Finite Group Databasebeta

← #13284215

Subgroups — #13284215

Order 1792.

1290 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89751C211yesmaximal, characteristic
2C7×#1441C211yesmaximal, characteristic
3#11729320C212yesmaximal
4#11729317C211yesmaximal, characteristic
5#11727178C212yesmaximal
6#11727161C212yesmaximal
7#11727160C212yesmaximal
8#11727159C212yesmaximal
9Dic7⋊#95C212yesmaximal
10C23×Dic14C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C22×Dic7⋊C4C2212yes—
15C22×Dic7⋊C4C2212yes—
16C22×C4×Dic7C2212yes—
17C22×C4×Dic7C2211yescharacteristic
18C2×C7×#95C2212yes—
19C2×C7×#95C2211yescharacteristic
20C2×C7×#95C2212yes—
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31#89751C418yescomplemented
32#14601—71nomaximal, Sylow
33C23×C4×C7C2312yes—
34C23×C4×C7C2311yescharacteristic
35C23×C4×C7C2311yescharacteristic
36C23×C4×C7C2312yes—
37C23×C4×C7C2311yescharacteristic
38C23×Dic7C2312yes—
39C23×Dic7C2312yes—
40C23×Dic7C2312yes—
41C23×Dic7C2312yes—
42C22×Dic14—41no—
43C22×Dic14—41no—
44C22×Dic14—42no—
45C22×Dic14—42no—
46C22×Dic14C2×C414yes—
47C2×C7×C4⋊C4—41no—
48C2×C7×C4⋊C4—41no—
49C2×C7×C4⋊C4—42no—
50C2×C7×C4⋊C4—41no—
51C2×C7×C4⋊C4C2×C414yes—
52C2×C4⋊Dic7—41no—
53C2×C4⋊Dic7—41no—
54C2×C4⋊Dic7Q814yescomplemented
55C2×Dic7⋊C4—42no—
56C2×Dic7⋊C4—42no—
57C2×Dic7⋊C4—42no—
58C2×Dic7⋊C4—42no—
59C2×Dic7⋊C4C2×C418yes—
60C2×Dic7⋊C4Q818yescomplemented
61C2×C4×Dic7—22no—
62C2×C4×Dic7—24no—
63C2×C4×Dic7—42no—
64C2×C4×Dic7—42no—
65C2×C4×Dic7—41no—
66C2×C4×Dic7—41no—
67C7×#95—24no—
68C7×#95—41no—
69C7×#95—42no—
70C7⋊#95—24no—
71C7⋊#95—24no—
72C7⋊#95—42no—
73C7⋊#95—42no—
74C7⋊#95—42no—
75C7⋊#95—42no—
76C7⋊#95—42no—
77C7⋊#95—42no—
78C7⋊#95C2×C418yes—
79C7⋊#95C2×C414yes—
80C2×#395—71no—
81#1446—72no—
82#1441—72no—
83#1441D711yescharacteristic
84#1436—72no—
85#1436—72no—
86#1436—72no—
87C4⋊#95—72no—
88C4⋊#95—71no—
89C24×C7C2411yescharacteristic
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—22no—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—21no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7C22×C412yes—
116C22×C4×C7C22×C412yes—
117C22×C4×C7C22×C412yes—
118C22×C4×C7C2×Q814yes—
119C22×C4×C7C2×Q812yes—
120C22×C4×C7#4112yes—
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×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7C22×C414yes—
144C22×Dic7C22×C414yes—
145C22×Dic7C2×Q814yes—
146C22×Dic7C2×Q814yes—
147C22×Dic7C2×Q814yes—
148C22×Dic7#4114yes—
149C2×Dic14—48no—
150C2×Dic14—44no—
151C2×Dic14—48no—
152C2×Dic14—44no—
153C2×Dic14—44no—
154C7×C4⋊C4—44no—
155C7×C4⋊C4—44no—
156C4⋊Dic7—44no—
157Dic7⋊C4—48no—
158Dic7⋊C4—48no—
159C4×Dic7—48no—
160C4×Dic7—44no—
161C23×Q8—71no—
162C22×C4⋊C4—72no—
163C22×C4⋊C4—72no—
164C22×C4⋊C4—71no—
165C22×C4⋊C4C2×D711yescharacteristic
166C22×C4⋊C4C2×D711yescharacteristic
167C22×C42—72no—
168C22×C42—71no—
169#395—78no—
170C2×#95—71no—
171C2×#95—72no—
172C2×#95—72no—
173C2×#95—72no—
174C2×#95—72no—
175C2×#95—72no—
176C2×#95—72no—
177C2×#95—72no—
178C2×#95—72no—
179C2×#95—72no—
180C2×#95C2×D712yes—
181C2×#95C2×D711yescharacteristic
182C2×#95C2×D712yes—
183C23×C7C23×C411yescharacteristic
184C23×C7C22×Q812yes—
