Finite Group Databasebeta

← #13364190

Subgroups — #13364190

Order 1792.

830 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89609C211yesmaximal, characteristic
2#11731592C211yesmaximal, characteristic
3#11731198C211yesmaximal, characteristic
4C7×#1949C211yesmaximal, characteristic, complemented
5#11728385C211yesmaximal, characteristic
6#11728369C211yesmaximal, characteristic, complemented
7#11728332C211yesmaximal, characteristic
8#11728317C211yesmaximal, characteristic, complemented
9#11728209C211yesmaximal, characteristic, complemented
10#11727294C211yesmaximal, characteristic, complemented
11#11727222C211yesmaximal, characteristic, complemented
12#11726624C211yesmaximal, characteristic
13#11726624C211yesmaximal, characteristic
14#11726550C211yesmaximal, characteristic, complemented
15#11726549C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C22×#1131C2211yescharacteristic
18C2×#6435—21no—
19C2×C42×D7C2211yescharacteristic
20C2×C7×#107C2211yescharacteristic
21C2×C7×#106C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C7×#382C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6392C2211yescharacteristic
26C2×#6391C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29#89759C2211yescharacteristic
30#89759C2211yescharacteristic
31#89759C2211yescharacteristic
32#89759C2211yescharacteristic
33#89748C2211yescharacteristic
34#89748C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89708C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43C4×Dic7⋊C4—21no—
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45#89609C418yescomplemented
46#89439—21no—
47#89438—21no—
48#89436—21no—
49#89423—21no—
50#89422—21no—
51#89420—21no—
52C7×#340C2211yescharacteristic
53C7×#340C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21963—71nomaximal, Sylow
61C22×C4×D7D411yescharacteristic
62C22×C4×D7D411yescharacteristic
63C22×C4×D7C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C42×C7—21no—
71C2×C42×C7C2311yescharacteristic
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7D411yescharacteristic
88C2×C4×Dic7D411yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C2×#1131—41no—
93C2×#1131—41no—
94C2×#1131—41no—
95C2×#1131—41no—
96C2×#1131C2×C414yes—
97C2×#1131C2×C414yes—
98#6435—44no—
99C42×D7C2×C414yes—
100C42×D7Q814yes—
101C7×#107—41no—
102C7×#106—41no—
103C7×C4⋊C8C2×C414yes—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116#6392—41no—
117#6391—41no—
118Dic7⋊C8C2×C414yes—
119Dic7⋊C8C2×C414yes—
120#6382—41no—
121#6382—41no—
122#6381—41no—
123#6381—41no—
124C4⋊(C7⋊C8)C2×C414yes—
125C2×#423—71no—
126#2285—71no—
127#2283—71no—
128#1949—71no—
129#1949—71no—
130#1949—71no—
131#1949D711yescharacteristic, complemented
132#1948—71no—
133#1948—71no—
134#1947—71no—
135#1947—71no—
136#1809—71no—
137#1809—71no—
138#1809—71no—
139#1809—71no—
140C23×D7C2×D411yescharacteristic
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C22×Dic7C2×D411yescharacteristic
155C2×C4×D7—21no—
156C2×C4×D7—21no—
157C2×C4×D7—21no—
158C2×C4×D7—21no—
159C2×C4×D7—22no—
160C2×C4×D7—22no—
161C2×C4×D7—41no—
162C2×C4×D7—41no—
163C2×C4×D7C4⋊C414yes—
164C2×C4×D7C4⋊C414yes—
165C2×C4×D7C22×C412yes—
166C2×C4×D7C2×Q812yes—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8C22×C412yes—
172C2×C7×C8C22×C412yes—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C42×C7—42no—
182C42×C7C22×C412yes—
183C42×C7C2×Q812yes—
184(C2×D7)⋊C4—42no—
185(C2×D7)⋊C4—42no—
186(C2×D7)⋊C4—42no—
