Finite Group Databasebeta

← #13371527

Subgroups — #13371527

Order 1792.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89633C211yesmaximal, characteristic
2#11733646C211yesmaximal, characteristic
3#11731078C211yesmaximal, characteristic
4C7×#1919C211yesmaximal, characteristic, complemented
5#11728312C211yesmaximal, characteristic, complemented
6#11727374C211yesmaximal, characteristic, complemented
7#11727235C211yesmaximal, characteristic, complemented
8#11726625C211yesmaximal, characteristic
9#11726624C211yesmaximal, characteristic
10#11726613C211yesmaximal, characteristic, complemented
11#11726574C211yesmaximal, characteristic, complemented
12#11726567C211yesmaximal, characteristic
13#11726538C211yesmaximal, characteristic, complemented
14#11726404C211yesmaximal, characteristic
15#11726375C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C2×#6455C2211yescharacteristic
18C2×D7×C4⋊C4C2211yescharacteristic
19C2×C7×#106C2211yescharacteristic
20C7×#389C2211yescharacteristic
21C4×C7×C4⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C8C2211yescharacteristic
23C2×#6392C2211yescharacteristic
24C2×C8×Dic7C2211yescharacteristic
25#89759—21no—
26#89759C2211yescharacteristic
27#89759C2211yescharacteristic
28#89757C2211yescharacteristic
29#89750—21no—
30C4⋊C4×Dic7C2211yescharacteristic
31C4⋊C4×Dic7C2211yescharacteristic
32#89745C2211yescharacteristic
33C2×#6382C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6381C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37#89712—21no—
38C4×(C2×D7)⋊C4C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40#89704—21no—
41#89700C2211yescharacteristic
42C2×#6377C2211yescharacteristic
43#89633C418yescomplemented
44#89441—21no—
45#89438—21no—
46#89427—21no—
47#89422—21no—
48C7×#340C2211yescharacteristic
49C7×#340C2211yescharacteristic
50C7×#340C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22621—71nomaximal, Sylow
61C22×C4×D7—21no—
62C22×C4×D7C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
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)⋊C4D411yescharacteristic
76C2×(C2×D7)⋊C4D411yescharacteristic
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C2×#1131—22no—
93C2×#1131—22no—
94C2×#1131C2×C414yes—
95#6455—22no—
96#6455C2×C414yes—
97D7×C4⋊C4—22no—
98D7×C4⋊C4C2×C414yes—
99C7×#106C2×C414yes—
100C7×#95—21no—
101C7×#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95D411yescharacteristic
111C7⋊#95D411yescharacteristic
112(C2×D7)⋊C8—22no—
113#6392C2×C414yes—
114C8×Dic7—22no—
115#6382—22no—
116#6382C2×C414yes—
117#6381—22no—
118#6381C2×C414yes—
119#6377—22no—
120C2×#428—71no—
121#2346—71no—
122#2345—71no—
123#1943—71no—
124#1942—71no—
125#1921—71no—
126#1920—71no—
127#1919D711yescharacteristic, complemented
128#1918—71no—
129#1825—71no—
130#1823—71no—
131#1809—71no—
132#1776—71no—
133C4×#106—71no—
134#1753—71no—
135C23×D7C2×D411yescharacteristic
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2411yescharacteristic
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149C22×Dic7C2×D411yescharacteristic
150C2×C4×D7—21no—
151C2×C4×D7—21no—
152C2×C4×D7—21no—
153C2×C4×D7—21no—
154C2×C4×D7—24no—
155C2×C4×D7—81no—
156C2×C4×D7—81no—
157C2×C4×D7C22×C412yes—
158C2×C4×D7#4112yes—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8C22×C412yes—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—81no—
172C7×C4⋊C4C22×C412yes—
173C7×C4⋊C4C22×C412yes—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—42no—
177(C2×D7)⋊C4—24no—
178(C2×D7)⋊C4—22no—