185C23×C7C22×Q811yescharacteristic
186C23×C7C2×#4111yescharacteristic
187C23×C7C2×#4112yes—
188C23×C7H(5,𝔽2)11yescharacteristic
189C23×C7H(5,𝔽2)11yescharacteristic
190C23×C7H(5,𝔽2)12yes—
191C23×C7H(5,𝔽2)12yes—
192C23×C7H(5,𝔽2)11yescharacteristic
193C23×C72−1+411yescharacteristic
194C2×C4×C7—22no—
195C2×C4×C7—22no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7—22no—
199C2×C4×C7—22no—
200C2×C4×C7—24no—
201C2×C4×C7—22no—
202C2×C4×C7—22no—
203C2×C4×C7—24no—
204C2×C4×C7—24no—
205C2×C4×C7—24no—
206C2×C4×C7—24no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7—42no—
211C2×C4×C7—41no—
212C2×C4×C7—42no—
213C2×C4×C7—42no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7C4×Q814yes—
225C2×Dic7—24no—
226C2×Dic7—24no—
227C2×Dic7—24no—
228C2×Dic7—24no—
229C2×Dic7—24no—
230C2×Dic7—24no—
231C2×Dic7—24no—
232C2×Dic7—24no—
233C2×Dic7—24no—
234C2×Dic7—24no—
235C2×Dic7—24no—
236C2×Dic7—24no—
237C2×Dic7—24no—
238C2×Dic7—24no—
239C2×Dic7—24no—
240C2×Dic7—42no—
241C2×Dic7—42no—
242C2×Dic7—42no—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7—42no—
252C2×Dic7C4×Q818yes—
253Dic14—416no—
254C22×Q8—74no—
255C22×Q8—282no—
256C22×Q8—281no—
257C22×Q8—281no—
258C22×Q8—282no—
259C23×C4—72no—
260C23×C4—72no—
261C23×C4—72no—
262C23×C4—72no—
263C23×C4C22×D712yes—
264C23×C4C22×D711yescharacteristic
265C23×C4C22×D711yescharacteristic
266C23×C4C22×D712yes—
267C23×C4C22×D711yescharacteristic
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—78no—
273C2×C4⋊C4—78no—
274C2×C4⋊C4—74no—
275C2×C4⋊C4—282no—
276C2×C4⋊C4—282no—
277C2×C4⋊C4—282no—
278C2×C4⋊C4—282no—
279C2×C4⋊C4—281no—
280C2×C4⋊C4—281no—
281C2×C4⋊C4C4×D714yes—
282C2×C42—144no—
283C2×C42—142no—
284C2×C42—281no—
285C2×C42—281no—
286C2×C42—282no—
287C2×C42—282no—
288#95—24no—
289#95—41no—
290#95—42no—
291#95—74no—
292#95—78no—
293#95—144no—
294#95—144no—
295#95—282no—
296#95—282no—
297#95—282no—
298#95—282no—
299#95—282no—
300#95—282no—
301C22×C7C2×C4×Q811yescharacteristic
302C22×C7C2×C4×Q812yes—
303C22×C7#51811yescharacteristic
304C22×C7#51812yes—
305C22×C7#52011yescharacteristic
306C22×C7#53812yes—
307C22×C7#53811yescharacteristic
308C22×C7#53911yescharacteristic
309C22×C7#53912yes—
310C22×C7#54312yes—
311C22×C7#54311yescharacteristic
312C22×C7#54412yes—
313C22×C7#54411yescharacteristic
314C22×C7#55412yes—
315C22×C7#55412yes—
316C22×C7#55412yes—
317C22×C7#55512yes—
318C22×C7#55511yescharacteristic
319C22×C7Q8211yescharacteristic
320C22×C7Q8212yes—
321C22×C7#56012yes—
322C22×C7#56011yescharacteristic
323C22×C7H(3,𝔽4)11yescharacteristic
324C4×C7—24no—
325C4×C7—42no—
326C4×C7—42no—
327C4×C7—44no—
328C4×C7—82no—
329Dic7—28no—
330Dic7—44no—
331Dic7—44no—
332Dic7—44no—
333C24C23×D711yescharacteristic
334C2×Q8—284no—
335C2×Q8—288no—
336C2×Q8—284no—
337C2×Q8—288no—
338C2×Q8—284no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—74no—
365C22×C4—74no—
366C22×C4—74no—
367C22×C4—74no—
368C22×C4—74no—
369C22×C4—74no—
370C22×C4—142no—
371C22×C4—142no—
372C22×C4—142no—
373C22×C4—142no—
374C22×C4—142no—
375C22×C4—142no—
376C22×C4—142no—
377C22×C4—142no—
378C22×C4—142no—
379C22×C4—142no—
380C22×C4—142no—
381C22×C4—142no—
