Finite Group Databasebeta

← #13358665

Subgroups — #13358665

Order 1792.

914 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89605C211yesmaximal, characteristic
2#11731763C211yesmaximal, characteristic
3C4×#6455C211yesmaximal, characteristic
4C7×#1948C211yesmaximal, characteristic, complemented
5#11728334C212yesmaximal
6Dic7×#106C212yesmaximal, complemented
7#11728208C211yesmaximal, characteristic, complemented
8#11727321C211yesmaximal, characteristic, complemented
9#11727220C211yesmaximal, characteristic, complemented
10#11726630C212yesmaximal
11#11726539C212yesmaximal, complemented
12C2×#6455—21no—
13C2×#6455C2211yescharacteristic
14C2×#6455C2211yescharacteristic
15C2×C42×D7C2211yescharacteristic
16C2×C7×#106C2212yes—
17C2×C7×C4⋊C8C2211yescharacteristic
18C7×#384C2211yescharacteristic
19C4×C7×C4⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C8C2212yes—
21C2×#6391C2212yes—
22C2×C8×Dic7C2212yes—
23#89759—21no—
24#89759C2212yes—
25#89749—21no—
26C4⋊C4×Dic7—21no—
27C4⋊C4×Dic7C2212yes—
28C4⋊C4×Dic7C2211yescharacteristic
29C4⋊C4×Dic7C2211yescharacteristic
30C2×#6381C2212yes—
31C2×#6381C2212yes—
32C4×(C2×D7)⋊C4—21no—
33#89707C2211yescharacteristic
34C4×C4⋊Dic7C2211yescharacteristic
35C42×Dic7—21no—
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89605C418yescomplemented
38#89438—21no—
39#89420—21no—
40C7×#340C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21362—71nomaximal, Sylow
45C22×C4×D7D411yescharacteristic
46C22×C4×D7D411yescharacteristic
47C22×C4×D7C2311yescharacteristic
48C22×C7×C8C2312yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4C2312yes—
52C2×C7×C4⋊C4C2311yescharacteristic
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C42×C7—21no—
55C2×C42×C7C2311yescharacteristic
56C2×(C2×D7)⋊C4—21no—
57C2×(C2×D7)⋊C4—21no—
58C2×(C2×D7)⋊C4—22no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7C2312yes—
62C2×C4⋊Dic7C2311yescharacteristic
63C2×C4⋊Dic7C2311yescharacteristic
64C2×C4×Dic7—21no—
65C2×C4×Dic7—21no—
66C2×C4×Dic7—21no—
67C2×C4×Dic7—22no—
68C2×C4×Dic7—21no—
69C2×C4×Dic7—21no—
70C2×C4×Dic7D411yescharacteristic
71C2×C4×Dic7D411yescharacteristic
72C2×C4×Dic7C2311yescharacteristic
73C22×C7⋊C8C2312yes—
74#6455—22no—
75#6455—22no—
76#6455—22no—
77#6455—22no—
78#6455—44no—
79C42×D7C2×C414yes—
80C42×D7Q814yes—
81C7×#106—24no—
82C7×C4⋊C8C2×C414yes—
83C7×#95—21no—
84C7⋊#95—21no—
85C7⋊#95—21no—
86C7⋊#95—21no—
87C7⋊#95—22no—
88C7⋊#95—22no—
89(C2×D7)⋊C8C2×C418yes—
90#6391—24no—
91C8×Dic7C2×C418yes—
92#6381—24no—
93#6381—24no—
94C4⋊(C7⋊C8)C2×C414yes—
95C2×#431—71no—
96#2285—71no—
97C4×#117—71no—
98#1948—71no—
99#1948D711yescharacteristic, complemented
100#1947—71no—
101#1947—71no—
102#1854—72no—
103#1854—72no—
