Finite Group Databasebeta

← #11740129

Subgroups — #11740129

Order 896.

676 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×C7×#121C211yesmaximal, characteristic, complemented
2C2×#6502C211yesmaximal, characteristic, complemented
3C2×#6487C211yesmaximal, characteristic, complemented
4C2×#6448C211yesmaximal, characteristic, complemented
5#89995C211yesmaximal, characteristic, complemented
6#89995C211yesmaximal, characteristic, complemented
7#89994C211yesmaximal, characteristic, complemented
8#89993C211yesmaximal, characteristic, complemented
9#89992C211yesmaximal, characteristic, complemented
10#89985C211yesmaximal, characteristic, complemented
11#89766C211yesmaximal, characteristic, complemented
12#89750C211yesmaximal, characteristic, complemented
13#89733C211yesmaximal, characteristic, complemented
14#89722C211yesmaximal, characteristic, complemented
15#89719C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C23×C4×C7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×C22⋊D7C2211yescharacteristic
23C23×Dic7C2211yescharacteristic
24C22×Dic14C2211yescharacteristic
25C7×#121—24no—
26C2×C7×C4⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34C2×C22⋊Dic7C2211yescharacteristic
35C2×C22⋊Dic7C2211yescharacteristic
36C2×C22⋊Dic7C2211yescharacteristic
37#6502—24no—
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×(C2×D7)⋊C4C2211yescharacteristic
42C2×C4⋊Dic7C2211yescharacteristic
43C2×C4⋊Dic7C2211yescharacteristic
44#6487—24no—
45C2×Dic7⋊C4C2211yescharacteristic
46C2×Dic7⋊C4C2211yescharacteristic
47C2×C4×Dic7C2211yescharacteristic
48#6448—24no—
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2676—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C24×C7C2311yescharacteristic
59C23×D7C2311yescharacteristic
60C2×C7×D4—22no—
61C2×C7×D4—22no—
62C2×C7×D4—22no—
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—41no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—41no—
72C22×C4×C7C2311yescharacteristic
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C22×C4×C7C2311yescharacteristic
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⋊D7—41no—
83C2×C22⋊D7—41no—
84C2×C22⋊D7—41no—
85C22×Dic7—22no—
86C22×Dic7—41no—
87C22×Dic7—41no—
88C22×Dic7C2311yescharacteristic
89C22×Dic7C2311yescharacteristic
90C22×Dic7C2311yescharacteristic
91C22×Dic7C2311yescharacteristic
92C22×Dic7C2311yescharacteristic
93C22×Dic7C2311yescharacteristic
94C22×Dic7C2311yescharacteristic
95C2×Dic14—22no—
96C2×Dic14—22no—
97C2×Dic14—41no—
98C7×C4⋊C4—22no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4D414yescomplemented
101C22⋊Dic7—22no—
102C22⋊Dic7—22no—
103C22⋊Dic7—22no—
104C22⋊Dic7—22no—
105C22⋊Dic7—41no—
106C22⋊Dic7—41no—
107C22⋊Dic7—41no—
108C22⋊Dic7—41no—
109(C2×D7)⋊C4—22no—
110(C2×D7)⋊C4—22no—
111(C2×D7)⋊C4—22no—
112(C2×D7)⋊C4—41no—
113C4⋊Dic7—22no—
114C4⋊Dic7—41no—
115Dic7⋊C4—22no—
116Dic7⋊C4D414yescomplemented
117C4×Dic7—22no—
118C2×#124—71no—
119C2×C22⋊Q8—71no—
120C2×#121—71no—
121C2×#121D711yescharacteristic, complemented
122#396—71no—
123#394—71no—
124#394—71no—
125#393—71no—
126#392—71no—
127#392—71no—
128#390—71no—
129#387—71no—
130#386—71no—
131#386—71no—
132#385—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×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7C2×D412yes—
153C23×C7#4112yes—
154C23×C7C2411yescharacteristic
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—21no—
160C22×D7—21no—
161C22×D7—21no—
162C7×D4—44no—
163C7×D4—44no—
164C7×D4—44no—
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—44no—
175C2×C4×C7C2×D412yes—
176C2×C4×C7C2×D412yes—
177C2×C4×C7C2×D412yes—
178C22⋊D7—44no—
179C22⋊D7—44no—
180C22⋊D7—44no—
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—21no—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—21no—
198C2×Dic7—21no—
199C2×Dic7—21no—
200C2×Dic7—44no—
201C2×Dic7C2×D412yes—
202C2×Dic7C2×D412yes—
203Dic14—44no—
204C22×Q8—71no—
205C22×D4—71no—
206C22×D4—71no—
207C22×D4—71no—
208C22×D4C2×D711yescharacteristic
209C22×D4C2×D711yescharacteristic
210C22×D4C2×D711yescharacteristic
211C23×C4—71no—
212C23×C4C2×D711yescharacteristic
213#124—144no—
214C22⋊Q8—144no—
215#121—24no—
216#121—144no—
217C2×C4⋊C4—71no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4—71no—
220C2×C4⋊C4—71no—
221C2×C4⋊C4C2×D711yescharacteristic
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⋊C4—71no—
