Finite Group Databasebeta

← #13358285

Subgroups — #13358285

Order 1792.

1106 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
1#11743342C211yesmaximal, characteristic
2#11735959C211yesmaximal, characteristic
3#11731200C211yesmaximal, characteristic
4C7×#1986C211yesmaximal, characteristic, complemented
5#11729595C211yesmaximal, characteristic, complemented
6#11727622C211yesmaximal, characteristic, complemented
7#11727247C211yesmaximal, characteristic, complemented
8#11726507C212yesmaximal
9D7×#339C212yesmaximal
10#11726484C212yesmaximal, complemented
11#11726484C212yesmaximal, complemented
12C7×#533C2211yescharacteristic
13C2×D7×M16C2212yes—
14#90420C412yescomplemented
15#90420C412yescomplemented
16#90380C412yescomplemented
17#90380C412yescomplemented
18#90325—21no—
19#90325—21no—
20#90314—21no—
21#90314—21no—
22#90269—22no—
23#90269—22no—
24D7×#117C2212yes—
25#90208C2212yes—
26C2×#6435C2211yescharacteristic
27C2×#6435C2211yescharacteristic
28C2×C42×D7C2211yescharacteristic
29#90161C2211yescharacteristic
30C4×C7×M16C2211yescharacteristic
31C4×C7×C4⋊C4C2211yescharacteristic
32M16×Dic7C2212yes—
33#89761—21no—
34#89744—21no—
35C4×C4⋊Dic7C2211yescharacteristic
36C4×Dic7⋊C4C2211yescharacteristic
37C4×Dic7⋊C4C2211yescharacteristic
38C4×#1136C2211yescharacteristic
39#89480—22no—
40#89461—22no—
41#89440—21no—
42#89419—21no—
43C7×#339C2212yes—
44C7×#339C2212yes—
45#89350C2212yes—
46#89350C2212yes—
47#89327C2212yes—
48#89327C2212yes—
49#89318C2212yes—
50#89318C2212yes—
51#21307—71nomaximal, Sylow
52C22×C4×D7—21no—
53C22×C4×D7C2311yescharacteristic
54C2×C7×M16C2312yes—
55C7×C4⋊Q8C2×C411yescharacteristic, complemented
56C7×C4⋊Q8C2×C411yescharacteristic, complemented
57C7×#125C2×C412yescomplemented
58C7×C22⋊Q8—21no—
59C7×C22⋊Q8—21no—
60C4×C7×Q8—21no—
61C4×C7×Q8—21no—
62C7×#117C2312yes—
63C2×C7×C4⋊C4—21no—
64C2×C42×C7C2311yescharacteristic
65C2×C42×C7C2311yescharacteristic
66C2×C42×C7C2311yescharacteristic
67#6509—22no—
68#6509—22no—
69#6507—22no—
70#6507—22no—
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4D411yescharacteristic
73C2×(C2×D7)⋊C4D411yescharacteristic
74#6489C2312yes—
75C2×C4⋊Dic7—21no—
76#6487—21no—
77#6487—21no—
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4D411yescharacteristic
82C2×Dic7⋊C4D411yescharacteristic
83C2×C4×Dic7—21no—
84C2×C4×Dic7C2311yescharacteristic
85C2×C4×Dic7C2311yescharacteristic
86C2×C4×Dic7C2311yescharacteristic
87C2×#1136C2312yes—
88D7×M16—24no—
89D7×M16Q818yescomplemented
90C2×#1131—22no—
91C2×C8×D7—22no—
92#6460—22no—
93#6460—22no—
94#6459—22no—
95#6459—22no—
96#6459—22no—
97#6459—22no—
98#6455C2×C412yes—
99#6455C2×C412yes—
100D7×C4⋊C4C2×C412yes—
101D7×C4⋊C4C2×C412yes—
102#6453C2×C412yescomplemented
103#6453C2×C412yescomplemented
104#6452—22no—
105#6452—22no—
106#6452—22no—
107#6452—22no—
108#6451—22no—
109#6451—22no—
110#6451C2×C412yescomplemented
111#6451C2×C412yescomplemented
112D7×C22⋊C4—22no—
113#6441—22no—
114#6441—22no—
115#6440—22no—
116#6435—22no—
117#6435—24no—
118#6435C2×C412yes—
119#6435C2×C412yes—
120#6435D412yes—
121#6435D412yes—
122C42×D7—22no—
123C42×D7—24no—
124#6433C2×C412yescomplemented
125#6432C2×C411yescharacteristic, complemented
