Finite Group Databasebeta

← #11734194

Subgroups — #11734194

Order 896.

508 conjugacy classes of subgroups.

The subgroup lattice has 462 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#89994C211yesmaximal, characteristic, complemented
7#89993C211yesmaximal, characteristic, complemented
8#89733C211yesmaximal, characteristic, complemented
9#89733C211yesmaximal, characteristic, complemented
10#89731C211yesmaximal, characteristic, complemented
11#89725C211yesmaximal, characteristic, complemented
12#89724C211yesmaximal, characteristic, complemented
13#89723C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89432C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
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×C4⋊Dic7C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43C7×#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2828—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C24×C7C2311yescharacteristic
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C2×C7×D4—22no—
57C2×C7×D4—41no—
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—41no—
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×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C7×C22⋊C4—22no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—41no—
88C22⋊Dic7—41no—
89C22⋊Dic7—41no—
90C22⋊Dic7—41no—
91C22⋊Dic7—41no—
92C22⋊Dic7—41no—
93C22⋊Dic7—41no—
94C22⋊Dic7—41no—
95C22⋊Dic7—41no—
96C22⋊Dic7—41no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—41no—
99(C2×D7)⋊C4—41no—
100(C2×D7)⋊C4—41no—
101C4⋊Dic7—41no—
102C4⋊Dic7—41no—
103Dic7⋊C4—41no—
104Dic7⋊C4—41no—
105C4×Dic7—41no—
106#400—71no—
107#397—71no—
108#397—71no—
109#394—71no—
110#394—71no—
111#394—71no—
112#394—71no—
113#394—71no—
114#392—71no—
115#392—71no—
116#392—71no—
117#392D711yescharacteristic, complemented
118#390—71no—
119#388—71no—
120#388—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—21no—
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#4112yes—
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—
189C22×D4—71no—
190C22×D4—71no—
191C22×D4—71no—
192C22×D4C2×D711yescharacteristic
193C22×D4C2×D711yescharacteristic
194C22×D4C2×D711yescharacteristic
195C2×C4⋊C4—71no—
196C2×C4⋊C4—71no—
197C2×C4⋊C4—71no—
198C2×C4⋊C4—71no—
199C2×C22⋊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
215C2×C42—71no—
216#95—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95—71no—
221#95—71no—
222#95—71no—
223#95C2×D711yescharacteristic
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
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×C7C2×#4111yescharacteristic
246C22×C7H(5,𝔽2)11yescharacteristic
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
252C2×D7—41no—
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—41no—
256C2×D7—41no—
257C2×D7—41no—
258C2×D7—41no—
259C4×C7—22no—
260C4×C7—41no—
261C4×C7—41no—
262C4×C7—41no—
263Dic7—41no—
264Dic7—41no—
265Dic7—41no—
266Dic7—41no—
267Dic7—41no—
268Dic7—41no—
269Dic7—41no—
270C24—71no—
271C24C22×D711yescharacteristic
272C24C22×D711yescharacteristic
273C24C22×D711yescharacteristic
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—281no—
284C2×D4—281no—
285C2×D4—281no—
286C2×D4—281no—
287C2×D4—281no—
288C2×D4—281no—
289C2×D4—281no—
290C2×D4—281no—
291C2×D4—281no—
292C22×C4—71no—
293C22×C4—71no—
294C22×C4—71no—
295C22×C4—71no—
296C22×C4—71no—
297C22×C4—71no—
298C22×C4—71no—
299C22×C4C22×D711yescharacteristic
300C22×C4C22×D711yescharacteristic
301C22×C4C22×D711yescharacteristic
302C22×C4C22×D711yescharacteristic
303C4⋊C4—281no—
304C4⋊C4—281no—
305C4⋊C4—281no—
306C4⋊C4—281no—
307C22⋊C4—22no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—142no—
311C22⋊C4—281no—
312C22⋊C4—281no—
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—
323C42—281no—
324C2×C7—81no—
325C2×C7—81no—
326C2×C7—81no—
327C2×C7#53811yescharacteristic
328C2×C7#54011yescharacteristic
329C2×C7#54011yescharacteristic
330C2×C7#56011yescharacteristic
331C2×C7#56011yescharacteristic
332C2×C7#56011yescharacteristic
333C2×C7H(3,𝔽4)11yescharacteristic
334D7—81no—
335C23—21no—
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—
363C23C23×D711yescharacteristic
364D4—44no—
365D4—44no—
366D4—44no—
367D4—284no—
368D4—284no—
369D4—284no—
370C2×C4—21no—
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—141no—
382C2×C4—141no—
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#115912yes—
403C7#282811yescharacteristic, complemented, Sylow
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
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—281no—
426C22—281no—
427C22—281no—
428C22—281no—
429C22—281no—
430C22—281no—
431C22—281no—
432C22C2×#115911yescharacteristic
433C22#654811yescharacteristic
434C22#654811yescharacteristic
435C22#654811yescharacteristic
436C22#654811yescharacteristic
437C22#654811yescharacteristic
438C22#655311yescharacteristic
439C4—22no—
440C4—41no—
441C4—41no—
442C4—41no—
443C4—281no—
444C4—281no—
445C4—281no—
446C4—281no—
447C4—281no—
448C4—281no—
449C4—281no—
450C2—81no—
451C2—81no—
452C2—81no—
453C2—561no—
454C2#9028411yescharacteristic
455C2#9028511yescharacteristic
456C2#9028611yescharacteristic
457C2#9031011yescharacteristic
458C2#9031111yescharacteristic
459C2#9031111yescharacteristic
460C2#9041011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2828
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. #89994C2semidirect
7. #89993C2semidirect
8. #89733C2semidirect
9. #89733C2semidirect
10. #89731C2semidirect
11. #89725C2semidirect
12. #89724C2semidirect
13. #89723C2semidirect
14. #89441C2semidirect
15. #89432C2semidirect
117. #392D7semidirect
403. C7#2828semidirect

Maximal subgroups

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

subgroupindex
C7×#3922
#899952
#899942
#899932
#897332
#897312
#897252
#897242
#897232
#894412
#894322
#28287