Finite Group Databasebeta

← #13284801

Subgroups — #13284801

Order 1792.

1094 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89751C211yesmaximal, characteristic
2C7×#1445C211yesmaximal, characteristic
3#11729318C211yesmaximal, characteristic
4#11727178C212yesmaximal
5#11727175C212yesmaximal
6#11727172C212yesmaximal
7#11727171C212yesmaximal
8#11727171C212yesmaximal
9#11727159C212yesmaximal
10C23×Dic14C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×Dic7⋊C4C2212yes—
13C22×C4×Dic7C2211yescharacteristic
14C2×C7×#95C2212yes—
15C2×C7×#95C2211yescharacteristic
16C2×C7×#95C2212yes—
17C2×C7×#95C2211yescharacteristic
18C2×C7⋊#95C2212yes—
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⋊#95C2211yescharacteristic
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31#89751C418yescomplemented
32#14655—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⋊C4C2×C414yes—
50C2×Dic7⋊C4—42no—
51C2×Dic7⋊C4—42no—
52C2×Dic7⋊C4C2×C418yes—
53C2×C4×Dic7—42no—
54C2×C4×Dic7—41no—
55C7×#95—42no—
56C7×#95—41no—
57C7×#95—41no—
58C7×#95—42no—
59C7⋊#95—24no—
60C7⋊#95—24no—
61C7⋊#95—24no—
62C7⋊#95—42no—
63C7⋊#95—42no—
64C7⋊#95—42no—
65C7⋊#95—42no—
66C7⋊#95—42no—
67C7⋊#95—42no—
68C7⋊#95—42no—
69C7⋊#95—41no—
70C7⋊#95C2×C414yes—
71C7⋊#95C2×C418yes—
72C2×#395—71no—
73#1446—72no—
74#1445—72no—
75#1445D711yescharacteristic
76#1443—72no—
77#1443—72no—
78#1443—72no—
79#1436—72no—
80#1436—71no—
81C24×C7C2411yescharacteristic
82C22×C4×C7—22no—
83C22×C4×C7—21no—
84C22×C4×C7—22no—
85C22×C4×C7—22no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—22no—
89C22×C4×C7—22no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—22no—
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—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×C7C22×C412yes—
109C22×C4×C7C22×C412yes—
110C22×C4×C7C22×C412yes—
111C22×Dic7—22no—
112C22×Dic7—22no—
113C22×Dic7—22no—
114C22×Dic7—22no—
115C22×Dic7—22no—
116C22×Dic7—22no—
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×Dic7C22×C414yes—
137C22×Dic7C22×C414yes—
138C22×Dic7#4114yes—
139C2×Dic14—48no—
140C2×Dic14—44no—
141C2×Dic14—44no—
142C2×Dic14—48no—
143C2×Dic14—44no—
144C7×C4⋊C4—44no—
145Dic7⋊C4—48no—
146C4×Dic7—44no—
147C23×Q8—71no—
148C22×C4⋊C4—72no—
149C22×C4⋊C4C2×D711yescharacteristic
150C22×C42—71no—
151#395—78no—
152C2×#95—72no—
153C2×#95—71no—
154C2×#95—72no—
155C2×#95—72no—
156C2×#95—72no—
157C2×#95—72no—
158C2×#95—72no—
159C2×#95—72no—
160C2×#95—71no—
161C2×#95—72no—
162C2×#95—72no—
163C2×#95—72no—
164C2×#95—72no—
165C2×#95C2×D712yes—
166C2×#95C2×D711yescharacteristic
167C2×#95C2×D711yescharacteristic
168C2×#95C2×D712yes—
169C23×C7C23×C411yescharacteristic
170C23×C7C2×#4112yes—
171C23×C7H(5,𝔽2)11yescharacteristic
172C23×C7H(5,𝔽2)12yes—
173C23×C7H(5,𝔽2)11yescharacteristic
174C23×C7H(5,𝔽2)12yes—
175C23×C72−1+412yes—
176C23×C72−1+412yes—
177C23×C72−1+412yes—
178C2×C4×C7—22no—
179C2×C4×C7—24no—
180C2×C4×C7—22no—
181C2×C4×C7—22no—
182C2×C4×C7—22no—
183C2×C4×C7—24no—
184C2×C4×C7—22no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—41no—
188C2×C4×C7—42no—
189C2×C4×C7—41no—
190C2×C4×C7—42no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—42no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—42no—
199C2×C4×C7—42no—
200C2×C4×C7—42no—
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×Dic7—24no—
208C2×Dic7—24no—
209C2×Dic7—24no—
210C2×Dic7—24no—
211C2×Dic7—24no—
212C2×Dic7—24no—
213C2×Dic7—24no—
214C2×Dic7—24no—
215C2×Dic7—24no—
216C2×Dic7—42no—
217C2×Dic7—42no—
218C2×Dic7—42no—
219C2×Dic7—42no—
220C2×Dic7—42no—
221C2×Dic7—42no—
222C2×Dic7—42no—
223C2×Dic7—42no—
224C2×Dic7—42no—
225C2×Dic7—42no—
226C2×Dic7—42no—
227C2×Dic7—42no—
228C2×Dic7—42no—
229C2×Dic7—42no—
230C2×Dic7—42no—
231C2×Dic7—42no—
232C2×Dic7—42no—
233C2×Dic7—42no—
234C2×Dic7—42no—
235Dic14—416no—
236C22×Q8—74no—
237C22×Q8—282no—
238C22×Q8—281no—
239C22×Q8—281no—
240C22×Q8—282no—
241C23×C4—72no—
242C23×C4—72no—
243C23×C4—72no—
244C23×C4—72no—
245C23×C4C22×D712yes—
246C23×C4C22×D712yes—
