Finite Group Databasebeta

← #13365977

Subgroups — #13365977

Order 1792.

732 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
1C7×#2330C211yesmaximal, characteristic, complemented
2C2×#89838C211yesmaximal, characteristic
3#11730860C211yesmaximal, characteristic
4#11729492C211yesmaximal, characteristic, complemented
5#11727374C211yesmaximal, characteristic, complemented
6#11727372C211yesmaximal, characteristic, complemented
7#11727371C211yesmaximal, characteristic
8#11727369C211yesmaximal, characteristic
9#11727338C211yesmaximal, characteristic
10#11727322C211yesmaximal, characteristic, complemented
11#11727217C211yesmaximal, characteristic
12#11727217C211yesmaximal, characteristic
13#11727214C211yesmaximal, characteristic, complemented
14#11727212C211yesmaximal, characteristic, complemented
15#11727190C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×C4×D28C2211yescharacteristic
18C7×#389—21no—
19C7×#389—21no—
20C7×#389C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#387—21no—
23C7×#387—21no—
24C7×#384C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#376C2211yescharacteristic
28#89838C418yescomplemented
29C2×#6383C2211yescharacteristic
30C2×#6383C2211yescharacteristic
31C2×#6383C2211yescharacteristic
32C2×#6383C2211yescharacteristic
33C2×#6382C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37#89711C2211yescharacteristic
38#89711C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40#89700C2211yescharacteristic
41#89700C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44C2×#6377C2211yescharacteristic
45C2×C4×C7⋊C8C2211yescharacteristic
46#89441—21no—
47#89441—21no—
48#89439—21no—
49#89436—21no—
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22122—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C7×#125—22no—
64C7×#125C2×C414yes—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
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×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×C4×Dic7—21no—
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C4×D28—22no—
96C4×D28C2×C414yes—
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95D411yescharacteristic
102C7×#95D411yescharacteristic
103C7×#95D411yescharacteristic
104C7×#95D411yescharacteristic
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108#6383—41no—
109#6383—41no—
110#6383C2×C414yes—
111#6383C2×C414yes—
112#6382—41no—
113#6382—41no—
114#6382C2×C414yes—
115#6381C2×C414yes—
116C4⋊(C7⋊C8)—22no—
117C4⋊(C7⋊C8)C2×C414yes—
118#6377—41no—
119C4×C7⋊C8—41no—
120C2×#478—71no—
121#2330D711yescharacteristic, complemented
122#2319—71no—
123#1989—71no—
124#1948—71no—
125#1943—71no—
126#1941—71no—
127#1936—71no—
128#1934—71no—
129#1926—71no—
130#1926—71no—
131#1920—71no—
132#1918—71no—
133#1802—71no—
134#1768—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×D28C22×C412yes—
156C2×C4×D7—24no—
157C2×C4×D7—81no—
158C2×C4×D7—81no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—81no—
168C7×C4⋊C4—81no—
169C7×C4⋊C4—81no—
170C7×C4⋊C4—81no—
171C7×C4⋊C4C22×C412yes—
172C7×C4⋊C4C22×C412yes—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—21no—
177C42×C7—81no—
178C42×C7—81no—
179C42×C7C22×C412yes—
180C42×C7#4112yes—
181(C2×D7)⋊C4—24no—
182(C2×D7)⋊C4—81no—
183(C2×D7)⋊C4—81no—
184(C2×D7)⋊C4—81no—
185(C2×D7)⋊C4—81no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7C22×C412yes—
192Dic7⋊C4—81no—
193Dic7⋊C4—81no—
194C4×Dic7—81no—
195C2×C7⋊C8—21no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—21no—
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⋊C8C22×C412yes—
206C2×C7⋊C8C22×C412yes—
207C2×#125C2×D711yescharacteristic
208C2×C4×D4—71no—
209#478—78no—
210C2×#107—71no—
211C2×#106—71no—
212C2×#106—71no—
213C2×#106—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8—71no—
216C2×#102—71no—
217C2×#102—71no—
218C2×#102—71no—
219C2×#102—71no—
220C2×#97—71no—
221C2×C4×C8—71no—
222#390—71no—
223#390—71no—
224#389—21no—
225#389—21no—
226#389—71no—
227#389—71no—
228#389C2×D711yescharacteristic
229#389C2×D711yescharacteristic
230#388—141no—
231#388—141no—
232#387—21no—
233#387—21no—
234#384C2×D711yescharacteristic
235#381—141no—
236#380—141no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4C2×D711yescharacteristic
239C4×C4⋊C4C2×D711yescharacteristic
240#376C2×D711yescharacteristic
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#336—71no—
250#336—71no—
251C23×C7C22×D411yescharacteristic
252C22×D7—22no—
253C22×D7—41no—
254C22×D7—41no—
255C22×D7—41no—
256C22×D7—41no—
257C22×D7—41no—
258C22×D7—41no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
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×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7C4×D412yes—
288C2×C4×C7C4×D412yes—
289C2×C4×C7C23×C411yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7H(5,𝔽2)11yescharacteristic
292C2×C4×C72−1+411yescharacteristic
293C2×C4×C72−1+411yescharacteristic
294C2×C4×C72−1+411yescharacteristic
295C2×Dic7—22no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304D28—22no—
305D28—22no—
