Finite Group Databasebeta

← #11732198

Subgroups — #11732198

Order 896.

704 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6503C211yesmaximal, characteristic
2C2×C4×C22⋊D7C211yesmaximal, characteristic, complemented
3C2×#6448C211yesmaximal, characteristic, complemented
4C2×#6448C211yesmaximal, characteristic, complemented
5C7×#386C211yesmaximal, characteristic, complemented
6#89994C211yesmaximal, characteristic, complemented
7#89985C211yesmaximal, characteristic, complemented
8#89731C211yesmaximal, characteristic, complemented
9#89731C211yesmaximal, characteristic, complemented
10#89729C211yesmaximal, characteristic, complemented
11#89728C211yesmaximal, characteristic, complemented
12#89444C211yesmaximal, characteristic, complemented
13#89437C211yesmaximal, characteristic, complemented
14#89421C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×D28C2211yescharacteristic
23C22×C4×D7C2211yescharacteristic
24C22×Dic14C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32#6503—24no—
33C4×C22⋊D7—24no—
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×(C2×D7)⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×Dic7⋊C4C2211yescharacteristic
44C2×C4×Dic7C2211yescharacteristic
45C2×C4×Dic7C2211yescharacteristic
46C2×C4×Dic7C2211yescharacteristic
47#6448—24no—
48#6448—24no—
49C7×#95C2211yescharacteristic
50C7×#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2544—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C23×D7C2311yescharacteristic
59C23×D7C2311yescharacteristic
60C2×C7×D4—22no—
61C2×C7×D4—22no—
62C2×C7×D4—41no—
63C22×C4×C7—22no—
64C22×C4×C7—41no—
65C22×C4×C7—41no—
66C22×C4×C7C2311yescharacteristic
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—22no—
77C2×C22⋊D7—22no—
78C2×C22⋊D7—22no—
79C2×C22⋊D7—22no—
80C2×C22⋊D7—41no—
81C2×C22⋊D7—41no—
82C2×C22⋊D7D414yes—
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C22×Dic7C2311yescharacteristic
89C2×D28—22no—
90C2×D28—22no—
91C2×D28—41no—
92C2×C4×D7—22no—
93C2×C4×D7—41no—
94C2×C4×D7—41no—
95C2×Dic14—22no—
96C2×Dic14—22no—
97C2×Dic14—22no—
98C7×C22⋊C4—22no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4—41no—
101C22⋊Dic7—22no—
102C22⋊Dic7—22no—
103C22⋊Dic7—22no—
104C22⋊Dic7—41no—
105(C2×D7)⋊C4—22no—
106(C2×D7)⋊C4—22no—
107(C2×D7)⋊C4—22no—
108(C2×D7)⋊C4—22no—
109(C2×D7)⋊C4—22no—
110(C2×D7)⋊C4—41no—
111(C2×D7)⋊C4—41no—
112Dic7⋊C4—22no—
113Dic7⋊C4—22no—
114Dic7⋊C4—22no—
115C4×Dic7—22no—
116C4×Dic7—22no—
117C4×Dic7D414yescomplemented
118C2×#127—71no—
119C2×#124—71no—
120C2×#124—71no—
121C2×C4×D4—71no—
122#394—71no—
123#394—71no—
124#394—71no—
125#394—71no—
126#391—71no—
127#390—71no—
128#386—71no—
129#386—71no—
130#386—71no—
131#386D711yescharacteristic, complemented
132C4×C4⋊C4—71no—
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×C7C2×D412yes—
147C23×C7C2411yescharacteristic
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—21no—
153C22×D7—21no—
154C22×D7—21no—
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—21no—
160C22×D7—21no—
161C22×D7C2×D412yes—
162C7×D4—44no—
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—
175C2×C4×C7—21no—
176C2×C4×C7—44no—
177C2×C4×C7C2×D412yes—
178C2×C4×C7#4112yes—
179C22⋊D7—28no—
180C22⋊D7—44no—
181C22⋊D7—44no—
182C22⋊D7—44no—
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×Dic7C2×D412yes—
197C2×Dic7C2×D412yes—
198C2×Dic7C2×D412yes—
199C2×Dic7#4112yes—
200C2×Dic7#4112yes—
201D28—44no—
202C4×D7—44no—
203Dic14—44no—
204C22×Q8—71no—
205C22×D4—71no—
206C22×D4—71no—
207C22×D4—71no—
208C22×D4—71no—
209C22×D4—71no—
210C22×D4C2×D711yescharacteristic
211C23×C4—71no—
212C23×C4C2×D711yescharacteristic
213#127—144no—
214#124—144no—
215#124—144no—
216C4×D4—144no—
217C2×C4⋊C4—71no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4—71no—
220C2×C22⋊C4—71no—
221C2×C22⋊C4—71no—
222C2×C22⋊C4—71no—
223C2×C22⋊C4—71no—
224C2×C22⋊C4—71no—
225C2×C22⋊C4—71no—
226C2×C22⋊C4—71no—
227C2×C22⋊C4—71no—
228C2×C22⋊C4—71no—
229C2×C22⋊C4—71no—
230C2×C22⋊C4—71no—
231C2×C22⋊C4C2×D711yescharacteristic
232C2×C22⋊C4C2×D711yescharacteristic
233C2×C22⋊C4C2×D711yescharacteristic
234C2×C42—71no—
235C2×C42—71no—
236C2×C42—71no—
237#95—71no—
238#95—71no—
239#95—71no—
240#95—71no—
241#95C2×D711yescharacteristic
242#95C2×D711yescharacteristic
243C22×C7—22no—
244C22×C7—22no—
245C22×C7—22no—
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×C7—41no—
256C22×C7—41no—
257C22×C7C22×D411yescharacteristic
