Finite Group Databasebeta

← #13282502

Subgroups — #13282502

Order 1792.

1290 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
1C2×#89700C211yesmaximal, characteristic
2C2×#89426C212yesmaximal
3C7×#1434C211yesmaximal, characteristic
4#11729289C211yesmaximal, characteristic
5#11727174C212yesmaximal
6#11727165C212yesmaximal
7#11727163C212yesmaximal
8#11727161C212yesmaximal
9Dic7⋊#95C212yesmaximal
10C22×C42×C7C2211yescharacteristic
11C22×C4⋊Dic7C2212yes—
12C22×C4⋊Dic7C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C22×Dic7⋊C4C2212yes—
15C22×Dic7⋊C4C2212yes—
16C22×Dic7⋊C4C2212yes—
17C22×C4×Dic7C2212yes—
18C2×C7×#95C2212yes—
19C2×C7×#95C2211yescharacteristic
20C2×C7×#95C2212yes—
21C2×C7×#95C2211yescharacteristic
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31#89700C418yescomplemented
32#89426—28no—
33#14465—71nomaximal, Sylow
34C23×C4×C7C2312yes—
35C23×C4×C7C2311yescharacteristic
36C23×C4×C7C2312yes—
37C23×C4×C7C2311yescharacteristic
38C23×C4×C7C2311yescharacteristic
39C23×Dic7C2312yes—
40C23×Dic7C2312yes—
41C23×Dic7C2312yes—
42C23×Dic7C2312yes—
43C2×C42×C7—42no—
44C2×C42×C7C2×C414yes—
45C2×C4⋊Dic7—24no—
46C2×C4⋊Dic7—24no—
47C2×C4⋊Dic7—42no—
48C2×C4⋊Dic7—42no—
49C2×C4⋊Dic7C2×C414yes—
50C2×C4⋊Dic7C2×C414yes—
51C2×C4⋊Dic7D414yes—
52C2×C4⋊Dic7Q814yes—
53C2×Dic7⋊C4—24no—
54C2×Dic7⋊C4—24no—
55C2×Dic7⋊C4—42no—
56C2×Dic7⋊C4—42no—
57C2×Dic7⋊C4—42no—
58C2×Dic7⋊C4—42no—
59C2×Dic7⋊C4—42no—
60C2×Dic7⋊C4—42no—
61C2×Dic7⋊C4C2×C418yes—
62C2×C4×Dic7—24no—
63C2×C4×Dic7—24no—
64C2×C4×Dic7—42no—
65C7×#95—24no—
66C7×#95—22no—
67C7×#95—42no—
68C7×#95Q814yes—
69C7⋊#95—24no—
70C7⋊#95—24no—
71C7⋊#95—24no—
72C7⋊#95—41no—
73C7⋊#95—42no—
74C7⋊#95—42no—
75C7⋊#95—42no—
76C7⋊#95—41no—
77C7⋊#95C2×C418yes—
78C2×#389—71no—
79C2×#387—72no—
80#1444—72no—
81#1441—72no—
82#1441—72no—
83#1436—72no—
84C4⋊#95—71no—
85C4⋊#95—72no—
86#1434D711yescharacteristic
87C24×C7C2411yescharacteristic
88C22×C4×C7—22no—
89C22×C4×C7—22no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—21no—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—21no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7C22×C412yes—
110C22×C4×C7C22×C412yes—
111C22×C4×C7C22×C412yes—
112C22×C4×C7C2×D412yes—
113C22×C4×C7C2×Q814yes—
114C22×C4×C7C2×Q812yes—
115C22×C4×C7C2×Q812yes—
116C22×C4×C7#4112yes—
117C22×Dic7—22no—
118C22×Dic7—22no—
119C22×Dic7—22no—
120C22×Dic7—22no—
121C22×Dic7—22no—
122C22×Dic7—22no—
123C22×Dic7—22no—
124C22×Dic7—22no—
125C22×Dic7—22no—
126C22×Dic7—22no—
127C22×Dic7—22no—
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—22no—
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7C22×C414yes—
141C22×Dic7C22×C414yes—
142C22×Dic7C2×D414yes—
143C22×Dic7C2×Q814yes—
144C22×Dic7#4114yes—
145C42×C7—44no—
146C4⋊Dic7—44no—
147C4⋊Dic7—48no—
148C4⋊Dic7C4⋊C4116yescomplemented
149Dic7⋊C4—48no—
150Dic7⋊C4—48no—
151Dic7⋊C4—48no—
152C4×Dic7—48no—
153C22×C4⋊C4—71no—
154C22×C4⋊C4—72no—
155C22×C4⋊C4—71no—
156C22×C4⋊C4—72no—
157C22×C4⋊C4—72no—
158C22×C4⋊C4—72no—
159C22×C42—72no—
160C22×C42C2×D711yescharacteristic
161#389—78no—
162#387—148no—
163C2×#95—72no—
164C2×#95—72no—
165C2×#95—72no—
166C2×#95—72no—
167C2×#95—71no—
168C2×#95—72no—
169C2×#95—71no—
170C2×#95—72no—
171C2×#95—72no—
172C2×#95C2×D712yes—
173C2×#95C2×D711yescharacteristic