232C2×C22⋊C4—71no—
233C2×C22⋊C4—71no—
234C2×C22⋊C4C2×D711yescharacteristic
235C2×C22⋊C4C2×D711yescharacteristic
236C2×C42—71no—
237#95—71no—
238#95—71no—
239#95—71no—
240#95—71no—
241#95—71no—
242#95—71no—
243C22×C7—22no—
244C22×C7—22no—
245C22×C7—22no—
246C22×C7—22no—
247C22×C7—22no—
248C22×C7—22no—
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×C7—41no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7C22×D411yescharacteristic
265C22×C7C22×D411yescharacteristic
266C22×C7C2×#4111yescharacteristic
267C22×C7H(5,𝔽2)11yescharacteristic
268C22×C7H(5,𝔽2)11yescharacteristic
269C22×C7H(5,𝔽2)11yescharacteristic
270C22×C72−1+411yescharacteristic
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—22no—
281C4×C7—41no—
282Dic7—22no—
283Dic7—22no—
284Dic7—41no—
285Dic7—41no—
286Dic7—41no—
287Dic7—41no—
288Dic7—41no—
289C24—71no—
290C24C22×D711yescharacteristic
291C24C22×D711yescharacteristic
292C24C22×D711yescharacteristic
293C2×Q8—142no—
294C2×Q8—142no—
295C2×Q8—281no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—142no—
306C2×D4—142no—
307C2×D4—142no—
308C2×D4—142no—
309C2×D4—281no—
310C2×D4—281no—
311C2×D4—281no—
312C2×D4—281no—
313C2×D4—281no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—71no—
318C22×C4—71no—
319C22×C4—71no—
320C22×C4—71no—
321C22×C4—71no—
322C22×C4—71no—
323C22×C4—71no—
324C22×C4—142no—
325C22×C4—281no—
326C22×C4—281no—
327C22×C4C22×D711yescharacteristic
328C22×C4C22×D711yescharacteristic
329C22×C4C22×D711yescharacteristic
330C22×C4C22×D711yescharacteristic
331C4⋊C4—22no—
332C4⋊C4—74no—
333C4⋊C4—142no—
334C4⋊C4—142no—
335C4⋊C4—281no—
336C22⋊C4—22no—
337C22⋊C4—142no—
338C22⋊C4—142no—
339C22⋊C4—142no—
340C22⋊C4—142no—
341C22⋊C4—142no—
342C22⋊C4—142no—
343C22⋊C4—142no—
344C22⋊C4—281no—
345C22⋊C4—281no—
346C22⋊C4—281no—
347C22⋊C4—281no—
348C22⋊C4—281no—
349C22⋊C4C22⋊D714yescomplemented
350C42—142no—
351C2×C7—42no—
352C2×C7—42no—
353C2×C7—81no—
354C2×C7#53511yescharacteristic
355C2×C7#53711yescharacteristic
356C2×C7#53811yescharacteristic
357C2×C7D4211yescharacteristic
358C2×C7#54611yescharacteristic
359C2×C7#54811yescharacteristic
360C2×C7#56211yescharacteristic
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—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—141no—
382C23—141no—
383C23—141no—
384C23—141no—
385C23—141no—
386C23—141no—
387C23—141no—
388C23#115912yes—
389C23C2×C22⋊D712yes—
390C23C23×D711yescharacteristic
391Q8—284no—
392D4—44no—
393D4—44no—
394D4—44no—
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—44no—
408C2×C4—72no—
409C2×C4—72no—
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—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—284no—
430C2×C4D4×D712yes—
431C2×C4C2×C22⋊D712yes—
432C2×C4C2×C22⋊D712yes—
433C7#267611yescharacteristic, complemented, Sylow
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—281no—
456C22—281no—
457C22—281no—
458C22—281no—
459C22—281no—
460C22—281no—
461C22—281no—
462C22C2×D4×D711yescharacteristic
463C22C2×#115911yescharacteristic
464C22#654811yescharacteristic
465C22#654811yescharacteristic
466C22#655311yescharacteristic
467C22#655411yescharacteristic
468C22C22×C22⋊D711yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—142no—
474C4—142no—
475C4—281no—
476C4—281no—
477C4—281no—
478C4—281no—
479C4—281no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2—561no—
484C2#9028911yescharacteristic
485C2#9029811yescharacteristic
486C2#9030811yescharacteristic
487C2#9031211yescharacteristic
488C2D4×C22⋊D711yescharacteristic
489C2#9049711yescharacteristic
490C2#9052411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2676
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C7×#121C2semidirect
2. C2×#6502C2semidirect
3. C2×#6487C2semidirect
4. C2×#6448C2semidirect
5. #89995C2semidirect
6. #89995C2semidirect
7. #89994C2semidirect
8. #89993C2semidirect
9. #89992C2semidirect
10. #89985C2semidirect
11. #89766C2semidirect
12. #89750C2semidirect
13. #89733C2semidirect
14. #89722C2semidirect
15. #89719C2semidirect
100. C7×C22⋊C4D4semidirect
116. Dic7⋊C4D4semidirect
121. C2×#121D7semidirect
349. C22⋊C4C22⋊D7semidirect
433. C7#2676semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1212
C2×#65022
C2×#64872
C2×#64482
#899952
#899942
#899932
#899922
#899852
#897662
#897502
#897332
#897222
#897192
#26767