Finite Group Databasebeta

← #13284608

Subgroups — #13284608

Order 1792.

1138 conjugacy classes of subgroups.

The subgroup lattice has 520 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#11727175C212yesmaximal
7#11727172C212yesmaximal
8#11727169C212yesmaximal
9#11727155C212yesmaximal
10C23×Dic14C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C22×Dic7⋊C4C2212yes—
15C22×C4×Dic7C2211yescharacteristic
16C2×C7×#95C2212yes—
17C2×C7×#95C2212yes—
18C2×C7×#95C2211yescharacteristic
19C2×C7⋊#95C2212yes—
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⋊#95C2212yes—
30C2×C7⋊#95C2211yescharacteristic
31#89751C418yescomplemented
32#14638—71nomaximal, Sylow
33C23×C4×C7C2312yes—
34C23×C4×C7C2312yes—
35C23×C4×C7C2311yescharacteristic
36C23×C4×C7C2311yescharacteristic
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—22no—
53C2×C4⋊Dic7—41no—
54C2×C4⋊Dic7—41no—
55C2×Dic7⋊C4—42no—
56C2×Dic7⋊C4—42no—
57C2×Dic7⋊C4C2×C418yes—
58C2×C4×Dic7—41no—
59C2×C4×Dic7—41no—
60C2×C4×Dic7Q814yescomplemented
61C7×#95—24no—
62C7×#95—41no—
63C7×#95—42no—
64C7⋊#95—42no—
65C7⋊#95—42no—
66C7⋊#95—42no—
67C7⋊#95—42no—
68C7⋊#95—42no—
69C7⋊#95—42no—
70C7⋊#95—42no—
71C7⋊#95—42no—
72C7⋊#95—42no—
73C7⋊#95—42no—
74C7⋊#95C2×C414yes—
75C7⋊#95C2×C418yes—
76C2×#395—71no—
77#1446—72no—
78#1445—72no—
79#1443—72no—
80#1442—72no—
81#1441—72no—
82#1441D711yescharacteristic
83C4⋊#95—71no—
84#1434—72no—
85C24×C7C2411yescharacteristic
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—21no—
93C22×C4×C7—22no—
94C22×C4×C7—22no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
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×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7C22×C412yes—
113C22×C4×C7C22×C412yes—
114C22×C4×C7C22×C412yes—
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×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7C22×C414yes—
143C22×Dic7C22×C414yes—
144C22×Dic7C2×Q814yes—
145C2×Dic14—48no—
146C2×Dic14—44no—
147C2×Dic14—48no—
148C2×Dic14—44no—
149C2×Dic14—44no—
150C7×C4⋊C4—44no—
151C7×C4⋊C4—44no—
152C4⋊Dic7—44no—
153Dic7⋊C4—48no—
154C4×Dic7—44no—
155C23×Q8—71no—
156C22×C4⋊C4—72no—
157C22×C4⋊C4—71no—
158C22×C4⋊C4C2×D711yescharacteristic
159C22×C4⋊C4C2×D711yescharacteristic
160C22×C42—71no—
161#395—78no—
162C2×#95—72no—
163C2×#95—71no—
164C2×#95—72no—
165C2×#95—72no—
166C2×#95—72no—
167C2×#95—72no—
168C2×#95—72no—
169C2×#95—72no—
170C2×#95—72no—
171C2×#95—72no—
172C2×#95—72no—
173C2×#95—72no—
174C2×#95C2×D712yes—
175C2×#95C2×D712yes—
176C2×#95C2×D711yescharacteristic
177C23×C7C23×C411yescharacteristic
178C23×C7C22×Q811yescharacteristic
179C23×C7C2×#4111yescharacteristic
180C23×C7H(5,𝔽2)11yescharacteristic
181C23×C7H(5,𝔽2)11yescharacteristic
182C23×C7H(5,𝔽2)12yes—
183C23×C7H(5,𝔽2)12yes—
184C23×C7H(5,𝔽2)11yescharacteristic
185C23×C72−1+412yes—
186C23×C72−1+412yes—
187C23×C72−1+411yescharacteristic
188C2×C4×C7—22no—
189C2×C4×C7—22no—
190C2×C4×C7—22no—
191C2×C4×C7—22no—
192C2×C4×C7—22no—
193C2×C4×C7—22no—
194C2×C4×C7—24no—
195C2×C4×C7—22no—
196C2×C4×C7—22no—
197C2×C4×C7—24no—
198C2×C4×C7—42no—
199C2×C4×C7—41no—
200C2×C4×C7—42no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
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×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7C4×Q814yes—
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×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247Dic14—416no—
248C22×Q8—74no—
249C22×Q8—282no—
250C22×Q8—281no—
251C22×Q8—281no—
252C22×Q8—282no—
253C23×C4—72no—
254C23×C4—72no—
255C23×C4—72no—
256C23×C4—72no—
257C23×C4C22×D712yes—
258C23×C4C22×D712yes—
259C23×C4C22×D711yescharacteristic
260C23×C4C22×D711yescharacteristic
261C23×C4C22×D711yescharacteristic
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—78no—
267C2×C4⋊C4—142no—
268C2×C4⋊C4—282no—
269C2×C4⋊C4—282no—
270C2×C4⋊C4—281no—
271C2×C4⋊C4—281no—
272C2×C4⋊C4C4×D714yes—
273C2×C42—74no—
274C2×C42—281no—
275C2×C42—281no—
276#95—24no—