174C2×#95C2×D712yes—
175C2×#95C2×D711yescharacteristic
176C23×C7C23×C411yescharacteristic
177C23×C7C22×D411yescharacteristic
178C23×C7C22×Q811yescharacteristic
179C23×C7C22×Q811yescharacteristic
180C23×C7C2×#4111yescharacteristic
181C23×C7C2×#4112yes—
182C23×C7H(5,𝔽2)12yes—
183C23×C7H(5,𝔽2)12yes—
184C23×C7H(5,𝔽2)11yescharacteristic
185C23×C72−1+412yes—
186C23×C72−1+411yescharacteristic
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—22no—
190C2×C4×C7—22no—
191C2×C4×C7—24no—
192C2×C4×C7—24no—
193C2×C4×C7—24no—
194C2×C4×C7—24no—
195C2×C4×C7—22no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7—24no—
199C2×C4×C7—24no—
200C2×C4×C7—24no—
201C2×C4×C7—42no—
202C2×C4×C7—42no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7—42no—
206C2×C4×C7—42no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7—42no—
214C2×C4×C7C2×C4⋊C414yes—
215C2×C4×C7C4×Q814yes—
216C2×Dic7—24no—
217C2×Dic7—24no—
218C2×Dic7—24no—
219C2×Dic7—24no—
220C2×Dic7—24no—
221C2×Dic7—24no—
222C2×Dic7—24no—
223C2×Dic7—24no—
224C2×Dic7—24no—
225C2×Dic7—24no—
226C2×Dic7—24no—
227C2×Dic7—24no—
228C2×Dic7—42no—
229C2×Dic7—42no—
230C2×Dic7—42no—
231C2×Dic7—42no—
232C2×Dic7—42no—
233C2×Dic7—42no—
234C2×Dic7—42no—
235C2×Dic7—42no—
236C2×Dic7—42no—
237C2×Dic7—42no—
238C2×Dic7—42no—
239C2×Dic7—42no—
240C2×Dic7—42no—
241C2×Dic7—42no—
242C2×Dic7—42no—
243C2×Dic7C2×C4⋊C418yes—
244C23×C4—72no—
245C23×C4—72no—
246C23×C4—72no—
247C23×C4—72no—
248C23×C4C22×D712yes—
249C23×C4C22×D712yes—
250C23×C4C22×D711yescharacteristic
251C23×C4C22×D711yescharacteristic
252C23×C4C22×D711yescharacteristic
253C2×C4⋊C4—74no—
254C2×C4⋊C4—74no—
255C2×C4⋊C4—74no—
256C2×C4⋊C4—78no—
257C2×C4⋊C4—74no—
258C2×C4⋊C4—144no—
259C2×C4⋊C4—144no—
260C2×C4⋊C4—144no—
261C2×C4⋊C4—144no—
262C2×C4⋊C4—282no—
263C2×C4⋊C4—282no—
264C2×C4⋊C4—282no—
265C2×C4⋊C4—282no—
266C2×C4⋊C4—282no—
267C2×C4⋊C4—282no—
268C2×C4⋊C4—282no—
269C2×C4⋊C4—282no—
270C2×C42—42no—
271C2×C42—144no—
272C2×C42—144no—
273C2×C42—282no—
274C2×C42C4×D714yes—
275#95—22no—
276#95—24no—
277#95—42no—
278#95—78no—
279#95—144no—
280#95—144no—
281#95—144no—
282#95—282no—
283#95—281no—
284#95—281no—
285#95—282no—
286#95—282no—
287#95Dic1414yes—
288C22×C7C22×C4⋊C411yescharacteristic
289C22×C7C2×C4×Q811yescharacteristic
290C22×C7C4×#4111yescharacteristic
291C22×C7#51811yescharacteristic
292C22×C7#51811yescharacteristic
293C22×C7#51911yescharacteristic
294C22×C7#52011yescharacteristic
295C22×C7#53712yes—
296C22×C7#53912yes—
297C22×C7#54112yes—
298C22×C7#54212yes—
299C22×C7#54412yes—
300C22×C7#54412yes—
301C22×C7#54612yes—
302C22×C7D4×Q811yescharacteristic
303C22×C7#55011yescharacteristic
304C22×C7#55112yes—
305C22×C7#55412yes—
306C22×C7#55412yes—
307C22×C7#55711yescharacteristic
308C22×C7Q8211yescharacteristic
309C22×C7#56012yes—
310C22×C7#56212yes—
311C4×C7—24no—
312C4×C7—24no—
313C4×C7—42no—
314C4×C7—44no—
315C4×C7—82no—
316Dic7—28no—
317Dic7—44no—
318Dic7—44no—
319Dic7—82no—
320C24C23×D711yescharacteristic
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—74no—
343C22×C4—74no—
344C22×C4—74no—
345C22×C4—74no—
346C22×C4—74no—
347C22×C4—142no—
348C22×C4—142no—
349C22×C4—142no—
350C22×C4—142no—
351C22×C4—142no—
352C22×C4—142no—
353C22×C4—142no—
354C22×C4—142no—
355C22×C4—142no—
