Finite Group Databasebeta

← #13380620

Subgroups — #13380620

Order 1792.

1950 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic
2C2×#90516C211yesmaximal, characteristic
3C23×#1144C211yesmaximal, characteristic, complemented
4C2×#90006C211yesmaximal, characteristic
5C2×#89997C211yesmaximal, characteristic, complemented
6C2×#89776C211yesmaximal, characteristic, complemented
7#11729454C218yesmaximal, complemented
8#11729334C211yesmaximal, characteristic, complemented
9C22×C7×#41C2211yescharacteristic
10C23×C7×Q8C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C7×#536—24no—
13C7×#536—24no—
14C2×C7×C4⋊Q8—21no—
15C2×C7×#124—21no—
16C2×C7×#123—22no—
17C2×C7×C22⋊Q8C2214yes—
18C2×C7×#117C2211yescharacteristic
19#90516—24no—
20#90516—24no—
21C2×#6507—21no—
22C22×#1144—24no—
23C22×#1144—24no—
24C22×#1144—24no—
25C22×#1144C2214yes—
26C2×#6499—21no—
27C2×#6498—22no—
28C2×#6489C2211yescharacteristic
29C2×#6487C2214yes—
30C22×#1136C2211yescharacteristic
31C23×C7⋊C8C2211yescharacteristic
32#90006—24no—
33#89997—24no—
34C2×#6409C2214yes—
35C2×C22⋊(C7⋊C8)C2214yes—
36#89776—24no—
37C2×#6384C2214yes—
38#89324C2214yes—
39#23389—71nomaximal, Sylow
40C2×C7×#41—22no—
41C2×C7×#41—22no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C22×C7×Q8—21no—
47C22×C7×Q8—22no—
48C22×C7×Q8—22no—
49C22×C7×Q8—22no—
50C22×C7×Q8—22no—
51C22×C7×Q8—22no—
52C22×C7×Q8D414yes—
53C22×C7×Q8D411yescharacteristic
54C22×C7×Q8C2312yes—
55C22×C7×D4D412yes—
56C22×C7×D4D411yescharacteristic
57C23×C4×C7—21no—
58C23×C4×C7C2312yes—
59C23×C4×C7C2311yescharacteristic
60C23×Dic7—21no—
61C22×Dic14—22no—
62C22×Dic14—22no—
63C22×Dic14—22no—
64C22×Dic14—22no—
65C22×Dic14—21no—
66C22×Dic14—22no—
67C22×Dic14D414yes—
68C22×Dic14C2312yes—
69C7×C4⋊Q8—28no—
70C7×#124—28no—
71C7×#123—216no—
72C7×C22⋊Q8—28no—
73C7×C22⋊Q8—28no—
74C7×#117—22no—
75C7×#117D414yes—
76C2×C7×C4⋊C4—24no—
77C2×C7×C4⋊C4C2312yes—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—24no—
80C2×C42×C7—21no—
81C2×C22⋊Dic7—24no—
82C2×C22⋊Dic7—21no—
83#6507—28no—
84C2×#1144—232no—
85C2×#1144—28no—
86C2×#1144—28no—
87C2×#1144—28no—
88#6499—28no—
89#6498—216no—
90#6489—22no—
91#6489D414yes—
92C2×C4⋊Dic7C2312yes—
93#6487—28no—
94#6487—28no—
95C2×Dic7⋊C4—24no—
96C2×C4×Dic7—21no—
97C2×#1136—22no—
98C2×#1136—22no—
99C2×#1136—22no—
100C22×C7⋊C8—22no—
101C22×C7⋊C8—22no—
102C22×C7⋊C8D414yes—
103C22×C7⋊C8C2312yes—
104C22×C7⋊C8C2312yes—
105#6409—28no—
106C22⋊(C7⋊C8)—28no—
107#6384—28no—
108C23×Q16—71no—
109C2×#536—71no—
110C2×#536D711yescharacteristic
111C2×#419—71no—
112C2×#419—71no—
113C2×#408—71no—
114#2001—78no—
115#1735—71no—
116C24×C7C2×D412yes—
117C24×C7C2×D411yescharacteristic
118C7×#41—44no—
119C7×#41—44no—
120C7×#41—44no—
121C7×#41—44no—
122C2×C7×Q8—22no—
123C2×C7×Q8—24no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—28no—
128C2×C7×Q8—28no—
129C2×C7×Q8—22no—
130C2×C7×Q8—22no—
131C2×C7×Q8—22no—
132C2×C7×Q8—22no—
133C2×C7×Q8—42no—
134C2×C7×Q8—42no—
135C2×C7×Q8—44no—
136C2×C7×Q8—44no—
137C2×C7×Q8C2×D414yes—
138C2×C7×D4—24no—
139C2×C7×D4—24no—
140C2×C7×D4—24no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C22×C4×C7—24no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7#4112yes—
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—24no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—42no—
168C22×Dic7—42no—
169C2×Dic14—28no—
170C2×Dic14—28no—
171C2×Dic14—22no—
172C2×Dic14—22no—
173C2×Dic14—22no—
