Finite Group Databasebeta

← #11739813

Subgroups — #11739813

Order 896.

668 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×D4C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D7C211yesmaximal, characteristic, complemented
3C2×#6487C211yesmaximal, characteristic, complemented
4C2×#6459C211yesmaximal, characteristic, complemented
5#89995C211yesmaximal, characteristic
6#89994C211yesmaximal, characteristic, complemented
7#89985C211yesmaximal, characteristic, complemented
8#89982C211yesmaximal, characteristic, complemented
9#89753C211yesmaximal, characteristic, complemented
10#89733C211yesmaximal, characteristic, complemented
11#89728C211yesmaximal, characteristic, complemented
12#89719C211yesmaximal, characteristic, complemented
13#89717C211yesmaximal, characteristic, complemented
14#89712C211yesmaximal, characteristic, complemented
15#89705C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C22×Dic14C2211yescharacteristic
23C4×C7×D4—24no—
24C2×C7×C4⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C42×C7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34C4×C22⋊D7—24no—
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×C4⋊Dic7C2211yescharacteristic
40#6487—24no—
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×Dic7⋊C4C2211yescharacteristic
44C2×Dic7⋊C4C2211yescharacteristic
45C2×C4×Dic7C2211yescharacteristic
46#6459—24no—
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2463—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C23×D7C2311yescharacteristic
59C2×C7×D4—22no—
60C2×C7×D4—22no—
61C2×C7×D4—22no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—41no—
67C22×C4×C7—41no—
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C2×C22⋊D7—22no—
74C2×C22⋊D7—22no—
75C2×C22⋊D7—22no—
76C2×C22⋊D7—22no—
77C2×C22⋊D7—22no—
78C2×C22⋊D7—41no—
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C2×C4×D7—22no—
87C2×C4×D7—22no—
88C2×C4×D7—41no—
89C2×Dic14—22no—
90C2×Dic14—22no—
91C2×Dic14—41no—
92C7×C4⋊C4D414yescomplemented
93C7×C22⋊C4—22no—
94C7×C22⋊C4—22no—
95C42×C7—22no—
96C22⋊Dic7—22no—
97C22⋊Dic7—22no—
98C22⋊Dic7—22no—
99C22⋊Dic7—22no—
100C22⋊Dic7—22no—
101C22⋊Dic7—41no—
102(C2×D7)⋊C4—22no—
103(C2×D7)⋊C4—22no—
104(C2×D7)⋊C4—22no—
105(C2×D7)⋊C4—22no—
106C4⋊Dic7—22no—
107Dic7⋊C4—22no—
108Dic7⋊C4—22no—
109Dic7⋊C4—22no—
110Dic7⋊C4D414yescomplemented
111C4×Dic7—22no—
112C2×C22⋊Q8—71no—
113C2×C22⋊Q8—71no—
114C2×C4×D4—71no—
115C2×C4×D4D711yescharacteristic, complemented
116#394—71no—
117#394—71no—
118#393—71no—
119#392—71no—
120#388—71no—
121#388—71no—
122#387—71no—
123#387—71no—
124#386—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#4112yes—
140C23×C7#4112yes—
141C23×C7C2411yescharacteristic
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7#4112yes—
149C7×D4—28no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
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—44no—
162C2×C4×C7—44no—
163C2×C4×C7C2×D412yes—
164C2×C4×C7C2×D412yes—
165C2×C4×C7C2×D412yes—
166C2×C4×C7C2×D412yes—
167C22⋊D7—28no—
168C22⋊D7—44no—
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×Dic7C2×D412yes—
189C2×Dic7C2×D412yes—
190C4×D7—44no—
191Dic14—44no—
192C22×Q8—71no—
193C22×D4—71no—
194C22×D4—71no—
195C22×D4C2×D711yescharacteristic
196C23×C4—71no—
197C23×C4C2×D711yescharacteristic
198C23×C4C2×D711yescharacteristic
199C22⋊Q8—144no—
200C22⋊Q8—144no—
201C4×D4—24no—
202C4×D4—144no—
203C2×C4⋊C4—71no—
204C2×C4⋊C4—71no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—71no—
207C2×C4⋊C4—71no—
208C2×C4⋊C4C2×D711yescharacteristic
209C2×C22⋊C4—71no—
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⋊C4C2×D711yescharacteristic
220C2×C22⋊C4C2×D711yescharacteristic
221C2×C42—71no—
222C2×C42C2×D711yescharacteristic
223#95—71no—
224#95—71no—
225#95—71no—
226#95—71no—
227#95—71no—
228#95—71no—
229#95—71no—
230#95—71no—
231C22×C7—22no—
232C22×C7—22no—
233C22×C7—22no—
234C22×C7—22no—
235C22×C7—22no—
236C22×C7—22no—