356C22×C4—142no—
357C22×C4—142no—
358C22×C4—142no—
359C22×C4—142no—
360C22×C4—142no—
361C22×C4—142no—
362C22×C4—142no—
363C22×C4—142no—
364C22×C4—142no—
365C22×C4—142no—
366C22×C4—142no—
367C22×C4—142no—
368C22×C4—142no—
369C22×C4—142no—
370C22×C4C2×Dic1414yes—
371C22×C4C2×Dic1412yes—
372C22×C4C2×C4×D712yes—
373C22×C4C2×C4×D712yes—
374C22×C4C2×C4×D712yes—
375C22×C4#115712yes—
376C22×C4D4×D712yes—
377C22×C4Q8×D712yes—
378C4⋊C4—716no—
379C4⋊C4—288no—
380C4⋊C4—284no—
381C4⋊C4—288no—
382C4⋊C4—288no—
383C4⋊C4—288no—
384C42—44no—
385C42—288no—
386C2×C7#231611yescharacteristic
387C2×C7#233111yescharacteristic
388C2×C7#233611yescharacteristic
389C2×C7#234911yescharacteristic
390C2×C7Q8×C4⋊C411yescharacteristic
391C2×C7#236811yescharacteristic
392C2×C7#237711yescharacteristic
393C2×C7#270412yes—
394C2×C7#270512yes—
395C2×C7#275512yes—
396C2×C7#279212yes—
397C23C22×Dic1411yescharacteristic
398C23C22×C4×D711yescharacteristic
399C23C2×#115711yescharacteristic
400C23C2×D4×D711yescharacteristic
401C23#654812yes—
402C23C2×Q8×D711yescharacteristic
403C23D7×#4112yes—
404C23#655312yes—
405C23#655311yescharacteristic
406C23#655411yescharacteristic
407C23#655412yes—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—78no—
436C2×C4—144no—
437C2×C4—144no—
438C2×C4—144no—
439C2×C4—144no—
440C2×C4—144no—
441C2×C4—144no—
442C2×C4—144no—
443C2×C4—144no—
444C2×C4—144no—
445C2×C4—144no—
446C2×C4—144no—
447C2×C4—144no—
448C2×C4—282no—
449C2×C4—282no—
450C2×C4—282no—
451C2×C4—282no—
452C2×C4—282no—
453C2×C4—282no—
454C2×C4—282no—
455C2×C4—282no—
456C2×C4—282no—
457C2×C4—282no—
458C2×C4—282no—
459C2×C4—282no—
460C2×C4—282no—
461C2×C4—282no—
462C2×C4—282no—
463C2×C4C4×Dic1414yes—
464C2×C4D7×C4⋊C414yes—
465C7#1446511yescharacteristic, complemented, Sylow
466C22C2×C4×Dic1411yescharacteristic
467C22#9017711yescharacteristic
468C22C2×D7×C4⋊C411yescharacteristic
469C22#9020711yescharacteristic
470C22#9021012yes—
471C22#9021211yescharacteristic
472C22#9021311yescharacteristic
473C22#9021411yescharacteristic
474C22#9022512yes—
475C22D4×Dic1411yescharacteristic
476C22#9023012yes—
477C22#9024411yescharacteristic
478C22#9024712yes—
479C22Q8×Dic1411yescharacteristic
480C22#9025811yescharacteristic
481C22#9028812yes—
482C22#9031412yes—
483C22#9035712yes—
484C22#9035812yes—
485C22#9036712yes—
486C22#9038112yes—
487C22#9039212yes—
488C22#9039812yes—
489C4—24no—
490C4—24no—
491C4—42no—
492C4—44no—
493C4—82no—
494C4—148no—
495C4—284no—
496C4—284no—
497C4—562no—
498C2#1173083611yescharacteristic
499C2#1173091811yescharacteristic
500C2C4⋊C4×Dic1411yescharacteristic
501C2#1173131411yescharacteristic
502C2#1173156911yescharacteristic
503C2#1173167311yescharacteristic
504C2#1173170611yescharacteristic
505C2#1173263412yes—
506C2#1173270112yes—
507C2#1173313012yes—
508C2#1173313112yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1434C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1434D7
hypocenterC7#14465
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
31. #89700C4semidirect
148. C4⋊Dic7C4⋊C4semidirect
465. C7#14465semidirect

Maximal subgroups

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

subgroupindex
C2×#897002
C2×#894262
C7×#14342
#117292892
#117271742
#117271652
#117271632
#117271612
Dic7⋊#952
#144657