Finite Group Databasebeta

← #13426387

Subgroups — #13426387

Order 1792.

726 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2370C211yesmaximal, characteristic, complemented
2#11731120C211yesmaximal, characteristic
3#11729497C211yesmaximal, characteristic
4#11729493C211yesmaximal, characteristic, complemented
5#11727383C211yesmaximal, characteristic
6#11727380C211yesmaximal, characteristic
7#11727374C211yesmaximal, characteristic, complemented
8#11727359C211yesmaximal, characteristic, complemented
9#11727244C211yesmaximal, characteristic, complemented
10#11727237C211yesmaximal, characteristic, complemented
11#11727228C211yesmaximal, characteristic
12#11727217C211yesmaximal, characteristic
13#11727216C211yesmaximal, characteristic
14#11727214C211yesmaximal, characteristic, complemented
15#11727214C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×#6459—21no—
18C2×C4×D28C2211yescharacteristic
19C2×#6432C2211yescharacteristic
20C7×#389—21no—
21C7×#389C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#387—21no—
24C7×#387—21no—
25C4×C7×C4⋊C4—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30#89766—21no—
31C2×#6383C2211yescharacteristic
32C2×#6383C2211yescharacteristic
33C2×#6383C2211yescharacteristic
34C2×#6383C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89711C2211yescharacteristic
40#89711C2211yescharacteristic
41#89700C2211yescharacteristic
42#89700C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44C2×#6377C2211yescharacteristic
45#89445—21no—
46#89440—21no—
47#89324C2211yescharacteristic
48#89324C2211yescharacteristic
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89319C2211yescharacteristic
56#89319C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#27026—71nomaximal, Sylow
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×Dic14—21no—
63C7×#125—22no—
64C7×#125—22no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C42×C7—21no—
76C2×C42×C7—21no—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96#6459—44no—
97C4×D28—22no—
98C4×D28—41no—
99#6432—22no—
100#6432—41no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95D411yescharacteristic
107C7×#95D411yescharacteristic
108C7⋊#95—21no—
109C7⋊#95—21no—
110#6383—22no—
111#6383—22no—
112#6383—41no—
113#6383Q814yescomplemented
114#6382—22no—
115#6382—22no—
116#6382—41no—
117#6381—22no—
118C4⋊(C7⋊C8)—41no—
119#6377—41no—
120#2452—71no—
121#2370D711yescharacteristic, complemented
122#2065—71no—
123#2059—71no—
124#2058—71no—
125#2025—71no—
126#1988—71no—
127#1943—71no—
128#1926—71no—
129#1925—71no—
130#1920—71no—
131#1920—71no—
132#1917—71no—
133#1864—71no—
134#1762—71no—
135C23×D7—21no—
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×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C2×D28—21no—
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28C2×Q812yes—
156C2×C4×D7—42no—
157C2×C4×D7—42no—
158C2×C4×D7—81no—
159C2×Dic14—42no—
160C2×Dic14—42no—
161C2×Dic14—81no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—41no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—81no—
173C7×C4⋊C4C2×Q812yes—
174C7×C4⋊C4#4112yes—
175C7×C4⋊C4#4112yes—
176C42×C7—21no—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—42no—
183C42×C7—81no—
184C42×C7—81no—
185(C2×D7)⋊C4—42no—
186(C2×D7)⋊C4—42no—
187(C2×D7)⋊C4—42no—
188(C2×D7)⋊C4—42no—
189(C2×D7)⋊C4—81no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—42no—
196C4⋊Dic7#4112yes—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8C2×Q812yes—
211C2×C4⋊Q8—71no—
212C2×#125C2×D711yescharacteristic
213C2×C22⋊Q8—141no—
214C2×C4×D4—71no—
215C2×#107—71no—
216C2×#106—71no—
217C2×#106—71no—
218C2×#106—71no—
219C2×C4⋊C8—71no—
220C2×#102—71no—
221C2×#102—71no—
222C2×#102—71no—
223C2×#102—71no—
224C2×#97—71no—
225#396—141no—
226#393—141no—
227#390—71no—
228#390—71no—
229#389—21no—
230#389—71no—
231#389—71no—
232#389C2×D711yescharacteristic
233#389C2×D711yescharacteristic
234#387—21no—
235#387—21no—
236#385—141no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4C2×D711yescharacteristic
239C4×C4⋊C4C2×D711yescharacteristic
240C4×C4⋊C4C2×D711yescharacteristic
241C4×C4⋊C4C2×D711yescharacteristic
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#336—71no—
251#336—71no—
252#336—71no—
253#336—71no—
254C23×C7C22×D411yescharacteristic
255C22×D7—22no—
256C22×D7—41no—
257C22×D7—41no—
258C22×D7—41no—
259C22×D7—41no—
260C22×D7—41no—
261C22×D7—41no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7AGL(1,ℤ/8)12yes—
292C2×C4×C7C22×Q811yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7H(5,𝔽2)11yescharacteristic
297C2×C4×C72−1+411yescharacteristic
298C2×Dic7—22no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307D28—22no—
308D28—22no—
309D28—41no—
310D28—41no—
311C4×D7—84no—
312Dic14—84no—
313C7⋊C8—22no—
314C7⋊C8—41no—
315C7⋊C8—41no—
316C7⋊C8—41no—
