Finite Group Databasebeta

← #11734192

Subgroups — #11734192

Order 896.

522 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#392C211yesmaximal, characteristic, complemented
2#89995C211yesmaximal, characteristic, complemented
3#89995C211yesmaximal, characteristic, complemented
4#89995C211yesmaximal, characteristic, complemented
5#89994C211yesmaximal, characteristic, complemented
6#89992C211yesmaximal, characteristic, complemented
7#89992C211yesmaximal, characteristic, complemented
8#89733C211yesmaximal, characteristic, complemented
9#89731C211yesmaximal, characteristic, complemented
10#89730C211yesmaximal, characteristic, complemented
11#89725C211yesmaximal, characteristic, complemented
12#89724C211yesmaximal, characteristic, complemented
13#89720C211yesmaximal, characteristic, complemented
14#89447C211yesmaximal, characteristic, complemented
15C7⋊#401C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C23×Dic7C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×C4⋊Dic7C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C7×#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2825—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C24×C7C2311yescharacteristic
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C2×C7×D4—22no—
57C2×C7×D4—22no—
58C2×C7×D4—41no—
59C2×C7×D4—41no—
60C2×C7×D4—41no—
61C2×C7×D4—41no—
62C2×C7×D4—41no—
63C2×C7×D4—41no—
64C2×C7×D4—41no—
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—22no—
70C2×C22⋊D7—41no—
71C2×C22⋊D7—41no—
72C2×C22⋊D7—41no—
73C2×C22⋊D7—41no—
74C2×C22⋊D7—41no—
75C2×C22⋊D7—41no—
76C2×C22⋊D7—41no—
77C2×C22⋊D7—41no—
78C22×Dic7—41no—
79C22×Dic7—41no—
80C22×Dic7—41no—
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C22⋊Dic7—41no—
92C22⋊Dic7—41no—
93C22⋊Dic7—41no—
94C22⋊Dic7—41no—
95C22⋊Dic7—41no—
96C22⋊Dic7—41no—
97C22⋊Dic7—41no—
98C22⋊Dic7—41no—
99(C2×D7)⋊C4—41no—
100(C2×D7)⋊C4—41no—
101(C2×D7)⋊C4—41no—
102(C2×D7)⋊C4—41no—
103C4⋊Dic7—41no—
104Dic7⋊C4—41no—
105Dic7⋊C4—41no—
106#401—71no—
107#397—71no—
108#397—71no—
109#397—71no—
110#396—71no—
111#394—71no—
112#394—71no—
113#394—71no—
114#392—71no—
115#392—71no—
116#392—71no—
117#392D711yescharacteristic, complemented
118#386—71no—
119#386—71no—
120#386—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
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#4112yes—
142C23×C7C2411yescharacteristic
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C7×D4—44no—
151C7×D4—44no—
152C7×D4—44no—
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—
165C22⋊D7—44no—
166C22⋊D7—44no—
167C22⋊D7—44no—
168C2×Dic7—21no—
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—21no—
187C2×Dic7—21no—
188C2×Dic7—21no—
189C2×Dic7—44no—
190C22×D4—71no—
191C22×D4—71no—
192C22×D4—71no—
193C22×D4C2×D711yescharacteristic
194C22×D4C2×D711yescharacteristic
195C22×D4C2×D711yescharacteristic
196C23×C4—71no—
197C2×C4⋊C4—71no—
198C2×C4⋊C4—71no—
199C2×C4⋊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⋊C4C2×D711yescharacteristic
213C2×C22⋊C4C2×D711yescharacteristic
214C2×C22⋊C4C2×D711yescharacteristic
215#95—71no—
216#95—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95—71no—
221#95—71no—
222#95—71no—
223#95—71no—
224#95C2×D711yescharacteristic
225C22×C7—22no—
226C22×C7—22no—
227C22×C7—22no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
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×C7C2×#4111yescharacteristic
247C22×C7H(5,𝔽2)11yescharacteristic
248C22×C7H(5,𝔽2)11yescharacteristic
249C22×C7H(5,𝔽2)11yescharacteristic
250C22×C7H(5,𝔽2)11yescharacteristic
251C22×C7H(5,𝔽2)11yescharacteristic
