Finite Group Databasebeta

← #11734481

Subgroups — #11734481

Order 896.

598 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6447C211yesmaximal, characteristic, complemented
2C2×#6442C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic, complemented
4#89995C211yesmaximal, characteristic, complemented
5#89995C211yesmaximal, characteristic, complemented
6#89760C211yesmaximal, characteristic, complemented
7#89750C211yesmaximal, characteristic, complemented
8#89733C211yesmaximal, characteristic, complemented
9#89732C211yesmaximal, characteristic, complemented
10#89729C211yesmaximal, characteristic, complemented
11#89729C211yesmaximal, characteristic, complemented
12#89728C211yesmaximal, characteristic, complemented
13#89716C211yesmaximal, characteristic, complemented
14#89447C211yesmaximal, characteristic, complemented
15#89440C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×D28C2211yescharacteristic
22C22×C4×D7C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×C4⋊Dic7C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×C4×Dic7C2211yescharacteristic
44C2×C4×Dic7C2211yescharacteristic
45#6447—24no—
46#6442—24no—
47C7×#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2714—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C2×C7×D4—41no—
59C2×C7×D4—41no—
60C2×C7×D4—41no—
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—22no—
67C2×C22⋊D7—22no—
68C2×C22⋊D7—22no—
69C2×C22⋊D7—22no—
70C2×C22⋊D7—22no—
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
84C2×D28—22no—
85C2×D28—22no—
86C2×D28—41no—
87C2×C4×D7—22no—
88C2×C4×D7—41no—
89C2×C4×D7—41no—
90C7×C4⋊C4—41no—
91C7×C22⋊C4—22no—
92C7×C22⋊C4—22no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C22⋊Dic7—22no—
96C22⋊Dic7—22no—
97C22⋊Dic7—22no—
98C22⋊Dic7—41no—
99(C2×D7)⋊C4—22no—
100(C2×D7)⋊C4—22no—
101(C2×D7)⋊C4—41no—
102(C2×D7)⋊C4—41no—
103(C2×D7)⋊C4—41no—
104(C2×D7)⋊C4—41no—
105(C2×D7)⋊C4—41no—
106C4⋊Dic7—22no—
107Dic7⋊C4—22no—
108Dic7⋊C4—41no—
109Dic7⋊C4D414yescomplemented
110C4×Dic7—22no—
111C4×Dic7—41no—
112C2×#126—71no—
113C2×#121—71no—
114#397—71no—
115#394—71no—
116#394D711yescharacteristic, complemented
117#392—71no—
118#392—71no—
119#392—71no—
120#390—71no—
121#390—71no—
122#390—71no—
123#388—71no—
124#387—71no—
125#386—71no—
126#385—71no—
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×C7C2411yescharacteristic
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—21no—
153C22×D7—21no—
154C22×D7—21no—
155C22×D7#4112yes—
156C7×D4—44no—
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—
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×C7C2×D412yes—
172C22⋊D7—44no—
173C22⋊D7—44no—
174C22⋊D7—44no—
175C22⋊D7—44no—
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—21no—
190C2×Dic7—21no—
191C2×Dic7C2×D412yes—
192C2×Dic7C2×D412yes—
193C2×Dic7#4112yes—
194D28—44no—
195C4×D7—44no—
196C22×D4—71no—
197C22×D4—71no—
198C22×D4—71no—
199C22×D4—71no—
200C22×D4—71no—
201C22×D4C2×D711yescharacteristic
202C23×C4—71no—
203#126—144no—
204#121—144no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—71no—
207C2×C4⋊C4—71no—
208C2×C4⋊C4—71no—
209C2×C4⋊C4C2×D711yescharacteristic
210C2×C22⋊C4—71no—
211C2×C22⋊C4—71no—
212C2×C22⋊C4—71no—
213C2×C22⋊C4—71no—
214C2×C22⋊C4—71no—
215C2×C22⋊C4—71no—
216C2×C22⋊C4—71no—
217C2×C22⋊C4—71no—
218C2×C22⋊C4—71no—
219C2×C22⋊C4—71no—
220C2×C22⋊C4—71no—
221C2×C22⋊C4C2×D711yescharacteristic
222C2×C22⋊C4C2×D711yescharacteristic
223C2×C22⋊C4C2×D711yescharacteristic
224C2×C22⋊C4C2×D711yescharacteristic
225C2×C42—71no—
226C2×C42—71no—
227#95—71no—
228#95—71no—
229#95—71no—
230#95—71no—
231#95—71no—
232#95C2×D711yescharacteristic
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×C7—41no—
247C22×C7C22×D411yescharacteristic
248C22×C7C2×#4111yescharacteristic
249C22×C7C2×#4111yescharacteristic
250C22×C7H(5,𝔽2)11yescharacteristic
251C22×C7H(5,𝔽2)11yescharacteristic
252C22×C7H(5,𝔽2)11yescharacteristic
253C22×C7H(5,𝔽2)11yescharacteristic
254C2×D7—22no—
255C2×D7—22no—
256C2×D7—22no—
257C2×D7—41no—