104C4×#107—72no—
105C4×#106—72no—
106C23×D7C2×D411yescharacteristic
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—22no—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2411yescharacteristic
113C22×Dic7—21no—
114C22×Dic7—21no—
115C22×Dic7—22no—
116C22×Dic7C2×D411yescharacteristic
117C22×Dic7C2×D411yescharacteristic
118C22×Dic7C2×D411yescharacteristic
119C2×C4×D7—22no—
120C2×C4×D7—22no—
121C2×C4×D7—21no—
122C2×C4×D7—21no—
123C2×C4×D7—21no—
124C2×C4×D7—21no—
125C2×C4×D7—22no—
126C2×C4×D7—22no—
127C2×C4×D7C4⋊C414yes—
128C2×C4×D7C4⋊C414yes—
129C2×C4×D7C22×C412yes—
130C2×C4×D7C2×Q812yes—
131C2×C7×C8—22no—
132C2×C7×C8—22no—
133C2×C7×C8C22×C414yes—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—42no—
139C7×C4⋊C4—42no—
140C7×C4⋊C4#4112yes—
141C7×C4⋊C4#4112yes—
142C42×C7—42no—
143C42×C7C22×C412yes—
144C42×C7C2×Q812yes—
145(C2×D7)⋊C4—24no—
146(C2×D7)⋊C4—24no—
147(C2×D7)⋊C4—44no—
148C4⋊Dic7—21no—
149C4⋊Dic7—21no—
150C4⋊Dic7—22no—
151C4⋊Dic7—22no—
152C4⋊Dic7—42no—
153C4⋊Dic7—42no—
154C4⋊Dic7#4112yes—
155C4⋊Dic7#4112yes—
156C4×Dic7—24no—
157C4×Dic7—24no—
158C4×Dic7—42no—
159C4×Dic7—44no—
160C4×Dic7—42no—
161C4×Dic7—42no—
162C4×Dic7C4⋊C414yes—
163C4×Dic7C4⋊C414yes—
164C4×Dic7C22×C412yes—
165C4×Dic7C2×Q812yes—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—22no—
168C2×C7⋊C8C22×C414yes—
169C2×#117—71no—
170C2×#117—71no—
171C2×#117—141no—
172C22×C42—71no—
173#431—78no—
174C2×#107—72no—
175C2×#107—72no—
176C2×#106—72no—
177C2×#106C2×D712yes—
178C2×C4⋊C8—71no—
179C2×C4⋊C8C2×D711yescharacteristic
180C2×C22⋊C8—72no—
181C2×C4×C8—72no—
182#384—71no—
183#384C2×D711yescharacteristic
184#382—141no—
185#380—72no—
186#380—141no—
187C4×C4⋊C4—71no—
188C4×C4⋊C4—71no—
189C4×C4⋊C4—72no—
190C4×C4⋊C4—71no—
191C4×C4⋊C4—141no—
192C4×C4⋊C4C2×D711yescharacteristic
193C4×C22⋊C4—141no—
194C4×C22⋊C4—141no—
195#376—141no—
196C43—141no—
197#340—72no—
198#340—72no—
199#340—72no—
200#340C2×D712yes—
201C23×C7C22×D411yescharacteristic
202C22×D7—21no—
203C22×D7—21no—
204C22×D7—21no—
205C22×D7—21no—
206C22×D7—21no—
207C22×D7—21no—
208C22×D7C2×C4⋊C412yes—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7C2×C4⋊C412yes—
221C2×C4×C7C2×C4⋊C412yes—
222C2×C4×C7#13512yes—
223C2×C4×C7#13512yes—
224C2×C4×C7C23×C411yescharacteristic
225C2×C4×C7C22×Q811yescharacteristic
226C2×C4×C7C2×#4111yescharacteristic
227C2×C4×C7C2×#4111yescharacteristic
228C2×C4×C7C2×#4111yescharacteristic
229C2×C4×C7C2×#4111yescharacteristic
230C2×Dic7—21no—
231C2×Dic7—21no—
232C2×Dic7—21no—
233C2×Dic7—21no—
234C2×Dic7—22no—
235C2×Dic7—22no—
236C2×Dic7—41no—
237C2×Dic7—41no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7—42no—
241C2×Dic7—42no—
242C2×Dic7—42no—
243C2×Dic7C2×C4⋊C412yes—
244C2×Dic7C2×C4⋊C412yes—
