Finite Group Databasebeta

← #13403941

Subgroups — #13403941

Order 1792.

894 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740685C211yesmaximal, characteristic
2C2×#89866C211yesmaximal, characteristic, complemented
3C2×C8×D28C211yesmaximal, characteristic
4#11731814C211yesmaximal, characteristic
5C7×#1990C211yesmaximal, characteristic, complemented
6C4×#6390C211yesmaximal, characteristic, complemented
7#11727322C211yesmaximal, characteristic, complemented
8#11726645C212yesmaximal, complemented
9#11726636C212yesmaximal, complemented
10#11726622C212yesmaximal, complemented
11#11726578C212yesmaximal, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×#6510—21no—
14C22×C8×D7C2212yes—
15C2×#6460C2212yes—
16C2×C4×D28C2211yescharacteristic
17C2×C4×Dic14C2211yescharacteristic
18C2×C4×C7×C8C2211yescharacteristic
19C7×#384C2211yescharacteristic
20#90002—21no—
21C2×(C2×D7)⋊C8C2212yes—
22C2×#6390C2211yescharacteristic
23C2×#6390C2211yescharacteristic
24#89866—24no—
25#89766—22no—
26#89765—22no—
27#89760—21no—
28#89759C2212yes—
29#89759C2212yes—
30#89759C2212yes—
31#89757—21no—
32C2×#6384C2212yes—
33C2×#6382C2212yes—
34C2×#6382C2212yes—
35C4×C4⋊Dic7C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C8×D28—24no—
38C7×#340C2212yes—
39C7×#340C2212yes—
40#89346C2211yescharacteristic
41#89346C2211yescharacteristic
42#89324C2212yes—
43#25341—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×D28D411yescharacteristic
46C22×C4×D7C2312yes—
47C22×Dic14C2311yescharacteristic
48C22×C7×C8C2311yescharacteristic
49C22×C7×C8C2311yescharacteristic
50C7×C4⋊Q8—22no—
51C7×C4⋊Q8—22no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4C2312yes—
56C2×C7×C4⋊C4C2312yes—
57C2×C42×C7C2311yescharacteristic
58#6510—44no—
59C2×(C2×D7)⋊C4—22no—
60C2×(C2×D7)⋊C4—22no—
61C2×(C2×D7)⋊C4—22no—
62C2×(C2×D7)⋊C4—22no—
63C2×(C2×D7)⋊C4D412yes—
64C2×C4⋊Dic7—22no—
65C2×C4⋊Dic7D411yescharacteristic
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2311yescharacteristic
68C2×C4⋊Dic7C2311yescharacteristic
69C2×Dic7⋊C4—21no—
70C2×C4×Dic7—21no—
71C2×C4×Dic7—21no—
72C22×C7⋊C8C2312yes—
73C2×C8×D7—24no—
74C2×C8×D7—24no—
75C2×C8×D7—42no—
76#6460—24no—
77#6460—42no—
78C4×D28—22no—
79C4×D28—22no—
80C4×Dic14—22no—
81C4×Dic14—22no—
82C4×C7×C8—22no—
83C7⋊#95—21no—
84C7⋊#95—22no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87(C2×D7)⋊C8—24no—
88#6390—22no—
89#6390—22no—
90#6384—24no—
91#6382—24no—
92#6382—42no—
93C4⋊(C7⋊C8)D414yes—
94C2×#475—71no—
95C2×C8×D4—71no—
96#2547—71no—
97#2538—71no—
98#2077—72no—
99#2035—72no—
100#1990D711yescharacteristic, complemented
101#1948—71no—
102#1892—72no—
103#1807—72no—
104C4×#103—71no—
105C23×D7C2×D412yes—
106C2×C7×Q8—42no—
107C2×C7×Q8—42no—
108C2×C7×Q8—42no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2411yescharacteristic
114C22×Dic7—21no—
115C22×Dic7—21no—
116C22×Dic7—21no—
117C22×Dic7—21no—
118C22×Dic7C2×D412yes—
119C2×D28—22no—
120C2×D28—22no—
121C2×D28SD1614yes—
122C2×C4×D7—22no—
123C2×C4×D7—22no—
124C2×C4×D7—22no—
125C2×C4×D7—22no—
126C2×C4×D7—22no—
127C2×C4×D7—22no—
128C2×Dic14—21no—
129C2×Dic14—21no—
130C2×Dic14—21no—
131C2×Dic14—21no—
132C2×Dic14—21no—
133C2×Dic14#4112yes—
134C2×C7×C8—21no—
135C2×C7×C8—21no—
136C2×C7×C8—21no—
137C2×C7×C8—21no—
138C2×C7×C8—21no—
139C2×C7×C8#4112yes—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—42no—
145C7×C4⋊C4—42no—
146C7×C4⋊C4—81no—
147C42×C7C2×D412yes—
148C42×C7#4112yes—
149(C2×D7)⋊C4—24no—
150(C2×D7)⋊C4—44no—
151(C2×D7)⋊C4—44no—
152(C2×D7)⋊C4—44no—
153(C2×D7)⋊C4—82no—
154C4⋊Dic7—22no—
155C4⋊Dic7—21no—
