Finite Group Databasebeta

← #13372201

Subgroups — #13372201

Order 1792.

1030 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
1D4×C4⋊Dic7C211yesmaximal, characteristic
2C2×C4×#1142C211yesmaximal, characteristic
3C2×#89530C212yesmaximal
4#11731768C211yesmaximal, characteristic, complemented
5C7×#1802C211yesmaximal, characteristic
6#11728239C211yesmaximal, characteristic, complemented
7#11728219C211yesmaximal, characteristic, complemented
8#11727923C212yesmaximal
9#11727330C211yesmaximal, characteristic, complemented
10#11726661C212yesmaximal, complemented
11#11726612C212yesmaximal, complemented
12C2×C4×C7×D4C2211yescharacteristic
13C2×D4×Dic7C2212yes—
14C22×#1142C2211yescharacteristic
15C22×C4⋊Dic7—21no—
16C2×#6450C2212yes—
17C2×C4×Dic14C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×#102C2212yes—
20C7×#384C2211yescharacteristic
21C2×#6406C2211yescharacteristic
22#89984—21no—
23C2×#6390C2212yes—
24C2×C8×Dic7C2212yes—
25C4×#1142C418yescomplemented
26#89757C2212yes—
27#89748—21no—
28C4⋊C4×Dic7—21no—
29#89745C2212yes—
30#89744—22no—
31C2×#6384C2211yescharacteristic
32C2×#6382C2212yes—
33C2×#6382C2212yes—
34C2×#6382C2211yescharacteristic
35#89722—22no—
36#89707C2211yescharacteristic
37C4×C4⋊Dic7C2211yescharacteristic
38C2×C4×C7⋊C8C2211yescharacteristic
39#89530—28no—
40C7×#340C2212yes—
41#89346C2212yes—
42#89345C2212yes—
43C4⋊C4×SD16—71nomaximal, Sylow
44C22×C7×D4C2311yescharacteristic
45C23×C4×C7—21no—
46C23×Dic7—22no—
47C22×Dic14C2311yescharacteristic
48C22×C7×C8C2312yes—
49C4×C7×D4—22no—
50C4×C7×D4C2×C414yes—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C7×C22⋊C4—21no—
56C2×C42×C7C2311yescharacteristic
57C2×C22⋊Dic7—22no—
58D4×Dic7—24no—
59D4×Dic7—42no—
60C2×#1142C2×C414yes—
61C2×#1142D414yes—
62C2×#1142Q814yes—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—22no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—44no—
67C2×C4⋊Dic7D411yescharacteristic
68C2×C4⋊Dic7D411yescharacteristic
69C2×C4⋊Dic7C2312yes—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—22no—
73C2×Dic7⋊C4—22no—
74C2×C4×Dic7—21no—
75C2×C4×Dic7—22no—
76C2×C4×Dic7C2312yes—
77C22×C7⋊C8C2311yescharacteristic
78C22×C7⋊C8C2311yescharacteristic
79#6450—24no—
80#6450—42no—
81C4×Dic14—22no—
82C4×Dic14C2×C414yescomplemented
83C7×C4⋊C8—22no—
84C7×#102—24no—
85#6406C2×C414yes—
86C7⋊#95—22no—
87C7⋊#95—22no—
88C7⋊#95D412yes—
89#6390—24no—
90C8×Dic7—24no—
91#6384C2×C414yescomplemented
92#6382—24no—
93#6382—42no—
94#6382C2×C414yescomplemented
95C4×C7⋊C8C2×C414yescomplemented
96C2×C4×SD16—72no—
97C2×C4×SD16—71no—
98Q8×C4⋊C4—71no—
99D4×C4⋊C4—71no—
100#1941—72no—
101#1865—72no—
102#1864—72no—
103#1844—71no—
104#1803—71no—
105#1802D711yescharacteristic
106C8×C4⋊C4—71no—
107C24×C7—21no—
108C2×C7×D4—21no—
109C2×C7×D4—21no—
110C2×C7×D4—21no—