252C22×C7H(5,𝔽2)11yescharacteristic
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—41no—
256C2×D7—41no—
257C2×D7—41no—
258C2×D7—41no—
259C2×D7—41no—
260C4×C7—41no—
261C4×C7—41no—
262C4×C7—41no—
263C4×C7—41no—
264Dic7—41no—
265Dic7—41no—
266Dic7—41no—
267Dic7—41no—
268Dic7—41no—
269Dic7—41no—
270Dic7—41no—
271C24—71no—
272C24C22×D711yescharacteristic
273C24C22×D711yescharacteristic
274C24C22×D711yescharacteristic
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—142no—
285C2×D4—281no—
286C2×D4—281no—
287C2×D4—281no—
288C2×D4—281no—
289C2×D4—281no—
290C2×D4—281no—
291C2×D4—281no—
292C2×D4—281no—
293C22×C4—71no—
294C22×C4—71no—
295C22×C4—71no—
296C22×C4—71no—
297C22×C4—71no—
298C22×C4—71no—
299C22×C4—71no—
300C22×C4—281no—
301C22×C4—281no—
302C22×C4—281no—
303C22×C4C22×D711yescharacteristic
304C22×C4C22×D711yescharacteristic
305C22×C4C22×D711yescharacteristic
306C22×C4C22×D711yescharacteristic
307C4⋊C4—281no—
308C4⋊C4—281no—
309C4⋊C4—281no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—281no—
314C22⋊C4—281no—
315C22⋊C4—281no—
316C22⋊C4—281no—
317C22⋊C4—281no—
318C22⋊C4—281no—
319C22⋊C4—281no—
320C22⋊C4—281no—
321C22⋊C4—281no—
322C22⋊C4—281no—
323C22⋊C4—281no—
324C22⋊C4—281no—
325C2×C7—42no—
326C2×C7—81no—
327C2×C7—81no—
328C2×C7#53811yescharacteristic
329C2×C7#53811yescharacteristic
330C2×C7#53811yescharacteristic
331C2×C7#56011yescharacteristic
332C2×C7#56011yescharacteristic
333C2×C7#56011yescharacteristic
334C2×C7#56011yescharacteristic
335D7—81no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—141no—
357C23—141no—
358C23—141no—
359C23—141no—
360C23—141no—
361C23—141no—
362C23—141no—
363C23#115912yes—
364C23C23×D711yescharacteristic
365D4—44no—
366D4—44no—
367D4—44no—
368D4—284no—
369D4—284no—
370D4—284no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
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—141no—
394C2×C4—141no—
395C2×C4—141no—
396C2×C4—141no—
397C2×C4—141no—
398C2×C4—141no—
399C2×C4—141no—
400C2×C4—141no—
401C2×C4—141no—
402C2×C4—141no—
403C2×C4—141no—
404C2×C4—284no—
405C7#282511yescharacteristic, complemented, Sylow
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—281no—
428C22—281no—
429C22—281no—
430C22—281no—
431C22—281no—
432C22—281no—
433C22—281no—
434C22C2×#115911yescharacteristic
435C22#654811yescharacteristic
436C22#654811yescharacteristic
437C22#654811yescharacteristic
438C22#654811yescharacteristic
439C22#654811yescharacteristic
440C22#655311yescharacteristic
441C4—41no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—281no—
446C4—281no—
447C4—281no—
448C4—281no—
449C4—281no—
450C4—281no—
451C4—281no—
452C2—42no—
453C2—81no—
454C2—81no—
455C2—561no—
456C2#9028411yescharacteristic
457C2#9028411yescharacteristic
458C2#9028611yescharacteristic
459C2#9030811yescharacteristic
460C2#9031111yescharacteristic
461C2#9031111yescharacteristic
462C2#9041111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2825
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C7×#392C2semidirect
2. #89995C2semidirect
3. #89995C2semidirect
4. #89995C2semidirect
5. #89994C2semidirect
6. #89992C2semidirect
7. #89992C2semidirect
8. #89733C2semidirect
9. #89731C2semidirect
10. #89730C2semidirect
11. #89725C2semidirect
12. #89724C2semidirect
13. #89720C2semidirect
14. #89447C2semidirect
15. C7⋊#401C2semidirect
117. #392D7semidirect
405. C7#2825semidirect

Maximal subgroups

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

subgroupindex
C7×#3922
#899952
#899942
#899922
#897332
#897312
#897302
#897252
#897242
#897202
#894472
C7⋊#4012
#28257