Finite Group Databasebeta

← #13453640

Subgroups — #13453640

Order 1792.

770 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740697C211yesmaximal, characteristic, complemented
2#11733737C211yesmaximal, characteristic
3C7×#2023C211yesmaximal, characteristic
4#11729548C211yesmaximal, characteristic, complemented
5#11727912C212yesmaximal
6#11727908C212yesmaximal
7#11727392C211yesmaximal, characteristic
8#11726659C212yesmaximal, complemented
9#11726602C212yesmaximal, complemented
10#11726443C211yesmaximal, characteristic, complemented
11#11726356C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×#6499C2211yescharacteristic
14C2×D4×Dic7C2211yescharacteristic
15C2×#6450C2211yescharacteristic
16C2×#6432C2211yescharacteristic
17C2×C7×#102C2212yes—
18C7×#390C2211yescharacteristic
19#90002—21no—
20#90002—21no—
21C2×#6406C2212yes—
22C2×#6390C2212yes—
23C2×Dic7⋊C8C2212yes—
24C2×C8×Dic7C2212yes—
25#89749C2211yescharacteristic
26#89749C2212yes—
27#89749C2212yes—
28#89746—21no—
29#89746—21no—
30C2×#6384C2212yes—
31#89716—21no—
32#89716—21no—
33C22⋊C4×Dic7—21no—
34C22⋊C4×Dic7—21no—
35C42×Dic7C2211yescharacteristic
36#89701C2211yescharacteristic
37C7×#340C2212yes—
38C7×#336C2211yescharacteristic
39#89346C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#89319C2211yescharacteristic
43#29142—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×C7×D4C2311yescharacteristic
46C23×Dic7—21no—
47C22×Dic14—21no—
48C22×Dic14C2311yescharacteristic
49C22×C7×C8C2312yes—
50C7×C4⋊Q8—42no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4C2311yescharacteristic
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—21no—
56C2×C42×C7C2311yescharacteristic
57C2×C22⋊Dic7—21no—
58C2×C22⋊Dic7—21no—
59C2×C22⋊Dic7—21no—
60#6499—22no—
61#6499Q814yescomplemented
62D4×Dic7—24no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7C2311yescharacteristic
65C2×C4⋊Dic7C2312yes—
66C2×Dic7⋊C4—21no—
67C2×C4×Dic7—21no—
68C2×C4×Dic7D411yescharacteristic
69C2×C4×Dic7D411yescharacteristic
70C2×C4×Dic7D411yescharacteristic
71C2×C4×Dic7D411yescharacteristic
72C2×C4×Dic7C2311yescharacteristic
73C2×C4×Dic7C2311yescharacteristic
74C22×C7⋊C8C2312yes—
75#6450—24no—
76#6432—42no—
77C7×#102—42no—
78#6406—42no—
79C7⋊#95—21no—
80C7⋊#95—21no—
81C7⋊#95—21no—
82C7⋊#95—21no—
83C7⋊#95—22no—
84C7⋊#95—22no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87#6390—42no—
88Dic7⋊C8—24no—
89C8×Dic7—24no—
90#6384—42no—
91#2862—71no—
92C4×#124—71no—
93#2024—71no—
94#2024—71no—
95#2023—71no—
96#2023D711yescharacteristic
97#1958—72no—
98#1835—72no—
99#1805—72no—
100#1804—72no—
101C4×C4⋊C8—71no—
102C24×C7—21no—
103C2×C7×Q8—42no—
104C2×C7×Q8—42no—
105C2×C7×Q8—42no—
106C2×C7×D4—22no—
107C2×C7×D4—21no—
108C2×C7×D4—22no—
109C2×C7×D4C2×Q812yes—
110C22×C4×C7—22no—
111C22×C4×C7—21no—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2411yescharacteristic
115C22×Dic7—21no—
116C22×Dic7—22no—
117C22×Dic7—21no—
118C22×Dic7—42no—
119C22×Dic7—42no—
120C22×Dic7—42no—
121C22×Dic7C2×D411yescharacteristic
122C22×Dic7C2×D411yescharacteristic
123C22×Dic7C2×D411yescharacteristic
124C22×Dic7C2×D411yescharacteristic
125C2×Dic14—22no—
126C2×Dic14—22no—
127C2×Dic14—21no—
128C2×Dic14—42no—
129C2×Dic14—42no—
130C2×Dic14—42no—
131C2×Dic14C2×Q812yes—
132C2×C7×C8—22no—
133C2×C7×C8—22no—
134C2×C7×C8—22no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—42no—