187(C2×D7)⋊C4—42no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196Dic7⋊C4—42no—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200C4×Dic7—42no—
201C4×Dic7—42no—
202C4×Dic7—42no—
203C4×Dic7C4⋊C414yes—
204C4×Dic7C4⋊C414yes—
205C4×Dic7C22×C412yes—
206C4×Dic7C2×Q812yes—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8C22×C412yes—
212C2×C7⋊C8C22×C412yes—
213#1131—44no—
214#1131—44no—
215C22×M16—71no—
216C22×M16—71no—
217C2×#117—141no—
218C22×C42—71no—
219C2×#107—71no—
220C2×#107—71no—
221C2×#107—71no—
222C2×#107C2×D711yescharacteristic
223C2×#106—71no—
224C2×#106—71no—
225C2×#106—71no—
226C2×#106C2×D711yescharacteristic
227#423—78no—
228C2×C4⋊C8—71no—
229C2×C4⋊C8—71no—
230C2×C4⋊C8—71no—
231C2×C4⋊C8C2×D711yescharacteristic
232#384—71no—
233#384—71no—
234#383—141no—
235#382—71no—
236#382—141no—
237#382C2×D711yescharacteristic
238#381—141no—
239#380—71no—
240#380—71no—
241#380—71no—
242#380—71no—
243#380—141no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—71no—
247C4×C4⋊C4—141no—
248C4×C4⋊C4C2×D711yescharacteristic
249C4×C22⋊C4—141no—
250#376—141no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340C2×D711yescharacteristic
258#340C2×D711yescharacteristic
259C23×C7C22×D411yescharacteristic
260C22×D7—21no—
261C22×D7—21no—
262C22×D7—21no—
263C22×D7—21no—
264C22×D7—21no—
265C22×D7—21no—
266C22×D7C2×C4⋊C412yes—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7C2×C4⋊C412yes—
284C2×C4×C7C2×C4⋊C412yes—
285C2×C4×C7C23×C411yescharacteristic
286C2×C4×C7C22×Q811yescharacteristic
287C2×C4×C7H(5,𝔽2)11yescharacteristic
288C2×C4×C7H(5,𝔽2)11yescharacteristic
289C2×C4×C72−1+411yescharacteristic
290C2×C4×C72−1+411yescharacteristic
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7C2×C4⋊C412yes—
308C2×Dic7C2×C4⋊C412yes—
309C2×Dic7C2×C4⋊C412yes—
310C2×Dic7AGL(1,ℤ/8)12yes—
311C2×Dic7#13712yes—
312C4×D7—22no—
313C4×D7—22no—
314C4×D7—22no—
315C4×D7—22no—
316C4×D7—28no—
317C4×D7—44no—
318C7×C8—22no—
319C7×C8—22no—
320C7⋊C8—22no—
321C7⋊C8—22no—
322C23×C4—71no—
323C23×C4—71no—
324C23×C4—71no—
325C2×M16—74no—
326C2×M16—74no—
327C2×M16—281no—
328C2×M16—281no—
329C2×M16—281no—
330C2×M16—281no—
331C22×C8—71no—
332C22×C8—71no—
333C22×C8C22×D711yescharacteristic
334C22×C8C22×D711yescharacteristic
335#117—284no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C42—21no—
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—71no—
356C2×C42—74no—
357C2×C42—74no—
358C2×C42—141no—
359C2×C42—141no—
360C2×C42—141no—
361C2×C42C22×D711yescharacteristic
362#107—41no—
363#107—281no—
364#107—281no—
365#107—281no—
366#106—41no—
367#106—281no—
368#106—281no—
369#106—281no—
370C4⋊C8—74no—
371C4⋊C8—74no—
372C4⋊C8—74no—
373C4⋊C8C4×D714yes—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386C22×C7C22×C4⋊C411yescharacteristic
387C22×C7#53711yescharacteristic
388C22×C7#53711yescharacteristic
389C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
390C22×C7C2×#13711yescharacteristic
391C22×C7#57511yescharacteristic
392C22×C7#57511yescharacteristic
393C2×D7—22no—
394C2×D7—22no—
395C2×D7—22no—
396C2×D7—41no—
397C2×D7—41no—
398C2×D7—41no—
399C2×D7—41no—
400C4×C7—22no—
401C4×C7—22no—
402C4×C7—81no—
403C4×C7—81no—
404C4×C7—81no—
405C4×C7—81no—
406C4×C7#51811yescharacteristic
407C4×C7#51911yescharacteristic
408C4×C7#54411yescharacteristic
409C4×C7#54411yescharacteristic
410Dic7—22no—
411Dic7—22no—
412Dic7—81no—
413Dic7—81no—
414Dic7—81no—
415Dic7—81no—
416Dic7#42814yes—
