Finite Group Databasebeta

← #13396333

Subgroups — #13396333

Order 1792.

858 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2929C211yesmaximal, characteristic
2C2×#89915C211yesmaximal, characteristic, complemented
3C2×#89888C211yesmaximal, characteristic, complemented
4C22×#6393C211yesmaximal, characteristic, complemented
5#11740370C212yesmaximal
6#11729770C211yesmaximal, characteristic, complemented
7#11728298C212yesmaximal, complemented
8#11727758C212yesmaximal, complemented
9#11727740C212yesmaximal, complemented
10#11727658C211yesmaximal, characteristic, complemented
11#11727648C211yesmaximal, characteristic, complemented
12C2×C7×#131C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C7×#525C2211yescharacteristic
15C2×#6514C2212yes—
16C2×#6487C2212yes—
17C22×#1136C2212yes—
18C22×Dic28C2211yescharacteristic
19C7×#435—21no—
20C7×#435—21no—
21C7×C8×D4—21no—
22C7×C8×D4—21no—
23C7×#424C2211yescharacteristic
24C7×#408C2211yescharacteristic
25C7×#407C2211yescharacteristic
26C7×#405C2211yescharacteristic
27#90004C2212yes—
28#89915—24no—
29C2×#6398C2212yes—
30C2×#6398C2212yes—
31#89888—21no—
32#89888—21no—
33#89888—21no—
34#89888—21no—
35C2×#6393—24no—
36C2×#6393C2211yescharacteristic
37C2×#6393C2212yes—
38C2×#6393C2211yescharacteristic
39C2×#6392C2211yescharacteristic
40C2×#6390C2212yes—
41#89824—22no—
42#89823—22no—
43#89814—22no—
44#89813—22no—
45#89776C2212yes—
46#89767C2212yes—
47#89349C2212yes—
48#89349C2212yes—
49#24804—71nomaximal, Sylow
50C2×C7×#41C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C22×Dic14C2312yes—
53C7×#131—22no—
54C7×#131—22no—
55C2×C7×M16—21no—
56C2×C7×M16C2311yescharacteristic
57C2×C7×M16C2311yescharacteristic
58C22×C7×C8—22no—
59C22×C7×C8—22no—
60C22×C7×C8—21no—
61C22×C7×C8D411yescharacteristic
62C22×C7×C8D411yescharacteristic
63C22×C7×C8D411yescharacteristic
64C22×C7×C8D411yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×#123—21no—
68C7×C22⋊Q8D411yescharacteristic
69C7×C22⋊Q8D411yescharacteristic
70C7×#121D411yescharacteristic
71C7×#121D411yescharacteristic
72C7×#120—21no—
73C4×C7×D4—21no—
74C4×C7×D4—21no—
75C7×#117C2311yescharacteristic
76C2×C22⋊Dic7—22no—
77#6514—24no—
78#6514—42no—
79C2×#1144—22no—
80C2×#1142—22no—
81C2×C4⋊Dic7C2312yes—
82#6487—22no—
83#6487—22no—
84#6487—22no—
85#6487—22no—
86C2×Dic7⋊C4—22no—
87C2×#1136—22no—
88C2×#1136—24no—
89C2×#1136—42no—
90C2×#1136C2312yes—
91C2×#1136C2312yes—
92C22×C7⋊C8—22no—
93C2×Dic28—21no—
94C2×Dic28—21no—
95C2×Dic28—22no—
96C2×Dic28—22no—
97C7×C4⋊C8—21no—
98C7×C4⋊C8—21no—
99C7×C22⋊C8—21no—
100C7×C22⋊C8—21no—
101C7×C22⋊C8—21no—
102C7×C22⋊C8—21no—
103C7×#97—21no—
104C4×C7×C8—21no—
105#6409—22no—
106#6406—22no—
107#6398—24no—
108#6398—42no—
109#6393—22no—
110#6393—24no—
111#6393D414yescomplemented
112#6392—21no—
113#6392—21no—
114#6390—22no—
115#6390—22no—
116#6384—22no—
117#6384—22no—
118#6382—22no—
119#6381—22no—
120C2×#473—71no—
121#3056—72no—
122C2×#467—71no—
123#2929D711yescharacteristic
124C22×#108—71no—
125#2064—72no—
126#2052—72no—
127#1979—71no—
128#1952—71no—
129#1901—72no—
130#1816—71no—
131C24×C7#4111yescharacteristic
132C7×#41—21no—
133C7×#41—21no—
134C7×#41—21no—
135C7×#41—21no—
136C2×C7×Q8C2×D411yescharacteristic
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4C2×D411yescharacteristic
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7#4111yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—22no—
153C22×Dic7—22no—
154C2×Dic14—22no—
155C2×Dic14—22no—
156C2×Dic14—22no—
157C2×Dic14—22no—
158C2×Dic14—22no—
159C2×Dic14—22no—
160C7×M16—21no—
161C7×M16—21no—
162C7×M16—21no—
163C7×M16—21no—
164C7×M16—42no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—42no—
174C2×C7×C8C2×D411yescharacteristic
175C2×C7×C8C2×D411yescharacteristic
176C2×C7×C8C2×D412yes—
177C2×C7×C8C2×D411yescharacteristic
178C2×C7×C8C2×D411yescharacteristic
