Finite Group Databasebeta

← #13387755

Subgroups — #13387755

Order 1792.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90270C211yesmaximal, characteristic
2C2×C4×C7×Q16C211yesmaximal, characteristic, complemented
3#11739945C211yesmaximal, characteristic
4C2×#89793C211yesmaximal, characteristic, complemented
5#11729825C212yesmaximal, complemented
6#11729380C211yesmaximal, characteristic, complemented
7#11728386C212yesmaximal, complemented
8#11728029C212yesmaximal, complemented
9#11728022C212yesmaximal, complemented
10#11727541C211yesmaximal, characteristic
11#11727531C211yesmaximal, characteristic, complemented
12C22×C7×Q16C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×Q8C2211yescharacteristic
15C2×#6509C2212yes—
16C2×#6509C2212yes—
17#90270—24no—
18#90270—24no—
19C2×#6459—22no—
20C2×D7×C4⋊C4C2212yes—
21C2×C4×D28—21no—
22C2×C4×D28C2211yescharacteristic
23C2×#6433—21no—
24C2×#6432C2211yescharacteristic
25C4×C7×Q16—24no—
26C2×C7×#107C2211yescharacteristic
27C2×C7×#103C2211yescharacteristic
28C2×C7×#103C2211yescharacteristic
29C2×C4×C7×C8C2211yescharacteristic
30C2×#6409C2212yes—
31C2×#6409C2212yes—
32C2×(C2×D7)⋊C8C2212yes—
33C2×(C2×D7)⋊C8C2212yes—
34#89793—24no—
35#89766—22no—
36#89759C2212yes—
37C2×#6381C2212yes—
38#89711—21no—
39#89707C2211yescharacteristic
40#89700—21no—
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42#89346C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#24122—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C22×C7×Q8C2311yescharacteristic
48C22×D28D411yescharacteristic
49C22×C4×D7—22no—
50C22×C4×D7C2312yes—
51C22×Dic14—21no—
52C2×C7×Q16—22no—
53C2×C7×Q16—22no—
54C2×C7×Q16—22no—
55C22×C7×C8C2311yescharacteristic
56C22×C7×C8C2311yescharacteristic
57C4×C7×Q8—22no—
58C4×C7×Q8—22no—
59C4×C7×Q8—22no—
60C4×C7×Q8D414yes—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C42×C7—21no—
66C2×C42×C7—21no—
67C2×C42×C7C2311yescharacteristic
68#6509—24no—
69#6509—24no—
70#6509—24no—
71#6509—42no—
72C2×(C2×D7)⋊C4—22no—
73C2×(C2×D7)⋊C4—22no—
74C2×(C2×D7)⋊C4—22no—
75C2×(C2×D7)⋊C4D412yes—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7D411yescharacteristic
80C2×C4⋊Dic7C2312yes—
81C2×C4⋊Dic7C2312yes—
82C2×Dic7⋊C4—22no—
83C2×Dic7⋊C4—22no—
84C2×Dic7⋊C4—22no—
85C22×C7⋊C8C2312yes—
86#6459—216no—
87D7×C4⋊C4—24no—
88D7×C4⋊C4—24no—
89C4×D28—28no—
90C4×D28—22no—
91C4×D28—22no—
92#6433—28no—
93#6432—22no—
94#6432D414yes—
95C7×#107—22no—
96C7×#103—22no—
97C7×#103—22no—
98C4×C7×C8—22no—
99#6409—24no—
100#6409—42no—
101C7⋊#95—22no—
102(C2×D7)⋊C8—24no—
103(C2×D7)⋊C8—42no—
104#6381—24no—
105C4⋊(C7⋊C8)—22no—
106C2×#554—71no—
107C2×#477—71no—
108C2×C4×Q16D711yescharacteristic, complemented
109#2538—71no—
110#2036—72no—
111#2001—72no—
112#1990—71no—
113#1954—71no—
114#1872—72no—
115#1852—72no—
116#1803—71no—
117C23×D7C2×D412yes—
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8C2×D412yes—
129C2×C7×Q8#4112yes—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2411yescharacteristic
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7C2×D412yes—
140C2×D28—22no—
141C2×D28—22no—
142C2×D28Q1614yescomplemented
143C2×C4×D7—22no—
144C2×C4×D7—22no—
145C2×C4×D7—22no—
146C2×C4×D7—22no—
147C2×C4×D7—22no—
148C2×C4×D7—22no—
149C2×C4×D7—28no—
150C2×C4×D7—82no—
151C2×C4×D7—82no—
152C2×Dic14—24no—
153C2×Dic14—81no—
154C2×Dic14—81no—
155C7×Q16—28no—
156C2×C7×C8—21no—
157C2×C7×C8—21no—
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8#4112yes—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—42no—
167C7×C4⋊C4C2×D412yes—
168C42×C7—24no—
169C42×C7—42no—
170C42×C7C2×D412yes—
171C42×C7C2×D412yes—
172(C2×D7)⋊C4—28no—
173(C2×D7)⋊C4—24no—
174(C2×D7)⋊C4—44no—
175(C2×D7)⋊C4—82no—