417C24—71no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—71no—
423C22×C4—71no—
424C22×C4—71no—
425C22×C4—74no—
426C22×C4—72no—
427C22×C4—72no—
428C22×C4—74no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—281no—
440C22×C4—281no—
441C22×C4D4×D711yescharacteristic
442C22×C4D4×D711yescharacteristic
443C22×C4C23×D711yescharacteristic
444M16—284no—
445M16—284no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—72no—
451C2×C8—72no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8C2×C4×D712yes—
457C2×C8C2×C4×D712yes—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—282no—
477C4⋊C4—282no—
478C22⋊C4—282no—
479C22⋊C4—282no—
480C22⋊C4—282no—
481C22⋊C4—282no—
482C42—42no—
483C42—74no—
484C42—74no—
485C42—72no—
486C42—72no—
487C42—282no—
488C42—282no—
489C42—282no—
490C42C2×C4×D712yes—
491C42Q8×D712yes—
492C2×C7#231611yescharacteristic
493C2×C7C2×#42811yescharacteristic
494C2×C7#291111yescharacteristic
495C2×C7#325111yescharacteristic
496C2×C7#325111yescharacteristic
497C2×C7#325211yescharacteristic
498C2×C7#325211yescharacteristic
499D7—42no—
500C23—72no—
501C23—141no—
502C23—141no—
503C23—141no—
504C23—141no—
505C23—141no—
506C23—141no—
507C23C2×D4×D711yescharacteristic
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—72no—
525C2×C4—72no—
526C2×C4—72no—
527C2×C4—72no—
528C2×C4—72no—
529C2×C4—148no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—142no—
535C2×C4—142no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—284no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4D7×C4⋊C412yes—
552C2×C4D7×C4⋊C412yes—
553C2×C4C22×C4×D711yescharacteristic
554C2×C4#654811yescharacteristic
555C2×C4C2×Q8×D711yescharacteristic
556C2×C4#655111yescharacteristic
557C2×C4#655311yescharacteristic
558C2×C4#655411yescharacteristic
559C8—22no—
560C8—22no—
561C8—142no—
562C8—142no—
563C7#2196311yescharacteristic, complemented, Sylow
564C22—142no—
565C22—142no—
566C22—142no—
567C22—281no—
568C22—281no—
569C22—281no—
570C22—281no—
571C22C2×D7×C4⋊C411yescharacteristic
572C22#9029911yescharacteristic
573C22#9032011yescharacteristic
574C22#9045911yescharacteristic
575C22#9045911yescharacteristic
576C22D7×AGL(1,ℤ/8)11yescharacteristic
577C22D7×#13711yescharacteristic
578C4—22no—
579C4—22no—
580C4—74no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—142no—
586C4—142no—
587C4—561no—
588C4—561no—
589C4—561no—
590C4—561no—
591C4#9023611yescharacteristic
592C4#9026311yescharacteristic
593C4#9038011yescharacteristic
594C4#9042011yescharacteristic
595C2—282no—
596C2#1173364011yescharacteristic
597C2#1173662211yescharacteristic
598C2D7×#42811yescharacteristic
599C2#1173795811yescharacteristic
600C2#1173802211yescharacteristic
601C2#1173859411yescharacteristic
602C2#1173865811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1949C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1949D7
hypocenterC7#21963
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. C7×#1949C2semidirect
6. #11728369C2semidirect
8. #11728317C2semidirect
9. #11728209C2semidirect
10. #11727294C2semidirect
11. #11727222C2semidirect
14. #11726550C2semidirect
15. #11726549C2semidirect
45. #89609C4semidirect
131. #1949D7semidirect
563. C7#21963semidirect

Maximal subgroups

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

subgroupindex
C2×#896092
#117315922
#117311982
C7×#19492
#117283852
#117283692
#117283322
#117283172
#117282092
#117272942
#117272222
#117266242
#117265502
#117265492
#219637