Finite Group Databasebeta

← #13363350

Subgroups — #13363350

Order 1792.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90380C211yesmaximal, characteristic
2C2×#89462C211yesmaximal, characteristic
3#11731669C211yesmaximal, characteristic
4C7×#1991C211yesmaximal, characteristic, complemented
5#11729495C211yesmaximal, characteristic, complemented
6#11727531C211yesmaximal, characteristic, complemented
7#11727309C211yesmaximal, characteristic, complemented
8#11726627C211yesmaximal, characteristic
9#11726626C211yesmaximal, characteristic
10#11726625C211yesmaximal, characteristic
11#11726567C211yesmaximal, characteristic
12#11726556C211yesmaximal, characteristic, complemented
13#11726555C211yesmaximal, characteristic, complemented
14#11726554C211yesmaximal, characteristic, complemented
15#11726553C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17#90380—24no—
18#90380C418yescomplemented
19C2×#6460—21no—
20C2×#6459—21no—
21C2×#6455C2211yescharacteristic
22C2×D7×C4⋊C4C2211yescharacteristic
23C2×#6453C2211yescharacteristic
24C2×#6452—21no—
25C2×#6451—21no—
26C2×#6451C2211yescharacteristic
27C2×#6435C2211yescharacteristic
28C2×#6432C2211yescharacteristic
29C2×C4×C7×C8C2211yescharacteristic
30C7×#382C2211yescharacteristic
31C2×(C2×D7)⋊C8C2211yescharacteristic
32C2×(C2×D7)⋊C8C2211yescharacteristic
33C2×Dic7⋊C8C2211yescharacteristic
34C2×Dic7⋊C8C2211yescharacteristic
35#89759C2211yescharacteristic
36#89759C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39#89707C2211yescharacteristic
40C2×#6377C2211yescharacteristic
41#89462—24no—
42#89442—21no—
43#89442—21no—
44#89427—21no—
45#89427—21no—
46C7×#340C2211yescharacteristic
47C7×#340C2211yescharacteristic
48C7×#340C2211yescharacteristic
49C7×#340C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21843—71nomaximal, Sylow
63C22×C4×D7—21no—
64C22×C4×D7C2311yescharacteristic
65C22×Dic14—21no—
66C22×Dic14—21no—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C7×#125—22no—
70C7×#125C2×C414yes—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—21no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4D411yescharacteristic
82C2×(C2×D7)⋊C4D411yescharacteristic
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4D411yescharacteristic
95C2×Dic7⋊C4D411yescharacteristic
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6460—28no—
103#6459—28no—
104#6455—22no—
105#6455C2×C414yes—
106D7×C4⋊C4—22no—
107D7×C4⋊C4C2×C414yes—
108#6453—22no—
109#6453C2×C414yes—
110#6452—28no—
111#6451—28no—
112#6451—22no—
113#6451C2×C414yes—
114#6435C2×C414yes—
115#6435D414yes—
116#6432—22no—
117#6432C2×C414yes—
118C4×C7×C8—22no—
119C7×#95—21no—
120C7×#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125(C2×D7)⋊C8—22no—
126(C2×D7)⋊C8—22no—
127Dic7⋊C8—22no—
128Dic7⋊C8—22no—
129#6377—22no—
130C2×#544—71no—
131C2×#433—71no—
132#2314—71no—
133#1993—71no—
134#1992—71no—
135#1991D711yescharacteristic, complemented
136#1990—71no—
137#1989—71no—
138#1989—71no—
139#1988—71no—
140#1987—71no—
141#1825—71no—
142#1825—71no—
143#1824—71no—
144#1823—71no—
145C23×D7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C2×C4×D7—21no—
161C2×C4×D7—21no—
162C2×C4×D7—21no—
163C2×C4×D7—21no—
164C2×C4×D7—24no—
165C2×C4×D7—81no—
166C2×C4×D7—81no—
167C2×C4×D7C22×C412yes—
168C2×C4×D7C2×D412yes—
169C2×Dic14—24no—
170C2×Dic14—24no—
171C2×Dic14—81no—
172C2×Dic14—81no—
173C2×Dic14—81no—
174C2×Dic14—81no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4C22×C412yes—
190C7×C4⋊C4C22×C412yes—
