Finite Group Databasebeta

← #13364202

Subgroups — #13364202

Order 1792.

754 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89611C211yesmaximal, characteristic
2#11731803C211yesmaximal, characteristic
3#11731205C211yesmaximal, characteristic
4C7×#1948C211yesmaximal, characteristic, complemented
5#11728332C211yesmaximal, characteristic
6#11728331C211yesmaximal, characteristic
7#11728318C211yesmaximal, characteristic, complemented
8#11728318C211yesmaximal, characteristic, complemented
9#11728209C211yesmaximal, characteristic, complemented
10#11727321C211yesmaximal, characteristic, complemented
11#11727222C211yesmaximal, characteristic, complemented
12#11726624C211yesmaximal, characteristic
13#11726623C211yesmaximal, characteristic
14#11726550C211yesmaximal, characteristic, complemented
15#11726549C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C22×C8×D7C2211yescharacteristic
18C2×#6435C2211yescharacteristic
19C2×C7×#106C2211yescharacteristic
20C2×C7×#106C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C7×#384C2211yescharacteristic
23C4×C7×C4⋊C4C2211yescharacteristic
24C2×#6392C2211yescharacteristic
25C2×#6391C2211yescharacteristic
26C2×Dic7⋊C8C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28#89762—21no—
29#89762—21no—
30#89759C2211yescharacteristic
31#89759C2211yescharacteristic
32#89759C2211yescharacteristic
33#89759C2211yescharacteristic
34#89750—21no—
35#89750—21no—
36#89749C2211yescharacteristic
37#89748C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89712—21no—
45#89708C2211yescharacteristic
46C4×C4⋊Dic7C2211yescharacteristic
47#89705—21no—
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49#89611C418yescomplemented
50#89441—21no—
51#89424—21no—
52C7×#340C2211yescharacteristic
53C7×#340C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21966—71nomaximal, Sylow
61C22×C4×D7C2311yescharacteristic
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
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×(C2×D7)⋊C4D411yescharacteristic
77C2×(C2×D7)⋊C4D411yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4D411yescharacteristic
85C2×Dic7⋊C4D411yescharacteristic
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C2×#1131—41no—
93C2×#1131—41no—
94C2×#1131C2×C414yes—
95C2×C8×D7—41no—
96C2×C8×D7—41no—
97C2×C8×D7C2×C414yes—
98#6435C2×C414yes—
99#6435Q814yes—
100C7×#106—41no—
101C7×#106—41no—
102C7×C4⋊C8C2×C414yes—
103C7×#95—21no—
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—
115#6392—41no—
116#6391—41no—
117Dic7⋊C8C2×C414yes—
118Dic7⋊C8C2×C414yes—
119#6382—41no—
120#6381—41no—
121#6381—41no—
122#6381—41no—
123C4⋊(C7⋊C8)C2×C414yes—
124C2×#424—71no—
125#2314—71no—
126#2314—71no—
127#1949—71no—
128#1949—71no—
129#1949—71no—
130#1949—71no—
131#1948—71no—
132#1948D711yescharacteristic, complemented
133#1947—71no—
134#1947—71no—
135#1809—71no—
136#1809—71no—
137#1808—71no—
138#1807—71no—
139C23×D7C2×D411yescharacteristic
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C2×C4×D7—21no—
155C2×C4×D7—21no—
156C2×C4×D7—21no—
157C2×C4×D7—21no—
158C2×C4×D7C22×C412yes—
159C2×C4×D7C2×Q812yes—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8C22×C412yes—
165C2×C7×C8C22×C412yes—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—81no—
175C7×C4⋊C4—81no—
176C42×C7—81no—
177C42×C7C22×C412yes—
178C42×C7C2×Q812yes—
179(C2×D7)⋊C4—81no—
180(C2×D7)⋊C4—81no—
181(C2×D7)⋊C4—81no—
182(C2×D7)⋊C4—81no—
