Finite Group Databasebeta

← #13403927

Subgroups — #13403927

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89622C211yesmaximal, characteristic, complemented
2C2×#89606C211yesmaximal, characteristic
3#11731814C211yesmaximal, characteristic
4#11731252C211yesmaximal, characteristic
5C7×#1947C211yesmaximal, characteristic, complemented
6#11728389C212yesmaximal, complemented
7#11728384C212yesmaximal, complemented
8#11727336C211yesmaximal, characteristic, complemented
9C4×#6384C211yesmaximal, characteristic, complemented
10#11726636C212yesmaximal, complemented
11#11726585C212yesmaximal, complemented
12C22×C8×D7C2212yes—
13C2×#6460C2212yes—
14C2×C4×D28C2211yescharacteristic
15C2×#6432C2211yescharacteristic
16C2×C4×Dic14C2211yescharacteristic
17C2×C7×#107C2212yes—
18C2×C7×C4⋊C8C2211yescharacteristic
19C7×#384C2211yescharacteristic
20C4×C7×C4⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C8C2212yes—
22C2×#6392C2212yes—
23C2×#6390C2212yes—
24#89766—22no—
25#89760—21no—
26#89759C2212yes—
27#89759C2212yes—
28#89759C2212yes—
29#89757—21no—
30C2×#6384C2211yescharacteristic
31C2×#6384C2211yescharacteristic
32#89711—21no—
33#89708—21no—
34C2×C4×C7⋊C8C2211yescharacteristic
35#89622—24no—
36#89606—24no—
37#89445—22no—
38C7×#340C2212yes—
39#89324C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89324C2211yescharacteristic
43#25340—71nomaximal, Sylow
44C22×D28D411yescharacteristic
45C22×C4×D7C2312yes—
46C22×Dic14—21no—
47C22×Dic14C2311yescharacteristic
48C22×C7×C8C2312yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C7×C4⋊C4C2312yes—
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C42×C7—21no—
55C2×C42×C7C2311yescharacteristic
56C2×(C2×D7)⋊C4—22no—
57C2×(C2×D7)⋊C4—22no—
58C2×(C2×D7)⋊C4—22no—
59C2×(C2×D7)⋊C4—22no—
60C2×(C2×D7)⋊C4D412yes—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7—22no—
63C2×C4⋊Dic7D411yescharacteristic
64C2×C4⋊Dic7C2312yes—
65C2×C4⋊Dic7C2311yescharacteristic
66C2×Dic7⋊C4—21no—
67C2×C4×Dic7—21no—
68C22×C7⋊C8C2311yescharacteristic
69C22×C7⋊C8C2311yescharacteristic
70C2×C8×D7—24no—
71C2×C8×D7—24no—
72C2×C8×D7—42no—
73#6460—24no—
74#6460—42no—
75C4×D28—22no—
76C4×D28—22no—
77#6432—22no—
78#6432—22no—
79C4×Dic14—22no—
80C4×Dic14—22no—
81C7×#107—42no—
82C7×C4⋊C8D414yes—
83C7×#95—21no—
84C7⋊#95—22no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87(C2×D7)⋊C8—24no—
88#6392—24no—
89#6390—24no—
90#6384—22no—
91#6384—22no—
92C4×C7⋊C8—22no—
93C2×#477—71no—
94C2×C8×D4—71no—
95#2563—71no—
96#2538—71no—
97#2076—72no—
98#2036—72no—
99#1990—71no—
100#1947D711yescharacteristic, complemented
101#1892—72no—
102#1808—72no—
103C4×#103—71no—