317C22×Q8—141no—
318C22×D4—71no—
319C23×C4—141no—
320C22×C8—71no—
321C22×C8—71no—
322C22×C8—71no—
323C22×C8—71no—
324C4⋊Q8—142no—
325C4⋊Q8—281no—
326#125—22no—
327#125—22no—
328C22⋊Q8—284no—
329C4×D4—142no—
330C4×D4—281no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4C22⋊D711yescharacteristic
342C2×C4⋊C4C22⋊D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C22⋊C4—141no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42C22×D711yescharacteristic
356C2×C42C22×D711yescharacteristic
357C2×C42C22×D711yescharacteristic
358#107—142no—
359#106—142no—
360#106—142no—
361#106—281no—
362C4⋊C8—281no—
363#102—74no—
364#102—142no—
365#102—142no—
366#102—281no—
367#97—281no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—141no—
374#95—141no—
375#95C22⋊D711yescharacteristic
376#95C22⋊D711yescharacteristic
377C22×C7#52511yescharacteristic
378C22×C7#54611yescharacteristic
379C22×C7D4×Q811yescharacteristic
380C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
381C22×C7#57511yescharacteristic
382C22×C7#57711yescharacteristic
383C22×C7#57711yescharacteristic
384C2×D7—42no—
385C2×D7—42no—
386C2×D7—42no—
387C2×D7—81no—
388C2×D7—81no—
389C2×D7—81no—
390C2×D7—81no—
391C4×C7—22no—
392C4×C7—41no—
393C4×C7—42no—
394C4×C7—41no—
395C4×C7—42no—
396C4×C7—42no—
397C4×C7—41no—
398C4×C7—41no—
399C4×C7—41no—
400C4×C7—81no—
401C4×C7#53311yescharacteristic
402C4×C7#54211yescharacteristic
403C4×C7#55411yescharacteristic
404C4×C7#55611yescharacteristic
405Dic7—42no—
406Dic7—81no—
407Dic7—81no—
408C24—141no—
409C2×Q8—282no—
410C2×Q8—282no—
411C2×Q8—561no—
412C2×D4—72no—
413C2×D4—141no—
414C2×D4—141no—
415C2×D4—141no—
416C2×D4—141no—
417C2×D4—141no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—282no—
426C22×C4—282no—
427C22×C4—561no—
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C2×C22⋊D711yescharacteristic
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C2×C22⋊D711yescharacteristic
434C22×C4C23×D711yescharacteristic
435C2×C8—72no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—72no—
458C4⋊C4—81no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—282no—
465C4⋊C4—282no—
466C4⋊C4—282no—
467C4⋊C4#115712yes—
468C4⋊C4#115712yes—
469C4⋊C4Q8×D712yes—
470C22⋊C4—282no—
471C22⋊C4—282no—
472C22⋊C4—282no—
473C22⋊C4—282no—
474C22⋊C4—561no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—42no—
482C42—81no—
483C42—81no—
484C2×C7#246511yescharacteristic
485C2×C7#311511yescharacteristic
486C2×C7#312511yescharacteristic
487C2×C7#331011yescharacteristic
488C2×C7#332611yescharacteristic
489C2×C7#338411yescharacteristic
490C2×C7#338811yescharacteristic
491D7—82no—
492C23—142no—
493C23—281no—
494C23—281no—
495C23—281no—
496C23—281no—
497C23—281no—
498C23—281no—
499C23C22×C22⋊D711yescharacteristic
500Q8—564no—
501D4—142no—
502D4—142no—
503D4—281no—
504D4—281no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—142no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—564no—
544C2×C4#651212yes—
545C2×C4C2×#115711yescharacteristic
546C2×C4C2×#115711yescharacteristic
547C2×C4C2×Q8×D711yescharacteristic
548C2×C4#655111yescharacteristic
549C2×C4D7×#4111yescharacteristic
550C2×C4#655311yescharacteristic
551C8—142no—
552C8—281no—
553C8—281no—
554C8—281no—
555C7#2702611yescharacteristic, complemented, Sylow
556C22—282no—
557C22—282no—
558C22—282no—
559C22—561no—
560C22—561no—
561C22—561no—
562C22—561no—
563C22#9048211yescharacteristic
564C22Q8×C22⋊D711yescharacteristic
565C22C2×#651211yescharacteristic
566C22#9051311yescharacteristic
567C22#9052011yescharacteristic
568C22#9052711yescharacteristic
569C22#9052811yescharacteristic
570C4—22no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—41no—
578C4—42no—
579C4—81no—
580C4—282no—
581C4—561no—
582C4—561no—
583C4#9025911yescharacteristic
584C4#9026911yescharacteristic
585C4#9027011yescharacteristic
586C4#9027311yescharacteristic
587C2—562no—
588C2#1173994311yescharacteristic
589C2#1173997611yescharacteristic
590C2#1174002111yescharacteristic
591C2#1174104411yescharacteristic
592C2#1174106611yescharacteristic
593C2#1174107511yescharacteristic
594C2#1174114011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2370C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2370D7
hypocenterC7#27026
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
1. C7×#2370C2semidirect
4. #11729493C2semidirect
7. #11727374C2semidirect
8. #11727359C2semidirect
9. #11727244C2semidirect
10. #11727237C2semidirect
14. #11727214C2semidirect
15. #11727214C2semidirect
113. #6383Q8semidirect
121. #2370D7semidirect
555. C7#27026semidirect

Maximal subgroups

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

subgroupindex
C7×#23702
#117311202
#117294972
#117294932
#117273832
#117273802
#117273742
#117273592
#117272442
#117272372
#117272282
#117272172
#117272162
#117272142
#270267