258C2×D7—41no—
259C2×D7—41no—
260C2×D7—41no—
261C2×D7—41no—
262C2×D7—41no—
263C2×D7—41no—
264C2×D7—41no—
265C2×D7—41no—
266C2×D7—41no—
267C2×D7—41no—
268C4×C7—22no—
269C4×C7—41no—
270C4×C7—41no—
271C4×C7—41no—
272C4×C7—41no—
273Dic7—22no—
274Dic7—22no—
275Dic7—22no—
276Dic7—41no—
277Dic7—41no—
278Dic7—41no—
279C24—71no—
280C24—71no—
281C24C22×D711yescharacteristic
282C24C22×D711yescharacteristic
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—142no—
287C2×D4—142no—
288C2×D4—142no—
289C2×D4—142no—
290C2×D4—142no—
291C2×D4—142no—
292C2×D4—142no—
293C2×D4—281no—
294C2×D4—281no—
295C2×D4—281no—
296C2×D4—281no—
297C2×D4—281no—
298C2×D4—281no—
299C2×D4—281no—
300C2×D4—281no—
301C22×C4—71no—
302C22×C4—71no—
303C22×C4—71no—
304C22×C4—71no—
305C22×C4—71no—
306C22×C4—71no—
307C22×C4—142no—
308C22×C4—281no—
309C22×C4—281no—
310C22×C4C22×D711yescharacteristic
311C22×C4C22×D711yescharacteristic
312C22×C4C22×D711yescharacteristic
313C22×C4C22×D711yescharacteristic
314C22×C4C22×D711yescharacteristic
315C4⋊C4—41no—
316C4⋊C4—74no—
317C4⋊C4—142no—
318C4⋊C4—142no—
319C4⋊C4—281no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—142no—
325C22⋊C4—142no—
326C22⋊C4—142no—
327C22⋊C4—142no—
328C22⋊C4—142no—
329C22⋊C4—281no—
330C22⋊C4—281no—
331C22⋊C4—281no—
332C22⋊C4—281no—
333C22⋊C4—281no—
334C22⋊C4—281no—
335C42—142no—
336C42—281no—
337C2×C7—81no—
338C2×C7—81no—
339C2×C7#53511yescharacteristic
340C2×C7#53811yescharacteristic
341C2×C7#53811yescharacteristic
342C2×C7#54611yescharacteristic
343C2×C7#55011yescharacteristic
344C2×C7#55311yescharacteristic
345C2×C7#56011yescharacteristic
346D7—42no—
347D7—81no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—72no—
363C23—141no—
364C23—141no—
365C23—141no—
366C23—141no—
367C23—141no—
368C23—141no—
369C23—141no—
370C23—141no—
371C23—141no—
372C23—141no—
373C23—141no—
374C23—141no—
375C23—141no—
376C23C23×D711yescharacteristic
377D4—44no—
378D4—284no—
379D4—284no—
380D4—284no—
381D4—284no—
382D4—284no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—72no—
398C2×C4—72no—
399C2×C4—72no—
400C2×C4—141no—
401C2×C4—141no—
402C2×C4—141no—
403C2×C4—141no—
404C2×C4—141no—
405C2×C4—141no—
406C2×C4—141no—
407C2×C4—141no—
408C2×C4—141no—
409C2×C4—141no—
410C2×C4—141no—
411C2×C4—141no—
412C2×C4—141no—
413C2×C4—141no—
414C2×C4—141no—
415C2×C4—284no—
416C2×C4D4×D712yes—
417C7#271411yescharacteristic, complemented, Sylow
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—142no—
433C22—142no—
434C22—142no—
435C22—281no—
436C22—281no—
437C22—281no—
438C22—281no—
439C22—281no—
440C22—281no—
441C22—281no—
442C22—281no—
443C22—281no—
444C22—281no—
445C22—281no—
446C22C2×D4×D711yescharacteristic
447C22#654811yescharacteristic
448C22#654811yescharacteristic
449C22#654811yescharacteristic
450C22D7×#4111yescharacteristic
451C22D7×#4111yescharacteristic
452C22#655311yescharacteristic
453C4—22no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—142no—
459C4—142no—
460C4—142no—
461C4—281no—
462C4—281no—
463C4—281no—
464C2—81no—
465C2—81no—
466C2—282no—
467C2—561no—
468C2#9028611yescharacteristic
469C2#9029011yescharacteristic
470C2#9029811yescharacteristic
471C2#9030111yescharacteristic
472C2#9034811yescharacteristic
473C2#9035211yescharacteristic
474C2#9036911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
hypercenter#394D7
hypocenterC7#2714
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#6447C2semidirect
2. C2×#6442C2semidirect
3. C7×#394C2semidirect
4. #89995C2semidirect
5. #89995C2semidirect
6. #89760C2semidirect
7. #89750C2semidirect
8. #89733C2semidirect
9. #89732C2semidirect
10. #89729C2semidirect
11. #89729C2semidirect
12. #89728C2semidirect
13. #89716C2semidirect
14. #89447C2semidirect
15. #89440C2semidirect
109. Dic7⋊C4D4semidirect
116. #394D7semidirect
417. C7#2714semidirect

Maximal subgroups

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

subgroupindex
C2×#64472
C2×#64422
C7×#3942
#899952
#897602
#897502
#897332
#897322
#897292
#897282
#897162
#894472
#894402
#27147