Finite Group Databasebeta

← #13373209

Subgroups — #13373209

Order 1792.

790 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×#89561C211yesmaximal, characteristic
2#11733935C211yesmaximal, characteristic
3#11733426C211yesmaximal, characteristic
4C7×#1935C211yesmaximal, characteristic
5#11728035C211yesmaximal, characteristic
6#11728007C211yesmaximal, characteristic
7#11727975C211yesmaximal, characteristic
8#11727567C211yesmaximal, characteristic
9#11727565C211yesmaximal, characteristic
10#11727444C211yesmaximal, characteristic
11#11727355C211yesmaximal, characteristic
12#11726659C211yesmaximal, characteristic
13#11726603C211yesmaximal, characteristic
14#11726573C211yesmaximal, characteristic
15#11726535C211yesmaximal, characteristic
16C2×Q8×Dic7C2211yescharacteristic
17C22×#1144C2211yescharacteristic
18C2×#6450C2211yescharacteristic
19C2×#6450C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C7×#391C2211yescharacteristic
25C7×#389C2211yescharacteristic
26#90002C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6390C2211yescharacteristic
29C2×#6390C2211yescharacteristic
30C2×#6389C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89757C2211yescharacteristic
34#89749C2211yescharacteristic
35#89748—21no—
36#89748C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38#89744—21no—
39C2×#6384C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C4×C4⋊Dic7C2211yescharacteristic
42C4×Dic7⋊C4C2211yescharacteristic
43#89561C418yescomplemented
44#89426—21no—
45#89425—21no—
46#89423—21no—
47#89421—21no—
48#89420—21no—
49#89419—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#22779—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×Dic14C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C42×C7C2311yescharacteristic
70Q8×Dic7—22no—
71Q8×Dic7C2×C414yes—
72C2×#1144—22no—
73C2×#1144—22no—
74C2×#1144C2×C414yes—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4D411yescharacteristic
86C2×Dic7⋊C4D411yescharacteristic
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7C2311yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97#6450—22no—
98#6450—22no—
99#6450—41no—
100#6450C2×C414yes—
101C4×Dic14—22no—
102C4×Dic14—41no—
103C7×#103—41no—
104C7×#103C2×C414yes—
105C7×#97—41no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109#6409—41no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119#6390—22no—
120#6390C2×C414yes—
121#6389C2×C414yes—
122Dic7⋊C8—22no—
123C8×Dic7—41no—
124#6384—22no—
125#6382C2×C414yes—
126C2×#441—71no—
127#2356—71no—
128Q8×C4⋊C4—71no—
129#1935D711yescharacteristic
130#1933—71no—
131#1920—71no—
132#1919—71no—
133#1878—71no—
134#1878—71no—
135#1836—71no—
136#1805—71no—
137#1803—71no—
138#1771—71no—
139#1763—71no—
140C4×#103—71no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8C22×C412yes—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2411yescharacteristic
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C2×Dic14—21no—
163C2×Dic14—21no—
164C2×Dic14—21no—
165C2×Dic14C22×C412yes—
166C2×Dic14#4112yes—
167C2×Dic14#4112yes—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8C22×C412yes—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—81no—
180C7×C4⋊C4C22×C412yes—
181C42×C7—21no—
182C42×C7—21no—
183#1144—28no—
184C4⋊Dic7—21no—
185C4⋊Dic7—24no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—81no—
191C4⋊Dic7C22×C412yes—
192Dic7⋊C4—24no—
193Dic7⋊C4—22no—
194Dic7⋊C4—41no—
195Dic7⋊C4—42no—
196Dic7⋊C4—42no—
197Dic7⋊C4—42no—
198Dic7⋊C4—81no—
199C4×Dic7—21no—
200C4×Dic7—21no—
201C4×Dic7—24no—
202C4×Dic7—24no—
203C4×Dic7—42no—
204C4×Dic7—42no—
205C4×Dic7—42no—
206C4×Dic7—42no—
207C4×Dic7C22×C412yes—
208C4×Dic7#4112yes—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8C22×C412yes—
215C22×Q16—71no—
216C2×C4×Q8—71no—
217C2×C4×Q8—71no—
218C2×C4×Q8—71no—
219C2×C4×Q8—71no—
220#441—78no—
221C2×#106—71no—
222C2×C4⋊C8—71no—
223C2×#103—71no—
224C2×#103—71no—
225C2×#103—71no—
226C2×#103—71no—
227C2×#103C2×D711yescharacteristic
228C2×#103C2×D711yescharacteristic
229C2×#97—71no—
230C2×#97C2×D711yescharacteristic
231C2×C4×C8—71no—
232#391—71no—
233#391C2×D711yescharacteristic
234#389—71no—
235#389—141no—
236#389—141no—
237#389C2×D711yescharacteristic
238#387—141no—
239#387—141no—
240#384—71no—
241#384—141no—
242#383—141no—
243#382—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—71no—
247C4×C4⋊C4—141no—
248#376—141no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340C2×D711yescharacteristic
256#340C2×D711yescharacteristic
257#336—71no—
258#336—71no—
259C23×C7C22×D411yescharacteristic
260C7×Q8—22no—
261C7×Q8—22no—
262C7×Q8—41no—
263C7×Q8—41no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
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—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7C23×C411yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7H(5,𝔽2)11yescharacteristic
287C2×C4×C72−1+411yescharacteristic
288C2×Dic7—22no—
