Finite Group Databasebeta

← #13282560

Subgroups — #13282560

Order 1792.

1254 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90002C211yesmaximal, characteristic
2C7×#1446C211yesmaximal, characteristic
3#11729888C211yesmaximal, characteristic
4#11727168C212yesmaximal
5#11727167C212yesmaximal
6#11727161C212yesmaximal
7#11727160C212yesmaximal
8#11727159C212yesmaximal
9Dic7⋊#95C212yesmaximal
10C23×C7×Q8C2211yescharacteristic
11C22×C4⋊Dic7C2211yescharacteristic
12C22×Dic7⋊C4C2212yes—
13C22×Dic7⋊C4C2212yes—
14C22×C4×Dic7C2212yes—
15C22×C4×Dic7C2211yescharacteristic
16C2×C7×#95C2212yes—
17C2×C7×#95C2212yes—
18C2×C7×#95C2211yescharacteristic
19C2×C7×#95C2211yescharacteristic
20#90002C418yescomplemented
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2211yescharacteristic
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32#14471—71nomaximal, Sylow
33C22×C7×Q8—24no—
34C22×C7×Q8—42no—
35C22×C7×Q8—41no—
36C22×C7×Q8—41no—
37C22×C7×Q8C2×C414yes—
38C23×C4×C7C2312yes—
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×Dic7C2312yes—
44C23×Dic7C2312yes—
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C2×C4⋊Dic7—41no—
48C2×C4⋊Dic7—41no—
49C2×C4⋊Dic7C2×C414yes—
50C2×Dic7⋊C4—24no—
51C2×Dic7⋊C4—42no—
52C2×Dic7⋊C4—42no—
53C2×Dic7⋊C4—42no—
54C2×Dic7⋊C4—42no—
55C2×Dic7⋊C4Q818yescomplemented
56C2×C4×Dic7—42no—
57C2×C4×Dic7—42no—
58C2×C4×Dic7—42no—
59C2×C4×Dic7—42no—
60C2×C4×Dic7C2×C414yes—
61C7×#95—24no—
62C7×#95—41no—
63C7×#95—42no—
64C7×#95—41no—
65C7⋊#95—24no—
66C7⋊#95—24no—
67C7⋊#95—24no—
68C7⋊#95—41no—
69C7⋊#95—42no—
70C7⋊#95—42no—
71C7⋊#95—42no—
72C7⋊#95—42no—
73C7⋊#95—41no—
74C7⋊#95C2×C418yes—
75C7⋊#95C2×C418yes—
76C2×#391—71no—
77#1446—71no—
78#1446D711yescharacteristic
79#1442—72no—
80#1442—72no—
81#1436—72no—
82#1436—72no—
83#1436—72no—
84C4⋊#95—72no—
85C24×C7C2411yescharacteristic
86C2×C7×Q8—28no—
87C2×C7×Q8—44no—
88C2×C7×Q8—44no—
89C2×C7×Q8—48no—
90C2×C7×Q8—48no—
91C22×C4×C7—22no—
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—21no—
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—22no—
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×C7#4114yes—
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×Dic7—22no—
143C22×Dic7C22×C414yes—
144C22×Dic7C22×C414yes—
145C22×Dic7C2×Q814yes—
146C22×Dic7C2×Q814yes—
147C22×Dic7#4114yes—
148C4⋊Dic7—44no—
149Dic7⋊C4—48no—
150Dic7⋊C4—48no—
151C4×Dic7—44no—
152C4×Dic7—48no—
153C23×Q8C2×D711yescharacteristic
154C22×C4⋊C4—72no—
155C22×C4⋊C4—71no—
156C22×C4⋊C4—72no—
157C22×C42—71no—
158C22×C42—72no—
159#391—78no—
160C2×#95—72no—
161C2×#95—72no—
162C2×#95—72no—
163C2×#95—72no—
164C2×#95—72no—
165C2×#95—72no—
166C2×#95—72no—
167C2×#95—71no—
168C2×#95—72no—
169C2×#95—72no—
170C2×#95—71no—
171C2×#95C2×D712yes—
172C2×#95C2×D711yescharacteristic
173C2×#95C2×D712yes—
174C2×#95C2×D711yescharacteristic
175C23×C7C23×C411yescharacteristic
176C23×C7C22×Q812yes—
177C23×C7C2×#4112yes—
178C23×C7C2×#4112yes—
179C23×C7H(5,𝔽2)12yes—
180C23×C7H(5,𝔽2)11yescharacteristic
181C23×C7H(5,𝔽2)11yescharacteristic
182C23×C7H(5,𝔽2)12yes—
183C23×C72−1+412yes—
184C7×Q8—416no—
185C2×C4×C7—22no—
186C2×C4×C7—22no—
187C2×C4×C7—24no—
188C2×C4×C7—24no—
189C2×C4×C7—22no—
190C2×C4×C7—22no—
191C2×C4×C7—22no—
192C2×C4×C7—22no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—24no—
196C2×C4×C7—24no—
197C2×C4×C7—24no—
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×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#11714yes—
214C2×Dic7—24no—
215C2×Dic7—24no—
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—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—
235C2×Dic7—42no—
236C2×Dic7—42no—
237C2×Dic7—42no—
238C2×Dic7—42no—
239C2×Dic7—42no—
240C2×Dic7—42no—
241C2×Dic7C4×Q818yes—
242C22×Q8—24no—
243C22×Q8—42no—
244C22×Q8—41no—
245C22×Q8—41no—
246C22×Q8C4×D714yes—
247C23×C4—72no—
248C23×C4—72no—
249C23×C4—72no—
250C23×C4—72no—
251C23×C4C22×D712yes—
252C23×C4C22×D711yescharacteristic
253C23×C4C22×D711yescharacteristic