111C2×C7×D4C22×C412yes—
112C2×C7×D4C2×D412yes—
113C2×C7×D4C2×Q812yes—
114C22×C4×C7—21no—
115C22×C4×C7—24no—
116C22×C4×C7—22no—
117C22×C4×C7—42no—
118C22×C4×C7—42no—
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2411yescharacteristic
122C22×Dic7—21no—
123C22×Dic7—21no—
124C22×Dic7—22no—
125C22×Dic7—44no—
126C22×Dic7—44no—
127C22×Dic7—82no—
128C22×Dic7C2×D412yes—
129C22×Dic7C2×D412yes—
130C2×Dic14—21no—
131C2×Dic14—21no—
132C2×Dic14—21no—
133C2×Dic14C22×C412yes—
134C2×Dic14C2×D412yes—
135C2×Dic14C2×Q812yes—
136C2×C7×C8—22no—
137C2×C7×C8—22no—
138C2×C7×C8—22no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—42no—
143C7×C4⋊C4—81no—
144C7×C4⋊C4C22×C412yes—
145C7×C22⋊C4—24no—
146C42×C7C22×C412yes—
147C42×C7#4112yes—
148C22⋊Dic7—44no—
149#1142C4⋊C4116yescomplemented
150C4⋊Dic7—22no—
151C4⋊Dic7—21no—
152C4⋊Dic7—22no—
153C4⋊Dic7—22no—
154C4⋊Dic7—42no—
155C4⋊Dic7—44no—
156C4⋊Dic7—81no—
157C4⋊Dic7SD1614yescomplemented
158C4⋊Dic7C22×C412yes—
159Dic7⋊C4—24no—
160Dic7⋊C4—44no—
161Dic7⋊C4—82no—
162C4×Dic7—24no—
163C4×Dic7—22no—
164C4×Dic7—22no—
165C4×Dic7—44no—
166C2×C7⋊C8—21no—
167C2×C7⋊C8—21no—
168C2×C7⋊C8C22×C412yes—
169C2×C7⋊C8C22×C412yes—
170C2×C7⋊C8C2×D412yes—
171C2×C7⋊C8C2×Q812yes—
172C22×SD16—71no—
173C2×C4×Q8—72no—
174C2×C4×Q8—71no—
175C2×C4×D4—72no—
176C2×C4×D4C2×D711yescharacteristic
177C22×C4⋊C4—141no—
178C4×SD16—78no—
179C4×SD16—148no—
180C2×#106—71no—
181C2×#106—72no—
182C2×#106—72no—
183C2×C4⋊C8C2×D711yescharacteristic
184C2×#103—71no—
185C2×#103—72no—
186C2×#102—71no—
187C2×#102C2×D712yes—
188C2×C4×C8—71no—
189C2×C4×C8—72no—
190#389—72no—
191#389—72no—
192#389—142no—
193#385—142no—
194#384—71no—
195#384—141no—
196#384C2×D711yescharacteristic
197#380—141no—
198C4×C4⋊C4—71no—
199C4×C4⋊C4—141no—
200#340—72no—
201#340C2×D712yes—
202#336—72no—
203C23×C7—22no—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7—41no—
207C23×C7—41no—
208C23×C7—41no—
209C23×C7—41no—
210C23×C7C22×D411yescharacteristic
211C7×D4—22no—
212C7×D4—22no—
213C7×D4—22no—
214C7×D4C2×C4⋊C414yes—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—22no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—48no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7C4×D412yes—
225C2×C4×C7C4×D412yes—
226C2×C4×C7C2×SD1612yes—
227C2×C4×C7#13512yes—
228C2×C4×C7C23×C411yescharacteristic
229C2×C4×C7C22×D411yescharacteristic
230C2×C4×C7C22×Q811yescharacteristic
231C2×C4×C7C2×#4111yescharacteristic
232C2×C4×C7H(5,𝔽2)11yescharacteristic
233C2×C4×C72−1+411yescharacteristic
234C2×Dic7—22no—
235C2×Dic7—22no—
236C2×Dic7—22no—
237C2×Dic7—22no—
238C2×Dic7—22no—
239C2×Dic7—22no—
240C2×Dic7—22no—
241C2×Dic7—22no—
242C2×Dic7—22no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—88no—
249C2×Dic7C2×SD1614yes—
250Dic14—22no—
251Dic14—22no—
252Dic14—22no—
253Dic14C2×C4⋊C414yescomplemented