179(C2×D7)⋊C4—41no—
180(C2×D7)⋊C4—42no—
181(C2×D7)⋊C4—81no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—81no—
189C4⋊Dic7C22×C412yes—
190C4⋊Dic7C22×C412yes—
191Dic7⋊C4—24no—
192Dic7⋊C4—81no—
193Dic7⋊C4—81no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—24no—
197C4×Dic7—42no—
198C4×Dic7—81no—
199C4×Dic7—81no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8C22×C412yes—
204C2×C7⋊C8#4112yes—
205C2×C7⋊C8#4112yes—
206#1131—28no—
207C22×M16—71no—
208C2×#117—71no—
209C22×C4⋊C4—71no—
210#428—78no—
211C2×#107—71no—
212C2×#107—71no—
213C2×#107—71no—
214C2×#106—71no—
215C2×#106—71no—
216C2×#106C2×D711yescharacteristic
217C2×C22⋊C8—71no—
218C2×#97—71no—
219C2×C4×C8—71no—
220#389—71no—
221#389—71no—
222#389—71no—
223#389C2×D711yescharacteristic
224#388—141no—
225#388—141no—
226#387—141no—
227#387—141no—
228#382—141no—
229#380—71no—
230#380—71no—
231#380—141no—
232C4×C4⋊C4—71no—
233C4×C4⋊C4—71no—
234C4×C4⋊C4—71no—
235C4×C4⋊C4C2×D711yescharacteristic
236C4×C22⋊C4—71no—
237C4×C22⋊C4—141no—
238#376—141no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340C2×D711yescharacteristic
247#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
249#336—71no—
250#336C2×D711yescharacteristic
251C23×C7C22×D411yescharacteristic
252C22×D7—21no—
253C22×D7—21no—
254C22×D7—21no—
255C22×D7—21no—
256C22×D7—21no—
257C22×D7—21no—
258C22×D7C4×D412yes—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
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×C7C23×C411yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C72−1+411yescharacteristic
282C2×C4×C72−1+411yescharacteristic
283C2×Dic7—21no—
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—22no—
288C2×Dic7—22no—
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×Dic7C4×D412yes—
303C2×Dic7#13712yes—
304C4×D7—22no—
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—22no—
308C4×D7—84no—
309C7×C8—22no—
310C7×C8—41no—
311C7⋊C8—41no—
312C7⋊C8#11714yes—
313C23×C4—71no—
314C23×C4—141no—
315C2×M16—74no—
316C2×M16—142no—
317C2×M16—142no—
318C22×C8—71no—
319C22×C8—71no—
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322#117—74no—
323#117—142no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—74no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C22⋊C4—71no—
340C2×C22⋊C4—71no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C42—21no—
345C2×C42—71no—
346C2×C42—141no—
347C2×C42—141no—
348C2×C42—141no—
349C2×C42—141no—
350C2×C42C22×D711yescharacteristic
351#107—74no—
352#107—74no—
353#107—142no—
354#106—74no—
355#106—142no—
356#106C4×D714yes—
357C22⋊C8—142no—
358#97—142no—
359C4×C8—142no—
360#95—21no—
361#95—21no—
362#95—71no—
363#95—71no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372C22×C7C2×C4×D411yescharacteristic
373C22×C7#54711yescharacteristic
374C22×C7#54711yescharacteristic
375C22×C7C2×#13711yescharacteristic
376C22×C7#57511yescharacteristic
377C22×C7#57611yescharacteristic
378C22×C7#57711yescharacteristic
379C2×D7—22no—
380C2×D7—22no—
381C2×D7—22no—
382C2×D7—41no—
383C2×D7—41no—
384C2×D7—41no—
385C2×D7—41no—
386C4×C7—41no—
387C4×C7—41no—
388C4×C7—42no—
389C4×C7—42no—
390C4×C7—81no—
391C4×C7—81no—
392C4×C7C2×#11711yescharacteristic
393C4×C7#51911yescharacteristic
394C4×C7#55211yescharacteristic
395C4×C7#55211yescharacteristic
396Dic7—22no—
397Dic7—22no—
398Dic7—41no—
399Dic7—42no—
400Dic7—42no—
401Dic7—81no—
402Dic7—81no—
403C24—71no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—71no—
409C22×C4—71no—
410C22×C4—72no—
411C22×C4—72no—
412C22×C4—71no—