191C42×C7C22×C412yes—
192C42×C7C2×D412yes—
193(C2×D7)⋊C4—24no—
194(C2×D7)⋊C4—81no—
195(C2×D7)⋊C4—81no—
196(C2×D7)⋊C4C22⋊C414yes—
197(C2×D7)⋊C4C22⋊C414yes—
198C4⋊Dic7—24no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—24no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—81no—
207C4⋊Dic7—81no—
208C4⋊Dic7C22×C412yes—
209C4⋊Dic7C22×C412yes—
210Dic7⋊C4—24no—
211Dic7⋊C4—24no—
212Dic7⋊C4—24no—
213Dic7⋊C4C22⋊C414yes—
214Dic7⋊C4C22⋊C414yes—
215C4×Dic7—24no—
216C4×Dic7—81no—
217C4×Dic7—81no—
218C4×Dic7C22×C412yes—
219C4×Dic7C2×D412yes—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226#544—78no—
227#544—144no—
228C2×C4⋊Q8—71no—
229C2×C4⋊Q8—71no—
230C2×C4⋊Q8—141no—
231C2×#125—71no—
232C2×#125—141no—
233C2×#125C2×D711yescharacteristic
234C2×C22⋊Q8—141no—
235C2×C22⋊Q8—141no—
236C2×#117—71no—
237C2×#117—71no—
238C22×C4⋊C4—71no—
239#433—144no—
240C2×C4⋊C8—71no—
241C2×C4⋊C8—71no—
242C2×C22⋊C8—71no—
243C2×C22⋊C8—71no—
244C2×#97—71no—
245C2×C4×C8C2×D711yescharacteristic
246#388—141no—
247#388—141no—
248#387—141no—
249#387—141no—
250#384—71no—
251#382C2×D711yescharacteristic
252#380—71no—
253#380—71no—
254C4×C4⋊C4—71no—
255C4×C4⋊C4—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340C2×D711yescharacteristic
269#340C2×D711yescharacteristic
270#340C2×D711yescharacteristic
271#340C2×D711yescharacteristic
272C23×C7C22×D411yescharacteristic
273C22×D7—21no—
274C22×D7—21no—
275C22×D7—21no—
276C22×D7—21no—
277C22×D7—21no—
278C22×D7—21no—
279C22×D7C2×C22⋊C412yes—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7C2×C22⋊C412yes—
297C2×C4×C7C2×C22⋊C412yes—
298C2×C4×C7C23×C411yescharacteristic
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C72−1+411yescharacteristic
301C2×C4×C72−1+411yescharacteristic
302C2×C4×C72−1+411yescharacteristic
303C2×C4×C72−1+411yescharacteristic
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7C2×C22⋊C412yes—
323C2×Dic7C2×C22⋊C412yes—
324C2×Dic7C2×C22⋊C412yes—
325C4×D7—22no—
326C4×D7—22no—
327C4×D7—22no—
328C4×D7—22no—
329C4×D7—84no—
330Dic14—84no—
331Dic14—84no—
332C7×C8—41no—
333C7×C8—41no—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C22×Q8—141no—
337C22×Q8—141no—
338C23×C4—71no—
339C23×C4—141no—
340C22×C8—71no—
341C22×C8—71no—
342C22×C8C22×D711yescharacteristic
343C22×C8C22×D711yescharacteristic
344C4⋊Q8—74no—
345C4⋊Q8—74no—
346C4⋊Q8—142no—
347C4⋊Q8—148no—
348C4⋊Q8—142no—
349#125—22no—
350#125—74no—
351#125—142no—
352#125—148no—
353#125C4×D714yes—
354C22⋊Q8—148no—
355C22⋊Q8—148no—
356#117—74no—
357#117—74no—
358#117—74no—
359#117—142no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—74no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—142no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C22⋊C4—71no—
382C2×C22⋊C4—71no—
383C2×C22⋊C4—141no—
384C2×C22⋊C4—141no—
385C2×C22⋊C4—141no—
386C2×C42—71no—
387C2×C42—141no—
388C2×C42—141no—
389C2×C42—141no—
390C2×C42C22×D711yescharacteristic
391C4⋊C8—142no—
392C4⋊C8—142no—
393C22⋊C8—142no—
394C22⋊C8—142no—
395#97—142no—
396C4×C8—22no—
397#95—21no—
398#95—21no—
399#95—141no—
400#95—141no—
401#95—141no—
402#95—141no—
403C22×C7C22×C22⋊C411yescharacteristic
404C22×C7#53611yescharacteristic
405C22×C7#53611yescharacteristic
406C22×C7#57611yescharacteristic
407C22×C7#57711yescharacteristic
408C22×C7#57711yescharacteristic
409C22×C7#57811yescharacteristic
410C2×D7—22no—
411C2×D7—22no—
412C2×D7—22no—
413C2×D7—41no—
414C2×D7—41no—
415C2×D7—41no—
416C2×D7—41no—
417C4×C7—22no—
418C4×C7—22no—
419C4×C7—42no—
420C4×C7—42no—
421C4×C7—81no—