289C2×Dic7—21no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—22no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7C4×D412yes—
310C2×Dic7C4×D412yes—
311C2×Dic7#13712yes—
312Dic14—22no—
313Dic14—22no—
314Dic14—22no—
315Dic14#11714yes—
316C7×C8—22no—
317C7×C8—41no—
318C7⋊C8—22no—
319C7⋊C8—41no—
320C22×Q8—71no—
321C22×Q8C22×D711yescharacteristic
322C2×Q16—74no—
323C2×Q16—142no—
324C2×Q16—142no—
325C22×C8—71no—
326C22×C8—71no—
327C22×C8C22×D711yescharacteristic
328C22×C8C22×D711yescharacteristic
329C4×Q8—74no—
330C4×Q8—74no—
331C4×Q8—142no—
332C4×Q8—142no—
333C4×Q8—142no—
334C4×Q8—142no—
335C4×Q8—281no—
336C4×Q8—281no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—141no—
356C2×C42—141no—
357C2×C42—141no—
358C2×C42—141no—
359C2×C42—141no—
360C2×C42—141no—
361C2×C42C22×D711yescharacteristic
362#106—74no—
363C4⋊C8—142no—
364#103—41no—
365#103—74no—
366#103—142no—
367#103—142no—
368#103—281no—
369#103C4×D714yes—
370#97—41no—
371#97—74no—
372C4×C8—281no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—71no—
377#95—71no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385C22×C7C2×C4×D411yescharacteristic
386C22×C7#54611yescharacteristic
387C22×C7#54811yescharacteristic
388C22×C7C2×#13711yescharacteristic
389C22×C7#57511yescharacteristic
390C22×C7#57711yescharacteristic
391C22×C7#57811yescharacteristic
392C4×C7—42no—
393C4×C7—42no—
394C4×C7—41no—
395C4×C7—41no—
396C4×C7—81no—
397C4×C7—81no—
398C4×C7C2×#11711yescharacteristic
399C4×C7#52011yescharacteristic
400C4×C7#55211yescharacteristic
401C4×C7#55311yescharacteristic
402Dic7—24no—
403Dic7—22no—
404Dic7—22no—
405Dic7—22no—
406Dic7—42no—
407Dic7—41no—
408Dic7—81no—
409Dic7—81no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—72no—
416C2×Q8—72no—
417C2×Q8—72no—
418C2×Q8—141no—
419C2×Q8—141no—
420C2×Q8—141no—
421C2×Q8C2×C4×D712yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—71no—
427C22×C4—71no—
428C22×C4—71no—
429C22×C4—71no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4D4×D711yescharacteristic
435C22×C4D4×D711yescharacteristic
436C22×C4C23×D711yescharacteristic
437Q16—148no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—72no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8C2×C4×D712yes—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—72no—
456C4⋊C4—81no—
457C4⋊C4—142no—
458C4⋊C4—144no—
459C4⋊C4—144no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—281no—
466C4⋊C4—282no—
467C4⋊C4—282no—
468C4⋊C4—282no—
469C4⋊C4—561no—
470C4⋊C4—561no—
471C4⋊C4C2×C4×D712yes—
472C42—21no—
473C42—21no—
474C42—72no—
475C42—72no—
476C42—144no—
477C42—144no—
478C42—141no—
479C42—141no—
480C42—282no—
481C42—282no—
482C42—282no—
483C42—282no—
484C2×C7#235211yescharacteristic
485C2×C7C4×#13711yescharacteristic
486C2×C7#294711yescharacteristic
487C2×C7#331211yescharacteristic
488C2×C7#331711yescharacteristic
489C2×C7#336911yescharacteristic
490C2×C7#337711yescharacteristic
491C23C2×D4×D711yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496Q8—74no—
497Q8—142no—
498Q8—142no—
499Q8—142no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—72no—
519C2×C4—72no—
520C2×C4—72no—
521C2×C4—142no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—142no—
528C2×C4—141no—
529C2×C4—142no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—142no—
533C2×C4—141no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4C22×C4×D711yescharacteristic
543C2×C4#654811yescharacteristic
544C2×C4D7×#4111yescharacteristic
545C2×C4D7×#4111yescharacteristic
546C2×C4D7×#4111yescharacteristic
547C2×C4#655411yescharacteristic
548C8—22no—
549C8—41no—
550C8—142no—
551C8—281no—
552C7#2277911yescharacteristic, complemented, Sylow
553C22C4×D4×D711yescharacteristic
554C22#9030111yescharacteristic
555C22#9036811yescharacteristic
556C22#9046611yescharacteristic
557C22D7×#13711yescharacteristic
558C22#9046811yescharacteristic
559C22#9047011yescharacteristic
560C4—42no—
561C4—42no—
562C4—41no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—144no—
567C4—142no—
568C4—142no—
569C4—142no—
570C4—281no—
571C4—282no—
572C4—561no—
573C4—561no—
574C4D7×#11711yescharacteristic
575C4#9023711yescharacteristic
576C4#9035211yescharacteristic
577C4#9039911yescharacteristic
578C2#1173327911yescharacteristic
579C2#1173722111yescharacteristic
580C2#1173727411yescharacteristic
581C2#1173826911yescharacteristic
582C2#1173836011yescharacteristic
583C2#1173911611yescharacteristic
584C2#1173918011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1935C2
socle, prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#1935D7
hypocenterC7#22779
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1935

Complements

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

NH ≅ G/Ntype
43. #89561C4semidirect
552. C7#22779semidirect

Maximal subgroups

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

subgroupindex
C2×#895612
#117339352
#117334262
C7×#19352
#117280352
#117280072
#117279752
#117275672
#117275652
#117274442
#117273552
#117266592
#117266032
#117265732
#117265352
#227797