306D28—41no—
307D28—41no—
308C4×D7—84no—
309C7⋊C8—22no—
310C7⋊C8—22no—
311C7⋊C8—41no—
312C7⋊C8—41no—
313C22×D4—71no—
314C23×C4—141no—
315C22×C8—71no—
316C22×C8—71no—
317C22×C8—71no—
318C22×C8—71no—
319#125—22no—
320#125C4×D714yes—
321C4×D4—74no—
322C4×D4—142no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C22⋊C4—141no—
339C2×C22⋊C4—141no—
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—141no—
346C2×C42C22×D711yescharacteristic
347C2×C42C22×D711yescharacteristic
348C2×C42C22×D711yescharacteristic
349#107—74no—
350#106—74no—
351#106—281no—
352#106—281no—
353C4⋊C8—74no—
354C4⋊C8—142no—
355#102—74no—
356#102—74no—
357#102—281no—
358#102—281no—
359#97—281no—
360C4×C8—281no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95C22⋊D711yescharacteristic
369#95C22⋊D711yescharacteristic
370#95C22⋊D711yescharacteristic
371#95C22⋊D711yescharacteristic
372C22×C7C2×C4×D411yescharacteristic
373C22×C7#53611yescharacteristic
374C22×C7#53711yescharacteristic
375C22×C7#57611yescharacteristic
376C22×C7#57711yescharacteristic
377C22×C7#57711yescharacteristic
378C22×C7#57711yescharacteristic
379C2×D7—42no—
380C2×D7—42no—
381C2×D7—42no—
382C2×D7—81no—
383C2×D7—81no—
384C2×D7—81no—
385C2×D7—81no—
386C4×C7—22no—
387C4×C7—22no—
388C4×C7—42no—
389C4×C7—42no—
390C4×C7—41no—
391C4×C7—41no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7—81no—
395C4×C7—81no—
396C4×C7#51911yescharacteristic
397C4×C7#52011yescharacteristic
398C4×C7#54111yescharacteristic
399C4×C7#54211yescharacteristic
400Dic7—42no—
401Dic7—81no—
402Dic7—81no—
403C24—141no—
404C2×D4—72no—
405C2×D4—141no—
406C2×D4—141no—
407C2×D4—141no—
408C2×D4—141no—
409C2×D4—141no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—141no—
415C22×C4—144no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—561no—
419C22×C4—561no—
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C2×C22⋊D711yescharacteristic
423C22×C4C2×C22⋊D711yescharacteristic
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C23×D711yescharacteristic
427C2×C8—72no—
428C2×C8—72no—
429C2×C8—141no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—72no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—561no—
458C4⋊C4—561no—
459C4⋊C4C2×C4×D712yes—
460C4⋊C4C2×C4×D712yes—
461C22⋊C4—144no—
462C22⋊C4—561no—
463C22⋊C4—561no—
464C22⋊C4—561no—
465C22⋊C4—561no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—81no—
471C42—81no—
472C42—561no—
473C42C2×C4×D712yes—
474C42#115712yes—
475C2×C7#232311yescharacteristic
476C2×C7#294611yescharacteristic
477C2×C7#295211yescharacteristic
478C2×C7#324111yescharacteristic
479C2×C7#324411yescharacteristic
480C2×C7#325711yescharacteristic
481C2×C7#326011yescharacteristic
482D7—82no—
483C23—142no—
484C23—281no—
485C23—281no—
486C23—281no—
487C23—281no—
488C23—281no—
489C23—281no—
490C23C22×C22⋊D711yescharacteristic
491D4—142no—
492D4—142no—
493D4—281no—
494D4—281no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
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—142no—
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—564no—
533C2×C4C4×C22⋊D712yes—
534C2×C4C4×C22⋊D712yes—
535C2×C4C22×C4×D711yescharacteristic
536C2×C4C2×#115711yescharacteristic
537C2×C4#655111yescharacteristic
538C2×C4#655111yescharacteristic
539C2×C4#655311yescharacteristic
540C2×C4#655411yescharacteristic
541C8—142no—
542C8—142no—
543C8—281no—
544C8—281no—
545C7#2212211yescharacteristic, complemented, Sylow
546C22—282no—
547C22—282no—
548C22—282no—
549C22—561no—
550C22—561no—
551C22—561no—
552C22—561no—
553C22C2×C4×C22⋊D711yescharacteristic
554C22#9050711yescharacteristic
555C22#9052411yescharacteristic
556C22#9052611yescharacteristic
557C22#9052711yescharacteristic
558C22#9052811yescharacteristic
559C22#9052811yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—81no—
569C4—81no—
570C4—282no—
571C4—561no—
572C4—561no—
573C4#9026311yescharacteristic
574C4#9026511yescharacteristic
575C4#9027311yescharacteristic
576C4#9027411yescharacteristic
577C2—562no—
578C2#1173993311yescharacteristic
579C2#1174000311yescharacteristic
580C2#1174000911yescharacteristic
581C2#1174088211yescharacteristic
582C2#1174089511yescharacteristic
583C2#1174092211yescharacteristic
584C2#1174092411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2330C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#2330D7
hypocenterC7#22122
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #2330

Complements

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

NH ≅ G/Ntype
1. C7×#2330C2semidirect
4. #11729492C2semidirect
5. #11727374C2semidirect
6. #11727372C2semidirect
10. #11727322C2semidirect
13. #11727214C2semidirect
14. #11727212C2semidirect
15. #11727190C2semidirect
28. #89838C4semidirect
121. #2330D7semidirect
545. C7#22122semidirect

Maximal subgroups

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

subgroupindex
C7×#23302
C2×#898382
#117308602
#117294922
#117273742
#117273722
#117273712
#117273692
#117273382
#117273222
#117272172
#117272142
#117272122
#117271902
#221227