126#6432C2×C411yescharacteristic, complemented
127C4×Dic14—21no—
128C4×Dic14—21no—
129C4×Dic14—22no—
130C4×Dic14—22no—
131C7×#97—21no—
132C4×C7×C8—21no—
133C7×#95—21no—
134C7⋊#95—21no—
135C7⋊#95—21no—
136C7⋊#95—21no—
137#6389—22no—
138C8×Dic7—22no—
139#6377—21no—
140C4×C7⋊C8—21no—
141#3458—71no—
142#2873—71no—
143#2296—71no—
144#1986—71no—
145#1986—71no—
146#1986—72no—
147#1986—72no—
148#1986—71no—
149#1986D711yescharacteristic, complemented
150#1733—72no—
151C2×#339—72no—
152C23×D7C2×D411yescharacteristic
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C2×C4×D7—22no—
166C2×C4×D7—22no—
167C2×C4×D7—42no—
168C2×C4×D7—42no—
169C2×C4×D7—42no—
170C2×C4×D7C22×C412yes—
171C2×C4×D7C2×D412yes—
172C2×C4×D7C2×Q814yes—
173C2×C4×D7#4114yes—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14—22no—
177C2×Dic14—22no—
178C7×M16—22no—
179C7×M16C2×Q814yes—
180C2×C7×C8—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4C22×C412yes—
187C7×C4⋊C4C22×C412yes—
188C7×C22⋊C4—22no—
189C42×C7—22no—
190C42×C7—22no—
191C42×C7—22no—
192C42×C7C22×C411yescharacteristic
193C42×C7C22×C411yescharacteristic
194C42×C7C2×D411yescharacteristic
195C42×C7C2×D411yescharacteristic
196C22⋊Dic7—22no—
197(C2×D7)⋊C4—22no—
198(C2×D7)⋊C4—22no—
199(C2×D7)⋊C4—42no—
200(C2×D7)⋊C4C22⋊C412yes—
201(C2×D7)⋊C4C22⋊C412yes—
202(C2×D7)⋊C4C22⋊C412yes—
203(C2×D7)⋊C4C22⋊C412yes—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—22no—
207C4⋊Dic7—22no—
208C4⋊Dic7—42no—
209C4⋊Dic7C22×C412yes—
210C4⋊Dic7C22×C412yes—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—22no—
219Dic7⋊C4—42no—
220Dic7⋊C4—42no—
221Dic7⋊C4—42no—
222Dic7⋊C4C22⋊C412yes—
223Dic7⋊C4C22⋊C412yes—
224Dic7⋊C4C22⋊C412yes—
225Dic7⋊C4C22⋊C412yes—
226C4×Dic7—22no—
227C4×Dic7—22no—
228C4×Dic7—22no—
229C4×Dic7—22no—
230C4×Dic7—22no—
231C4×Dic7—42no—
232C4×Dic7C22×C412yes—
233C4×Dic7C2×D412yes—
234#1136—22no—
235#1136C2×Q814yes—
236C2×C7⋊C8—22no—
237#1131—28no—
238C8×D7—28no—
239C22×M16—72no—
240#544—72no—
241#544—72no—
242#544—72no—
243#544—72no—
244#544—141no—
245#544—141no—
246#543—141no—
247#543—141no—
248#533—72no—
249#533—71no—
250#533—142no—
251#533—142no—
252#533C2×D711yescharacteristic
253C2×#117—71no—
254C2×#117—71no—
255C2×#117—72no—
256C22×C42—71no—
257#405—142no—
258#405—142no—
259C4×M16—72no—
260C4×M16—71no—
261C4×M16C2×D711yescharacteristic
262#389—141no—
263#389—141no—
264#385—141no—
265#385—141no—
266C4×C4⋊C4—71no—
267C4×C4⋊C4—71no—
268C4×C4⋊C4—71no—
269C4×C4⋊C4C2×D711yescharacteristic
270#339—72no—
271#339—72no—
272#339—72no—
273#339—72no—
274#339—72no—
275#339—72no—
276#339C2×D712yes—
277#339C2×D712yes—
278C23×C7C22×D411yescharacteristic
279C22×D7—41no—
280C22×D7—41no—
281C22×D7C2×C22⋊C412yes—
282C22×D7#12412yes—
283C22×D7C4⋊Q812yes—
284C7×Q8—81no—
285C7×Q8—81no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
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⋊C411yescharacteristic
297C2×C4×C7C2×C22⋊C411yescharacteristic
298C2×C4×C7C2×C22⋊C411yescharacteristic
299C2×C4×C7C2×C22⋊C411yescharacteristic
300C2×C4×C7C23×C411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C22×Q812yes—
303C2×C4×C7C2×#4112yes—
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7C2×C22⋊C412yes—
315C2×Dic7C2×C22⋊C412yes—