179C2×C7×C8#4112yes—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4C2×D411yescharacteristic
183C7×C4⋊C4C2×D411yescharacteristic
184C7×C22⋊C4—21no—
185C7×C22⋊C4—21no—
186C7×C22⋊C4—21no—
187C7×C22⋊C4—21no—
188C7×C22⋊C4—21no—
189C42×C7—21no—
190C22⋊Dic7—42no—
191C22⋊Dic7—42no—
192#1144—44no—
193#1142—44no—
194C4⋊Dic7—22no—
195C4⋊Dic7—22no—
196Dic7⋊C4—42no—
197Dic7⋊C4—42no—
198#1136—22no—
199#1136—22no—
200#1136—22no—
201#1136—44no—
202#1136C2×D414yes—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—44no—
206Dic28—42no—
207Dic28—42no—
208C2×#137—72no—
209C22×Q16—71no—
210C2×#131C2×D711yescharacteristic
211C22×M16—72no—
212C23×C8C2×D711yescharacteristic
213#525C2×D711yescharacteristic
214C2×C22⋊Q8—72no—
215#473—144no—
216#470—142no—
217#468—142no—
218#467—141no—
219#467—141no—
220#467—141no—
221#467—141no—
222#467—142no—
223#465—142no—
224#435—21no—
225#435—21no—
226C8×D4—21no—
227C8×D4—21no—
228C2×#108—71no—
229C2×#108—72no—
230C2×#108—71no—
231C2×#108—144no—
232#428—72no—
233C2×#107—71no—
234#424C2×D711yescharacteristic
235#419—72no—
236#417—72no—
237C2×#103—72no—
238C2×#101—72no—
239C2×#101—72no—
240#408C2×D711yescharacteristic
241#407C2×D711yescharacteristic
242#405C2×D711yescharacteristic
243#343—72no—
244#343—72no—
245C23×C7—21no—
246C23×C7—21no—
247C23×C7—22no—
248C23×C7—21no—
249C23×C7—41no—
250C23×C7#12111yescharacteristic
251C23×C7#12111yescharacteristic
252C23×C7C2×#4111yescharacteristic
253C7×Q8—21no—
254C7×Q8—21no—
255C7×D4—21no—
256C7×D4—21no—
257C7×D4—42no—
258C7×D4—41no—
259C7×D4—41no—
260C7×D4—42no—
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—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—41no—
275C2×C4×C7—41no—
276C2×C4×C7#12111yescharacteristic
277C2×C4×C7#12111yescharacteristic
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—82no—
289Dic14—42no—
290Dic14—42no—
291Dic14—42no—
292Dic14—42no—
293C7×C8—21no—
294C7×C8—21no—
295C7×C8—41no—
296C7×C8—41no—
297C7×C8#12112yes—
298C7×C8#12112yes—
299C7⋊C8—24no—
300C7⋊C8—42no—
301C2×#41C22×D711yescharacteristic
302C22×Q8—72no—
303C23×C4C22×D711yescharacteristic
304#137—144no—
305#137—282no—
306C2×Q16—142no—
307C2×Q16—142no—
308C2×Q16—142no—
309C2×Q16—141no—
310C2×Q16—141no—
311C2×SD16—142no—
312#131—22no—
313#131—22no—
314C2×M16—21no—
315C2×M16—72no—
316C2×M16—72no—
317C2×M16—142no—
318C2×M16—144no—
319C2×M16—282no—
320C2×M16C22×D711yescharacteristic
321C2×M16C22×D711yescharacteristic
322C22×C8—22no—
323C22×C8—22no—
324C22×C8—21no—
325C22×C8—142no—
326C22×C8C22⋊D711yescharacteristic
327C22×C8C22⋊D711yescharacteristic
328C22×C8C22⋊D711yescharacteristic
329C22×C8C22⋊D711yescharacteristic
330C22×C8C22×D711yescharacteristic
331C22×C8C22×D711yescharacteristic
332#123—21no—
333C22⋊Q8—142no—
334C22⋊Q8—142no—
335C22⋊Q8—142no—
336C22⋊Q8—142no—
337C22⋊Q8D2811yescharacteristic
338C22⋊Q8D2811yescharacteristic
339#121D2811yescharacteristic
340#121D2811yescharacteristic
341#120—21no—
342C4×D4—21no—
343C4×D4—21no—
344#117C22×D711yescharacteristic
345C2×C4⋊C4—72no—
346C2×C4⋊C4—142no—
347C2×C22⋊C4—142no—
348#108—74no—
349#108—144no—
350#108—142no—
351#107—141no—
352#107—141no—
353#107—142no—
354#106—142no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357#103—142no—
358#103—142no—
359#103—142no—
360#103—142no—
361#103—142no—
362#102—142no—
363#101—144no—
364#101—282no—
365C22⋊C8—21no—
366C22⋊C8—21no—
367C22⋊C8—21no—
368C22⋊C8—21no—
369#97—21no—
370C4×C8—21no—
371C22×C7—21no—
372C22×C7—21no—
373C22×C7—21no—
374C22×C7—21no—
375C22×C7—21no—
376C22×C7—21no—
377C22×C7—21no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—41no—
381C22×C7—41no—
382C22×C7—41no—
383C22×C7#47312yescomplemented
384C22×C7C2×#12111yescharacteristic
385C22×C7#54811yescharacteristic
386C22×C7#55311yescharacteristic
387C4×C7—21no—
388C4×C7—21no—
389C4×C7—41no—
390C4×C7—41no—
391C4×C7—41no—
392C4×C7C2×#12111yescharacteristic
