89985C211yesmaximal, characteristic, complemented
6#89733C211yesmaximal, characteristic, complemented
7#89732C211yesmaximal, characteristic, complemented
8#89732C211yesmaximal, characteristic, complemented
9#89730C211yesmaximal, characteristic, complemented
10#89725C211yesmaximal, characteristic, complemented
11#89715C211yesmaximal, characteristic, complemented
12#89446C211yesmaximal, characteristic, complemented
13#89440C211yesmaximal, characteristic, complemented
14#89437C211yesmaximal, characteristic, complemented
15#89431C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C23×Dic7C2211yescharacteristic
23C22×D28C2211yescharacteristic
24C22×C4×D7C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31#6502—24no—
32#6501—24no—
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
41C2×C4⋊Dic7C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C7×#95C2211yescharacteristic
44C7×#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2688—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C2×C7×D4—41no—
59C2×C7×D4—41no—
60C2×C7×D4D414yescomplemented
61C22×C4×C7—41no—
62C22×C4×C7—41no—
63C22×C4×C7—41no—
64C22×C4×C7C2311yescharacteristic
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—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—
80C2×C22⋊D7—41no—
81C22×Dic7—22no—
82C22×Dic7—22no—
83C22×Dic7—41no—
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C22×Dic7C2311yescharacteristic
89C22×Dic7C2311yescharacteristic
90C2×D28—22no—
91C2×D28—41no—
92C2×D28—41no—
93C2×C4×D7—22no—
94C2×C4×D7—41no—
95C2×C4×D7—41no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C22⋊Dic7—22no—
100C22⋊Dic7—22no—
101C22⋊Dic7—22no—
102(C2×D7)⋊C4—22no—
103(C2×D7)⋊C4—22no—
104(C2×D7)⋊C4—41no—
105(C2×D7)⋊C4—41no—
106(C2×D7)⋊C4—41no—
107(C2×D7)⋊C4—41no—
108(C2×D7)⋊C4—41no—
109C4⋊Dic7—22no—
110C4⋊Dic7—41no—
111Dic7⋊C4—22no—
112C2×#121—71no—
113C2×#121—71no—
114#400—71no—
115#397—71no—
116#397—71no—
117#394—71no—
118#392—71no—
119#392—71no—
120#386—71no—
121#386—71no—
122#386—71no—
123#386—71no—
124#386D711yescharacteristic, complemented
125#385—71no—
126#381—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×C7C2×D412yes—
140C23×C7C2×D412yes—
141C23×C7C2411yescharacteristic
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—21no—
153C22×D7—21no—
154C22×D7—21no—
155C22×D7#4112yes—
156C7×D4—44no—
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—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—44no—
172C2×C4×C7C2×D412yes—
173C22⋊D7—44no—
174C22⋊D7—44no—
175C22⋊D7—44no—
176C22⋊D7—44no—
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—21no—
194C2×Dic7—44no—
195C2×Dic7#4112yes—
196D28—44no—
197C4×D7—44no—
198C22×D4—71no—
199C22×D4—71no—
200C22×D4—71no—
201C22×D4—71no—
202C22×D4—71no—
203C22×D4C2×D711yescharacteristic
204C23×C4—71no—
205C23×C4—71no—
206C23×C4C2×D711yescharacteristic
207#121—144no—
208#121—144no—
209C2×C4⋊C4—71no—
210C2×C4⋊C4—71no—
211C2×C4⋊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
225#95—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
234#95C2×D711yescharacteristic
235C22×C7—22no—
236C22×C7—22no—
237C22×C7—22no—
238C22×C7—22no—
239C22×C7—22no—
240C22×C7—22no—
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×C7C22×D411yescharacteristic
250C22×C7C2×#4111yescharacteristic
251C22×C7C2×#4111yescharacteristic
252C22×C7H(5,𝔽2)11yescharacteristic
253C22×C7H(5,𝔽2)11yescharacteristic
254C22×C7H(5,𝔽2)11yescharacteristic
255C22×C7H(5,𝔽2)11yescharacteristic
256C2×D7—22no—
257C2×D7—22no—
258C2×D7—22no—
259C2×D7—41no—
260C2×D7—41no—
261C2×D7—41no—
262C2×D7—41no—
263C2×D7—41no—
264C2×D7—41no—
265C2×D7—41no—
266C2×D7—41no—
267C2×D7—41no—
268C2×D7—41no—
269C2×D7—41no—
270C4×C7—22no—
271C4×C7—41no—
272C4×C7—41no—
273C4×C7—41no—
274C4×C7—41no—
275Dic7—22no—
276Dic7—41no—