104C23×D7C2×D412yes—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—22no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2411yescharacteristic
111C22×Dic7—21no—
112C22×Dic7—21no—
113C22×Dic7—22no—
114C22×Dic7C2×D412yes—
115C2×D28—22no—
116C2×D28—22no—
117C2×D28Q1614yes—
118C2×C4×D7—22no—
119C2×C4×D7—22no—
120C2×C4×D7—22no—
121C2×C4×D7—22no—
122C2×C4×D7—22no—
123C2×C4×D7—22no—
124C2×Dic14—21no—
125C2×Dic14—21no—
126C2×Dic14—21no—
127C2×Dic14—21no—
128C2×Dic14—21no—
129C2×Dic14—42no—
130C2×Dic14—81no—
131C2×Dic14—81no—
132C2×Dic14#4112yes—
133C2×C7×C8—22no—
134C2×C7×C8—22no—
135C2×C7×C8C2×D414yes—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—21no—
142C7×C4⋊C4—42no—
143C7×C4⋊C4—81no—
144C42×C7—42no—
145C42×C7C2×D412yes—
146C42×C7#4112yes—
147(C2×D7)⋊C4—24no—
148(C2×D7)⋊C4—44no—
149(C2×D7)⋊C4—82no—
150(C2×D7)⋊C4—82no—
151(C2×D7)⋊C4—82no—
152C4⋊Dic7—22no—
153C4⋊Dic7—21no—
154C4⋊Dic7—22no—
155C4⋊Dic7—21no—
156C4⋊Dic7—22no—
157C4⋊Dic7—44no—
158C4⋊Dic7—42no—
159Dic7⋊C4—42no—
160C4×Dic7—42no—
161C2×C7⋊C8—21no—
162C2×C7⋊C8—21no—
163C2×C7⋊C8—21no—
164C2×C7⋊C8—21no—
165C2×C7⋊C8—21no—
166C2×C7⋊C8#4112yes—
167C8×D7—48no—
168C23×C8—72no—
169C2×C4⋊Q8—71no—
170C2×C22⋊Q8—72no—
171C2×C4×Q8—71no—
172C2×C4×D4—71no—
173#477—144no—
174C8×D4—144no—
175C2×#107—72no—
176C2×#107C2×D712yes—
177C2×C4⋊C8C2×D711yescharacteristic
178C2×#103—72no—
179C2×#103—71no—
180C2×#103—71no—
181C2×C22⋊C8—72no—
182C2×C4×C8—71no—
183#396—142no—
184#393—142no—
185#390—141no—
186#390—141no—
187#389—141no—
188#384C2×D711yescharacteristic
189#382—141no—
190#380—72no—
191#380—72no—
192#380—72no—
193C4×C4⋊C4C2×D711yescharacteristic
194#340—71no—
195#340—72no—
196#340—71no—
197#340—72no—
198#340C2×D712yes—
199C23×C7C22×D411yescharacteristic
200C22×D7—22no—
201C22×D7—22no—
202C22×D7—22no—
203C22×D7—22no—
204C22×D7—22no—
205C22×D7—22no—
206C22×D7C2×Q1614yes—
207C2×C4×C7—21no—
208C2×C4×C7—21no—
209C2×C4×C7—42no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—42no—
215C2×C4×C7—41no—
216C2×C4×C7—42no—
217C2×C4×C7—41no—
218C2×C4×C7#12112yes—
219C2×C4×C7#12112yes—
220C2×C4×C7C2×Q1612yes—
221C2×C4×C7#13512yes—
222C2×C4×C7C22×D411yescharacteristic
223C2×C4×C7C2×#4111yescharacteristic
224C2×C4×C7C2×#4111yescharacteristic
225C2×C4×C7C2×#4111yescharacteristic
226C2×C4×C7H(5,𝔽2)11yescharacteristic
227C2×C4×C72−1+411yescharacteristic
228C2×Dic7—22no—
229C2×Dic7—22no—
230C2×Dic7—22no—
231C2×Dic7—22no—
232C2×Dic7—41no—
233C2×Dic7—41no—
234C2×Dic7—41no—
235C2×Dic7—42no—
236C2×Dic7—41no—
237C2×Dic7—42no—
238C2×Dic7—42no—
239C2×Dic7—41no—
240D28—28no—
241C4×D7—42no—
242C4×D7—42no—
243C4×D7—42no—
244C4×D7—42no—
