Finite Group Databasebeta

← #13515161

Subgroups — #13515161

Order 1792.

2486 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90410C211yesmaximal, characteristic
2C7×#2328C211yesmaximal, characteristic, complemented
3Dic7×#127C211yesmaximal, characteristic, complemented
4C4×#6501C212yesmaximal, complemented
5#11739762C212yesmaximal, complemented
6#11733875C212yesmaximal, complemented
7#11733734C212yesmaximal, complemented
8#11732126C214yesmaximal, complemented
9#11732123C214yesmaximal, complemented
10#11732122C214yesmaximal, complemented
11#11732120C214yesmaximal, complemented
12#11731669C211yesmaximal, characteristic, complemented
13#11731595C211yesmaximal, characteristic, complemented
14#11730835C211yesmaximal, characteristic, complemented
15#11730827C211yesmaximal, characteristic, complemented
16C2×C7×#127C2211yescharacteristic
17C2×C4×C7×D4C2212yes—
18C2×#6502C2214yes—
19C2×#6501C2212yes—
20C2×#6498C2214yes—
21C2×D4×Dic7C2212yes—
22C2×D4×Dic7C2214yes—
23C2×D4×Dic7C2212yes—
24C2×C4×C22⋊D7C2214yes—
25#90410C4116yescomplemented
26C2×#6440C2214yes—
27C2×#6435C2211yescharacteristic
28C2×#6435C2211yescharacteristic
29C2×C42×D7C2211yescharacteristic
30C2×#6433C2211yescharacteristic
31C7×#390C2212yes—
32C7×#386C2214yes—
33C7×#386C2214yes—
34C7×#382C2211yescharacteristic
35C7×#376C2211yescharacteristic
36#89993C2214yes—
37#89993C2212yes—
38C4×C22⋊Dic7C2214yes—
39C4×C22⋊Dic7C2214yes—
40C2×C7⋊#95C2214yes—
41#89759C2212yes—
42C4⋊C4×Dic7C2212yes—
43C4⋊C4×Dic7C2212yes—
44#89745C2212yes—
45#89730C2214yes—
46#89729C2214yes—
47#89728C2214yes—
48#89718C2214yes—
49#89717C2214yes—
50#89716C2214yes—
51#89716C2214yes—
52#89716C2214yes—
53C22⋊C4×Dic7C2214yes—
54C22⋊C4×Dic7C2214yes—
55C22⋊C4×Dic7C2214yes—
56#89707C2211yescharacteristic
57C4×C4⋊Dic7C2212yes—
58#89704C2211yescharacteristic
59C42×Dic7C2211yescharacteristic
60C42×Dic7C2211yescharacteristic
61#89700C2212yes—
62#89442C2214yes—
63#89438C2212yes—
64#89438C2212yes—
65#89427C2214yes—
66#89427C2214yes—
67#89424C2214yes—
68#89421C2214yes—
69#89421C2212yes—
70#89421C2212yes—
71#89419C2214yes—
72#34280—71nomaximal, Sylow
73C22×C7×D4C2312yes—
74C22×C7×D4C2314yes—
75C23×C4×C7C2314yes—
76C22×C22⋊D7C2314yes—
77C23×Dic7C2314yes—
78C23×Dic7C2314yes—
79C22×C4×D7C2312yes—
80C22×C4×D7C2311yescharacteristic
81C7×#127C2×C418yescomplemented
82C4×C7×D4—28no—
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C22⋊C4C2314yes—
85C2×C7×C22⋊C4C2314yes—
86C2×C7×C22⋊C4C2314yes—
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2311yescharacteristic
90C2×C22⋊Dic7C2314yes—
91C2×C22⋊Dic7C2314yes—
92C2×C22⋊Dic7C2314yes—
93C2×C22⋊Dic7C2314yes—
94C2×C22⋊Dic7C2314yes—
95#6502C2×C4132yescomplemented