393C4×C7#53711yescharacteristic
394C4×C7D4211yescharacteristic
395C4×C7#54711yescharacteristic
396Dic7—82no—
397Dic7—82no—
398C24#115711yescharacteristic
399#41—21no—
400#41—21no—
401#41—21no—
402#41—21no—
403C2×Q8—142no—
404C2×Q8—142no—
405C2×Q8—142no—
406C2×Q8—142no—
407C2×Q8—142no—
408C2×Q8—142no—
409C2×Q8C2×D2811yescharacteristic
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C2×D2811yescharacteristic
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—142no—
420C22×C4—142no—
421C22×C4C2×D2811yescharacteristic
422C22×C4C2×D2811yescharacteristic
423C22×C4#115711yescharacteristic
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C23×D711yescharacteristic
427Q16—284no—
428Q16—282no—
429Q16—282no—
430SD16—284no—
431M16—21no—
432M16—21no—
433M16—21no—
434M16—21no—
435M16—42no—
436M16—74no—
437M16—142no—
438M16—142no—
439M16—142no—
440M16—284no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—42no—
450C2×C8—142no—
451C2×C8—142no—
452C2×C8—284no—
453C2×C8D4×D712yes—
454C2×C8#115912yes—
455C2×C8C2×C22⋊D711yescharacteristic
456C2×C8C2×C22⋊D711yescharacteristic
457C2×C8C2×C22⋊D711yescharacteristic
458C2×C8C2×C22⋊D711yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—142no—
462C4⋊C4—142no—
463C4⋊C4—282no—
464C4⋊C4—282no—
465C4⋊C4C2×D2811yescharacteristic
466C4⋊C4C2×D2811yescharacteristic
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—282no—
473C22⋊C4—282no—
474C42—21no—
475C2×C7—21no—
476C2×C7—21no—
477C2×C7—22no—
478C2×C7—41no—
479C2×C7—81no—
480C2×C7#241611yescharacteristic
481C2×C7C2×#47311yescharacteristic
482C2×C7#306711yescharacteristic
483C23—22no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—41no—
488C23#649311yescharacteristic
489C23#649311yescharacteristic
490C23C2×#115711yescharacteristic
491Q8—21no—
492Q8—21no—
493Q8—282no—
494Q8—282no—
495Q8—282no—
496Q8—282no—
497D4—21no—
498D4—21no—
499D4—42no—
500D4—42no—
501D4—41no—
502D4—41no—
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—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4—282no—
521C2×C4—282no—
522C2×C4—562no—
523C2×C4#649311yescharacteristic
524C2×C4#649311yescharacteristic
525C2×C4C22×D2811yescharacteristic
526C2×C4C2×D4×D711yescharacteristic
527C2×C4C2×#115911yescharacteristic
528C2×C4#655311yescharacteristic
529C2×C4#655411yescharacteristic
530C2×C4C22×C22⋊D711yescharacteristic
531C8—21no—
532C8—21no—
533C8—41no—
534C8—41no—
535C8—144no—
536C8—282no—
537C8#650212yes—
538C8#650212yes—
539C7#2480411yescharacteristic, complemented, Sylow
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22#8991512yescomplemented
553C22#9024311yescharacteristic
554C22#9025611yescharacteristic
555C22C2×#649311yescharacteristic
556C4—21no—
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—562no—
562C4—562no—
563C4D4×D2811yescharacteristic
564C4#9024611yescharacteristic
565C4C2×#650211yescharacteristic
566C4#9052411yescharacteristic
567C2—22no—
568C2—21no—
569C2—21no—
570C2—41no—
571C2—81no—
572C2#1173979711yescharacteristic
573C2#1174125011yescharacteristic
574C2C2×#8991511yescharacteristic
subgroupG/H
centerC22C2×#6493
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2929C2
socleC22×C7C2×#121
prime radicalC7×#525C22
hypercenter#2929D7
hypocenterC7#24804
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. C2×#89915C2semidirect
3. C2×#89888C2semidirect
4. C22×#6393C2semidirect
6. #11729770C2semidirect
7. #11728298C2semidirect
8. #11727758C2semidirect
9. #11727740C2semidirect
10. #11727658C2semidirect
11. #11727648C2semidirect
111. #6393D4semidirect
383. C22×C7#473semidirect
539. C7#24804semidirect
552. C22#89915semidirect

Maximal subgroups

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

subgroupindex
C7×#29292
C2×#899152
C2×#898882
C22×#63932
#117403702
#117297702
#117282982
#117277582
#117277402
#117276582
#117276482
#248047