176C4⋊Dic7—24no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—22no—
181C4⋊Dic7—42no—
182C4⋊Dic7—81no—
183C4⋊Dic7C2×D414yes—
184Dic7⋊C4—28no—
185Dic7⋊C4—28no—
186Dic7⋊C4—44no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C22×Q16C2×D711yescharacteristic
191#554—144no—
192#554—144no—
193C2×C4⋊Q8—71no—
194C2×#125—141no—
195C2×C22⋊Q8—72no—
196C2×C22⋊Q8—72no—
197C2×C22⋊Q8—142no—
198C2×C4×Q8C2×D711yescharacteristic
199C2×C4×Q8C2×D711yescharacteristic
200C2×C4×D4—71no—
201C2×C4×D4—141no—
202C22×C4⋊C4—72no—
203#477—144no—
204C4×Q16—24no—
205C2×#107—72no—
206C2×#107C2×D711yescharacteristic
207C2×C4⋊C8—71no—
208C2×#103—72no—
209C2×#103—72no—
210C2×#103C2×D711yescharacteristic
211C2×#103C2×D711yescharacteristic
212C2×C22⋊C8—72no—
213C2×C22⋊C8—72no—
214C2×C4×C8C2×D711yescharacteristic
215#396—142no—
216#390—141no—
217#389—141no—
218#384—71no—
219#380—72no—
220#340—72no—
221#340—72no—
222#340—72no—
223C23×C7C22×D411yescharacteristic
224C22×D7—22no—
225C22×D7—22no—
226C22×D7—22no—
227C22×D7—22no—
228C22×D7—22no—
229C22×D7—22no—
230C22×D7C2×Q1614yes—
231C7×Q8—22no—
232C7×Q8—22no—
233C7×Q8—22no—
234C7×Q8—22no—
235C7×Q8—41no—
236C7×Q8—41no—
237C7×Q8—41no—
238C7×Q8—41no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—41no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7#12012yes—
254C2×C4×C7#12012yes—
255C2×C4×C7C2×Q1612yes—
256C2×C4×C7#13712yes—
257C2×C4×C7C22×D411yescharacteristic
258C2×C4×C7C22×D411yescharacteristic
259C2×C4×C7C2×#4111yescharacteristic
260C2×C4×C7C2×#4111yescharacteristic
261C2×C4×C7H(5,𝔽2)11yescharacteristic
262C2×C4×C72−1+411yescharacteristic
263C2×Dic7—22no—
264C2×Dic7—24no—
265C2×Dic7—22no—
266C2×Dic7—22no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272D28—28no—
273C4×D7—42no—
274C4×D7—42no—
275C4×D7—42no—
276C4×D7—42no—
277C4×D7—88no—
278Dic14—84no—
279C7×C8—22no—
280C7×C8—41no—
281C7⋊C8—42no—
282C22×Q8—141no—
283C22×Q8C22×D711yescharacteristic
284C22×Q8C22×D711yescharacteristic
285C22×D4—71no—
286C23×C4—72no—
287C23×C4—142no—
288C2×Q16—22no—
289C2×Q16—22no—
290C2×Q16—22no—
291C22×C8—72no—
292C22×C8C22×D711yescharacteristic
293C22×C8C22×D711yescharacteristic
294C4⋊Q8—74no—
295C4⋊Q8—142no—
296#125—148no—
297C22⋊Q8—1416no—
298C22⋊Q8—144no—
299C22⋊Q8—144no—
300C22⋊Q8—144no—
301C22⋊Q8—282no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8C22⋊D714yes—
306C4×D4—148no—
307C4×D4—142no—
308C4×D4—142no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4—71no—
313C2×C4⋊C4—72no—
314C2×C4⋊C4—144no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—144no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—141no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4C22×D711yescharacteristic
323C2×C4⋊C4C22×D711yescharacteristic
324C2×C22⋊C4—72no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4—142no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42C22×D711yescharacteristic
331#107—22no—
332#107—144no—
333C4⋊C8—142no—
334#103—22no—
335#103—22no—
336#103—144no—
337#103—282no—
338C22⋊C8—144no—
339C22⋊C8—282no—
340C4×C8—22no—
341#95—142no—
342C22×C7C2×#12011yescharacteristic
343C22×C7#54611yescharacteristic
344C22×C7#54711yescharacteristic
345C22×C7C22×Q1611yescharacteristic
346C22×C7C2×#13711yescharacteristic
347C22×C7#57611yescharacteristic
348C22×C7#57711yescharacteristic
349C2×D7—24no—
350C2×D7—24no—
351C2×D7—24no—
352C2×D7—42no—
353C2×D7—42no—
354C2×D7—42no—
355C2×D7—42no—
356C4×C7—22no—
357C4×C7—42no—
358C4×C7—42no—
359C4×C7—42no—
360C4×C7—81no—
361C4×C7#45114yes—
362C4×C7#52511yescharacteristic
363C4×C7#53711yescharacteristic
364C4×C7#54811yescharacteristic
365C4×C7#55011yescharacteristic
366Dic7—42no—
367Dic7—44no—
368Dic7—82no—
369C24—72no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—144no—