96#6501C2×C4116yescomplemented
97#6498C2×C4132yescomplemented
98D4×Dic7—28no—
99D4×Dic7—48no—
100D4×Dic7C2×C4116yescomplemented
101C4×C22⋊D7—48no—
102C2×(C2×D7)⋊C4C2311yescharacteristic
103C2×(C2×D7)⋊C4C2311yescharacteristic
104C2×(C2×D7)⋊C4C2314yes—
105C2×(C2×D7)⋊C4C2312yes—
106C2×C4⋊Dic7C2312yes—
107C2×C4⋊Dic7C2314yes—
108C2×C4⋊Dic7C2312yes—
109C2×Dic7⋊C4C2311yescharacteristic
110C2×Dic7⋊C4C2311yescharacteristic
111C2×Dic7⋊C4C2314yes—
112C2×Dic7⋊C4C2314yes—
113C2×Dic7⋊C4C2314yes—
114C2×Dic7⋊C4C2312yes—
115C2×C4×Dic7C2312yes—
116C2×C4×Dic7C2311yescharacteristic
117C2×C4×Dic7C2312yes—
118C2×C4×Dic7C2312yes—
119C2×C4×Dic7C2314yes—
120C2×C4×Dic7C2314yes—
121C2×C4×Dic7C2312yes—
122C2×C4×Dic7C2314yes—
123C2×C4×Dic7C2312yes—
124C2×C4×Dic7C2311yescharacteristic
125C2×C4×Dic7C2311yescharacteristic
126#6440—48no—
127#6435—24no—
128#6435C2×C418yescomplemented
129C42×D7—24no—
130#6433C2×C418yescomplemented
131C7×#95C2312yes—
132C7×#95C2314yes—
133C7×#95C2312yes—
134C7⋊#95C2314yes—
135C7⋊#95C2314yes—
136C7⋊#95C2312yes—
137C7⋊#95C2312yes—
138C7⋊#95C2314yes—
139C7⋊#95C2314yes—
140C7⋊#95C2314yes—
141C7⋊#95C2314yes—
142C2×#540—71no—
143#2374—74no—
144#2364—74no—
145#2363—74no—
146#2359—74no—
147#2330—71no—
148#2328D711yescharacteristic, complemented
149#2324—72no—
150#2314—71no—
151C4×#127—71no—
152C4×#125—71no—
153C4×#121—72no—
154#2293—72no—
155#2275—72no—
156C4×#117—71no—
157C24×C7C2414yes—
158C23×D7C2411yescharacteristic
159C2×C7×D4—24no—
160C2×C7×D4—24no—
161C2×C7×D4—44no—
162C2×C7×D4—44no—
163C2×C7×D4C22×C418yes—
164C2×C7×D4C22×C4116yes—
165C22×C4×C7—28no—
166C22×C4×C7—44no—
167C22×C4×C7—44no—
168C22×C4×C7C2411yescharacteristic
169C22×C4×C7C2414yes—
170C22×C4×C7C2411yescharacteristic
171C22×C4×C7C2412yes—
172C22×C4×C7C2412yes—
173C22×C4×C7C2411yescharacteristic
174C2×C22⋊D7—44no—
175C2×C22⋊D7—44no—
176C2×C22⋊D7C22×C4116yes—
177C22×Dic7—28no—
178C22×Dic7—44no—
179C22×Dic7—44no—
180C22×Dic7—44no—
181C22×Dic7—44no—
182C22×Dic7C22×C4116yes—
183C22×Dic7C2414yes—
184C22×Dic7C2412yes—
185C22×Dic7C2414yes—
186C22×Dic7C2411yescharacteristic
187C22×Dic7C2411yescharacteristic
188C22×Dic7C2411yescharacteristic
189C22×Dic7C2412yes—
190C2×C4×D7—22no—
191C2×C4×D7—22no—
192C2×C4×D7—24no—
193C2×C4×D7—24no—
194C2×C4×D7—42no—
195C2×C4×D7C22×C414yes—
196C7×C4⋊C4—24no—
197C7×C22⋊C4—28no—
198C7×C22⋊C4—44no—
199C7×C22⋊C4—44no—
200C42×C7—22no—
201C42×C7C22×C414yes—
202C42×C7#4114yes—
203C22⋊Dic7—28no—
204C22⋊Dic7—28no—
205C22⋊Dic7—28no—
206C22⋊Dic7C22×C4116yes—
207C22⋊Dic7C22×C4116yes—
208(C2×D7)⋊C4—22no—
209(C2×D7)⋊C4—22no—
210(C2×D7)⋊C4—44no—
211(C2×D7)⋊C4C22×C418yes—
212C4⋊Dic7—24no—
213C4⋊Dic7—44no—
214C4⋊Dic7C22×C418yes—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—28no—