183(C2×D7)⋊C4C4⋊C414yes—
184(C2×D7)⋊C4C4⋊C414yes—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193Dic7⋊C4—81no—
194Dic7⋊C4—81no—
195Dic7⋊C4C4⋊C414yes—
196Dic7⋊C4C4⋊C414yes—
197C4×Dic7—81no—
198C4×Dic7—81no—
199C4×Dic7—81no—
200C4×Dic7C22×C412yes—
201C4×Dic7C2×Q812yes—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8C22×C412yes—
207C2×C7⋊C8C22×C412yes—
208#1131—44no—
209C8×D7—44no—
210C22×M16—71no—
211C23×C8—71no—
212C2×#117—71no—
213C2×#107—71no—
214C2×#107—71no—
215C2×#107—71no—
216C2×#107—71no—
217C2×#106—71no—
218C2×#106—71no—
219C2×#106C2×D711yescharacteristic
220C2×#106C2×D711yescharacteristic
221#424—78no—
222C2×C4⋊C8—71no—
223C2×C4⋊C8—71no—
224C2×C4⋊C8—71no—
225C2×C4⋊C8C2×D711yescharacteristic
226#388—141no—
227#388—141no—
228#388—141no—
229#388—141no—
230#387—141no—
231#387—141no—
232#387—141no—
233#387—141no—
234#384—71no—
235#384C2×D711yescharacteristic
236#382—71no—
237#382—71no—
238#380—71no—
239#380—71no—
240#380—71no—
241#380—71no—
242C4×C4⋊C4—71no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4C2×D711yescharacteristic
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340C2×D711yescharacteristic
253#340C2×D711yescharacteristic
254C23×C7C22×D411yescharacteristic
255C22×D7—21no—
256C22×D7—21no—
257C22×D7—21no—
258C22×D7—21no—
259C22×D7—21no—
260C22×D7—21no—
261C22×D7C2×C4⋊C412yes—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
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×C7C2×C4⋊C412yes—
279C2×C4×C7C2×C4⋊C412yes—
280C2×C4×C7C23×C411yescharacteristic
281C2×C4×C7C22×Q811yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C7H(5,𝔽2)11yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C2×C4×C72−1+411yescharacteristic
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
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×Dic7C2×C4⋊C412yes—
305C2×Dic7C2×C4⋊C412yes—
306C2×Dic7C2×C4⋊C412yes—
307C4×D7—22no—
308C4×D7—22no—
309C4×D7—22no—
310C4×D7—22no—
311C7×C8—22no—
312C7×C8—22no—
313C7⋊C8—22no—
314C7⋊C8—22no—
315C23×C4—71no—
316C2×M16—74no—
317C2×M16—281no—
318C2×M16—281no—
319C22×C8—71no—
320C22×C8—71no—
321C22×C8—74no—
322C22×C8—281no—
323C22×C8—281no—
324C22×C8C22×D711yescharacteristic
325C22×C8C22×D711yescharacteristic
326#117—74no—
327#117—74no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C22⋊C4—71no—
343C2×C22⋊C4—71no—
344C2×C22⋊C4—141no—
345C2×C22⋊C4—141no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C42—21no—
349C2×C42—71no—
350C2×C42—141no—
351C2×C42—141no—
352C2×C42—141no—
353C2×C42C22×D711yescharacteristic
354#107—281no—
355#107—281no—
356#107—281no—
357#107—281no—
358#106—41no—
359#106—41no—
360#106—281no—
361#106—281no—
362C4⋊C8—74no—
363C4⋊C8—74no—
364C4⋊C8—74no—
365C4⋊C8C4×D714yes—
366#95—21no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378C22×C7C22×C4⋊C411yescharacteristic
379C22×C7#53711yescharacteristic
380C22×C7#53711yescharacteristic
381C22×C7#57611yescharacteristic
382C22×C7#57711yescharacteristic
383C22×C7#57711yescharacteristic
384C22×C7#57811yescharacteristic
385C2×D7—22no—
386C2×D7—22no—
387C2×D7—22no—
388C2×D7—41no—
389C2×D7—41no—
390C2×D7—41no—
391C2×D7—41no—
392C4×C7—22no—
393C4×C7—22no—
394C4×C7—81no—
395C4×C7—81no—
396C4×C7—81no—
397C4×C7—81no—
398C4×C7#51811yescharacteristic
399C4×C7#51911yescharacteristic
400C4×C7#54411yescharacteristic
401C4×C7#54411yescharacteristic
402Dic7—22no—
403Dic7—22no—
404Dic7—22no—
405Dic7—81no—
