Finite Group Databasebeta

← #13366004

Subgroups — #13366004

Order 1792.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740673C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic
3#11731206C211yesmaximal, characteristic
4C7×#1988C211yesmaximal, characteristic, complemented
5#11728234C211yesmaximal, characteristic
6#11728196C211yesmaximal, characteristic, complemented
7#11727222C211yesmaximal, characteristic, complemented
8#11726615C211yesmaximal, characteristic, complemented
9#11726612C211yesmaximal, characteristic, complemented
10#11726611C211yesmaximal, characteristic
11#11726607C211yesmaximal, characteristic
12#11726546C211yesmaximal, characteristic
13#11726545C211yesmaximal, characteristic
14#11726543C211yesmaximal, characteristic, complemented
15#11726542C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C4×D28C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C4×C7×C4⋊C4C2211yescharacteristic
20#90002—21no—
21#90002—21no—
22C2×#6395C2211yescharacteristic
23C2×#6395C2211yescharacteristic
24C2×#6389C2211yescharacteristic
25C2×C8×Dic7C2211yescharacteristic
26#89856C418yescomplemented
27#89760C2211yescharacteristic
28#89760C2211yescharacteristic
29#89757C2211yescharacteristic
30#89757C2211yescharacteristic
31#89753—21no—
32#89753—21no—
33C4⋊C4×Dic7C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35#89745C2211yescharacteristic
36#89745C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43#89712—21no—
44#89712—21no—
45#89708C2211yescharacteristic
46#89704C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48#89439—21no—
49#89436—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52C7×#340C2211yescharacteristic
53C7×#340C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22123—71nomaximal, Sylow
61C22×C7×Q8—21no—
62C22×D28C2311yescharacteristic
63C22×C4×D7—21no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8C2×C414yes—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C42×C7—21no—
74C2×C42×C7C2311yescharacteristic
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7C2311yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C4×D28—22no—
94C4×D28C2×C414yes—
95C7×C4⋊C8—22no—
96C7×#95—21no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95D411yescharacteristic
106C7⋊#95D411yescharacteristic
107C7⋊#95D411yescharacteristic
108C7⋊#95D411yescharacteristic
109#6395—41no—
110#6395—41no—
111#6389—41no—
112C8×Dic7—41no—
113#6383C2×C414yes—
114#6383C2×C414yes—
115#6382—41no—
116#6382C2×C414yes—
117#6382C2×C414yes—
118#6381—41no—
119C4⋊(C7⋊C8)C2×C414yes—
120C2×#476—71no—
121#2331—71no—
122#2319—71no—
123#1988D711yescharacteristic, complemented
124#1949—71no—
125#1944—71no—
126#1941—71no—
127#1936—71no—
128#1934—71no—
129#1926—71no—
130#1925—71no—
131#1920—71no—
132#1919—71no—
133#1804—71no—
134#1768—71no—
135C23×D7—21no—
136C2×C7×Q8—24no—
137C2×C7×Q8—81no—
138C2×C7×Q8—81no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2411yescharacteristic
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7C2×D411yescharacteristic
150C22×Dic7C2×D411yescharacteristic
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28—21no—
157C2×D28—21no—
158C2×D28C22×C412yes—
159C2×C4×D7—24no—
160C2×C4×D7—81no—
161C2×C4×D7—81no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4C22×C412yes—
176C7×C4⋊C4C22×C412yes—
177C42×C7—81no—
178C42×C7C22×C412yes—
179C42×C7#4112yes—
180(C2×D7)⋊C4—24no—
181(C2×D7)⋊C4—81no—
182(C2×D7)⋊C4—81no—
183(C2×D7)⋊C4—81no—
184(C2×D7)⋊C4—81no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—81no—
191C4⋊Dic7—81no—
192C4⋊Dic7C22×C412yes—
193Dic7⋊C4—81no—
194Dic7⋊C4—81no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—21no—
198C4×Dic7—21no—
199C4×Dic7—81no—
200C4×Dic7—81no—
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×C4⋊Q8C2×D711yescharacteristic
208C2×C4×D4—71no—
209#476—78no—
210C2×#107—71no—
211C2×#106—71no—
212C2×#106—71no—
213C2×#106—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8C2×D711yescharacteristic
216C2×#102—71no—
217C2×#102—71no—
218C2×#102—71no—
219C2×#102—71no—
220C2×#97—71no—
221C2×C4×C8—71no—
222#391—141no—
223#391—141no—
224#390—71no—
225#390—71no—
226#389—71no—
227#389—71no—
228#389—71no—
229#389—71no—
230#388—141no—
231#388—141no—
232#387—141no—
233#387—141no—
234#382—71no—
235#381—141no—
236#380—141no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4C2×D711yescharacteristic
240#376—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340C2×D711yescharacteristic
246#340C2×D711yescharacteristic
247#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
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—
259C7×Q8—84no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
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×C7C4×D412yes—
277C2×C4×C7C4×D412yes—
278C2×C4×C7C23×C411yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7H(5,𝔽2)11yescharacteristic
281C2×C4×C72−1+411yescharacteristic
282C2×C4×C72−1+411yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×Dic7—22no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305D28—22no—
306D28—22no—