245C2×Dic7C2×C4⋊C412yes—
246C2×Dic7AGL(1,ℤ/8)12yes—
247C2×Dic7#13712yes—
248C4×D7—22no—
249C4×D7—22no—
250C4×D7—22no—
251C4×D7—22no—
252C4×D7—28no—
253C4×D7—28no—
254C7×C8—24no—
255C7⋊C8—24no—
256C23×C4—71no—
257C23×C4—71no—
258C23×C4—71no—
259C22×C8—72no—
260C22×C8C22×D712yes—
261#117—142no—
262#117—142no—
263#117—142no—
264#117—142no—
265#117—284no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—71no—
269C2×C4⋊C4—71no—
270C2×C4⋊C4—72no—
271C2×C4⋊C4—141no—
272C2×C4⋊C4—141no—
273C2×C4⋊C4C22×D712yes—
274C2×C4⋊C4C22×D711yescharacteristic
275C2×C4⋊C4C22×D711yescharacteristic
276C2×C22⋊C4—141no—
277C2×C22⋊C4—141no—
278C2×C22⋊C4—142no—
279C2×C42—21no—
280C2×C42—71no—
281C2×C42—71no—
282C2×C42—71no—
283C2×C42—74no—
284C2×C42—74no—
285C2×C42—141no—
286C2×C42—141no—
287C2×C42—142no—
288C2×C42—141no—
289C2×C42—141no—
290C2×C42—141no—
291C2×C42C22×D711yescharacteristic
292#107—144no—
293#107—144no—
294#106—24no—
295#106—144no—
296C4⋊C8—74no—
297C4⋊C8C4×D714yes—
298C22⋊C8—78no—
299C4×C8—78no—
300#95—21no—
301#95—141no—
302#95—141no—
303#95—142no—
304#95—142no—
305#95—141no—
306C22×C7C22×C4⋊C411yescharacteristic
307C22×C7#52511yescharacteristic
308C22×C7#52511yescharacteristic
309C22×C7C2×#13511yescharacteristic
310C22×C7C2×#13511yescharacteristic
311C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C7C2×#13711yescharacteristic
313C2×D7—22no—
314C2×D7—22no—
315C2×D7—22no—
316C2×D7—41no—
317C2×D7—41no—
318C2×D7—41no—
319C2×D7—41no—
320C4×C7—22no—
321C4×C7—42no—
322C4×C7—42no—
323C4×C7—82no—
324C4×C7#42714yes—
325C4×C7C4×#4111yescharacteristic
326C4×C7C4×#4111yescharacteristic
327C4×C7#53311yescharacteristic
328C4×C7#53311yescharacteristic
329Dic7—22no—
330Dic7—22no—
331Dic7—42no—
332Dic7—42no—
333Dic7—82no—
334Dic7#42814yes—
335C24—71no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—71no—
340C22×C4—71no—
341C22×C4—71no—
342C22×C4—74no—
343C22×C4—74no—
344C22×C4—72no—
345C22×C4—72no—
346C22×C4—142no—
347C22×C4—142no—
348C22×C4—142no—
349C22×C4—142no—
350C22×C4—141no—
351C22×C4—141no—
352C22×C4—141no—
353C22×C4—141no—
354C22×C4—141no—
355C22×C4—141no—
356C22×C4—142no—
357C22×C4D4×D711yescharacteristic
358C22×C4D4×D711yescharacteristic
359C22×C4C23×D711yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—74no—
363C2×C8—142no—
364C2×C8—142no—
365C2×C8C2×C4×D714yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—72no—
373C4⋊C4—72no—
374C4⋊C4—141no—
375C4⋊C4—141no—
376C4⋊C4—142no—
377C4⋊C4—142no—
378C4⋊C4—282no—
379C4⋊C4—282no—
380C4⋊C4#115912yes—
381C4⋊C4#116112yes—
382C22⋊C4—144no—
383C22⋊C4—144no—
384C22⋊C4—284no—
385C42—42no—
386C42—74no—
387C42—74no—
388C42—72no—
389C42—72no—
390C42—144no—
391C42—144no—
392C42—282no—
393C42—282no—
394C42—284no—