413C22×C4—144no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—561no—
423C22×C4—561no—
424C22×C4D4×D711yescharacteristic
425C22×C4D4×D711yescharacteristic
426C22×C4C23×D711yescharacteristic
427M16—148no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—72no—
434C2×C8—72no—
435C2×C8—72no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8C2×C4×D712yes—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—72no—
447C4⋊C4—72no—
448C4⋊C4—81no—
449C4⋊C4—144no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—561no—
457C4⋊C4—561no—
458C4⋊C4—561no—
459C4⋊C4C2×C4×D712yes—
460C4⋊C4C2×C4×D712yes—
461C22⋊C4—142no—
462C22⋊C4—144no—
463C22⋊C4—281no—
464C22⋊C4—282no—
465C22⋊C4—561no—
466C42—21no—
467C42—21no—
468C42—42no—
469C42—144no—
470C42—141no—
471C42—141no—
472C42—282no—
473C42—561no—
474C42—561no—
475C2×C7#234411yescharacteristic
476C2×C7C4×#13711yescharacteristic
477C2×C7#294611yescharacteristic
478C2×C7#334411yescharacteristic
479C2×C7#334711yescharacteristic
480C2×C7#335111yescharacteristic
481C2×C7#335711yescharacteristic
482D7—42no—
483C23—72no—
484C23—141no—
485C23—141no—
486C23—141no—
487C23—141no—
488C23—141no—
489C23—141no—
490C23C2×D4×D711yescharacteristic
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
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—72no—
510C2×C4—72no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—142no—
516C2×C4—142no—
517C2×C4—142no—
518C2×C4—142no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—142no—
522C2×C4—142no—
523C2×C4—141no—
524C2×C4—281no—
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—564no—
535C2×C4C22×C4×D711yescharacteristic
536C2×C4#655111yescharacteristic
537C2×C4D7×#4111yescharacteristic
538C2×C4D7×#4111yescharacteristic
539C2×C4D7×#4111yescharacteristic
540C2×C4#655411yescharacteristic
541C8—22no—
542C8—41no—
543C8—74no—
544C8—281no—
545C7#2262111yescharacteristic, complemented, Sylow
546C22—142no—
547C22—142no—
548C22—142no—
549C22—281no—
550C22—281no—
551C22—281no—
552C22—281no—
553C22C4×D4×D711yescharacteristic
554C22#9032211yescharacteristic
555C22#9034611yescharacteristic
556C22#9045311yescharacteristic
557C22#9046011yescharacteristic
558C22#9046411yescharacteristic
559C22D7×#13711yescharacteristic
560C4—42no—
561C4—41no—
562C4—42no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—142no—
567C4—142no—
568C4—282no—
569C4—282no—
570C4—281no—
571C4—561no—
572C4—561no—
573C4D7×#11711yescharacteristic
574C4#9026311yescharacteristic
575C4#9038411yescharacteristic
576C4#9039911yescharacteristic
577C2—282no—
578C2#1173240611yescharacteristic
579C2#1173709211yescharacteristic
580C2#1173721911yescharacteristic
581C2#1173847511yescharacteristic
582C2#1173864311yescharacteristic
583C2#1173878211yescharacteristic
584C2#1173895811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1919C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1919D7
hypocenterC7#22621
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. C7×#1919C2semidirect
5. #11728312C2semidirect
6. #11727374C2semidirect
7. #11727235C2semidirect
10. #11726613C2semidirect
11. #11726574C2semidirect
13. #11726538C2semidirect
15. #11726375C2semidirect
43. #89633C4semidirect
127. #1919D7semidirect
545. C7#22621semidirect

Maximal subgroups

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

subgroupindex
C2×#896332
#117336462
#117310782
C7×#19192
#117283122
#117273742
#117272352
#117266252
#117266242
#117266132
#117265742
#117265672
#117265382
#117264042
#117263752
#226217