139C7×C22⋊C4—42no—
140C7×C22⋊C4—42no—
141C42×C7—22no—
142C22⋊Dic7—24no—
143C22⋊Dic7—24no—
144C22⋊Dic7—42no—
145C4⋊Dic7—22no—
146C4⋊Dic7—22no—
147C4⋊Dic7—22no—
148C4⋊Dic7—42no—
149Dic7⋊C4—42no—
150C4×Dic7—22no—
151C4×Dic7—22no—
152C4×Dic7—41no—
153C4×Dic7—41no—
154C4×Dic7—42no—
155C4×Dic7C2×Q812yes—
156C4×Dic7#4112yes—
157C4×Dic7#4114yes—
158C2×C7⋊C8—22no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8—22no—
161C2×C4⋊Q8—71no—
162C2×C4⋊Q8C2×D711yescharacteristic
163C2×#124—71no—
164C2×C4×Q8—71no—
165C2×C4×D4—71no—
166C2×C4⋊C8—72no—
167C2×#103—72no—
168C2×#103—72no—
169C2×#102—72no—
170C2×#102C2×D712yes—
171C2×C4×C8—72no—
172#391—71no—
173#391—141no—
174#391—141no—
175#391—141no—
176#391—141no—
177#390C2×D711yescharacteristic
178#388—141no—
179#388—141no—
180#382—72no—
181#382—71no—
182#382—72no—
183C4×C22⋊C4—141no—
184C4×C22⋊C4—141no—
185C43—71no—
186#340—72no—
187#340—72no—
188#340—72no—
189#340C2×D712yes—
190#336—71no—
191#336C2×D711yescharacteristic
192C23×C7—22no—
193C23×C7—41no—
194C23×C7—41no—
195C23×C7—41no—
196C23×C7—41no—
197C23×C7—41no—
198C23×C7—41no—
199C23×C7C22×D411yescharacteristic
200C7×Q8—84no—
201C7×D4—24no—
202C7×D4—42no—
203C2×C4×C7—21no—
204C2×C4×C7—21no—
205C2×C4×C7—21no—
206C2×C4×C7—21no—
207C2×C4×C7—41no—
208C2×C4×C7—42no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7#13512yes—
214C2×C4×C7#13512yes—
215C2×C4×C7C22×Q811yescharacteristic
216C2×C4×C7C2×#4112yes—
217C2×C4×C7C2×#4111yescharacteristic
218C2×C4×C7H(5,𝔽2)12yes—
219C2×Dic7—21no—
220C2×Dic7—21no—
221C2×Dic7—21no—
222C2×Dic7—21no—
223C2×Dic7—21no—
224C2×Dic7—21no—
225C2×Dic7—21no—
226C2×Dic7—21no—
227C2×Dic7—22no—
228C2×Dic7—41no—
229C2×Dic7—41no—
230C2×Dic7—41no—
231C2×Dic7—42no—
232C2×Dic7—42no—
233C2×Dic7—42no—
234C2×Dic7—41no—
235C2×Dic7—41no—
236C2×Dic7—84no—
237C2×Dic7C22⋊Q812yes—
238C2×Dic7C22⋊Q812yes—
239C2×Dic7#12412yes—
240C2×Dic7#12412yes—
241Dic14—24no—
242Dic14—42no—
243Dic14—84no—
244C7×C8—42no—
245C7⋊C8—42no—
246C22×Q8—21no—
247C22×Q8—71no—
248C22×Q8—141no—
249C22×D4C22×D711yescharacteristic
250C23×C4—141no—
251C22×C8—72no—
252C22×C8C22×D712yes—
253C4⋊Q8—42no—
254C4⋊Q8—282no—
255#124—74no—
256#124—142no—
257C4×Q8—144no—
258C4×D4—144no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—71no—
261C2×C4⋊C4—72no—
262C2×C4⋊C4—141no—
263C2×C4⋊C4—141no—
264C2×C4⋊C4C22×D711yescharacteristic
265C2×C4⋊C4C22×D712yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—141no—
269C2×C22⋊C4—141no—
270C2×C22⋊C4—141no—
271C2×C42—71no—
272C2×C42—71no—
273C2×C42—71no—
274C2×C42—71no—
275C2×C42—71no—
276C2×C42—71no—
277C2×C42—141no—
278C2×C42C22×D711yescharacteristic
279C4⋊C8—144no—
280#103—282no—
281#103—282no—
282#102—42no—
283#102—282no—
284C4×C8—144no—
285#95—141no—
286#95—141no—
287#95—142no—
288#95—142no—
289#95—142no—
290#95—142no—
291#95—141no—
292#95—141no—
293C22×C7—42no—
294C22×C7—42no—
295C22×C7—42no—
296C22×C7—81no—
297C22×C7—81no—
298C22×C7—81no—
299C22×C7—81no—
300C22×C7C2×C22⋊Q811yescharacteristic
301C22×C7C2×#12411yescharacteristic
302C22×C7#53511yescharacteristic
303C22×C7C2×#13511yescharacteristic
304C22×C7C2×#13511yescharacteristic
305C22×C7#57511yescharacteristic
306C22×C7#57511yescharacteristic
307C4×C7—22no—
308C4×C7—22no—
309C4×C7—81no—
310C4×C7—82no—
311C4×C7#55412yes—
312C4×C7#55512yes—
313Dic7—22no—
314Dic7—22no—
315Dic7—22no—
316Dic7—22no—
317Dic7—42no—