395C42—282no—
396C42C2×C4×D712yes—
397C42Q8×D712yes—
398C2×C7#229611yescharacteristic
399C2×C7C2×#42711yescharacteristic
400C2×C7C2×#42811yescharacteristic
401C2×C7#310611yescharacteristic
402C2×C7#310611yescharacteristic
403C2×C7#310711yescharacteristic
404C2×C7#310711yescharacteristic
405D7—42no—
406C23—72no—
407C23—141no—
408C23—141no—
409C23—141no—
410C23—141no—
411C23—141no—
412C23—141no—
413C23C2×D4×D711yescharacteristic
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—72no—
426C2×C4—72no—
427C2×C4—72no—
428C2×C4—72no—
429C2×C4—72no—
430C2×C4—148no—
431C2×C4—141no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—142no—
436C2×C4—148no—
437C2×C4—142no—
438C2×C4—142no—
439C2×C4—142no—
440C2×C4—142no—
441C2×C4—142no—
442C2×C4—281no—
443C2×C4—281no—
444C2×C4—281no—
445C2×C4—281no—
446C2×C4—282no—
447C2×C4—282no—
448C2×C4—282no—
449C2×C4D7×C4⋊C412yes—
450C2×C4D7×C4⋊C412yes—
451C2×C4#647912yes—
452C2×C4#647912yes—
453C2×C4C22×C4×D711yescharacteristic
454C2×C4C2×#115911yescharacteristic
455C2×C4C2×Q8×D711yescharacteristic
456C2×C4C2×#116111yescharacteristic
457C2×C4D7×#4111yescharacteristic
458C2×C4D7×#4111yescharacteristic
459C8—24no—
460C8—144no—
461C7#2136211yescharacteristic, complemented, Sylow
462C22—142no—
463C22—142no—
464C22—142no—
465C22—281no—
466C22—281no—
467C22—281no—
468C22—281no—
469C22C2×D7×C4⋊C411yescharacteristic
470C22#9029711yescharacteristic
471C22#9031811yescharacteristic
472C22C2×#647911yescharacteristic
473C22C2×#647911yescharacteristic
474C22D7×AGL(1,ℤ/8)11yescharacteristic
475C22D7×#13711yescharacteristic
476C4—22no—
477C4—42no—
478C4—42no—
479C4—74no—
480C4—82no—
481C4—142no—
482C4—142no—
483C4—282no—
484C4—282no—
485C4—562no—
486C4#8963214yes—
487C4C4×#115911yescharacteristic
488C4C4×#116111yescharacteristic
489C4#9037911yescharacteristic
490C4#9041911yescharacteristic
491C2—282no—
492C2#1173340811yescharacteristic
493C2C2×#8963211yescharacteristic
494C2D7×#42811yescharacteristic
495C2#1173780611yescharacteristic
496C2#1173785411yescharacteristic
497C2#1173844211yescharacteristic
498C2#1173849011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1948C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1948D7
hypocenterC7#21362
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. C7×#1948C2semidirect
6. Dic7×#106C2semidirect
7. #11728208C2semidirect
8. #11727321C2semidirect
9. #11727220C2semidirect
11. #11726539C2semidirect
37. #89605C4semidirect
99. #1948D7semidirect
461. C7#21362semidirect

Maximal subgroups

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

subgroupindex
C2×#896052
#117317632
C4×#64552
C7×#19482
#117283342
Dic7×#1062
#117282082
#117273212
#117272202
#117266302
#117265392
#213627