174C2×Dic14—22no—
175C2×Dic14—24no—
176C2×Dic14—22no—
177C2×Dic14—42no—
178C2×Dic14—44no—
179C2×Dic14—42no—
180C2×Dic14—44no—
181C2×Dic14C2×D414yes—
182C7×C4⋊C4—216no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4C2×D414yes—
185C7×C22⋊C4—216no—
186C7×C22⋊C4—24no—
187C42×C7—24no—
188C22⋊Dic7—216no—
189C22⋊Dic7—24no—
190#1144—232no—
191C4⋊Dic7—22no—
192C4⋊Dic7C2×D414yes—
193Dic7⋊C4—216no—
194C4×Dic7—24no—
195#1136—44no—
196C2×C7⋊C8—28no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8C2×D414yes—
203C22×#41C2×D711yescharacteristic
204C23×Q8—71no—
205C23×Q8C2×D711yescharacteristic
206C22×Q16—74no—
207C22×Q16—144no—
208C22×Q16—144no—
209C22×Q16—144no—
210C22×M16—71no—
211C23×C8—71no—
212#536—24no—
213#536—24no—
214#536—144no—
215#536—144no—
216C2×C4⋊Q8—21no—
217C2×C4⋊Q8—141no—
218C2×#124—21no—
219C2×#124—141no—
220C2×#123—22no—
221C2×#123—142no—
222C2×C22⋊Q8—74no—
223C2×C22⋊Q8C2×D714yes—
224C2×#117—71no—
225C2×#117C2×D711yescharacteristic
226#419—144no—
227#419—144no—
228C2×#103—74no—
229C2×#103—74no—
230#408—144no—
231C2×C22⋊C8—74no—
232#340—74no—
233C23×C7—24no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—28no—
238C23×C7#12012yes—
239C23×C7#12012yes—
240C23×C7#12112yes—
241C23×C7C22×D411yescharacteristic
242C7×Q8—24no—
243C7×Q8—24no—
244C7×Q8—42no—
245C7×Q8—42no—
246C7×Q8—48no—
247C7×Q8—44no—
248C7×D4—48no—
249C7×D4—44no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—44no—
258C2×C4×C7—48no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7#12014yes—
264C2×C4×C7#12012yes—
265C2×C4×C7#12012yes—
266C2×C4×C7#12114yes—
267C2×C4×C7#12114yes—
268C2×C4×C7#12112yes—
269C2×C4×C7#12714yes—
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×Dic7—24no—
277C2×Dic7—24no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—44no—
281C2×Dic7—48no—
282Dic14—24no—
283Dic14—24no—
284Dic14—42no—
285Dic14—42no—
286Dic14—48no—
287C7⋊C8—24no—
288C7⋊C8—42no—
289C2×#41—22no—
290C2×#41—22no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C22×Q8—21no—
296C22×Q8—22no—
297C22×Q8—22no—
298C22×Q8—22no—
299C22×Q8—22no—
300C22×Q8—22no—
301C22×Q8—74no—
302C22×Q8—72no—
303C22×Q8—142no—
304C22×Q8—141no—
305C22×Q8—142no—
306C22×Q8—142no—
307C22×Q8—142no—
308C22×Q8—142no—
309C22×Q8C22⋊D714yes—
310C22×Q8C22⋊D711yescharacteristic
311C22×Q8C22×D712yes—
312C22×D4C22⋊D712yes—
313C22×D4C22⋊D711yescharacteristic
314C23×C4—21no—
315C23×C4—141no—
316C23×C4C22×D712yes—
317C23×C4C22×D711yescharacteristic
318C2×Q16—148no—
319C2×Q16—148no—
320C2×Q16—1432no—
321C2×Q16—148no—
322C2×M16—142no—
323C2×M16—142no—
324C2×M16—142no—
325C22×C8—74no—
326C22×C8—72no—
327C22×C8—72no—
328C22×C8—142no—
329C22×C8—142no—
330C4⋊Q8—28no—
331C4⋊Q8—148no—
332#124—28no—
333#124—148no—
334#123—216no—
335#123—1416no—
336C22⋊Q8—28no—
337C22⋊Q8—28no—
338C22⋊Q8—148no—
339C22⋊Q8—148no—
340#117—22no—
341#117—74no—
342#117—142no—
343#117C22⋊D714yes—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—72no—
346C2×C4⋊C4—144no—
347C2×C4⋊C4C22×D712yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—144no—
352C2×C42—21no—
353C2×C42—141no—
354#103—148no—
355#103—148no—
356C22⋊C8—148no—
357C22×C7—24no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—42no—
361C22×C7—44no—
362C22×C7—44no—
363C22×C7—44no—
364C22×C7#39214yes—
365C22×C7C2×#12011yescharacteristic
366C22×C7C2×#12011yescharacteristic
367C22×C7C2×#12111yescharacteristic
368C22×C7#57814yes—
369C4×C7—42no—