245Dic14—22no—
246Dic14—22no—
247Dic14—41no—
248Dic14—41no—
249Dic14—84no—
250C7×C8—24no—
251C7⋊C8—22no—
252C7⋊C8—41no—
253C22×Q8—71no—
254C22×Q8—141no—
255C22×D4—71no—
256C23×C4—72no—
257C22×C8—71no—
258C22×C8—71no—
259C22×C8—144no—
260C22×C8—144no—
261C22×C8—282no—
262C22×C8C22×D712yes—
263C4⋊Q8—142no—
264C4⋊Q8—142no—
265C22⋊Q8—144no—
266C22⋊Q8—282no—
267C4×Q8—142no—
268C4×Q8—142no—
269C4×D4—142no—
270C4×D4—142no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—71no—
274C2×C4⋊C4—71no—
275C2×C4⋊C4—72no—
276C2×C4⋊C4—142no—
277C2×C4⋊C4—141no—
278C2×C4⋊C4—141no—
279C2×C4⋊C4C22×D712yes—
280C2×C4⋊C4C22×D711yescharacteristic
281C2×C4⋊C4C22×D711yescharacteristic
282C2×C22⋊C4—72no—
283C2×C22⋊C4—142no—
284C2×C22⋊C4—142no—
285C2×C22⋊C4—142no—
286C2×C22⋊C4—142no—
287C2×C42—21no—
288C2×C42—141no—
289C2×C42C22×D711yescharacteristic
290#107—42no—
291#107—144no—
292C4⋊C8D2814yes—
293#103—142no—
294#103—142no—
295#103—144no—
296C22⋊C8—144no—
297C4×C8—142no—
298#95—21no—
299#95—142no—
300#95—142no—
301#95—142no—
302C22×C7C2×#12111yescharacteristic
303C22×C7#54611yescharacteristic
304C22×C7#54711yescharacteristic
305C22×C7C22×Q1611yescharacteristic
306C22×C7C2×#13511yescharacteristic
307C22×C7#57611yescharacteristic
308C22×C7#57711yescharacteristic
309C2×D7—24no—
310C2×D7—24no—
311C2×D7—24no—
312C2×D7—42no—
313C2×D7—42no—
314C2×D7—42no—
315C2×D7—42no—
316C4×C7—22no—
317C4×C7—81no—
318C4×C7—82no—
319C4×C7—81no—
320C4×C7#46714yes—
321C4×C7#52911yescharacteristic
322C4×C7#54211yescharacteristic
323C4×C7#54611yescharacteristic
324C4×C7#54811yescharacteristic
325Dic7—42no—
326Dic7—42no—
327Dic7—82no—
328Dic7—81no—
329C24—72no—
330C2×Q8—72no—
331C2×Q8—141no—
332C2×Q8—141no—
333C2×Q8—141no—
334C2×Q8—141no—
335C2×Q8—141no—
336C2×Q8—282no—
337C2×Q8—561no—
338C2×Q8—561no—
339C2×D4—74no—
340C2×D4—142no—
341C2×D4—142no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—72no—
346C22×C4—141no—
347C22×C4—142no—
348C22×C4—142no—
349C22×C4—142no—
350C22×C4—142no—
351C22×C4—142no—
352C22×C4—142no—
353C22×C4—141no—
354C22×C4—142no—
355C22×C4D4×D711yescharacteristic
356C22×C4D4×D711yescharacteristic
357C22×C4C23×D711yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—72no—
361C2×C8—141no—
362C2×C8—141no—
363C2×C8—141no—
364C2×C8—141no—
365C2×C8—141no—
366C2×C8—288no—
367C2×C8C2×D2814yes—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—81no—
376C4⋊C4—142no—
377C4⋊C4—142no—
378C4⋊C4—141no—
379C4⋊C4—141no—
380C4⋊C4—142no—
381C4⋊C4—284no—
382C4⋊C4—282no—
383C4⋊C4—282no—
384C22⋊C4—144no—
385C22⋊C4—284no—
386C22⋊C4—562no—
387C22⋊C4—562no—
388C22⋊C4—562no—
389C42—42no—
390C42—282no—
391C42C2×D2812yes—