381C2×Q8—561no—
382C2×Q8—561no—
383C2×Q8#115712yes—
384C2×Q8C2×C22⋊D712yes—
385C2×D4—74no—
386C2×D4—142no—
387C2×D4—142no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—72no—
393C22×C4—148no—
394C22×C4—142no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4—142no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4—562no—
403C22×C4—562no—
404C22×C4D4×D711yescharacteristic
405C22×C4D4×D711yescharacteristic
406C22×C4C23×D711yescharacteristic
407Q16—28no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—142no—
414C2×C8—142no—
415C2×C8—142no—
416C2×C8#115712yes—
417C4⋊C4—24no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—42no—
422C4⋊C4—74no—
423C4⋊C4—148no—
424C4⋊C4—148no—
425C4⋊C4—142no—
426C4⋊C4—144no—
427C4⋊C4—142no—
428C4⋊C4—142no—
429C4⋊C4—142no—
430C4⋊C4—284no—
431C4⋊C4—282no—
432C4⋊C4—561no—
433C4⋊C4C2×C22⋊D712yes—
434C22⋊C4—148no—
435C22⋊C4—144no—
436C22⋊C4—284no—
437C22⋊C4—562no—
438C42—24no—
439C42—42no—
440C42D4×D712yes—
441C42C2×C22⋊D712yes—
442C2×C7#240711yescharacteristic
443C2×C7C2×#45111yescharacteristic
444C2×C7#301211yescharacteristic
445C2×C7#329911yescharacteristic
446C2×C7#333311yescharacteristic
447C2×C7#333811yescharacteristic
448C2×C7#335611yescharacteristic
449D7—44no—
450C23—74no—
451C23—142no—
452C23—142no—
453C23—142no—
454C23—142no—
455C23—142no—
456C23—142no—
457C23C2×D4×D711yescharacteristic
458Q8—22no—
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8—41no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—564no—
467D4—148no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—144no—
483C2×C4—142no—
484C2×C4—142no—
485C2×C4—142no—
486C2×C4—282no—
487C2×C4—282no—
488C2×C4—282no—
489C2×C4—282no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—282no—
493C2×C4—282no—
494C2×C4—282no—
495C2×C4—568no—
496C2×C4D7×Q1612yes—
497C2×C4#648112yes—
498C2×C4#650012yes—
499C2×C4#650012yes—
500C2×C4C2×#115711yescharacteristic
501C2×C4C2×#115711yescharacteristic
502C2×C4C2×D4×D711yescharacteristic
503C2×C4#655311yescharacteristic
504C2×C4#655411yescharacteristic
505C2×C4C22×C22⋊D711yescharacteristic
506C8—22no—
507C8—41no—
508C8—282no—
509C7#2412211yescharacteristic, complemented, Sylow
510C22—144no—
511C22—144no—
512C22—144no—
513C22—282no—
514C22—282no—
515C22—282no—
516C22—282no—
517C22#9024111yescharacteristic
518C22#9024211yescharacteristic
519C22C2×D7×Q1611yescharacteristic
520C22C2×#648111yescharacteristic
521C22#9046011yescharacteristic
522C22#9046111yescharacteristic
523C22C2×#650011yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—81no—
529C4—284no—
530C4—282no—
531C4—562no—
532C4#8995814yes—
533C4#9024311yescharacteristic
534C4#9024411yescharacteristic
535C4#9048211yescharacteristic
536C4#9052411yescharacteristic
537C2—284no—
538C2#1173718011yescharacteristic
539C2#1173718511yescharacteristic
540C2#1173718711yescharacteristic
541C2#1173719011yescharacteristic
542C2#1173984711yescharacteristic
543C2C2×#8995811yescharacteristic
544C2#1174135711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C4×C7×Q16C2
socleC23×C7C22×D4
prime radicalC22×D28D4
hypercenterC2×C4×Q16D7
hypocenterC7#24122
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×C4×C7×Q16C2semidirect
4. C2×#89793C2semidirect
5. #11729825C2semidirect
6. #11729380C2semidirect
7. #11728386C2semidirect
8. #11728029C2semidirect
9. #11728022C2semidirect
11. #11727531C2semidirect
108. C2×C4×Q16D7semidirect
142. C2×D28Q16semidirect
509. C7#24122semidirect

Maximal subgroups

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

subgroupindex
C2×#902702
C2×C4×C7×Q162
#117399452
C2×#897932
#117298252
#117293802
#117283862
#117280292
#117280222
#117275412
#117275312
#241227