247C23×C4C22×D711yescharacteristic
248C23×C4C22×D711yescharacteristic
249C23×C4C22×D711yescharacteristic
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—78no—
253C2×C4⋊C4—282no—
254C2×C4⋊C4—282no—
255C2×C4⋊C4C4×D714yes—
256C2×C42—281no—
257C2×C42—282no—
258#95—41no—
259#95—41no—
260#95—42no—
261#95—42no—
262#95—74no—
263#95—78no—
264#95—144no—
265#95—144no—
266#95—144no—
267#95—281no—
268#95—282no—
269#95—282no—
270#95—282no—
271#95—282no—
272#95—282no—
273#95—282no—
274#95—282no—
275C22×C7C4×#4111yescharacteristic
276C22×C7#51811yescharacteristic
277C22×C7#51812yes—
278C22×C7#51911yescharacteristic
279C22×C7#51912yes—
280C22×C7#53912yes—
281C22×C7#53912yes—
282C22×C7#53912yes—
283C22×C7#54212yes—
284C22×C7#54212yes—
285C22×C7#54212yes—
286C22×C7#56011yescharacteristic
287C22×C7#56012yes—
288C22×C7H(3,𝔽4)11yescharacteristic
289C22×C7#56312yes—
290C22×C7#56312yes—
291C22×C7#56311yescharacteristic
292C22×C7#56312yes—
293C22×C7#56312yes—
294C22×C7#56312yes—
295C22×C7#56311yescharacteristic
296C4×C7—42no—
297C4×C7—42no—
298C4×C7—42no—
299C4×C7—82no—
300C4×C7—82no—
301Dic7—44no—
302Dic7—44no—
303Dic7—44no—
304Dic7—82no—
305C24C23×D711yescharacteristic
306C2×Q8—284no—
307C2×Q8—288no—
308C2×Q8—284no—
309C2×Q8—284no—
310C2×Q8—288no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
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—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—74no—
338C22×C4—74no—
339C22×C4—74no—
340C22×C4—142no—
341C22×C4—142no—
342C22×C4—142no—
343C22×C4—142no—
344C22×C4—142no—
345C22×C4—142no—
346C22×C4—142no—
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×C4C2×C4×D712yes—
366C22×C4C2×C4×D712yes—
367C22×C4C2×C4×D712yes—
368C4⋊C4—44no—
369C4⋊C4—288no—
370C42—284no—
371C2×C7#233212yes—
372C2×C7#233211yescharacteristic
373C2×C7#237511yescharacteristic
374C2×C7#238111yescharacteristic
375C2×C7#238112yes—
376C2×C7#283512yes—
377C2×C7#283812yes—
378C2×C7#283812yes—
379C2×C7#283812yes—
380C23C22×C4×D711yescharacteristic
381C23#654811yescharacteristic
382C23#654812yes—
383C23#654811yescharacteristic
384C23#655112yes—
385C23D7×#4112yes—
386C23#655312yes—
387C23#655412yes—
388C23#655412yes—
389Q8—2816no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—144no—
420C2×C4—144no—
421C2×C4—144no—
422C2×C4—144no—
423C2×C4—144no—
424C2×C4—144no—
425C2×C4—144no—
426C2×C4—144no—
427C2×C4—144no—
428C2×C4—282no—
429C2×C4—282no—
430C2×C4—282no—
431C2×C4—282no—
432C2×C4—282no—
433C2×C4—282no—
434C2×C4—282no—
435C2×C4—282no—
436C2×C4—282no—
437C2×C4—282no—
438C2×C4—282no—
439C2×C4—282no—
440C2×C4—282no—
441C2×C4—282no—
442C2×C4—282no—
443C2×C4—282no—
444C2×C4—282no—
445C2×C4—282no—
446C2×C4—282no—
447C7#1465511yescharacteristic, complemented, Sylow
448C22#9017711yescharacteristic
449C22#9020711yescharacteristic
450C22#9021311yescharacteristic
451C22#9023612yes—
452C22#9026312yes—
453C22#9031112yes—
454C22#9033112yes—
455C22#9033712yes—
456C22#9033712yes—
457C22#9034212yes—
458C22#9035112yes—
459C22#9035312yes—
460C22#9035312yes—
461C22#9036311yescharacteristic
462C22#9037411yescharacteristic
463C22#9038312yes—
464C22#9039012yes—
465C22#9039812yes—
466C22#9040312yes—
467C22#9041111yescharacteristic
468C22#9042611yescharacteristic
469C4—42no—
470C4—42no—
471C4—42no—
472C4—82no—
473C4—82no—
474C4—284no—
475C4—284no—
476C4—284no—
477C4—562no—
478C2#1173150211yescharacteristic
479C2#1173242212yes—
480C2#1173274612yes—
481C2#1173387911yescharacteristic
482C2#1173394211yescharacteristic
483C2#1173511112yes—
484C2#1173538212yes—
485C2#1173554312yes—
486C2#1173585212yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1445C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1445D7
hypocenterC7#14655
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
31. #89751C4semidirect
447. C7#14655semidirect

Maximal subgroups

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

subgroupindex
C2×#897512
C7×#14452
#117293182
#117271782
#117271752
#117271722
#117271712
#117271592
#146557