218Dic7⋊C4—44no—
219Dic7⋊C4C22×C4116yes—
220Dic7⋊C4C22×C418yes—
221C4×Dic7—24no—
222C4×Dic7—22no—
223C4×Dic7—24no—
224C4×Dic7—24no—
225C4×Dic7—44no—
226C4×Dic7—42no—
227C4×Dic7—44no—
228C4×Dic7—42no—
229C4×Dic7—44no—
230C4×Dic7C22×C414yes—
231C4×Dic7#4114yes—
232#540—716no—
233C2×#127C2×D711yescharacteristic
234C2×#125—71no—
235C2×#123—74no—
236C2×#121—72no—
237C2×#121—74no—
238C2×C4×D4—72no—
239C2×C4×D4—72no—
240C2×C4×D4—74no—
241C2×C4×D4—74no—
242C2×C4×D4C2×D712yes—
243C2×#117—74no—
244C2×#117—71no—
245C2×#117—71no—
246C22×C42—71no—
247#390—74no—
248#390—72no—
249#390—74no—
250#390C2×D712yes—
251#389—72no—
252#389—72no—
253#389—74no—
254#388—74no—
255#388—74no—
256#388—74no—
257#388—74no—
258#388—74no—
259#387—74no—
260#387—74no—
261#387—74no—
262#386—74no—
263#386—74no—
264#386C2×D714yes—
265#386C2×D714yes—
266#385—74no—
267#384—71no—
268#382C2×D711yescharacteristic
269#380—72no—
270C4×C4⋊C4—72no—
271C4×C4⋊C4—72no—
272C4×C4⋊C4—72no—
273C4×C4⋊C4—74no—
274C4×C4⋊C4—72no—
275C4×C4⋊C4—72no—
276C4×C22⋊C4—74no—
277C4×C22⋊C4—74no—
278C4×C22⋊C4—74no—
279C4×C22⋊C4—74no—
280C4×C22⋊C4—74no—
281C4×C22⋊C4—72no—
282C4×C22⋊C4—72no—
283#376—71no—
284#376C2×D711yescharacteristic
285C2×#95—74no—
286C43—71no—
287C43—71no—
288C23×C7—24no—
289C23×C7—24no—
290C23×C7—24no—
291C23×C7—24no—
292C23×C7—24no—
293C23×C7—24no—
294C23×C7C23×C418yes—
295C23×C7C2511yescharacteristic
296C22×D7—21no—
297C22×D7—21no—
298C22×D7—22no—
299C22×D7—22no—
300C22×D7C23×C412yes—
301C7×D4—216no—
302C7×D4—416no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—22no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—24no—
310C2×C4×C7—22no—
311C2×C4×C7—24no—
312C2×C4×C7—22no—
313C2×C4×C7—24no—
314C2×C4×C7—88no—
315C2×C4×C7C23×C414yes—
316C2×C4×C7C23×C412yes—
317C2×C4×C7C2×#4114yes—
318C2×C4×C7C2×#4112yes—
319C2×C4×C7C2×#4112yes—
320C22⋊D7—416no—
321C2×Dic7—21no—
322C2×Dic7—22no—
323C2×Dic7—21no—
324C2×Dic7—22no—
325C2×Dic7—22no—
326C2×Dic7—24no—
327C2×Dic7—22no—
328C2×Dic7—22no—
329C2×Dic7—22no—
330C2×Dic7—24no—
331C2×Dic7—24no—
332C2×Dic7—24no—
333C2×Dic7—24no—
334C2×Dic7—416no—
335C2×Dic7—88no—
336C2×Dic7C23×C414yes—
337C2×Dic7C23×C418yes—
338C2×Dic7C23×C412yes—
339C2×Dic7C2×#4112yes—
340C2×Dic7C2×#4112yes—
341C2×Dic7H(5,𝔽2)14yes—
342C4×D7—28no—
343C4×D7—48no—
344C22×D4—74no—
345C22×D4C22×D712yes—
346C22×D4C22×D714yes—
347C23×C4—74no—
348C23×C4—71no—
349C23×C4—74no—
350C23×C4—72no—
351C23×C4C22×D714yes—
352#127C4×D718yescomplemented
353#125—78no—
354#123—732no—
355#121—716no—
356#121—732no—
357C4×D4—28no—
358C4×D4—716no—
359C4×D4—148no—
360C4×D4—288no—
361C4×D4—288no—
362#117—78no—
363#117—144no—
364#117—288no—
365C2×C4⋊C4—72no—
366C2×C4⋊C4—72no—
367C2×C4⋊C4—74no—