392C42#116112yes—
393C2×C7#242711yescharacteristic
394C2×C7C2×#46711yescharacteristic
395C2×C7#306211yescharacteristic
396C2×C7#329911yescharacteristic
397C2×C7#330511yescharacteristic
398C2×C7#333811yescharacteristic
399C2×C7#334011yescharacteristic
400D7—44no—
401C23—74no—
402C23—142no—
403C23—142no—
404C23—142no—
405C23—142no—
406C23—142no—
407C23—142no—
408C23C2×D4×D711yescharacteristic
409Q8—142no—
410Q8—142no—
411Q8—281no—
412Q8—281no—
413Q8—564no—
414D4—148no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—142no—
427C2×C4—142no—
428C2×C4—142no—
429C2×C4—142no—
430C2×C4—281no—
431C2×C4—281no—
432C2×C4—282no—
433C2×C4—282no—
434C2×C4—282no—
435C2×C4—282no—
436C2×C4—281no—
437C2×C4—282no—
438C2×C4—281no—
439C2×C4—282no—
440C2×C4—282no—
441C2×C4—281no—
442C2×C4#645812yes—
443C2×C4#645812yes—
444C2×C4#647512yes—
445C2×C4D7×Q1612yes—
446C2×C4C22×D2811yescharacteristic
447C2×C4#654811yescharacteristic
448C2×C4C2×#116111yescharacteristic
449C2×C4#655111yescharacteristic
450C2×C4D7×#4111yescharacteristic
451C2×C4D7×#4111yescharacteristic
452C8—24no—
453C8—142no—
454C8—281no—
455C7#2534011yescharacteristic, complemented, Sylow
456C22—144no—
457C22—144no—
458C22—144no—
459C22—282no—
460C22—282no—
461C22—282no—
462C22—282no—
463C22C2×#645811yescharacteristic
464C22#9030311yescharacteristic
465C22#9032211yescharacteristic
466C22C2×#647511yescharacteristic
467C22C2×D7×Q1611yescharacteristic
468C22#9046511yescharacteristic
469C22#9046911yescharacteristic
470C4—22no—
471C4—81no—
472C4—81no—
473C4—82no—
474C4—282no—
475C4—282no—
476C4—562no—
477C4—561no—
478C4#8965914yes—
479C4#9024511yescharacteristic
480C4#9026711yescharacteristic
481C4#9038211yescharacteristic
482C4#9042411yescharacteristic
483C2—284no—
484C2#1173366411yescharacteristic
485C2C2×#8965911yescharacteristic
486C2#1173665811yescharacteristic
487C2#1173779811yescharacteristic
488C2#1173793611yescharacteristic
489C2#1173843211yescharacteristic
490C2#1173857011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1947C2
socleC23×C7C22×D4
prime radicalC22×D28D4
hypercenter#1947D7
hypocenterC7#25340
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
1. C2×#89622C2semidirect
5. C7×#1947C2semidirect
6. #11728389C2semidirect
7. #11728384C2semidirect
8. #11727336C2semidirect
9. C4×#6384C2semidirect
10. #11726636C2semidirect
11. #11726585C2semidirect
100. #1947D7semidirect
455. C7#25340semidirect

Maximal subgroups

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

subgroupindex
C2×#896222
C2×#896062
#117318142
#117312522
C7×#19472
#117283892
#117283842
#117273362
C4×#63842
#117266362
#117265852
#253407