254C7×C8—42no—
255C7⋊C8—22no—
256C7⋊C8C2×C4⋊C414yescomplemented
257C22×Q8—71no—
258C22×D4C22×D711yescharacteristic
259C23×C4—21no—
260C23×C4—142no—
261C2×SD16—74no—
262C2×SD16—74no—
263C2×SD16—74no—
264C22×C8—71no—
265C22×C8—71no—
266C22×C8C22×D712yes—
267C4×Q8—74no—
268C4×Q8—142no—
269C4×Q8—144no—
270C4×Q8—282no—
271C4×D4—22no—
272C4×D4—144no—
273C4×D4—282no—
274C4×D4C4×D714yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—71no—
278C2×C4⋊C4—71no—
279C2×C4⋊C4—71no—
280C2×C4⋊C4—72no—
281C2×C4⋊C4—141no—
282C2×C4⋊C4—142no—
283C2×C4⋊C4—142no—
284C2×C4⋊C4—141no—
285C2×C4⋊C4—142no—
286C2×C4⋊C4—141no—
287C2×C4⋊C4—284no—
288C2×C4⋊C4C22×D712yes—
289C2×C4⋊C4C22×D711yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—142no—
292C2×C42—72no—
293C2×C42—141no—
294C2×C42—142no—
295C2×C42C22×D711yescharacteristic
296#106—74no—
297#106—144no—
298#106—282no—
299C4⋊C8—22no—
300#103—74no—
301#103—144no—
302#102—24no—
303#102—74no—
304C4×C8—74no—
305C4×C8—144no—
306#95—72no—
307#95—142no—
308#95—142no—
309C22×C7—24no—
310C22×C7—42no—
311C22×C7—42no—
312C22×C7—42no—
313C22×C7—42no—
314C22×C7—42no—
315C22×C7—42no—
316C22×C7C2×C4×D411yescharacteristic
317C22×C7D4211yescharacteristic
318C22×C7D4×Q811yescharacteristic
319C22×C7C22×SD1611yescharacteristic
320C22×C7C2×#13511yescharacteristic
321C22×C7#57711yescharacteristic
322C22×C7#57711yescharacteristic
323C4×C7—22no—
324C4×C7—42no—
325C4×C7—82no—
326C4×C7C4×SD1614yescomplemented
327C4×C7C22×C4⋊C411yescharacteristic
328C4×C7#52011yescharacteristic
329C4×C7#54711yescharacteristic
330C4×C7#55411yescharacteristic
331Dic7—24no—
332Dic7—24no—
333Dic7—42no—
334Dic7—42no—
335Dic7—82no—
336C24—21no—
337C2×Q8—72no—
338C2×Q8—72no—
339C2×Q8—72no—
340C2×Q8—141no—
341C2×Q8—141no—
342C2×Q8—141no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C2×C4×D712yes—
347C2×D4D4×D712yes—
348C2×D4Q8×D712yes—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—72no—
355C22×C4—72no—
356C22×C4—141no—
357C22×C4—141no—
358C22×C4—142no—
359C22×C4—284no—
360C22×C4—284no—
361C22×C4—562no—
362C22×C4D4×D711yescharacteristic
363C22×C4D4×D711yescharacteristic
364C22×C4C23×D711yescharacteristic
365SD16—716no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—72no—
370C2×C8—72no—
371C2×C8—72no—
372C2×C8—72no—
373C2×C8—141no—
374C2×C8—141no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—74no—
380C4⋊C4—72no—
381C4⋊C4—81no—
382C4⋊C4—144no—
383C4⋊C4—142no—
384C4⋊C4—141no—
385C4⋊C4—142no—
386C4⋊C4—142no—
387C4⋊C4—284no—
388C4⋊C4—284no—
389C4⋊C4—282no—
390C4⋊C4—561no—
391C4⋊C4—562no—
392C4⋊C4C2×C4×D712yes—
393C22⋊C4—24no—
394C22⋊C4—284no—
395C42—144no—
396C42—142no—
397C42—142no—
398C42—284no—
399C42C2×C4×D712yes—
400C42#115912yes—
401C2×C7—44no—
402C2×C7D4×C4⋊C411yescharacteristic
403C2×C7C2×C4×SD1611yescharacteristic