368C2×C4⋊C4—74no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—72no—
372C2×C4⋊C4—74no—
373C2×C4⋊C4—74no—
374C2×C4⋊C4C22×D712yes—
375C2×C22⋊C4—74no—
376C2×C22⋊C4—74no—
377C2×C22⋊C4—72no—
378C2×C22⋊C4—74no—
379C2×C22⋊C4—74no—
380C2×C22⋊C4—74no—
381C2×C22⋊C4—71no—
382C2×C22⋊C4—71no—
383C2×C22⋊C4—74no—
384C2×C22⋊C4C22×D714yes—
385C2×C22⋊C4C22×D714yes—
386C2×C22⋊C4C22×D714yes—
387C2×C42—71no—
388C2×C42—71no—
389C2×C42—72no—
390C2×C42—72no—
391C2×C42—71no—
392C2×C42—72no—
393C2×C42—74no—
394C2×C42—74no—
395C2×C42—72no—
396C2×C42—72no—
397C2×C42—74no—
398C2×C42—144no—
399C2×C42C22×D711yescharacteristic
400C2×C42C22×D711yescharacteristic
401C2×C42C22×D711yescharacteristic
402#95—72no—
403#95—74no—
404#95—74no—
405#95—74no—
406#95—72no—
407#95—74no—
408#95—74no—
409#95—74no—
410#95C22×D712yes—
411#95C22×D712yes—
412#95C22×D714yes—
413C22×C7—28no—
414C22×C7—28no—
415C22×C7—28no—
416C22×C7—44no—
417C22×C7—44no—
418C22×C7—44no—
419C22×C7—44no—
420C22×C7C24×C411yescharacteristic
421C22×C7C22×#4111yescharacteristic
422C22×C7C22×#4111yescharacteristic
423C22×C7C2×H(5,𝔽2)12yes—
424C22×C7#58512yes—
425C2×D7—22no—
426C2×D7—24no—
427C2×D7—42no—
428C2×D7—41no—
429C2×D7—41no—
430C4×C7—24no—
431C4×C7—24no—
432C4×C7—44no—
433C4×C7—41no—
434C4×C7—41no—
435C4×C7C4×#4114yes—
436Dic7—22no—
437Dic7—24no—
438Dic7—28no—
439Dic7—24no—
440Dic7—22no—
441Dic7—22no—
442Dic7—44no—
443C24—71no—
444C24C23×D714yes—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—716no—
450C2×D4—284no—
451C2×D4—284no—
452C2×D4C2×C4×D718yes—
453C2×D4C2×C4×D7116yes—
454C22×C4—28no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—71no—
458C22×C4—72no—
459C22×C4—71no—
460C22×C4—74no—
461C22×C4—71no—
462C22×C4—72no—
463C22×C4—716no—
464C22×C4—74no—
465C22×C4—74no—
466C22×C4—144no—
467C22×C4—148no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—144no—
471C22×C4—282no—
472C22×C4—284no—
473C22×C4—284no—
474C22×C4—284no—
475C22×C4—284no—
476C22×C4C23×D711yescharacteristic
477C22×C4C23×D714yes—
478C22×C4C23×D711yescharacteristic
479C22×C4C23×D712yes—
480C22×C4C23×D712yes—
481C22×C4C23×D711yescharacteristic
482C4⋊C4—24no—
483C4⋊C4—78no—
484C4⋊C4—716no—
485C4⋊C4—78no—
486C4⋊C4—142no—
487C4⋊C4—142no—
488C4⋊C4—148no—
489C4⋊C4—144no—
490C4⋊C4—284no—
491C4⋊C4—284no—
492C22⋊C4—28no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—716no—
496C22⋊C4—716no—
497C22⋊C4—78no—
498C22⋊C4—142no—
499C22⋊C4—142no—
500C22⋊C4—148no—
501C22⋊C4—148no—
502C22⋊C4—148no—
503C22⋊C4—284no—
504C42—22no—
505C42—74no—
506C42—74no—
507C42—144no—
508C42—144no—
509C42—144no—
510C42—142no—
511C42—282no—
512C42—282no—
513C42—284no—
514C42—284no—
515C42—284no—
516C42C2×C4×D714yes—
517C42#115914yes—
518C2×C7—48no—