316C2×Dic7C2×C22⋊C412yes—
317C2×Dic7#12412yes—
318C2×Dic7C4⋊Q812yes—
319C4×D7—48no—
320C4×D7—82no—
321C4×D7—82no—
322C4×D7C4×Q818yes—
323C4×D7C22⋊Q814yes—
324C4×D7C22⋊Q814yes—
325Dic14—82no—
326Dic14—82no—
327Dic14—81no—
328Dic14—81no—
329C7×C8—24no—
330C7⋊C8—24no—
331C23×C4—71no—
332C23×C4—141no—
333C2×M16—72no—
334C2×M16—78no—
335C2×M16—142no—
336C2×M16—144no—
337C2×M16C22×D712yes—
338C22×C8—142no—
339C4⋊Q8—72no—
340C4⋊Q8—72no—
341C4⋊Q8—71no—
342C4⋊Q8—71no—
343C4⋊Q8—142no—
344C4⋊Q8—142no—
345C4⋊Q8—142no—
346C4⋊Q8—142no—
347C4⋊Q8C4×D711yescharacteristic, complemented
348C4⋊Q8C4×D711yescharacteristic, complemented
349#125—72no—
350#125—72no—
351#125—72no—
352#125—142no—
353#125—142no—
354#125—142no—
355#125—142no—
356#125C4×D712yescomplemented
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—142no—
360C22⋊Q8—142no—
361C22⋊Q8—142no—
362C22⋊Q8—142no—
363C22⋊Q8—142no—
364C22⋊Q8—142no—
365C22⋊Q8—142no—
366C22⋊Q8—142no—
367C22⋊Q8—142no—
368C22⋊Q8—142no—
369C22⋊Q8—141no—
370C22⋊Q8—141no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—142no—
374C4×Q8—142no—
375C4×Q8—141no—
376C4×Q8—141no—
377#117—72no—
378#117—72no—
379#117—72no—
380#117—72no—
381#117—72no—
382#117—72no—
383#117—72no—
384#117—142no—
385#117—144no—
386#117—142no—
387#117C22×D712yes—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—71no—
390C2×C4⋊C4—71no—
391C2×C4⋊C4—72no—
392C2×C4⋊C4—72no—
393C2×C4⋊C4—141no—
394C2×C4⋊C4—141no—
395C2×C4⋊C4—141no—
396C2×C4⋊C4—141no—
397C2×C22⋊C4—71no—
398C2×C22⋊C4—71no—
399C2×C22⋊C4—142no—
400C2×C22⋊C4—141no—
401C2×C42—71no—
402C2×C42—71no—
403C2×C42—71no—
404C2×C42—142no—
405C2×C42—144no—
406C2×C42—141no—
407C2×C42C22×D711yescharacteristic
408C2×C42C22×D711yescharacteristic
409C2×C42C22×D711yescharacteristic
410#97—21no—
411#97—142no—
412#97—141no—
413C4×C8—21no—
414C4×C8—142no—
415C4×C8—141no—
416#95—21no—
417#95—141no—
418#95—141no—
419#95—141no—
420C22×C7—21no—
421C22×C7—21no—
422C22×C7C22×C22⋊C411yescharacteristic
423C22×C7C2×#12411yescharacteristic
424C22×C7C2×C4⋊Q811yescharacteristic
425C2×D7—22no—
426C2×D7—22no—
427C2×D7—42no—
428C2×D7—42no—
429C2×D7#39114yes—
430C4×C7—21no—
431C4×C7—21no—
432C4×C7—21no—
433C4×C7—21no—
434C4×C7—42no—
435C4×C7—42no—
436C4×C7—42no—
437C4×C7—81no—
438C4×C7C2×C4×Q812yes—
439C4×C7C2×C22⋊Q812yes—
440Dic7—22no—
441Dic7—22no—
442Dic7—42no—
443Dic7—42no—
444Dic7—42no—
445Dic7—81no—
446Dic7#39114yes—
447C24—71no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—142no—
451C2×Q8—142no—
452C2×Q8—141no—
453C2×Q8—141no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—71no—
457C22×C4—72no—
458C22×C4—72no—
459C22×C4—74no—
460C22×C4—74no—
461C22×C4—71no—
462C22×C4—71no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—282no—
468C22×C4—282no—
469C22×C4—282no—
470C22×C4D4×D711yescharacteristic
471C22×C4D4×D711yescharacteristic
472C22×C4C23×D711yescharacteristic
473M16—22no—
474M16—74no—
475M16—148no—
476M16—142no—
477M16Q8×D714yes—
478C2×C8—22no—
479C2×C8—148no—
480C2×C8—142no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—72no—
487C4⋊C4—72no—
488C4⋊C4—72no—
489C4⋊C4—72no—
490C4⋊C4—72no—
491C4⋊C4—72no—
492C4⋊C4—142no—
493C4⋊C4—142no—
494C4⋊C4—142no—
495C4⋊C4—142no—
496C4⋊C4—142no—