318Dic7—81no—
319Dic7—82no—
320C24—21no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—72no—
325C2×Q8—141no—
326C2×Q8—142no—
327C2×Q8—142no—
328C2×Q8—282no—
329C2×Q8—282no—
330C2×Q8—282no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4Q8×D712yes—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—71no—
338C22×C4—71no—
339C22×C4—71no—
340C22×C4—71no—
341C22×C4—141no—
342C22×C4—141no—
343C22×C4—142no—
344C22×C4—282no—
345C22×C4—282no—
346C22×C4—282no—
347C22×C4D4×D711yescharacteristic
348C22×C4D4×D711yescharacteristic
349C22×C4C23×D711yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—142no—
354C2×C8—142no—
355C2×C8—142no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—142no—
361C4⋊C4—142no—
362C4⋊C4—142no—
363C4⋊C4—282no—
364C4⋊C4—282no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—144no—
368C22⋊C4—144no—
369C22⋊C4—282no—
370C42—22no—
371C42—72no—
372C42—72no—
373C42—74no—
374C42—142no—
375C42—142no—
376C42—281no—
377C42—281no—
378C42—282no—
379C2×C7—82no—
380C2×C7#256311yescharacteristic
381C2×C7#307611yescharacteristic
382C2×C7#307611yescharacteristic
383C2×C7#313111yescharacteristic
384C2×C7#314011yescharacteristic
385C2×C7#321111yescharacteristic
386C2×C7#321111yescharacteristic
387C23—22no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23C2×D4×D711yescharacteristic
395Q8—84no—
396Q8—144no—
397Q8—282no—
398Q8—564no—
399D4—24no—
400D4—42no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—72no—
412C2×C4—72no—
413C2×C4—72no—
414C2×C4—72no—
415C2×C4—141no—
416C2×C4—141no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—141no—
420C2×C4—141no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4—142no—
424C2×C4—281no—
425C2×C4—281no—
426C2×C4—282no—
427C2×C4—281no—
428C2×C4—281no—
429C2×C4—282no—
430C2×C4—282no—
431C2×C4—281no—
432C2×C4—564no—
433C2×C4#647512yes—
434C2×C4#647912yes—
435C2×C4#654812yes—
436C2×C4C2×Q8×D711yescharacteristic
437C2×C4D7×#4112yes—
438C2×C4D7×#4111yescharacteristic
439C8—42no—
440C8—282no—
441C7#2914211yescharacteristic, complemented, Sylow
442C22—42no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22#9028311yescharacteristic
450C22D7×C22⋊Q811yescharacteristic
451C22D7×#12411yescharacteristic
452C22C2×#647511yescharacteristic
453C22C2×#647911yescharacteristic
454C22#9046411yescharacteristic
455C22#9046811yescharacteristic
456C4—22no—
457C4—22no—
458C4—82no—
459C4—81no—
460C4—142no—
461C4—142no—
462C4—142no—
463C4—142no—
464C4—282no—
465C4—562no—
466C4—561no—
467C4#9032712yes—
468C4#9036112yes—
469C2—82no—
470C2#1173424411yescharacteristic
471C2#1173744011yescharacteristic
472C2#1173750211yescharacteristic
473C2#1173840911yescharacteristic
474C2#1173850411yescharacteristic
475C2#1173900711yescharacteristic
476C2#1173912711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2023C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2023D7
hypocenterC7#29142
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #11740697C2semidirect
4. #11729548C2semidirect
8. #11726659C2semidirect
9. #11726602C2semidirect
10. #11726443C2semidirect
11. #11726356C2semidirect
61. #6499Q8semidirect
441. C7#29142semidirect

Maximal subgroups

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

subgroupindex
#117406972
#117337372
C7×#20232
#117295482
#117279122
#117279082
#117273922
#117266592
#117266022
#117264432
#117263562
#291427