406Dic7—81no—
407Dic7—81no—
408Dic7—81no—
409C24—71no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—71no—
415C22×C4—71no—
416C22×C4—71no—
417C22×C4—72no—
418C22×C4—72no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4D4×D711yescharacteristic
428C22×C4D4×D711yescharacteristic
429C22×C4C23×D711yescharacteristic
430M16—284no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—72no—
436C2×C8—72no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—284no—
442C2×C8C2×C4×D712yes—
443C2×C8C2×C4×D712yes—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—74no—
453C4⋊C4—74no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—561no—
465C4⋊C4—561no—
466C22⋊C4—74no—
467C22⋊C4—74no—
468C22⋊C4—561no—
469C22⋊C4—561no—
470C22⋊C4—561no—
471C22⋊C4—561no—
472C42—72no—
473C42—72no—
474C42—81no—
475C42—561no—
476C42—561no—
477C42—561no—
478C42C2×C4×D712yes—
479C42Q8×D712yes—
480C2×C7#231611yescharacteristic
481C2×C7#291211yescharacteristic
482C2×C7#291311yescharacteristic
483C2×C7#325711yescharacteristic
484C2×C7#325711yescharacteristic
485C2×C7#325811yescharacteristic
486C2×C7#325811yescharacteristic
487D7—42no—
488C23—72no—
489C23—141no—
490C23—141no—
491C23—141no—
492C23—141no—
493C23—141no—
494C23—141no—
495C23C2×D4×D711yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—72no—
513C2×C4—72no—
514C2×C4—72no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—142no—
522C2×C4—142no—
523C2×C4—142no—
524C2×C4—142no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4D7×C4⋊C412yes—
538C2×C4D7×C4⋊C412yes—
539C2×C4C22×C4×D711yescharacteristic
540C2×C4#654811yescharacteristic
541C2×C4C2×Q8×D711yescharacteristic
542C2×C4#655111yescharacteristic
543C2×C4#655311yescharacteristic
544C2×C4#655411yescharacteristic
545C8—22no—
546C8—22no—
547C8—142no—
548C8—142no—
549C7#2196611yescharacteristic, complemented, Sylow
550C22—142no—
551C22—142no—
552C22—142no—
553C22—281no—
554C22—281no—
555C22—281no—
556C22—281no—
557C22C2×D7×C4⋊C411yescharacteristic
558C22#9029911yescharacteristic
559C22#9032011yescharacteristic
560C22#9045311yescharacteristic
561C22#9045311yescharacteristic
562C22#9046511yescharacteristic
563C22#9047011yescharacteristic
564C4—22no—
565C4—22no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—142no—
571C4—142no—
572C4—142no—
573C4—561no—
574C4—561no—
575C4—561no—
576C4—561no—
577C4#9023611yescharacteristic
578C4#9026311yescharacteristic
579C4#9038011yescharacteristic
580C4#9042011yescharacteristic
581C2—282no—
582C2#1173364011yescharacteristic
583C2#1173656111yescharacteristic
584C2#1173665511yescharacteristic
585C2#1173784811yescharacteristic
586C2#1173788711yescharacteristic
587C2#1173844811yescharacteristic
588C2#1173849511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1948C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1948D7
hypocenterC7#21966
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. C7×#1948C2semidirect
7. #11728318C2semidirect
8. #11728318C2semidirect
9. #11728209C2semidirect
10. #11727321C2semidirect
11. #11727222C2semidirect
14. #11726550C2semidirect
15. #11726549C2semidirect
49. #89611C4semidirect
132. #1948D7semidirect
549. C7#21966semidirect

Maximal subgroups

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

subgroupindex
C2×#896112
#117318032
#117312052
C7×#19482
#117283322
#117283312
#117283182
#117282092
#117273212
#117272222
#117266242
#117266232
#117265502
#117265492
#219667