497C4⋊C4—142no—
498C4⋊C4—142no—
499C4⋊C4—142no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4—141no—
503C4⋊C4—141no—
504C4⋊C4—282no—
505C4⋊C4—282no—
506C4⋊C4—282no—
507C4⋊C4—282no—
508C4⋊C4C2×C4×D712yes—
509C4⋊C4C2×C4×D712yes—
510C22⋊C4—22no—
511C22⋊C4—72no—
512C22⋊C4—72no—
513C22⋊C4—72no—
514C22⋊C4—72no—
515C22⋊C4—142no—
516C22⋊C4—142no—
517C22⋊C4—142no—
518C22⋊C4—282no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—72no—
523C42—72no—
524C42—142no—
525C42—142no—
526C42—142no—
527C42—142no—
528C42—142no—
529C42—282no—
530C42C2×C4×D711yescharacteristic
531C42C2×C4×D711yescharacteristic
532C42D4×D711yescharacteristic
533C42D4×D711yescharacteristic
534C2×C7—21no—
535C2×C7—21no—
536C2×C7C2×#39111yescharacteristic
537C2×C7#290111yescharacteristic
538C2×C7#290111yescharacteristic
539D7—24no—
540C23—72no—
541C23—72no—
542C23—72no—
543C23—281no—
544C23—281no—
545C23C2×D4×D711yescharacteristic
546Q8—81no—
547Q8—81no—
548Q8—561no—
549Q8—561no—
550Q8—562no—
551Q8—562no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—72no—
563C2×C4—72no—
564C2×C4—72no—
565C2×C4—78no—
566C2×C4—74no—
567C2×C4—74no—
568C2×C4—72no—
569C2×C4—72no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—288no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—562no—
582C2×C4—562no—
583C2×C4D7×C22⋊C411yescharacteristic
584C2×C4D7×C22⋊C411yescharacteristic
585C2×C4D7×C22⋊C411yescharacteristic
586C2×C4D7×C22⋊C411yescharacteristic
587C2×C4C22×C4×D711yescharacteristic
588C2×C4C2×D4×D711yescharacteristic
589C2×C4C2×Q8×D712yes—
590C2×C4D7×#4112yes—
591C8—24no—
592C8—144no—
593C7#2130711yescharacteristic, complemented, Sylow
594C22—21no—
595C22—21no—
596C22—74no—
597C22—142no—
598C22—142no—
599C22—282no—
600C22—282no—
601C22C2×D7×C22⋊C411yescharacteristic
602C22D7×#12411yescharacteristic
603C22D7×C4⋊Q811yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—74no—
612C4—81no—
613C4—142no—
614C4—142no—
615C4—282no—
616C4—282no—
617C4—282no—
618C4—561no—
619C4C4×Q8×D712yes—
620C4D7×C22⋊Q812yes—
621C2—21no—
622C2—21no—
623C2—144no—
624C2D7×#39111yescharacteristic
625C2#1173642811yescharacteristic
626C2#1173643711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1986C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1986D7
hypocenterC7#21307
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
4. C7×#1986C2semidirect
5. #11729595C2semidirect
6. #11727622C2semidirect
7. #11727247C2semidirect
10. #11726484C2semidirect
11. #11726484C2semidirect
14. #90420C4semidirect
15. #90420C4semidirect
16. #90380C4semidirect
17. #90380C4semidirect
55. C7×C4⋊Q8C2×C4semidirect
56. C7×C4⋊Q8C2×C4semidirect
57. C7×#125C2×C4semidirect
89. D7×M16Q8semidirect
102. #6453C2×C4semidirect
103. #6453C2×C4semidirect
110. #6451C2×C4semidirect
111. #6451C2×C4semidirect
124. #6433C2×C4semidirect
125. #6432C2×C4semidirect
126. #6432C2×C4semidirect
149. #1986D7semidirect
347. C4⋊Q8C4×D7semidirect
348. C4⋊Q8C4×D7semidirect
356. #125C4×D7semidirect
593. C7#21307semidirect

Maximal subgroups

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

subgroupindex
#117433422
#117359592
#117312002
C7×#19862
#117295952
#117276222
#117272472
#117265072
D7×#3392
#117264842
#213077