Finite Group Databasebeta

← #13282308

Subgroups — #13282308

Order 1792.

1614 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6507C211yesmaximal, characteristic
2C2×#89418C212yesmaximal
3C7×#1446C211yesmaximal, characteristic
4#11729888C211yesmaximal, characteristic
5#11727163C212yesmaximal
6#11727163C212yesmaximal
7#11727160C212yesmaximal
8#11727160C212yesmaximal
9#11727156C212yesmaximal
10C23×C7×Q8C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C2×#6507—28no—
13C2×#6507C418yescomplemented
14C22×Dic7⋊C4C2212yes—
15C22×Dic7⋊C4C2212yes—
16C22×Dic7⋊C4C2212yes—
17C22×C4×Dic7C2211yescharacteristic
18C22×C4×Dic7C2212yes—
19C2×C7×#95C2212yes—
20C2×C7×#95C2211yescharacteristic
21C2×C7×#95C2212yes—
22C2×C7×#95C2211yescharacteristic
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2212yes—
33#89418—28no—
34#14450—71nomaximal, Sylow
35C22×C7×Q8—24no—
36C22×C7×Q8—42no—
37C22×C7×Q8—41no—
38C22×C7×Q8—41no—
39C22×C7×Q8C2×C414yes—
40C23×C4×C7C2312yes—
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C22×Dic14—41no—
50C22×Dic14—42no—
51C22×Dic14—41no—
52C22×Dic14C2×C414yes—
53C22×Dic14D418yes—
54#6507—232no—
55C2×Dic7⋊C4—24no—
56C2×Dic7⋊C4—24no—
57C2×Dic7⋊C4—24no—
58C2×Dic7⋊C4—24no—
59C2×Dic7⋊C4—24no—
60C2×Dic7⋊C4—42no—
61C2×Dic7⋊C4—42no—
62C2×Dic7⋊C4C2×C418yes—
63C2×Dic7⋊C4C2×C418yes—
64C2×C4×Dic7—24no—
65C2×C4×Dic7—24no—
66C2×C4×Dic7—42no—
67C2×C4×Dic7—42no—
68C2×C4×Dic7C2×C414yes—
69C7×#95—24no—
70C7×#95—41no—
71C7×#95—42no—
72C7×#95—41no—
73C7⋊#95—24no—
74C7⋊#95—24no—
75C7⋊#95—24no—
76C7⋊#95—24no—
77C7⋊#95—24no—
78C7⋊#95—24no—
79C7⋊#95—41no—
80C7⋊#95—42no—
81C7⋊#95—41no—
82C7⋊#95Q818yescomplemented
83C22×C4⋊Q8—71no—
84C2×#391—72no—
85#1446—71no—
86#1446D711yescharacteristic
87#1441—72no—
88#1441—72no—
89#1436—72no—
90#1436—72no—
91#1434—72no—
92C24×C7C2411yescharacteristic
93C2×C7×Q8—28no—
94C2×C7×Q8—44no—
95C2×C7×Q8—44no—
96C2×C7×Q8—48no—
97C2×C7×Q8—48no—
98C22×C4×C7—22no—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7C22×C412yes—
123C22×C4×C7C22×C412yes—
124C22×C4×C7C22×C412yes—
125C22×C4×C7C2×D414yes—
126C22×C4×C7C2×Q814yes—
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×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7C22×C414yes—
149C22×Dic7C22×C414yes—
150C22×Dic7C2×D414yes—
151C22×Dic7C2×D414yes—
152C22×Dic7C2×Q814yes—
153C22×Dic7C2×Q814yes—
154C22×Dic7#4114yes—
155C2×Dic14—44no—
156C2×Dic14—44no—
157C2×Dic14—48no—
158C2×Dic14—48no—
159C2×Dic14C22⋊C4116yes—
160Dic7⋊C4—216no—
161Dic7⋊C4—216no—
162Dic7⋊C4—48no—
163C4×Dic7—28no—
164C4×Dic7—48no—
165C23×Q8—71no—
166C23×Q8C2×D711yescharacteristic
167C2×C4⋊Q8—78no—
168C2×C4⋊Q8—148no—
169C22×C4⋊C4—72no—
170C22×C4⋊C4—72no—
171C22×C4⋊C4—72no—
172C22×C42—71no—
173C22×C42—72no—
174#391—148no—
175C2×#95—72no—
176C2×#95—72no—
177C2×#95—72no—
178C2×#95—72no—
179C2×#95—72no—
180C2×#95—72no—
181C2×#95—72no—
182C2×#95—71no—
183C2×#95—72no—
184C2×#95—71no—
185C2×#95C2×D712yes—
186C2×#95C2×D711yescharacteristic
187C2×#95C2×D712yes—
188C2×#95C2×D711yescharacteristic
189C23×C7C23×C411yescharacteristic
190C23×C7C22×D412yes—
191C23×C7C22×Q812yes—
192C23×C7C2×#4112yes—
193C23×C7H(5,𝔽2)12yes—
194C23×C7H(5,𝔽2)11yescharacteristic
195C23×C7H(5,𝔽2)11yescharacteristic
196C23×C72−1+412yes—
197C23×C72−1+412yes—
198C7×Q8—416no—
199C2×C4×C7—22no—
200C2×C4×C7—22no—
201C2×C4×C7—24no—
202C2×C4×C7—24no—
203C2×C4×C7—22no—
204C2×C4×C7—22no—
205C2×C4×C7—22no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—24no—
210C2×C4×C7—24no—
211C2×C4×C7—24no—
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×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7C2×C22⋊C414yes—
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—24no—
241C2×Dic7—24no—
242C2×Dic7—24no—
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×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7C2×C22⋊C418yes—
255C2×Dic7C4×Q818yes—
256Dic14—416no—
257C22×Q8—24no—
258C22×Q8—41no—
259C22×Q8—42no—
260C22×Q8—41no—
261C22×Q8—74no—
262C22×Q8—78no—
263C22×Q8—281no—
264C22×Q8—282no—
265C22×Q8—281no—
266C22×Q8C4×D714yes—
