Finite Group Databasebeta

← #11734113

Subgroups — #11734113

Order 896.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#1159C211yesmaximal, characteristic, complemented
2C2×#6502C211yesmaximal, characteristic, complemented
3C2×D4×Dic7C211yesmaximal, characteristic, complemented
4C2×#6441C211yesmaximal, characteristic, complemented
5C7×#392C211yesmaximal, characteristic, complemented
6#89995C211yesmaximal, characteristic, complemented
7#89994C211yesmaximal, characteristic, complemented
8#89992C211yesmaximal, characteristic, complemented
9#89733C211yesmaximal, characteristic, complemented
10#89729C211yesmaximal, characteristic, complemented
11#89728C211yesmaximal, characteristic, complemented
12#89722C211yesmaximal, characteristic, complemented
13#89714C211yesmaximal, characteristic, complemented
14#89440C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C23×Dic7C2211yescharacteristic
23C23×Dic7C2211yescharacteristic
24C23×Dic7C2211yescharacteristic
25C2×#1159—24no—
26C2×#1159—24no—
27C2×#1159—24no—
28C22×C4×D7C2211yescharacteristic
29C22×Dic14C2211yescharacteristic
30C2×C7×C22⋊C4C2211yescharacteristic
31C2×C7×C22⋊C4C2211yescharacteristic
32C2×C7×C22⋊C4C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34C2×C22⋊Dic7C2211yescharacteristic
35C2×C22⋊Dic7C2211yescharacteristic
36C2×C22⋊Dic7C2211yescharacteristic
37C2×C22⋊Dic7C2211yescharacteristic
38C2×C22⋊Dic7C2211yescharacteristic
39#6502—24no—
40D4×Dic7—24no—
41C2×(C2×D7)⋊C4C2211yescharacteristic
42C2×(C2×D7)⋊C4C2211yescharacteristic
43C2×(C2×D7)⋊C4C2211yescharacteristic
44C2×C4⋊Dic7C2211yescharacteristic
45C2×C4⋊Dic7C2211yescharacteristic
46C2×Dic7⋊C4C2211yescharacteristic
47C2×Dic7⋊C4C2211yescharacteristic
48C2×Dic7⋊C4C2211yescharacteristic
49C2×C4×Dic7C2211yescharacteristic
50#6441—24no—
51C7×#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55C7⋊#95C2211yescharacteristic
56C7⋊#95C2211yescharacteristic
57#2407—71nomaximal, Sylow
58C24×C7C2311yescharacteristic
59C24×C7C2311yescharacteristic
60C24×C7C2311yescharacteristic
61C23×D7C2311yescharacteristic
62C2×C7×D4—22no—
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—22no—
67C2×C7×D4—22no—
68C2×C7×D4—22no—
69C2×C7×D4—22no—
70C2×C7×D4—41no—
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C2×C22⋊D7—22no—
76C2×C22⋊D7—22no—
77C2×C22⋊D7—22no—
78C2×C22⋊D7—22no—
79C2×C22⋊D7—22no—
80C2×C22⋊D7—22no—
81C2×C22⋊D7—22no—
82C2×C22⋊D7—41no—
83C2×C22⋊D7D414yescomplemented
84C22×Dic7—22no—
85C22×Dic7—22no—
86C22×Dic7—22no—
87C22×Dic7—22no—
88C22×Dic7—22no—
89C22×Dic7—22no—
90C22×Dic7—22no—
91C22×Dic7—22no—
92C22×Dic7D414yescomplemented
93C22×Dic7C2311yescharacteristic
94C22×Dic7C2311yescharacteristic
95C22×Dic7C2311yescharacteristic
96C22×Dic7C2311yescharacteristic
97C22×Dic7C2311yescharacteristic
98C22×Dic7C2311yescharacteristic
99C22×Dic7C2311yescharacteristic
100#1159—232no—
101C2×C4×D7—22no—
102C2×C4×D7—22no—
103C2×C4×D7—22no—
104C2×Dic14—22no—
105C2×Dic14—22no—
106C2×Dic14D414yescomplemented
107C7×C22⋊C4—22no—
108C7×C22⋊C4—22no—
109C7×C22⋊C4—41no—
110C22⋊Dic7—22no—
111C22⋊Dic7—22no—
112C22⋊Dic7—22no—
113C22⋊Dic7—22no—
114C22⋊Dic7—41no—
115C22⋊Dic7—41no—
116(C2×D7)⋊C4—22no—
117(C2×D7)⋊C4—22no—
118(C2×D7)⋊C4—41no—
119C4⋊Dic7—22no—
120C4⋊Dic7—22no—
121Dic7⋊C4—22no—
122Dic7⋊C4—22no—
123Dic7⋊C4—22no—
124C4×Dic7—22no—
125C22×#41—71no—
126C2×C22⋊Q8—71no—
127C2×#121—71no—
128C2×C4×D4—71no—