307D28—41no—
308D28—41no—
309C4×D7—84no—
310C7×C8—41no—
311C7×C8—41no—
312C7⋊C8—22no—
313C7⋊C8—22no—
314C22×Q8—21no—
315C22×D4—71no—
316C23×C4—141no—
317C22×C8—71no—
318C22×C8—71no—
319C22×C8C22×D711yescharacteristic
320C22×C8C22×D711yescharacteristic
321C4⋊Q8—22no—
322C4⋊Q8C4×D714yes—
323C4×D4—74no—
324C4×D4—142no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4C22×D711yescharacteristic
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C22⋊C4—141no—
338C2×C22⋊C4—141no—
339C2×C22⋊C4—141no—
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C42—21no—
343C2×C42—71no—
344C2×C42—71no—
345C2×C42—141no—
346C2×C42—141no—
347C2×C42C22×D711yescharacteristic
348#107—281no—
349#106—74no—
350#106—74no—
351#106—281no—
352C4⋊C8—22no—
353C4⋊C8—74no—
354#102—74no—
355#102—74no—
356#102—281no—
357#102—281no—
358#97—281no—
359C4×C8—281no—
360#95—21no—
361#95—71no—
362#95—71no—
363#95—71no—
364#95—71no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373C22×C7C2×C4×D411yescharacteristic
374C22×C7#53611yescharacteristic
375C22×C7#53711yescharacteristic
376C22×C7#57611yescharacteristic
377C22×C7#57711yescharacteristic
378C22×C7#57711yescharacteristic
379C22×C7#57711yescharacteristic
380C2×D7—42no—
381C2×D7—42no—
382C2×D7—42no—
383C2×D7—81no—
384C2×D7—81no—
385C2×D7—81no—
386C2×D7—81no—
387C4×C7—22no—
388C4×C7—22no—
389C4×C7—42no—
390C4×C7—42no—
391C4×C7—81no—
392C4×C7—81no—
393C4×C7#51911yescharacteristic
394C4×C7#52011yescharacteristic
395C4×C7#54111yescharacteristic
396C4×C7#54211yescharacteristic
397Dic7—42no—
398Dic7—41no—
399Dic7—41no—
400Dic7—41no—
401Dic7—41no—
402Dic7—81no—
403Dic7—81no—
404C24—141no—
405C2×Q8—24no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×D4—72no—
409C2×D4—141no—
410C2×D4—141no—
411C2×D4—141no—
412C2×D4—141no—
413C2×D4—141no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—71no—
422C22×C4—141no—
423C22×C4—144no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—561no—
427C22×C4—561no—
428C22×C4D4×D711yescharacteristic
429C22×C4D4×D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—72no—
438C2×C8—72no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—72no—
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⋊C4—561no—
460C4⋊C4C2×C4×D712yes—
461C4⋊C4C2×C4×D712yes—
462C22⋊C4—144no—
463C22⋊C4—561no—
464C22⋊C4—561no—
465C22⋊C4—561no—
466C22⋊C4—561no—
467C42—81no—
468C42—141no—
469C42—141no—
470C42—141no—
471C42—141no—
472C42—561no—
473C42—561no—
474C42C2×C4×D712yes—
475C42#115912yes—
476C2×C7#232311yescharacteristic
477C2×C7#294611yescharacteristic
478C2×C7#294911yescharacteristic
479C2×C7#324111yescharacteristic
480C2×C7#324411yescharacteristic
481C2×C7#325711yescharacteristic
482C2×C7#326311yescharacteristic
483D7—82no—
484C23—142no—
485C23—281no—
486C23—281no—
487C23—281no—
488C23—281no—
489C23—281no—
490C23—281no—
491C23C2×D4×D711yescharacteristic
492Q8—84no—
493D4—142no—
494D4—142no—
495D4—281no—
496D4—281no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
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—41no—
513C2×C4—142no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
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×C4(C22⋊D7)⋊C412yes—
536C2×C4(C22⋊D7)⋊C412yes—
537C2×C4C22×C4×D711yescharacteristic
538C2×C4C2×#115911yescharacteristic
539C2×C4#655111yescharacteristic
540C2×C4#655111yescharacteristic
541C2×C4#655311yescharacteristic
542C2×C4#655411yescharacteristic
543C8—41no—
544C8—41no—
545C8—142no—
546C8—142no—
547C7#2212311yescharacteristic, complemented, Sylow
548C22—282no—
549C22—282no—
550C22—282no—
551C22—561no—
552C22—561no—
553C22—561no—
554C22—561no—
555C22C2×(C22⋊D7)⋊C411yescharacteristic
556C22#9031911yescharacteristic
557C22#9032011yescharacteristic
558C22#9044811yescharacteristic
559C22#9045311yescharacteristic
560C22#9046911yescharacteristic
561C22#9046911yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—281no—
569C4—281no—
570C4—282no—
571C4—281no—
572C4—281no—
573C4—561no—
574C4—561no—
575C4#9026311yescharacteristic
576C4#9026511yescharacteristic
577C4#9037311yescharacteristic
578C4#9042211yescharacteristic
579C2—562no—
580C2#1173405611yescharacteristic
581C2#1173617111yescharacteristic
582C2#1173639211yescharacteristic
583C2#1173839911yescharacteristic
584C2#1173841811yescharacteristic
585C2#1173846511yescharacteristic
586C2#1173848111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1988C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1988D7
hypocenterC7#22123
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
1. #11740673C2semidirect
4. C7×#1988C2semidirect
6. #11728196C2semidirect
7. #11727222C2semidirect
8. #11726615C2semidirect
9. #11726612C2semidirect
14. #11726543C2semidirect
15. #11726542C2semidirect
26. #89856C4semidirect
123. #1988D7semidirect
547. C7#22123semidirect

Maximal subgroups

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

subgroupindex
#117406732
C2×#898562
#117312062
C7×#19882
#117282342
#117281962
#117272222
#117266152
#117266122
#117266112
#117266072
#117265462
#117265452
#117265432
#117265422
#221237