519C2×C7C2×C4×#4111yescharacteristic
520C2×C7C4×H(5,𝔽2)12yes—
521C2×C7#349412yes—
522C2×C7#350811yescharacteristic
523C2×C7#351211yescharacteristic
524D7—42no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—72no—
532C23—141no—
533C23—141no—
534C23—142no—
535C23—142no—
536C23C22×C4×D718yes—
537C23C24×D711yescharacteristic
538D4—216no—
539D4—416no—
540D4—2816no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—74no—
553C2×C4—72no—
554C2×C4—78no—
555C2×C4—72no—
556C2×C4—72no—
557C2×C4—74no—
558C2×C4—88no—
559C2×C4—141no—
560C2×C4—142no—
561C2×C4—141no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—144no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—144no—
569C2×C4—148no—
570C2×C4—144no—
571C2×C4—144no—
572C2×C4—144no—
573C2×C4—288no—
574C2×C4—2816no—
575C2×C4—568no—
576C2×C4C22×C4×D714yes—
577C2×C4C22×C4×D712yes—
578C2×C4C2×#115914yes—
579C2×C4C2×#115912yes—
580C2×C4D7×#4112yes—
581C7#3428011yescharacteristic, complemented, Sylow
582C22—28no—
583C22—28no—
584C22—28no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—142no—
590C22—144no—
591C22—281no—
592C22—281no—
593C22—282no—
594C22C23×C4×D711yescharacteristic
595C22C22×#115911yescharacteristic
596C22C2×D7×#4111yescharacteristic
597C22D7×H(5,𝔽2)12yes—
598C22#9061812yes—
599C4—24no—
600C4—24no—
601C4—41no—
602C4—41no—
603C4—44no—
604C4—142no—
605C4—142no—
606C4—142no—
607C4—144no—
608C4—144no—
609C4—148no—
610C4—284no—
611C4C4×#115914yes—
612C2—48no—
613C2—282no—
614C2C2×C4×#115911yescharacteristic
615C2#1174292512yes—
616C2#1174340012yes—
617C2#1174353711yescharacteristic
618C2#1174355411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2328C2
socle, cosocleC23×C7C25
prime radicalC2×#90410C2
hypercenter#2328D7
hypocenterC7#34280
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. C7×#2328C2semidirect
3. Dic7×#127C2semidirect
4. C4×#6501C2semidirect
5. #11739762C2semidirect
6. #11733875C2semidirect
7. #11733734C2semidirect
8. #11732126C2semidirect
9. #11732123C2semidirect
10. #11732122C2semidirect
11. #11732120C2semidirect
12. #11731669C2semidirect
13. #11731595C2semidirect
14. #11730835C2semidirect
15. #11730827C2semidirect
25. #90410C4semidirect
81. C7×#127C2×C4semidirect
95. #6502C2×C4semidirect
96. #6501C2×C4semidirect
97. #6498C2×C4semidirect
100. D4×Dic7C2×C4semidirect
128. #6435C2×C4semidirect
130. #6433C2×C4semidirect
148. #2328D7semidirect
352. #127C4×D7semidirect
581. C7#34280semidirect

Maximal subgroups

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

subgroupindex
C2×#904102
C7×#23282
Dic7×#1272
C4×#65012
#117397622
#117338752
#117337342
#117321262
#117321232
#117321222
#117321202
#117316692
#117315952
#117308352
#117308272
#342807