129#394—71no—
130#394—71no—
131#392—71no—
132#392D711yescharacteristic, complemented
133#391—71no—
134#390—71no—
135#386—71no—
136#386—71no—
137#385—71no—
138#385—71no—
139#380—71no—
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×C7—21no—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—21no—
157C23×C7—21no—
158C23×C7C2×D412yes—
159C23×C7C2×D412yes—
160C23×C7#4112yescomplemented
161C23×C7C2411yescharacteristic
162C22×D7—21no—
163C22×D7—21no—
164C22×D7—21no—
165C22×D7—21no—
166C22×D7—21no—
167C22×D7—21no—
168C22×D7C2×D412yes—
169C7×D4—28no—
170C7×D4—28no—
171C7×D4—44no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7C2×D412yes—
183C2×C4×C7#4112yescomplemented
184C22⋊D7—28no—
185C22⋊D7—28no—
186C22⋊D7—44no—
187C2×Dic7—28no—
188C2×Dic7—21no—
189C2×Dic7—21no—
190C2×Dic7—21no—
191C2×Dic7—21no—
192C2×Dic7—28no—
193C2×Dic7—21no—
194C2×Dic7—21no—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—21no—
198C2×Dic7—21no—
199C2×Dic7—21no—
200C2×Dic7—21no—
201C2×Dic7—21no—
202C2×Dic7—21no—
203C2×Dic7—21no—
204C2×Dic7—21no—
205C2×Dic7—44no—
206C2×Dic7C2×D412yes—
207C2×Dic7C2×D412yes—
208C2×Dic7C2×D412yes—
209C2×Dic7C2×D412yes—
210C2×Dic7C2×D412yes—
211C4×D7—28no—
212Dic14—28no—
213C2×#41—144no—
214C2×#41—144no—
215C2×#41—144no—
216C22×Q8—71no—
217C22×D4—71no—
218C22×D4—71no—
219C22×D4—71no—
220C22×D4C2×D711yescharacteristic
221C22×D4C2×D711yescharacteristic
222C22×D4C2×D711yescharacteristic
223C23×C4—71no—
224C23×C4—71no—
225C23×C4—71no—
226C23×C4—71no—
227C22⋊Q8—144no—
228#121—144no—
229C4×D4—144no—
230C2×C4⋊C4—71no—
231C2×C4⋊C4—71no—
232C2×C4⋊C4—71no—
233C2×C4⋊C4—71no—
234C2×C4⋊C4—71no—
235C2×C22⋊C4—71no—
236C2×C22⋊C4—71no—
237C2×C22⋊C4—71no—
238C2×C22⋊C4—71no—
239C2×C22⋊C4—71no—
240C2×C22⋊C4—71no—
241C2×C22⋊C4—71no—
242C2×C22⋊C4—71no—
243C2×C22⋊C4—71no—
244C2×C22⋊C4C2×D711yescharacteristic
245C2×C22⋊C4C2×D711yescharacteristic
246C2×C22⋊C4C2×D711yescharacteristic
247C2×C42—71no—
248#95—71no—
249#95—71no—
250#95—71no—
251#95—71no—
252#95—71no—
253#95C2×D711yescharacteristic
254C22×C7—22no—
255C22×C7—22no—
256C22×C7—22no—
257C22×C7—22no—
258C22×C7—22no—
259C22×C7—22no—
260C22×C7—22no—
261C22×C7—22no—
262C22×C7—22no—
263C22×C7—41no—
264C22×C7—41no—
265C22×C7—41no—
266C22×C7—41no—
267C22×C7—41no—
268C22×C7—41no—
269C22×C7—41no—
270C22×C7—41no—
271C22×C7—41no—
272C22×C7—41no—
273C22×C7—41no—
274C22×C7—41no—
275C22×C7C22×D411yescharacteristic
276C22×C7C22×D411yescharacteristic
277C22×C7C22×D411yescharacteristic
278C22×C7C2×#4111yescharacteristic
279C22×C7C2×#4111yescharacteristic
280C22×C7H(5,𝔽2)11yescharacteristic
281C22×C72−1+411yescharacteristic
282C2×D7—22no—
283C2×D7—22no—
284C2×D7—22no—
285C2×D7—41no—
286C2×D7—41no—
287C2×D7—41no—
288C2×D7—41no—
289C4×C7—22no—
290C4×C7—22no—
291C4×C7—41no—
292C4×C7—41no—
293Dic7—22no—
294Dic7—22no—
295Dic7—41no—
296Dic7—41no—
297Dic7—41no—
298Dic7—41no—
299Dic7#12014yes—
300C24—71no—
301C24C22×D711yescharacteristic
302C24C22×D711yescharacteristic
303C24C22×D711yescharacteristic
304#41—1432no—
305C2×Q8—74no—
306C2×Q8—142no—
307C2×Q8—142no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—41no—
317C2×D4—74no—
318C2×D4—142no—
319C2×D4—142no—
320C2×D4—142no—
321C2×D4—142no—
322C2×D4—142no—
323C2×D4—142no—
324C2×D4—142no—
325C2×D4—281no—
326C22×C4—71no—
327C22×C4—74no—
328C22×C4—71no—
329C22×C4—71no—
330C22×C4—71no—
331C22×C4—71no—
332C22×C4—71no—
333C22×C4—71no—