370C4×C7—44no—
371C4×C7—44no—
372C4×C7#39214yes—
373C4×C7C2×#12011yescharacteristic
374C4×C7C2×#12111yescharacteristic
375C4×C7C2×#12111yescharacteristic
376C4×C7C2×#12711yescharacteristic
377Dic7—44no—
378Dic7—44no—
379C24C2×C22⋊D712yes—
380C24C2×C22⋊D711yescharacteristic
381#41—44no—
382#41—44no—
383#41—44no—
384#41—44no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—74no—
401C2×Q8—144no—
402C2×Q8—142no—
403C2×Q8—148no—
404C2×Q8—148no—
405C2×Q8—142no—
406C2×Q8—142no—
407C2×Q8—142no—
408C2×Q8—142no—
409C2×Q8—282no—
410C2×Q8—284no—
411C2×Q8—284no—
412C2×Q8—282no—
413C2×Q8C2×C22⋊D714yes—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—144no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—282no—
435C22×C4—282no—
436C22×C4D4×D712yes—
437C22×C4D4×D712yes—
438C22×C4D4×D712yes—
439C22×C4#115912yes—
440C22×C4C2×C22⋊D712yes—
441C22×C4C2×C22⋊D711yescharacteristic
442C22×C4C2×C22⋊D712yes—
443C22×C4C2×C22⋊D712yes—
444C22×C4C23×D711yescharacteristic
445Q16—1432no—
446M16—284no—
447C2×C8—74no—
448C2×C8—148no—
449C2×C8—142no—
450C2×C8—142no—
451C2×C8—142no—
452C2×C8—142no—
453C2×C8—142no—
454C4⋊C4—216no—
455C4⋊C4—22no—
456C4⋊C4—74no—
457C4⋊C4—1416no—
458C4⋊C4—142no—
459C4⋊C4C2×C22⋊D714yes—
460C22⋊C4—216no—
461C22⋊C4—24no—
462C22⋊C4—1416no—
463C22⋊C4—144no—
464C42—24no—
465C42—144no—
466C2×C7—24no—
467C2×C7—82no—
468C2×C7C2×#39211yescharacteristic
469C2×C7#301112yes—
470C2×C7#301112yes—
471C2×C7#305912yes—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—22no—
477C23#650012yes—
478C23#650212yes—
479C23#651612yes—
480C23C22×C22⋊D711yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—42no—
484Q8—42no—
485Q8—48no—
486Q8—44no—
487Q8—144no—
488Q8—144no—
489Q8—282no—
490Q8—282no—
491Q8—288no—
492D4—48no—
493D4—44no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—144no—
508C2×C4—144no—
509C2×C4—282no—
510C2×C4—288no—
511C2×C4—284no—
512C2×C4—282no—
513C2×C4#650014yes—
514C2×C4#650012yes—
515C2×C4#650114yes—
516C2×C4#650214yes—
517C2×C4#650212yes—
518C2×C4#650314yes—
519C2×C4#651612yes—
520C2×C4C2×D4×D711yescharacteristic
521C2×C4C2×D4×D711yescharacteristic
522C2×C4C2×D4×D711yescharacteristic
523C2×C4C2×#115911yescharacteristic
524C2×C4C22×C22⋊D711yescharacteristic
525C2×C4C22×C22⋊D711yescharacteristic
526C8—144no—
527C8—282no—
528C7#2338911yescharacteristic, complemented, Sylow
529C22—24no—
530C22—24no—
531C22—24no—
532C22—42no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22#8999514yes—
537C22C2×#650011yescharacteristic
538C22C2×#650211yescharacteristic
539C22#9052914yes—
540C22C2×#651611yescharacteristic
541C4—42no—
542C4—44no—
543C4—44no—
544C4—284no—
545C4—284no—
546C4#8999514yes—
547C4C2×#650011yescharacteristic
548C4C2×#650111yescharacteristic
549C4C2×#650211yescharacteristic
550C4C2×#650311yescharacteristic
551C2—24no—
552C2—82no—
553C2#1174084412yes—
554C2#1174085612yes—
555C2C2×#8999511yescharacteristic
556C2#1174156212yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#536C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenterC2×#536D7
hypocenterC7#23389
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
3. C23×#1144C2semidirect
5. C2×#89997C2semidirect
6. C2×#89776C2semidirect
7. #11729454C2semidirect
8. #11729334C2semidirect
528. C7#23389semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C2×#905162
C23×#11442
C2×#900062
C2×#899972
C2×#897762
#117294542
#117293342
#233897