Finite Group Databasebeta

← #13281959

Subgroups — #13281959

Order 1792.

1670 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C4⋊Q8C211yesmaximal, characteristic
2C2×#90002C212yesmaximal
3C2×#89753C212yesmaximal
4#11729888C212yesmaximal
5#11729323C212yesmaximal
6#11729319C212yesmaximal
7#11729318C212yesmaximal
8#11729288C211yesmaximal, characteristic
9#11729288C211yesmaximal, characteristic
10C23×C7×Q8C2212yes—
11C2×C7×C4⋊Q8—28no—
12C2×C7×C4⋊Q8C418yescomplemented
13C22×C7×C4⋊C4C2212yes—
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C42×C7C2211yescharacteristic
17C22×C4⋊Dic7C2212yes—
18C22×C4×Dic7C2212yes—
19#90002—28no—
20C2×C7⋊#95C2212yes—
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2211yescharacteristic
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2211yescharacteristic
34#89753—28no—
35#14420—71nomaximal, Sylow
36C22×C7×Q8—24no—
37C22×C7×Q8—24no—
38C22×C7×Q8—24no—
39C22×C7×Q8—24no—
40C22×C7×Q8—42no—
41C22×C7×Q8—42no—
42C22×C7×Q8C2×C418yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×Dic7C2312yes—
49C23×Dic7C2312yes—
50C23×Dic7C2312yes—
51C23×Dic7C2312yes—
52C7×C4⋊Q8—232no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4—24no—
56C2×C7×C4⋊C4C2×C414yes—
57C2×C7×C4⋊C4C2×C418yes—
58C2×C7×C4⋊C4C2×C414yes—
59C2×C7×C4⋊C4D418yes—
60C2×C42×C7—24no—
61C2×C42×C7C2×C414yes—
62C2×C4⋊Dic7—24no—
63C2×C4⋊Dic7—24no—
64C2×C4⋊Dic7—42no—
65C2×C4×Dic7—24no—
66C2×C4×Dic7—24no—
67C2×C4×Dic7—42no—
68C7⋊#95—24no—
69C7⋊#95—24no—
70C7⋊#95—24no—
71C7⋊#95—24no—
72C7⋊#95—24no—
73C7⋊#95—22no—
74C7⋊#95—24no—
75C7⋊#95—22no—
76C7⋊#95—24no—
77C7⋊#95—24no—
78C7⋊#95—42no—
79C7⋊#95—42no—
80C7⋊#95D414yes—
81C7⋊#95Q814yescomplemented
82C22×C4⋊Q8D711yescharacteristic
83C2×#391—72no—
84C2×#387—72no—
85#1446—72no—
86#1445—72no—
87#1441—72no—
88#1436—72no—
89#1434—71no—
90#1434—71no—
91C24×C7C2411yescharacteristic
92C2×C7×Q8—216no—
93C2×C7×Q8—216no—
94C2×C7×Q8—48no—
95C2×C7×Q8—48no—
96C2×C7×Q8—48no—
97C2×C7×Q8—48no—
98C2×C7×Q8—48no—
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—21no—
108C22×C4×C7—21no—
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×C7—22no—
116C22×C4×C7C22×C414yes—
117C22×C4×C7C22×C412yes—
118C22×C4×C7C22×C414yes—
119C22×C4×C7C22×C412yes—
120C22×C4×C7C22×C412yes—
121C22×C4×C7C2×D414yes—
122C22×C4×C7C2×D414yes—
123C22×C4×C7C2×D414yes—
124C22×C4×C7C2×D412yes—
125C22×C4×C7C2×Q812yes—
126C22×C4×C7#4112yes—
127C22×C4×C7#4112yes—
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×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7C2×D414yes—
155C22×Dic7C2×Q814yes—
156C7×C4⋊C4—216no—
157C7×C4⋊C4—28no—
158C7×C4⋊C4C22⋊C4116yes—
159C42×C7—28no—
160C4⋊Dic7—48no—
161C4×Dic7—48no—
162C23×Q8C2×D712yes—
163C2×C4⋊Q8—28no—
164C2×C4⋊Q8Dic718yescomplemented
165C22×C4⋊C4—72no—
166C22×C4⋊C4C2×D712yes—
167C22×C4⋊C4C2×D711yescharacteristic
168C22×C4⋊C4C2×D711yescharacteristic
169C22×C42—72no—
170C22×C42C2×D711yescharacteristic
171#391—148no—
172#387—148no—