277Dic7—41no—
278Dic7—41no—
279Dic7—41no—
280Dic7—41no—
281C24—71no—
282C24—71no—
283C24C22×D711yescharacteristic
284C24C22×D711yescharacteristic
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—142no—
288C2×D4—142no—
289C2×D4—142no—
290C2×D4—142no—
291C2×D4—142no—
292C2×D4—142no—
293C2×D4—142no—
294C2×D4—142no—
295C2×D4—281no—
296C2×D4—281no—
297C2×D4—281no—
298C2×D4—281no—
299C2×D4—281no—
300C2×D4—281no—
301C2×D4—281no—
302C2×D4D2814yescomplemented
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—71no—
307C22×C4—71no—
308C22×C4—71no—
309C22×C4—71no—
310C22×C4—71no—
311C22×C4—71no—
312C22×C4—142no—
313C22×C4—142no—
314C22×C4—142no—
315C22×C4—281no—
316C22×C4—281no—
317C22×C4—281no—
318C22×C4C22×D711yescharacteristic
319C22×C4C22×D711yescharacteristic
320C22×C4C22×D711yescharacteristic
321C22×C4C22×D711yescharacteristic
322C22×C4C22×D711yescharacteristic
323C4⋊C4—142no—
324C4⋊C4—142no—
325C4⋊C4—281no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—142no—
330C22⋊C4—142no—
331C22⋊C4—142no—
332C22⋊C4—142no—
333C22⋊C4—142no—
334C22⋊C4—281no—
335C22⋊C4—281no—
336C22⋊C4—281no—
337C22⋊C4—281no—
338C22⋊C4—281no—
339C2×C7—42no—
340C2×C7—42no—
341C2×C7#53411yescharacteristic
342C2×C7#53811yescharacteristic
343C2×C7#54011yescharacteristic
344C2×C7#54611yescharacteristic
345C2×C7#54611yescharacteristic
346C2×C7#55311yescharacteristic
347C2×C7#56011yescharacteristic
348D7—42no—
349D7—81no—
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—72no—
363C23—141no—
364C23—141no—
365C23—141no—
366C23—141no—
367C23—141no—
368C23—141no—
369C23—141no—
370C23—141no—
371C23—141no—
372C23—141no—
373C23—141no—
374C23—141no—
375C23—141no—
376C23C2×D2812yes—
377C23C2×D2812yes—
378C23C23×D711yescharacteristic
379D4—44no—
380D4—284no—
381D4—284no—
382D4—284no—
383D4—284no—
384D4—284no—
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—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—44no—
400C2×C4—72no—
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—141no—
417C2×C4—141no—
418C2×C4—284no—
419C2×C4—284no—
420C2×C4C2×D2812yes—
421C7#268811yescharacteristic, complemented, Sylow
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—142no—
437C22—142no—
438C22—142no—
439C22—281no—
440C22—281no—
441C22—281no—
442C22—281no—
443C22—281no—
444C22—281no—
445C22—281no—
446C22—281no—
447C22—281no—
448C22—281no—
449C22—281no—
450C22C22×D2811yescharacteristic
451C22#654811yescharacteristic
452C22#654811yescharacteristic
453C22#654811yescharacteristic
454C22#654811yescharacteristic
455C22D7×#4111yescharacteristic
456C22D7×#4111yescharacteristic
457C4—22no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—41no—
462C4—142no—
463C4—281no—
464C4—281no—
465C4—281no—
466C4—281no—
467C4—281no—
468C2—42no—
469C2—42no—
470C2—282no—
471C2—561no—
472C2#9018411yescharacteristic
473C2#9024511yescharacteristic
474C2#9024511yescharacteristic
475C2#9028611yescharacteristic
476C2#9030411yescharacteristic
477C2#9030611yescharacteristic
478C2#9035211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
hypercenter#386D7
hypocenterC7#2688
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#6502C2semidirect
2. C2×#6501C2semidirect
3. C7×#386C2semidirect
4. #89985C2semidirect
5. #89985C2semidirect
6. #89733C2semidirect
7. #89732C2semidirect
8. #89732C2semidirect
9. #89730C2semidirect
10. #89725C2semidirect
11. #89715C2semidirect
12. #89446C2semidirect
13. #89440C2semidirect
14. #89437C2semidirect
15. #89431C2semidirect
60. C2×C7×D4D4semidirect
124. #386D7semidirect
302. C2×D4D28semidirect
421. C7#2688semidirect

Maximal subgroups

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

subgroupindex
C2×#65022
C2×#65012
C7×#3862
#899852
#897332
#897322
#897302
#897252
#897152
#894462
#894402
#894372
#894312
#26887