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×C7C22×D411yescharacteristic
246C22×C7C22×D411yescharacteristic
247C22×C7C2×#4111yescharacteristic
248C22×C7C2×#4111yescharacteristic
249C22×C7C2×#4111yescharacteristic
250C22×C7H(5,𝔽2)11yescharacteristic
251C22×C72−1+411yescharacteristic
252C2×D7—22no—
253C2×D7—22no—
254C2×D7—22no—
255C2×D7—41no—
256C2×D7—41no—
257C2×D7—41no—
258C2×D7—41no—
259C4×C7—22no—
260C4×C7—22no—
261C4×C7—22no—
262C4×C7—22no—
263C4×C7—41no—
264Dic7—22no—
265Dic7—22no—
266Dic7—41no—
267Dic7—41no—
268Dic7—41no—
269Dic7—41no—
270Dic7—41no—
271C24—71no—
272C24C22×D711yescharacteristic
273C24C22×D711yescharacteristic
274C2×Q8—142no—
275C2×Q8—142no—
276C2×Q8—281no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—142no—
281C2×D4—142no—
282C2×D4—142no—
283C2×D4—142no—
284C2×D4—142no—
285C2×D4—281no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—71no—
293C22×C4—71no—
294C22×C4—71no—
295C22×C4—71no—
296C22×C4—71no—
297C22×C4—71no—
298C22×C4—71no—
299C22×C4—142no—
300C22×C4—142no—
301C22×C4—281no—
302C22×C4C22×D711yescharacteristic
303C22×C4C22×D711yescharacteristic
304C22×C4C22×D711yescharacteristic
305C22×C4C22×D711yescharacteristic
306C22×C4C22×D711yescharacteristic
307C4⋊C4—74no—
308C4⋊C4—142no—
309C4⋊C4—142no—
310C4⋊C4—142no—
311C4⋊C4—142no—
312C4⋊C4C22⋊D714yescomplemented
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—142no—
316C22⋊C4—142no—
317C22⋊C4—142no—
318C22⋊C4—142no—
319C22⋊C4—142no—
320C22⋊C4—142no—
321C22⋊C4—142no—
322C22⋊C4—142no—
323C22⋊C4—142no—
324C22⋊C4—281no—
325C42—22no—
326C42—142no—
327C2×C7—42no—
328C2×C7—42no—
329C2×C7#52511yescharacteristic
330C2×C7#52511yescharacteristic
331C2×C7#53911yescharacteristic
332C2×C7D4211yescharacteristic
333C2×C7#54811yescharacteristic
334C2×C7#54811yescharacteristic
335C2×C7#55111yescharacteristic
336D7—42no—
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—72no—
350C23—141no—
351C23—141no—
352C23—141no—
353C23—141no—
354C23—141no—
355C23—141no—
356C23#115712yes—
357C23#115712yes—
358C23C23×D711yescharacteristic
359Q8—284no—
360D4—28no—
361D4—148no—
362D4—284no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—72no—
377C2×C4—72no—
378C2×C4—141no—
379C2×C4—141no—
380C2×C4—141no—
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—284no—
398C2×C4D4×D712yes—
399C2×C4C2×C22⋊D712yes—
400C2×C4C2×C22⋊D712yes—
401C2×C4C2×C22⋊D712yes—
402C7#246311yescharacteristic, complemented, Sylow
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—142no—
418C22—142no—
419C22—142no—
420C22—281no—
421C22—281no—
422C22—281no—
423C22—281no—
424C22C2×#115711yescharacteristic
425C22C2×#115711yescharacteristic
426C22C2×D4×D711yescharacteristic
427C22#654811yescharacteristic
428C22D7×#4111yescharacteristic
429C22#655411yescharacteristic
430C22C22×C22⋊D711yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—142no—
437C4—142no—
438C4—281no—
439C4—281no—
440C4—281no—
441C4—281no—
442C4—281no—
443C2—42no—
444C2—42no—
445C2—282no—
446C2#9023811yescharacteristic
447C2#9024311yescharacteristic
448C2#9024711yescharacteristic
449C2#9025511yescharacteristic
450C2#9048211yescharacteristic
451C2D4×C22⋊D711yescharacteristic
452C2#9052211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C4×C7×D4C2
prime radical#89995C2
hypercenterC2×C4×D4D7
hypocenterC7#2463
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×D4C2semidirect
2. C2×C4×C22⋊D7C2semidirect
3. C2×#6487C2semidirect
4. C2×#6459C2semidirect
6. #89994C2semidirect
7. #89985C2semidirect
8. #89982C2semidirect
9. #89753C2semidirect
10. #89733C2semidirect
11. #89728C2semidirect
12. #89719C2semidirect
13. #89717C2semidirect
14. #89712C2semidirect
15. #89705C2semidirect
92. C7×C4⋊C4D4semidirect
110. Dic7⋊C4D4semidirect
115. C2×C4×D4D7semidirect
312. C4⋊C4C22⋊D7semidirect
402. C7#2463semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×D42
C2×C4×C22⋊D72
C2×#64872
C2×#64592
#899952
#899942
#899852
#899822
#897532
#897332
#897282
#897192
#897172
#897122
#897052
#24637