404C2×C7#295211yescharacteristic
405C2×C7D4×SD1611yescharacteristic
406C2×C7#328411yescharacteristic
407C2×C7Q8×SD1611yescharacteristic
408C2×C7#338111yescharacteristic
409C23—22no—
410C23—22no—
411C23—22no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23C2×D4×D711yescharacteristic
417Q8—74no—
418Q8—142no—
419Q8—142no—
420Q8—142no—
421D4—22no—
422D4—22no—
423D4—22no—
424D4D7×C4⋊C414yes—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—48no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—74no—
435C2×C4—142no—
436C2×C4—142no—
437C2×C4—142no—
438C2×C4—142no—
439C2×C4—142no—
440C2×C4—142no—
441C2×C4—142no—
442C2×C4—142no—
443C2×C4—142no—
444C2×C4—281no—
445C2×C4—281no—
446C2×C4—282no—
447C2×C4—282no—
448C2×C4—282no—
449C2×C4—568no—
450C2×C4(C22⋊D7)⋊C412yes—
451C2×C4(C22⋊D7)⋊C412yes—
452C2×C4#647512yes—
453C2×C4D7×SD1612yes—
454C2×C4C22×C4×D711yescharacteristic
455C2×C4C2×D4×D711yescharacteristic
456C2×C4C2×#115911yescharacteristic
457C2×C4C2×Q8×D711yescharacteristic
458C2×C4#655311yescharacteristic
459C2×C4#655411yescharacteristic
460C8—42no—
461C8—74no—
462C8—142no—
463C7C4⋊C4×SD1611yescharacteristic, complemented, Sylow
464C22—24no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22C2×(C22⋊D7)⋊C411yescharacteristic
472C22#9030011yescharacteristic
473C22#9032311yescharacteristic
474C22C2×#647511yescharacteristic
475C22C2×D7×SD1611yescharacteristic
476C22#9046611yescharacteristic
477C22#9046911yescharacteristic
478C4—22no—
479C4—42no—
480C4—82no—
481C4—144no—
482C4—144no—
483C4—282no—
484C4—282no—
485C4—562no—
486C4#8953014yescomplemented
487C4C2×D7×C4⋊C411yescharacteristic
488C4#9021411yescharacteristic
489C4#9030211yescharacteristic
490C4#9032611yescharacteristic
491C2—44no—
492C2#1173145511yescharacteristic
493C2C2×#8953011yescharacteristic
494C2#1173631411yescharacteristic
495C2#1173779611yescharacteristic
496C2#1173789311yescharacteristic
497C2#1173840311yescharacteristic
498C2#1173849811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1802C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1802D7
hypocenterC7C4⋊C4×SD16
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
4. #11731768C2semidirect
6. #11728239C2semidirect
7. #11728219C2semidirect
9. #11727330C2semidirect
10. #11726661C2semidirect
11. #11726612C2semidirect
25. C4×#1142C4semidirect
82. C4×Dic14C2×C4semidirect
91. #6384C2×C4semidirect
94. #6382C2×C4semidirect
95. C4×C7⋊C8C2×C4semidirect
149. #1142C4⋊C4semidirect
157. C4⋊Dic7SD16semidirect
253. Dic14C2×C4⋊C4semidirect
256. C7⋊C8C2×C4⋊C4semidirect
326. C4×C7C4×SD16semidirect
463. C7C4⋊C4×SD16semidirect
486. C4#89530semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Dic72
C2×C4×#11422
C2×#895302
#117317682
C7×#18022
#117282392
#117282192
#117279232
#117273302
#117266612
#117266122
C4⋊C4×SD167