156C4⋊Dic7—21no—
157C4⋊Dic7—21no—
158C4⋊Dic7—21no—
159C4⋊Dic7—21no—
160C4⋊Dic7—21no—
161C4⋊Dic7—44no—
162Dic7⋊C4—42no—
163C4×Dic7—42no—
164C4×Dic7—42no—
165C2×C7⋊C8—22no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8C2×D414yes—
168C8×D7—48no—
169C23×C8—72no—
170C2×C4⋊Q8C2×D711yescharacteristic
171C2×#124—141no—
172C2×C22⋊Q8—72no—
173C2×C4×Q8—71no—
174C2×C4×D4—71no—
175#475—144no—
176C8×D4—144no—
177C2×#106—72no—
178C2×#106—72no—
179C2×C4⋊C8—71no—
180C2×#103—71no—
181C2×#103—71no—
182C2×#103—72no—
183C2×C22⋊C8—72no—
184C2×C4×C8C2×D711yescharacteristic
185#397—142no—
186#396—142no—
187#391—141no—
188#390—141no—
189#389—141no—
190#384C2×D711yescharacteristic
191#380—72no—
192#380—72no—
193#380—72no—
194C4×C4⋊C4—71no—
195#340—71no—
196#340—71no—
197#340—72no—
198#340C2×D712yes—
199#340C2×D712yes—
200C23×C7C22×D411yescharacteristic
201C22×D7—22no—
202C22×D7—22no—
203C22×D7—22no—
204C22×D7—22no—
205C22×D7—22no—
206C22×D7—22no—
207C22×D7C2×SD1614yes—
208C7×Q8—84no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7—42no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7—42no—
217C2×C4×C7#12112yes—
218C2×C4×C7#12112yes—
219C2×C4×C7C2×SD1612yes—
220C2×C4×C7#13512yes—
221C2×C4×C7C22×D411yescharacteristic
222C2×C4×C7C2×#4111yescharacteristic
223C2×C4×C7C2×#4111yescharacteristic
224C2×C4×C7C2×#4111yescharacteristic
225C2×C4×C7H(5,𝔽2)11yescharacteristic
226C2×C4×C72−1+411yescharacteristic
227C2×Dic7—22no—
228C2×Dic7—22no—
229C2×Dic7—22no—
230C2×Dic7—22no—
231C2×Dic7—41no—
232C2×Dic7—41no—
233C2×Dic7—41no—
234C2×Dic7—41no—
235C2×Dic7—41no—
236C2×Dic7—41no—
237C2×Dic7—41no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7—41no—
241C2×Dic7—41no—
242D28—28no—
243C4×D7—42no—
244C4×D7—42no—
245C4×D7—42no—
246C4×D7—42no—
247Dic14—22no—
248Dic14—22no—
249Dic14—41no—
250Dic14—41no—
251C7×C8—22no—
252C7×C8—41no—
253C7⋊C8—24no—
254C22×Q8—21no—
255C22×Q8—71no—
256C22×D4—71no—
257C23×C4—72no—
258C22×C8—72no—
259C22×C8—144no—
260C22×C8—144no—
261C22×C8—282no—
262C22×C8C22×D711yescharacteristic
263C22×C8C22×D711yescharacteristic
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266#124—284no—
267C22⋊Q8—144no—
268C22⋊Q8—282no—
269C4×Q8—142no—
270C4×Q8—142no—
271C4×D4—142no—
272C4×D4—142no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—71no—
277C2×C4⋊C4—71no—
278C2×C4⋊C4—71no—
279C2×C4⋊C4—71no—
280C2×C4⋊C4—142no—
281C2×C4⋊C4—141no—
282C2×C4⋊C4C22×D712yes—
283C2×C4⋊C4C22×D712yes—
284C2×C22⋊C4—72no—
285C2×C22⋊C4—142no—
286C2×C22⋊C4—142no—
287C2×C22⋊C4—142no—
288C2×C22⋊C4—142no—
289C2×C42—141no—
290C2×C42—141no—
291C2×C42C22×D711yescharacteristic
292#106—144no—
293#106—282no—
294C4⋊C8—74no—
295#103—144no—
296#103—142no—
297#103—142no—
298C22⋊C8—144no—
299C4×C8—22no—
300#95—142no—
301#95—142no—
302#95—142no—
303#95—141no—
304C22×C7C2×#12111yescharacteristic
305C22×C7#54611yescharacteristic
306C22×C7#54711yescharacteristic
307C22×C7C22×SD1611yescharacteristic
308C22×C7C2×#13511yescharacteristic
309C22×C7#57711yescharacteristic
310C22×C7#57811yescharacteristic
311C2×D7—24no—
312C2×D7—24no—
313C2×D7—24no—
314C2×D7—42no—
315C2×D7—42no—
316C2×D7—42no—
317C2×D7—42no—
318C4×C7—22no—
319C4×C7—82no—
320C4×C7—82no—
321C4×C7#46514yes—
322C4×C7#52911yescharacteristic
323C4×C7#54211yescharacteristic
324C4×C7#54611yescharacteristic
325C4×C7#54811yescharacteristic
326Dic7—42no—
327Dic7—42no—
328Dic7—81no—
329Dic7—81no—
330Dic7—81no—
331C24—72no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—72no—
336C2×Q8—141no—
337C2×Q8—141no—
338C2×Q8—141no—
339C2×Q8—141no—
