Finite Group Databasebeta

← #13282621

Subgroups — #13282621

Order 1792.

1102 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×#89708C211yesmaximal, characteristic
2C7×#1434C211yesmaximal, characteristic
3#11729288C211yesmaximal, characteristic
4#11727172C212yesmaximal
5#11727168C212yesmaximal
6#11727165C212yesmaximal
7#11727161C212yesmaximal
8#11727159C212yesmaximal
9#11727157C212yesmaximal
10C22×C42×C7C2211yescharacteristic
11C22×C4⋊Dic7C2211yescharacteristic
12C22×C4⋊Dic7C2211yescharacteristic
13C22×Dic7⋊C4C2212yes—
14C22×C4×Dic7C2212yes—
15C2×C7×#95C2212yes—
16C2×C7×#95C2212yes—
17C2×C7×#95C2211yescharacteristic
18C2×C7×#95C2211yescharacteristic
19C2×C7⋊#95C2212yes—
20C2×C7⋊#95C2212yes—
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2211yescharacteristic
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2211yescharacteristic
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31#89708C418yescomplemented
32#14477—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—
42C2×C42×C7—24no—
43C2×C42×C7C2×C414yes—
44C2×C4⋊Dic7—42no—
45C2×C4⋊Dic7—42no—
46C2×C4⋊Dic7C2×C414yes—
47C2×C4⋊Dic7C2×C414yes—
48C2×Dic7⋊C4—24no—
49C2×Dic7⋊C4—42no—
50C2×Dic7⋊C4—42no—
51C2×C4×Dic7—24no—
52C2×C4×Dic7—42no—
53C2×C4×Dic7—42no—
54C7×#95—24no—
55C7×#95—24no—
56C7×#95—41no—
57C7×#95—41no—
58C7⋊#95—24no—
59C7⋊#95—24no—
60C7⋊#95—24no—
61C7⋊#95—24no—
62C7⋊#95—41no—
63C7⋊#95—42no—
64C7⋊#95—42no—
65C7⋊#95—42no—
66C7⋊#95—42no—
67C7⋊#95—41no—
68C7⋊#95C2×C418yes—
69C7⋊#95C2×C418yes—
70C2×#382—71no—
71#1443—72no—
72#1442—72no—
73#1441—72no—
74#1436—72no—
75#1436—72no—
76#1434—71no—
77#1434—72no—
78#1434D711yescharacteristic
79C24×C7C2411yescharacteristic
80C22×C4×C7—22no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—22no—
84C22×C4×C7—22no—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—22no—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7—21no—
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
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×C7C22×C412yes—
105C22×C4×C7C22×C412yes—
106C22×C4×C7C22×C412yes—
107C22×C4×C7#4114yes—
108C22×C4×C7#4114yes—
109C22×Dic7—22no—
110C22×Dic7—22no—
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×Dic7C22×C414yes—
135C22×Dic7C22×C414yes—
136C22×Dic7#4114yes—
137C42×C7—28no—
138C4⋊Dic7—44no—
139C4⋊Dic7—44no—
140Dic7⋊C4—48no—
141C4×Dic7—48no—
142C22×C4⋊C4—71no—
143C22×C4⋊C4—71no—
144C22×C4⋊C4—72no—
145C22×C42—72no—
146C22×C42C2×D711yescharacteristic
147#382—78no—
148C2×#95—72no—
149C2×#95—72no—
150C2×#95—72no—
151C2×#95—72no—
152C2×#95—72no—
153C2×#95—72no—
154C2×#95—71no—
155C2×#95—72no—
156C2×#95—71no—
157C2×#95—72no—
158C2×#95—72no—
159C2×#95—72no—
160C2×#95C2×D712yes—
161C2×#95C2×D712yes—
162C2×#95C2×D711yescharacteristic
163C2×#95C2×D711yescharacteristic
164C23×C7C23×C411yescharacteristic
165C23×C7C2×#4112yes—
166C23×C7C2×#4112yes—
167C23×C7C2×#4112yes—
168C23×C7H(5,𝔽2)11yescharacteristic
169C23×C7H(5,𝔽2)12yes—
170C23×C7H(5,𝔽2)11yescharacteristic
171C23×C72−1+411yescharacteristic
172C23×C72−1+411yescharacteristic
173C23×C72−1+411yescharacteristic
174C23×C72−1+411yescharacteristic
175C2×C4×C7—22no—
176C2×C4×C7—22no—
177C2×C4×C7—24no—
178C2×C4×C7—24no—
179C2×C4×C7—22no—
180C2×C4×C7—22no—
181C2×C4×C7—22no—
182C2×C4×C7—22no—
183C2×C4×C7—22no—
184C2×C4×C7—22no—
185C2×C4×C7—24no—
186C2×C4×C7—24no—
187C2×C4×C7—24no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
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#11714yes—
204C2×Dic7—24no—
205C2×Dic7—24no—
206C2×Dic7—24no—
207C2×Dic7—24no—
208C2×Dic7—24no—
209C2×Dic7—24no—
210C2×Dic7—24no—
211C2×Dic7—24no—
212C2×Dic7—24no—
213C2×Dic7—42no—
214C2×Dic7—42no—
215C2×Dic7—42no—
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—
232C23×C4—72no—
233C23×C4—72no—
234C23×C4—72no—
235C23×C4—72no—
236C23×C4C22×D712yes—
237C23×C4C22×D711yescharacteristic
238C23×C4C22×D711yescharacteristic
239C23×C4C22×D712yes—
240C23×C4C22×D711yescharacteristic
241C2×C4⋊C4—74no—
242C2×C4⋊C4—74no—
243C2×C4⋊C4—144no—
244C2×C4⋊C4—282no—
245C2×C4⋊C4—282no—
246C2×C4⋊C4—282no—
247C2×C4⋊C4—282no—
248C2×C42—24no—
249C2×C42—144no—