258C22×C7C22×D411yescharacteristic
259C22×C7C2×#4111yescharacteristic
260C22×C7C2×#4111yescharacteristic
261C22×C7C2×#4111yescharacteristic
262C22×C7H(5,𝔽2)11yescharacteristic
263C22×C7H(5,𝔽2)11yescharacteristic
264C2×D7—22no—
265C2×D7—22no—
266C2×D7—22no—
267C2×D7—41no—
268C2×D7—41no—
269C2×D7—41no—
270C2×D7—41no—
271C2×D7—41no—
272C2×D7—41no—
273C2×D7—41no—
274C2×D7—41no—
275C2×D7—41no—
276C2×D7—41no—
277C2×D7—41no—
278C4×C7—22no—
279C4×C7—22no—
280C4×C7—41no—
281C4×C7—41no—
282C4×C7—41no—
283Dic7—22no—
284Dic7—22no—
285Dic7—41no—
286Dic7—41no—
287Dic7—41no—
288Dic7#12114yes—
289C24—71no—
290C24—71no—
291C24C22×D711yescharacteristic
292C24C22×D711yescharacteristic
293C2×Q8—142no—
294C2×Q8—142no—
295C2×Q8—142no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—74no—
300C2×D4—142no—
301C2×D4—142no—
302C2×D4—142no—
303C2×D4—142no—
304C2×D4—142no—
305C2×D4—142no—
306C2×D4—142no—
307C2×D4—142no—
308C2×D4—142no—
309C2×D4—142no—
310C2×D4—142no—
311C2×D4—281no—
312C2×D4—281no—
313C2×D4—281no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—71no—
318C22×C4—71no—
319C22×C4—71no—
320C22×C4—71no—
321C22×C4—71no—
322C22×C4—71no—
323C22×C4—142no—
324C22×C4—281no—
325C22×C4—281no—
326C22×C4C22×D711yescharacteristic
327C22×C4C22×D711yescharacteristic
328C22×C4C22×D711yescharacteristic
329C22×C4C22×D711yescharacteristic
330C22×C4C22×D711yescharacteristic
331C4⋊C4—142no—
332C4⋊C4—142no—
333C4⋊C4—142no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—41no—
337C22⋊C4—142no—
338C22⋊C4—142no—
339C22⋊C4—142no—
340C22⋊C4—142no—
341C22⋊C4—142no—
342C22⋊C4—142no—
343C22⋊C4—142no—
344C22⋊C4—142no—
345C22⋊C4—281no—
346C22⋊C4—281no—
347C22⋊C4—281no—
348C42—74no—
349C42—142no—
350C42—142no—
351C2×C7—42no—
352C2×C7—81no—
353C2×C7C2×#12111yescharacteristic
354C2×C7#53211yescharacteristic
355C2×C7#53511yescharacteristic
356C2×C7#54611yescharacteristic
357C2×C7#54611yescharacteristic
358C2×C7#55911yescharacteristic
359C2×C7#55911yescharacteristic
360D7—42no—
361D7—81no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—72no—
376C23—141no—
377C23—141no—
378C23—141no—
379C23—141no—
380C23—141no—
381C23—141no—
382C23—141no—
383C23—141no—
384C23—141no—
385C23—141no—
386C23—141no—
387C23—141no—
388C23—141no—
389C23D4×D712yes—
390C23C23×D711yescharacteristic
391Q8—284no—
392D4—44no—
393D4—148no—
394D4—284no—
395D4—284no—
396D4—284no—
397D4—284no—
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—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—44no—
412C2×C4—72no—
413C2×C4—72no—
414C2×C4—72no—
415C2×C4—72no—
416C2×C4—72no—
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—141no—
428C2×C4—141no—
429C2×C4—141no—
430C2×C4—284no—
431C2×C4#115712yes—
432C2×C4D4×D712yes—
433C7#254411yescharacteristic, complemented, Sylow
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—142no—
449C22—142no—
450C22—142no—
451C22—281no—
452C22—281no—
453C22—281no—
454C22—281no—
455C22—281no—
456C22—281no—
457C22—281no—
458C22—281no—
459C22—281no—
460C22—281no—
461C22—281no—
462C22C2×#115711yescharacteristic
463C22C2×D4×D711yescharacteristic
464C22C2×D4×D711yescharacteristic
465C22#654811yescharacteristic
466C22#654811yescharacteristic
467C22D7×#4111yescharacteristic
468C22D7×#4111yescharacteristic
469C4—22no—
470C4—22no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—74no—
475C4—142no—
476C4—142no—
477C4—281no—
478C4—281no—
479C4—281no—
480C2—42no—
481C2—81no—
482C2—282no—
483C2—561no—
484C2#9018111yescharacteristic
485C2#9023911yescharacteristic
486C2#9025011yescharacteristic
487C2#9028311yescharacteristic
488C2D7×#12111yescharacteristic
489C2#9030111yescharacteristic
490C2#9035511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC2×#6503C2
hypercenter#386D7
hypocenterC7#2544
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D7C2semidirect
3. C2×#6448C2semidirect
4. C2×#6448C2semidirect
5. C7×#386C2semidirect
6. #89994C2semidirect
7. #89985C2semidirect
8. #89731C2semidirect
9. #89731C2semidirect
10. #89729C2semidirect
11. #89728C2semidirect
12. #89444C2semidirect
13. #89437C2semidirect
14. #89421C2semidirect
15. #89418C2semidirect
117. C4×Dic7D4semidirect
131. #386D7semidirect
433. C7#2544semidirect

Maximal subgroups

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

subgroupindex
C2×#65032
C2×C4×C22⋊D72
C2×#64482
C7×#3862
#899942
#899852
#897312
#897292
#897282
#894442
#894372
#894212
#894182
#25447