340C2×Q8—141no—
341C2×D4—74no—
342C2×D4—142no—
343C2×D4—142no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—72no—
347C22×C4—141no—
348C22×C4—142no—
349C22×C4—142no—
350C22×C4—142no—
351C22×C4—142no—
352C22×C4—142no—
353C22×C4—142no—
354C22×C4—141no—
355C22×C4—141no—
356C22×C4—141no—
357C22×C4D4×D711yescharacteristic
358C22×C4D4×D711yescharacteristic
359C22×C4C23×D711yescharacteristic
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—74no—
366C2×C8—142no—
367C2×C8—142no—
368C2×C8—288no—
369C2×C8#116112yes—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—81no—
377C4⋊C4—142no—
378C4⋊C4—141no—
379C4⋊C4—141no—
380C4⋊C4—141no—
381C4⋊C4—141no—
382C4⋊C4—141no—
383C4⋊C4—141no—
384C4⋊C4—284no—
385C4⋊C4—282no—
386C22⋊C4—144no—
387C22⋊C4—284no—
388C22⋊C4—284no—
389C22⋊C4—284no—
390C22⋊C4—562no—
391C42—282no—
392C42—282no—
393C42#115712yes—
394C42D4×D712yes—
395C2×C7#242711yescharacteristic
396C2×C7C2×#46511yescharacteristic
397C2×C7#306311yescharacteristic
398C2×C7#329811yescharacteristic
399C2×C7#330411yescharacteristic
400C2×C7#333511yescharacteristic
401C2×C7#334211yescharacteristic
402D7—44no—
403C23—74no—
404C23—142no—
405C23—142no—
406C23—142no—
407C23—142no—
408C23—142no—
409C23—142no—
410C23C2×D4×D711yescharacteristic
411Q8—84no—
412Q8—142no—
413Q8—142no—
414Q8—281no—
415Q8—281no—
416D4—148no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—142no—
426C2×C4—142no—
427C2×C4—142no—
428C2×C4—142no—
429C2×C4—281no—
430C2×C4—281no—
431C2×C4—282no—
432C2×C4—282no—
433C2×C4—282no—
434C2×C4—282no—
435C2×C4—281no—
436C2×C4—281no—
437C2×C4—281no—
438C2×C4—281no—
439C2×C4—281no—
440C2×C4—281no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4—281no—
444C2×C4#644712yes—
445C2×C4#644712yes—
446C2×C4#647512yes—
447C2×C4D7×SD1612yes—
448C2×C4C2×#115711yescharacteristic
449C2×C4C2×D4×D711yescharacteristic
450C2×C4C2×#116111yescharacteristic
451C2×C4#655111yescharacteristic
452C2×C4D7×#4111yescharacteristic
453C2×C4#655311yescharacteristic
454C8—22no—
455C8—41no—
456C8—144no—
457C7#2534111yescharacteristic, complemented, Sylow
458C22—144no—
459C22—144no—
460C22—144no—
461C22—282no—
462C22—282no—
463C22—282no—
464C22—282no—
465C22C2×#644711yescharacteristic
466C22#9036711yescharacteristic
467C22#9041711yescharacteristic
468C22C2×#647511yescharacteristic
469C22C2×D7×SD1611yescharacteristic
470C22#9045311yescharacteristic
471C22#9045711yescharacteristic
472C4—22no—
473C4—82no—
474C4—82no—
475C4—282no—
476C4—282no—
477C4—561no—
478C4—561no—
479C4—561no—
480C4#8955014yes—
481C4#9027111yescharacteristic
482C4#9027311yescharacteristic
483C4#9032111yescharacteristic
484C4#9032411yescharacteristic
485C2—284no—
486C2#1173406911yescharacteristic
487C2C2×#8955011yescharacteristic
488C2#1173624511yescharacteristic
489C2#1173902311yescharacteristic
490C2#1173905711yescharacteristic
491C2#1173934411yescharacteristic
492C2#1173939411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1990C2
socleC23×C7C22×D4
prime radicalC22×D28D4
hypercenter#1990D7
hypocenterC7#25341
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C2×#89866C2semidirect
5. C7×#1990C2semidirect
6. C4×#6390C2semidirect
7. #11727322C2semidirect
8. #11726645C2semidirect
9. #11726636C2semidirect
10. #11726622C2semidirect
11. #11726578C2semidirect
100. #1990D7semidirect
457. C7#25341semidirect

Maximal subgroups

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

subgroupindex
#117406852
C2×#898662
C2×C8×D282
#117318142
C7×#19902
C4×#63902
#117273222
#117266452
#117266362
#117266222
#117265782
#253417