250C2×C42—282no—
251C2×C42—282no—
252C2×C42C4×D714yes—
253#95—24no—
254#95—24no—
255#95—41no—
256#95—41no—
257#95—78no—
258#95—78no—
259#95—144no—
260#95—144no—
261#95—144no—
262#95—144no—
263#95—281no—
264#95—281no—
265#95—282no—
266#95—282no—
267#95—282no—
268#95—282no—
269C22×C7C2×#11711yescharacteristic
270C22×C7C4×#4111yescharacteristic
271C22×C7C4×#4111yescharacteristic
272C22×C7#51811yescharacteristic
273C22×C7#51911yescharacteristic
274C22×C7#52011yescharacteristic
275C22×C7#52011yescharacteristic
276C22×C7#53812yes—
277C22×C7#53812yes—
278C22×C7#53912yes—
279C22×C7#53912yes—
280C22×C7#54212yes—
281C22×C7#54212yes—
282C22×C7#55212yes—
283C22×C7#55212yes—
284C22×C7#55212yes—
285C22×C7#55312yes—
286C22×C7#55512yes—
287C22×C7#55612yes—
288C22×C7#56211yescharacteristic
289C22×C7#56311yescharacteristic
290C22×C7#56311yescharacteristic
291C22×C7#56411yescharacteristic
292C4×C7—24no—
293C4×C7—42no—
294C4×C7—42no—
295C4×C7—44no—
296C4×C7—82no—
297Dic7—44no—
298Dic7—44no—
299Dic7—44no—
300Dic7—82no—
301C24C23×D711yescharacteristic
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—74no—
327C22×C4—74no—
328C22×C4—74no—
329C22×C4—142no—
330C22×C4—142no—
331C22×C4—142no—
332C22×C4—142no—
333C22×C4—142no—
334C22×C4—142no—
335C22×C4—142no—
336C22×C4—142no—
337C22×C4—142no—
338C22×C4—142no—
339C22×C4—142no—
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×C4C2×C4×D712yes—
355C22×C4C2×C4×D712yes—
356C22×C4C2×C4×D712yes—
357C22×C4#115714yes—
358C22×C4#115914yes—
359C4⋊C4—284no—
360C4⋊C4—284no—
361C4⋊C4—288no—
362C42—28no—
363C42—288no—
364C2×C7#231311yescharacteristic
365C2×C7#233711yescharacteristic
366C2×C7#233711yescharacteristic
367C2×C7#234611yescharacteristic
368C2×C7#234711yescharacteristic
369C2×C7#237111yescharacteristic
370C2×C7#238211yescharacteristic
371C2×C7#276512yes—
372C2×C7#276712yes—
373C2×C7#277512yes—
374C2×C7#278112yes—
375C23C22×C4×D711yescharacteristic
376C23C2×#115712yes—
377C23C2×#115912yes—
378C23#654811yescharacteristic
379C23#655111yescharacteristic
380C23D7×#4112yes—
381C23#655311yescharacteristic
382C23#655312yes—
383C23#655411yescharacteristic
384C23#655411yescharacteristic
385C23#655411yescharacteristic
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—42no—
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—144no—
415C2×C4—144no—
416C2×C4—144no—
417C2×C4—144no—
418C2×C4—144no—
419C2×C4—144no—
420C2×C4—144no—
421C2×C4—144no—
422C2×C4—144no—
423C2×C4—282no—
424C2×C4—282no—
425C2×C4—282no—
426C2×C4—282no—
427C2×C4—282no—
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#644014yes—
443C7#1447711yescharacteristic, complemented, Sylow
444C22C4×#115711yescharacteristic
445C22C2×#644011yescharacteristic
446C22#9019511yescharacteristic
447C22#9020711yescharacteristic
448C22#9021211yescharacteristic
449C22#9021311yescharacteristic
450C22#9021411yescharacteristic
451C22#9022012yes—
452C22#9022212yes—
453C22#9023312yes—
454C22#9025112yes—
455C22#9025612yes—
456C22#9027312yes—
457C22#9030712yes—
458C22#9033012yes—
459C22#9035612yes—
460C22#9035612yes—
461C22#9036311yescharacteristic
462C22#9037511yescharacteristic
463C22C7⋊#56411yescharacteristic
464C22#9039011yescharacteristic
465C22#9039512yes—
466C22#9040112yes—
467C4—24no—
468C4—42no—
469C4—42no—
470C4—44no—
471C4—82no—
472C4—284no—
473C4—284no—
474C4—284no—
475C4—562no—
476C2#1173083911yescharacteristic
477C2#1173090511yescharacteristic
478C2#1173107711yescharacteristic
479C2#1173132611yescharacteristic
480C2#1173149911yescharacteristic
481C2#1173160611yescharacteristic
482C2#1173170711yescharacteristic
483C2#1173265512yes—
484C2#1173287612yes—
485C2#1173321012yes—
486C2#1173321912yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1434C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1434D7
hypocenterC7#14477
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
31. #89708C4semidirect
443. C7#14477semidirect

Maximal subgroups

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

subgroupindex
C2×#897082
C7×#14342
#117292882
#117271722
#117271682
#117271652
#117271612
#117271592
#117271572
#144777