254C23×C4C22×D712yes—
255C23×C4C22×D711yescharacteristic
256C2×C4⋊C4—74no—
257C2×C4⋊C4—78no—
258C2×C4⋊C4—144no—
259C2×C4⋊C4—282no—
260C2×C4⋊C4—282no—
261C2×C4⋊C4—282no—
262C2×C4⋊C4—282no—
263C2×C4⋊C4—281no—
264C2×C4⋊C4—281no—
265C2×C42—74no—
266C2×C42—282no—
267C2×C42—282no—
268C2×C42—282no—
269C2×C42—282no—
270#95—24no—
271#95—41no—
272#95—42no—
273#95—41no—
274#95—78no—
275#95—78no—
276#95—144no—
277#95—144no—
278#95—144no—
279#95—281no—
280#95—282no—
281#95—281no—
282#95—282no—
283#95—282no—
284#95—282no—
285C22×C7C2×#11711yescharacteristic
286C22×C7C2×C4×Q812yes—
287C22×C7#51811yescharacteristic
288C22×C7#51811yescharacteristic
289C22×C7#52012yes—
290C22×C7#53812yes—
291C22×C7#53812yes—
292C22×C7#53812yes—
293C22×C7#53912yes—
294C22×C7#54312yes—
295C22×C7#54412yes—
296C22×C7#55412yes—
297C22×C7#55412yes—
298C22×C7#55511yescharacteristic
299C22×C7#55612yes—
300C22×C7#55712yes—
301C22×C7Q8211yescharacteristic
302C22×C7#55912yes—
303C22×C7H(3,𝔽4)11yescharacteristic
304C22×C7#56212yes—
305C22×C7#56311yescharacteristic
306C4×C7—24no—
307C4×C7—42no—
308C4×C7—42no—
309C4×C7—44no—
310C4×C7—82no—
311Dic7—28no—
312Dic7—44no—
313Dic7—44no—
314Dic7—82no—
315C24C23×D711yescharacteristic
316C2×Q8—28no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—48no—
320C2×Q8—48no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—74no—
346C22×C4—74no—
347C22×C4—74no—
348C22×C4—74no—
349C22×C4—74no—
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×C4—142no—
371C22×C4—142no—
372C22×C4—142no—
373C22×C4C2×C4×D712yes—
374C22×C4C2×C4×D712yes—
375C22×C4C2×C4×D712yes—
376C22×C4#115914yes—
377C22×C4Q8×D714yes—
378C4⋊C4—288no—
379C4⋊C4—284no—
380C4⋊C4—288no—
381C42—284no—
382C42—288no—
383C2×C7#231211yescharacteristic
384C2×C7#232912yes—
385C2×C7#235512yes—
386C2×C7#236911yescharacteristic
387C2×C7#237911yescharacteristic
388C2×C7#280412yes—
389C2×C7#280512yes—
390C2×C7#280612yes—
391C2×C7#280812yes—
392C23C22×C4×D711yescharacteristic
393C23C2×#115912yes—
394C23#654812yes—
395C23#654811yescharacteristic
396C23#654811yescharacteristic
397C23C2×Q8×D712yes—
398C23#655112yes—
399C23D7×#4112yes—
400C23#655312yes—
401Q8—416no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
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—78no—
431C2×C4—144no—
432C2×C4—144no—
433C2×C4—144no—
434C2×C4—144no—
435C2×C4—144no—
436C2×C4—144no—
437C2×C4—144no—
438C2×C4—144no—
439C2×C4—144no—
440C2×C4—144no—
441C2×C4—144no—
442C2×C4—144no—
443C2×C4—282no—
444C2×C4—282no—
445C2×C4—282no—
446C2×C4—282no—
447C2×C4—282no—
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#644014yes—
459C7#1447111yescharacteristic, complemented, Sylow
460C22C2×#644011yescharacteristic
461C22#9017711yescharacteristic
462C22#9017711yescharacteristic
463C22C4×Q8×D712yes—
464C22#9026512yes—
465C22#9028412yes—
466C22#9028412yes—
467C22#9032512yes—
468C22#9032612yes—
469C22#9032712yes—
470C22#9032912yes—
471C22#9033212yes—
472C22#9033812yes—
473C22#9036111yescharacteristic
474C22#9036912yes—
475C22#9037212yes—
476C22#9037411yescharacteristic
477C22#9041212yes—
478C22#9041311yescharacteristic
479C22#9042012yes—
480C22#9042611yescharacteristic
481C4—24no—
482C4—44no—
483C4—42no—
484C4—42no—
485C4—82no—
486C4—148no—
487C4—284no—
488C4—284no—
489C4—562no—
490C2#1173126511yescharacteristic
491C2#1173393412yes—
492C2#1173393911yescharacteristic
493C2#1173394111yescharacteristic
494C2#1173405712yes—
495C2#1173429612yes—
496C2#1173429712yes—
497C2#1173468312yes—
498C2#1173468412yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1446C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1446D7
hypocenterC7#14471
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
20. #90002C4semidirect
55. C2×Dic7⋊C4Q8semidirect
459. C7#14471semidirect

Maximal subgroups

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

subgroupindex
C2×#900022
C7×#14462
#117298882
#117271682
#117271672
#117271612
#117271602
#117271592
Dic7⋊#952
#144717