173C2×#95—71no—
174C2×#95—71no—
175C2×#95—72no—
176C2×#95—71no—
177C2×#95—72no—
178C2×#95—72no—
179C2×#95—71no—
180C2×#95—72no—
181C2×#95—72no—
182C2×#95—72no—
183C2×#95—72no—
184C2×#95—72no—
185C2×#95—72no—
186C2×#95—72no—
187C23×C7C23×C411yescharacteristic
188C23×C7C22×D411yescharacteristic
189C23×C7C22×D412yes—
190C23×C7C22×Q811yescharacteristic
191C23×C7C2×#4111yescharacteristic
192C23×C7C2×#4111yescharacteristic
193C23×C7H(5,𝔽2)12yes—
194C23×C72−1+412yes—
195C23×C72−1+411yescharacteristic
196C23×C72−1+411yescharacteristic
197C23×C72−1+412yes—
198C7×Q8—432no—
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—24no—
208C2×C4×C7—24no—
209C2×C4×C7—24no—
210C2×C4×C7—24no—
211C2×C4×C7—24no—
212C2×C4×C7—24no—
213C2×C4×C7—24no—
214C2×C4×C7—24no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7C2×C22⋊C414yes—
225C2×C4×C7C2×C22⋊C418yes—
226C2×C4×C7C4×D414yes—
227C2×C4×C7C4×Q814yes—
228C2×Dic7—24no—
229C2×Dic7—24no—
230C2×Dic7—24no—
231C2×Dic7—24no—
232C2×Dic7—24no—
233C2×Dic7—24no—
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—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C22×Q8—24no—
257C22×Q8—24no—
258C22×Q8—24no—
259C22×Q8—24no—
260C22×Q8—42no—
261C22×Q8—42no—
262C22×Q8C2×Dic718yes—
263C23×C4—72no—
264C23×C4—72no—
265C23×C4—72no—
266C23×C4—72no—
267C23×C4C22×D712yes—
268C23×C4C22×D711yescharacteristic
269C23×C4C22×D711yescharacteristic
270C23×C4C22×D712yes—
271C23×C4C22×D711yescharacteristic
272C4⋊Q8—232no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—144no—
277C2×C4⋊C4—144no—
278C2×C4⋊C4—282no—
279C2×C4⋊C4C2×Dic714yes—
280C2×C4⋊C4C2×Dic718yes—
281C2×C4⋊C4C2×Dic714yes—
282C2×C4⋊C4C22⋊D718yes—
283C2×C42—24no—
284C2×C42—144no—
285C2×C42—144no—
286C2×C42—282no—
287C2×C42C2×Dic714yes—
288#95—74no—
289#95—74no—
290#95—144no—
291#95—144no—
292#95—144no—
293#95—142no—
294#95—144no—
295#95—142no—
296#95—144no—
297#95—144no—
298#95—144no—
299#95—144no—
300#95—282no—
301#95—282no—
302C22×C7C22×C22⋊C411yescharacteristic
303C22×C7C2×C4×D411yescharacteristic
304C22×C7C2×C4×Q811yescharacteristic
305C22×C7#51811yescharacteristic
306C22×C7#51911yescharacteristic
307C22×C7#51911yescharacteristic
308C22×C7#51911yescharacteristic
309C22×C7#53612yes—
310C22×C7#53612yes—
311C22×C7#53612yes—
312C22×C7#54112yes—
313C22×C7#54212yes—
314C22×C7#54412yes—
315C22×C7D4212yes—
316C22×C7#54612yes—
317C22×C7#54711yescharacteristic
318C22×C7#54812yes—
319C22×C7D4×Q811yescharacteristic
320C22×C7D4×Q812yes—
321C22×C7#55611yescharacteristic
322C22×C7#55711yescharacteristic
323C22×C7#56312yes—
324C22×C7#56312yes—
325C4×C7—24no—
326C4×C7—28no—
327C4×C7—24no—
328C4×C7—24no—
329C4×C7—44no—
330Dic7—44no—
331Dic7—44no—
332Dic7—82no—
333Dic7—82no—
334C24C23×D711yescharacteristic
335C2×Q8—216no—
336C2×Q8—216no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—74no—
360C22×C4—74no—
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×C4—142no—
384C22×C4—142no—
385C22×C4—142no—
386C22×C4—142no—