267C23×C4—72no—
268C23×C4—72no—
269C23×C4—72no—
270C23×C4—72no—
271C23×C4C22×D712yes—
272C23×C4C22×D711yescharacteristic
273C23×C4C22×D711yescharacteristic
274C23×C4C22×D712yes—
275C23×C4C22×D711yescharacteristic
276C4⋊Q8—1432no—
277C2×C4⋊C4—78no—
278C2×C4⋊C4—78no—
279C2×C4⋊C4—144no—
280C2×C4⋊C4—144no—
281C2×C4⋊C4—144no—
282C2×C4⋊C4—144no—
283C2×C4⋊C4—144no—
284C2×C4⋊C4—282no—
285C2×C4⋊C4—282no—
286C2×C42—74no—
287C2×C42—144no—
288C2×C42—144no—
289C2×C42—282no—
290C2×C42—282no—
291#95—24no—
292#95—41no—
293#95—42no—
294#95—41no—
295#95—78no—
296#95—144no—
297#95—144no—
298#95—144no—
299#95—144no—
300#95—144no—
301#95—144no—
302#95—281no—
303#95—282no—
304#95—281no—
305C22×C7C22×C22⋊C411yescharacteristic
306C22×C7C2×C4×Q812yes—
307C22×C7#51811yescharacteristic
308C22×C7#51811yescharacteristic
309C22×C7#51912yes—
310C22×C7#53412yes—
311C22×C7#53412yes—
312C22×C7#54212yes—
313C22×C7#54212yes—
314C22×C7#54412yes—
315C22×C7#54412yes—
316C22×C7#54812yes—
317C22×C7#54812yes—
318C22×C7D4×Q812yes—
319C22×C7D4×Q812yes—
320C22×C7#55412yes—
321C22×C7#55511yescharacteristic
322C22×C7Q8211yescharacteristic
323C22×C7#56311yescharacteristic
324C22×C7#56312yes—
325C22×C7#56311yescharacteristic
326C4×C7—24no—
327C4×C7—42no—
328C4×C7—42no—
329C4×C7—44no—
330C4×C7—82no—
331Dic7—28no—
332Dic7—28no—
333Dic7—44no—
334Dic7—82no—
335C24C23×D711yescharacteristic
336C2×Q8—28no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—716no—
342C2×Q8—284no—
343C2×Q8—284no—
344C2×Q8—288no—
345C2×Q8—288no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—74no—
371C22×C4—74no—
372C22×C4—74no—
373C22×C4—74no—
374C22×C4—74no—
375C22×C4—74no—
376C22×C4—74no—
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×C4—142no—
393C22×C4—142no—
394C22×C4—142no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4C2×C4×D712yes—
399C22×C4C2×C4×D712yes—
400C22×C4C2×C4×D712yes—
401C22×C4D4×D714yes—
402C22×C4Q8×D714yes—
403C4⋊C4—1416no—
404C4⋊C4—1416no—
405C4⋊C4—288no—
406C42—148no—
407C42—288no—
408C2×C7#230811yescharacteristic
409C2×C7#233112yes—
410C2×C7Q8×C22⋊C412yes—
411C2×C7#236911yescharacteristic
412C2×C7#238111yescharacteristic
413C2×C7#272912yes—
414C2×C7#272912yes—
415C2×C7#273112yes—
416C2×C7#273412yes—
417C23C22×C4×D711yescharacteristic
418C23C2×D4×D712yes—
419C23#654811yescharacteristic
420C23#654812yes—
421C23#654811yescharacteristic
422C23C2×Q8×D712yes—
423C23#655112yes—
424C23D7×#4112yes—
425C23#655412yes—
426Q8—416no—
427Q8—2816no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—78no—
457C2×C4—78no—
458C2×C4—144no—
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—282no—
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×C4D7×C22⋊C414yes—
485C7#1445011yescharacteristic, complemented, Sylow
486C22C2×D7×C22⋊C411yescharacteristic
487C22#9017711yescharacteristic
488C22#9017711yescharacteristic
489C22C4×Q8×D712yes—
490C22#9026312yes—
491C22#9028012yes—
492C22#9028012yes—
493C22#9031412yes—
494C22#9032312yes—
495C22#9032412yes—
496C22#9032712yes—
497C22#9033112yes—
498C22#9033712yes—
499C22#9036012yes—
500C22#9036111yescharacteristic
501C22#9036311yescharacteristic
502C22#9036812yes—
503C22#9041311yescharacteristic
504C22#9041812yes—
505C22#9042012yes—
506C22#9042611yescharacteristic
507C4—24no—
508C4—44no—
509C4—42no—
510C4—42no—
511C4—82no—
512C4—148no—
513C4—148no—
514C4—284no—
515C4—562no—
516C2#1173129411yescharacteristic
517C2#1173393112yes—
518C2#1173393911yescharacteristic
519C2#1173394211yescharacteristic
520C2#1173405912yes—
521C2#1173427812yes—
522C2#1173427912yes—
523C2#1173466812yes—
524C2#1173466912yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1446C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1446D7
hypocenterC7#14450
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
13. C2×#6507C4semidirect
82. C7⋊#95Q8semidirect
485. C7#14450semidirect

Maximal subgroups

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

subgroupindex
C22×#65072
C2×#894182
C7×#14462
#117298882
#117271632
#117271602
#117271562
#144507