382C22×C4—142no—
383C22×C4—142no—
384C22×C4—142no—
385C22×C4—142no—
386C22×C4—142no—
387C22×C4—142no—
388C22×C4—142no—
389C22×C4—142no—
390C22×C4—142no—
391C22×C4—142no—
392C22×C4C2×C4×D712yes—
393C22×C4C2×C4×D712yes—
394C22×C4C2×C4×D712yes—
395C22×C4#115912yes—
396C22×C4Q8×D714yes—
397C22×C4Q8×D712yes—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—288no—
401C4⋊C4—288no—
402C4⋊C4—284no—
403C42—284no—
404C42—288no—
405C2×C7#232912yes—
406C2×C7#232911yescharacteristic
407C2×C7#236911yescharacteristic
408C2×C7#236912yes—
409C2×C7#237511yescharacteristic
410C2×C7#278412yes—
411C2×C7#278411yescharacteristic
412C2×C7#278511yescharacteristic
413C2×C7#279211yescharacteristic
414C2×C7#279212yes—
415C2×C7#281111yescharacteristic
416C23C22×C4×D711yescharacteristic
417C23C2×#115911yescharacteristic
418C23#654811yescharacteristic
419C23#654812yes—
420C23#654811yescharacteristic
421C23C2×Q8×D712yes—
422C23C2×Q8×D711yescharacteristic
423C23D7×#4112yes—
424C23#655312yes—
425C23#655311yescharacteristic
426C23#655411yescharacteristic
427Q8—2816no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—78no—
459C2×C4—144no—
460C2×C4—144no—
461C2×C4—144no—
462C2×C4—144no—
463C2×C4—144no—
464C2×C4—144no—
465C2×C4—144no—
466C2×C4—144no—
467C2×C4—144no—
468C2×C4—144no—
469C2×C4—144no—
470C2×C4—144no—
471C2×C4—144no—
472C2×C4—144no—
473C2×C4—144no—
474C2×C4—282no—
475C2×C4—282no—
476C2×C4—282no—
477C2×C4—282no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4—282no—
481C2×C4—282no—
482C2×C4—282no—
483C2×C4—282no—
484C2×C4—282no—
485C2×C4—282no—
486C2×C4#645014yes—
487C7#1460111yescharacteristic, complemented, Sylow
488C22#9017711yescharacteristic
489C22C2×#645011yescharacteristic
490C22#9021411yescharacteristic
491C22#9023612yes—
492C22C4×Q8×D712yes—
493C22#9028511yescharacteristic
494C22#9029311yescharacteristic
495C22#9030712yes—
496C22#9030811yescharacteristic
497C22#9031112yes—
498C22#9031411yescharacteristic
499C22#9032512yes—
500C22#9032511yescharacteristic
501C22#9032612yes—
502C22#9032712yes—
503C22#9035112yes—
504C22#9035711yescharacteristic
505C22#9036111yescharacteristic
506C22#9038012yes—
507C22#9038112yes—
508C22#9039512yes—
509C22#9041311yescharacteristic
510C22#9041312yes—
511C4—24no—
512C4—42no—
513C4—42no—
514C4—44no—
515C4—82no—
516C4—148no—
517C4—284no—
518C4—284no—
519C4—284no—
520C2#1173131911yescharacteristic
521C2#1173232711yescharacteristic
522C2#1173274412yes—
523C2#1173343212yes—
524C2#1173393911yescharacteristic
525C2#1173451511yescharacteristic
526C2#1173468511yescharacteristic
527C2#1173489112yes—
528C2#1173535711yescharacteristic
529C2#1173535812yes—
530C2#1173573711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1441C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1441D7
hypocenterC7#14601
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
31. #89751C4semidirect
54. C2×C4⋊Dic7Q8semidirect
60. C2×Dic7⋊C4Q8semidirect
487. C7#14601semidirect

Maximal subgroups

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

subgroupindex
C2×#897512
C7×#14412
#117293202
#117293172
#117271782
#117271612
#117271602
#117271592
Dic7⋊#952
#146017