387C22×C4D4×D712yes—
388C22×C4#115912yes—
389C22×C4Q8×D712yes—
390C22×C4#116112yes—
391C22×C4C22×Dic714yes—
392C22×C4C22×Dic712yes—
393C22×C4C22×Dic712yes—
394C22×C4C22×Dic712yes—
395C22×C4C22×Dic714yes—
396C22×C4C2×C22⋊D714yes—
397C22×C4C2×C22⋊D714yes—
398C22×C4C2×C22⋊D714yes—
399C4⋊C4—216no—
400C4⋊C4—28no—
401C4⋊C4—288no—
402C4⋊C4C22⋊Dic7116yes—
403C42—28no—
404C42—288no—
405C2×C7#231011yescharacteristic
406C2×C7#232711yescharacteristic
407C2×C7#233111yescharacteristic
408C2×C7D4×C22⋊C411yescharacteristic
409C2×C7Q8×C22⋊C411yescharacteristic
410C2×C7#236511yescharacteristic
411C2×C7#238111yescharacteristic
412C2×C7#267412yes—
413C2×C7#267712yes—
414C2×C7#271312yes—
415C2×C7#271612yes—
416C23C2×D4×D711yescharacteristic
417C23C2×#115911yescharacteristic
418C23#654812yes—
419C23C2×Q8×D711yescharacteristic
420C23C2×#116111yescharacteristic
421C23#655111yescharacteristic
422C23#655112yes—
423C23#655112yes—
424C23#655111yescharacteristic
425C23C23×Dic711yescharacteristic
426C23C22×C22⋊D712yes—
427Q8—432no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—144no—
454C2×C4—144no—
455C2×C4—144no—
456C2×C4—144no—
457C2×C4—144no—
458C2×C4—144no—
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—282no—
475C2×C4—282no—
476C2×C4—282no—
477C2×C4—282no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4—282no—
481C2×C4D4×Dic714yes—
482C2×C4Q8×Dic714yes—
483C2×C4C2×C22⋊Dic714yes—
484C2×C4C2×C22⋊Dic718yes—
485C7#1442011yescharacteristic, complemented, Sylow
486C22#9041211yescharacteristic
487C22#9041512yes—
488C22#9041711yescharacteristic
489C22#9041811yescharacteristic
490C22#9042012yes—
491C22#9042212yes—
492C22#9042311yescharacteristic
493C22#9042412yes—
494C22#9042612yes—
495C22#9042612yes—
496C22C2×D4×Dic711yescharacteristic
497C22#9048911yescharacteristic
498C22D4×C22⋊D712yes—
499C22#9049712yes—
500C22C2×Q8×Dic711yescharacteristic
501C22#9050311yescharacteristic
502C22#9050311yescharacteristic
503C22#9050311yescharacteristic
504C22Q8×C22⋊D712yes—
505C22#9050712yes—
506C22#9050712yes—
507C22#9050812yes—
508C22C22×C22⋊Dic711yescharacteristic
509C4—24no—
510C4—28no—
511C4—24no—
512C4—24no—
513C4—44no—
514C4—284no—
515C4—284no—
516C4—562no—
517C4—562no—
518C2#1174067211yescharacteristic
519C2#1174067311yescharacteristic
520C2#1174067411yescharacteristic
521C2#1174067611yescharacteristic
522C2#1174069212yes—
523C2#1174069312yes—
524C2#1174071012yes—
525C2#1174071112yes—
526C2D4×C22⋊Dic711yescharacteristic
527C2#1174150711yescharacteristic
528C2Q8×C22⋊Dic711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC22×C7×C4⋊Q8C2
socle, prime radical, cosocleC24×C7C24
hypercenterC22×C4⋊Q8D7
hypocenterC7#14420
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
12. C2×C7×C4⋊Q8C4semidirect
81. C7⋊#95Q8semidirect
164. C2×C4⋊Q8Dic7semidirect
485. C7#14420semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C4⋊Q82
C2×#900022
C2×#897532
#117298882
#117293232
#117293192
#117293182
#117292882
#144207