422C4×C7—81no—
423C4×C7#51911yescharacteristic
424C4×C7#51911yescharacteristic
425C4×C7#53611yescharacteristic
426C4×C7#53611yescharacteristic
427Dic7—22no—
428Dic7—22no—
429Dic7—22no—
430Dic7—42no—
431Dic7—42no—
432Dic7—81no—
433Dic7—81no—
434C24—71no—
435C2×Q8—144no—
436C2×Q8—144no—
437C2×Q8—561no—
438C2×Q8—561no—
439C2×Q8—561no—
440C2×Q8—561no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—72no—
449C22×C4—72no—
450C22×C4—144no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—141no—
459C22×C4—561no—
460C22×C4—561no—
461C22×C4D4×D711yescharacteristic
462C22×C4D4×D711yescharacteristic
463C22×C4C23×D711yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—74no—
485C4⋊C4—74no—
486C4⋊C4—72no—
487C4⋊C4—72no—
488C4⋊C4—144no—
489C4⋊C4—144no—
490C4⋊C4—144no—
491C4⋊C4—144no—
492C4⋊C4—144no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—561no—
500C4⋊C4—561no—
501C4⋊C4C2×C4×D712yes—
502C4⋊C4C2×C4×D712yes—
503C22⋊C4—74no—
504C22⋊C4—74no—
505C22⋊C4—144no—
506C22⋊C4—561no—
507C22⋊C4—561no—
508C42—72no—
509C42—72no—
510C42—144no—
511C42—561no—
512C42—561no—
513C42C2×C4×D712yes—
514C42D4×D712yes—
515C2×C7#231011yescharacteristic
516C2×C7#289711yescharacteristic
517C2×C7#289811yescharacteristic
518C2×C7#324111yescharacteristic
519C2×C7#324211yescharacteristic
520C2×C7#324311yescharacteristic
521C2×C7#324411yescharacteristic
522D7—42no—
523C23—72no—
524C23—141no—
525C23—141no—
526C23—141no—
527C23—141no—
528C23—141no—
529C23—141no—
530C23C2×D4×D711yescharacteristic
531Q8—564no—
532Q8—564no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—72no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—564no—
575C2×C4D7×C22⋊C412yes—
576C2×C4D7×C22⋊C412yes—
577C2×C4C22×C4×D711yescharacteristic
578C2×C4C2×D4×D711yescharacteristic
579C2×C4#655111yescharacteristic
580C2×C4#655111yescharacteristic
581C2×C4#655411yescharacteristic
582C2×C4#655411yescharacteristic
583C8—41no—
584C8—41no—
585C8—281no—
586C8—281no—
587C7#2184311yescharacteristic, complemented, Sylow
588C22—142no—
589C22—142no—
590C22—142no—
591C22—281no—
592C22—281no—
593C22—281no—
594C22—281no—
595C22C2×D7×C22⋊C411yescharacteristic
596C22#9036611yescharacteristic
597C22#9041511yescharacteristic
598C22#9046011yescharacteristic
599C22#9046111yescharacteristic
600C22#9046111yescharacteristic
601C22#9046211yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—142no—
609C4—142no—
610C4—142no—
611C4—282no—
612C4—282no—
613C4—561no—
614C4—561no—
615C4#9026311yescharacteristic
616C4#9026311yescharacteristic
617C4#9031911yescharacteristic
618C4#9031911yescharacteristic
619C2—282no—
620C2#1173405511yescharacteristic
621C2#1173627611yescharacteristic
622C2#1173627711yescharacteristic
623C2#1173909511yescharacteristic
624C2#1173909611yescharacteristic
625C2#1173936711yescharacteristic
626C2#1173936811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1991C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1991D7
hypocenterC7#21843
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
4. C7×#1991C2semidirect
5. #11729495C2semidirect
6. #11727531C2semidirect
7. #11727309C2semidirect
12. #11726556C2semidirect
13. #11726555C2semidirect
14. #11726554C2semidirect
15. #11726553C2semidirect
18. #90380C4semidirect
135. #1991D7semidirect
587. C7#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#903802
C2×#894622
#117316692
C7×#19912
#117294952
#117275312
#117273092
#117266272
#117266262
#117266252
#117265672
#117265562
#117265552
#117265542
#117265532
#218437