277#95—41no—
278#95—42no—
279#95—74no—
280#95—78no—
281#95—282no—
282#95—282no—
283#95—282no—
284#95—282no—
285#95—282no—
286#95—282no—
287#95—282no—
288#95—282no—
289#95—282no—
290#95—282no—
291C22×C7C2×C4×Q811yescharacteristic
292C22×C7#51811yescharacteristic
293C22×C7#51812yes—
294C22×C7#51912yes—
295C22×C7#52011yescharacteristic
296C22×C7#53811yescharacteristic
297C22×C7#53911yescharacteristic
298C22×C7#54212yes—
299C22×C7#54212yes—
300C22×C7#54311yescharacteristic
301C22×C7#54412yes—
302C22×C7#54412yes—
303C22×C7#54411yescharacteristic
304C22×C7#56012yes—
305C22×C7#56011yescharacteristic
306C22×C7H(3,𝔽4)11yescharacteristic
307C22×C7#56212yes—
308C22×C7#56212yes—
309C22×C7#56312yes—
310C22×C7#56311yescharacteristic
311C22×C7#56312yes—
312C22×C7#56312yes—
313C22×C7#56311yescharacteristic
314C4×C7—24no—
315C4×C7—42no—
316C4×C7—42no—
317C4×C7—82no—
318C4×C7—82no—
319Dic7—44no—
320Dic7—44no—
321Dic7—44no—
322Dic7—82no—
323C24C23×D711yescharacteristic
324C2×Q8—284no—
325C2×Q8—288no—
326C2×Q8—284no—
327C2×Q8—288no—
328C2×Q8—284no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—74no—
356C22×C4—74no—
357C22×C4—74no—
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×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×C4C2×C4×D712yes—
384C22×C4C2×C4×D712yes—
385C22×C4C2×C4×D712yes—
386C22×C4#115912yes—
387C22×C4Q8×D712yes—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—288no—
391C4⋊C4—284no—
392C42—284no—
393C2×C7#232911yescharacteristic
394C2×C7#233112yes—
395C2×C7#237511yescharacteristic
396C2×C7#237812yes—
397C2×C7#238111yescharacteristic
398C2×C7#283011yescharacteristic
399C2×C7#283211yescharacteristic
400C2×C7#283812yes—
401C2×C7#284112yes—
402C2×C7#284311yescharacteristic
403C2×C7#284411yescharacteristic
404C23C22×C4×D711yescharacteristic
405C23C2×#115911yescharacteristic
406C23#654812yes—
407C23#654811yescharacteristic
408C23#654811yescharacteristic
409C23C2×Q8×D711yescharacteristic
410C23#655112yes—
411C23#655312yes—
412C23#655311yescharacteristic
413C23#655412yes—
414C23#655411yescharacteristic
415Q8—2816no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—144no—
447C2×C4—144no—
448C2×C4—144no—
449C2×C4—144no—
450C2×C4—144no—
451C2×C4—144no—
452C2×C4—144no—
453C2×C4—144no—
454C2×C4—144no—
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×C4—282no—
464C2×C4—282no—
465C2×C4—282no—
466C2×C4—282no—
467C2×C4—282no—
468C2×C4—282no—
469C2×C4—282no—
470C2×C4—282no—
471C2×C4—282no—
472C2×C4—282no—
473C2×C4—282no—
474C2×C4#645014yes—
475C7#1463811yescharacteristic, complemented, Sylow
476C22#9017711yescharacteristic
477C22C2×#645011yescharacteristic
478C22#9021411yescharacteristic
479C22#9023612yes—
480C22#9026312yes—
481C22#9028511yescharacteristic
482C22#9029311yescharacteristic
483C22#9029412yes—
484C22#9030811yescharacteristic
485C22#9031112yes—
486C22#9031411yescharacteristic
487C22#9032511yescharacteristic
488C22#9033412yes—
489C22#9033712yes—
490C22#9033712yes—
491C22#9035711yescharacteristic
492C22#9035812yes—
493C22#9036311yescharacteristic
494C22#9038012yes—
495C22#9039112yes—
496C22#9040312yes—
497C22#9042012yes—
498C22#9042611yescharacteristic
499C4—24no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—82no—
504C4—284no—
505C4—284no—
506C4—284no—
507C4—562no—
508C2#1173131911yescharacteristic
509C2#1173232711yescharacteristic
510C2#1173242112yes—
511C2#1173341912yes—
512C2#1173394211yescharacteristic
513C2#1173454511yescharacteristic
514C2#1173469211yescharacteristic
515C2#1173485612yes—
516C2#1173545711yescharacteristic
517C2#1173546112yes—
518C2#1173578311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1441C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1441D7
hypocenterC7#14638
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
31. #89751C4semidirect
60. C2×C4×Dic7Q8semidirect
475. C7#14638semidirect

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
#117271752
#117271722
#117271692
#117271552
#146387