334C22×C4—142no—
335C22×C4—142no—
336C22×C4—142no—
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×C4C22×D711yescharacteristic
346C22×C4C22×D711yescharacteristic
347C22×C4C22×D711yescharacteristic
348C22×C4C22×D711yescharacteristic
349C4⋊C4—142no—
350C4⋊C4—142no—
351C4⋊C4—142no—
352C4⋊C4—142no—
353C4⋊C4—142no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—41no—
357C22⋊C4—142no—
358C22⋊C4—142no—
359C22⋊C4—142no—
360C22⋊C4—142no—
361C22⋊C4—142no—
362C22⋊C4—142no—
363C22⋊C4—281no—
364C22⋊C4—281no—
365C22⋊C4—281no—
366C42—142no—
367C2×C7—42no—
368C2×C7—42no—
369C2×C7—42no—
370C2×C7C2×#12011yescharacteristic
371C2×C7#52511yescharacteristic
372C2×C7#53711yescharacteristic
373C2×C7#54611yescharacteristic
374C2×C7#54711yescharacteristic
375C2×C7#54811yescharacteristic
376C2×C7#55011yescharacteristic
377D7—42no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—72no—
397C23—141no—
398C23—141no—
399C23—141no—
400C23—141no—
401C23—141no—
402C23—141no—
403C23D4×D712yes—
404C23D4×D712yes—
405C23#115912yescomplemented
406C23C23×D711yescharacteristic
407Q8—148no—
408D4—28no—
409D4—28no—
410D4—44no—
411D4—148no—
412D4—148no—
413D4—284no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—72no—
425C2×C4—72no—
426C2×C4—72no—
427C2×C4—72no—
428C2×C4—72no—
429C2×C4—148no—
430C2×C4—148no—
431C2×C4—148no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—141no—
436C2×C4—141no—
437C2×C4—141no—
438C2×C4—141no—
439C2×C4—141no—
440C2×C4—141no—
441C2×C4—141no—
442C2×C4—141no—
443C2×C4—141no—
444C2×C4—141no—
445C2×C4—141no—
446C2×C4—141no—
447C2×C4—141no—
448C2×C4—284no—
449C2×C4D4×D712yes—
450C2×C4#115912yescomplemented
451C7#240711yescharacteristic, complemented, Sylow
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—142no—
474C22—142no—
475C22—142no—
476C22—281no—
477C22—281no—
478C22—281no—
479C22—281no—
480C22C2×D4×D711yescharacteristic
481C22C2×D4×D711yescharacteristic
482C22C2×D4×D711yescharacteristic
483C22C2×#115911yescharacteristic
484C22C2×#115911yescharacteristic
485C22#654811yescharacteristic
486C22#655411yescharacteristic
487C4—22no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—74no—
492C4—142no—
493C4—142no—
494C4—281no—
495C4—281no—
496C4—281no—
497C4—281no—
498C2—42no—
499C2—42no—
500C2—42no—
501C2—282no—
502C2#9027711yescharacteristic
503C2D7×#12011yescharacteristic
504C2#9028211yescharacteristic
505C2#9028811yescharacteristic
506C2#9028911yescharacteristic
507C2#9030211yescharacteristic
508C2#9040911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2407
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#1159C2semidirect
2. C2×#6502C2semidirect
3. C2×D4×Dic7C2semidirect
4. C2×#6441C2semidirect
5. C7×#392C2semidirect
6. #89995C2semidirect
7. #89994C2semidirect
8. #89992C2semidirect
9. #89733C2semidirect
10. #89729C2semidirect
11. #89728C2semidirect
12. #89722C2semidirect
13. #89714C2semidirect
14. #89440C2semidirect
15. #89418C2semidirect
83. C2×C22⋊D7D4semidirect
92. C22×Dic7D4semidirect
106. C2×Dic14D4semidirect
132. #392D7semidirect
160. C23×C7#41semidirect
183. C2×C4×C7#41semidirect
405. C23#1159semidirect
450. C2×C4#1159semidirect
451. C7#2407semidirect

Maximal subgroups

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

subgroupindex
C22×#11592
C2×#65022
C2×D4×Dic72
C2×#64412
C7×#3922
#899952
#899942
#899922
#897332
#897292
#897282
#897222
#897142
#894402
#894182
#24077