Finite Group Databasebeta

← #99×#64425 #72246201

Subgroups — #99×#64425 #72246201

Order 9408.

1050 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72246200C212yesmaximal, complemented
2#72246199C211yesmaximal, characteristic, complemented
3#99×#3862C211yesmaximal, characteristic, complemented
4#99×C7⋊D7C311yesmaximal, characteristic, complemented, Hall
5C22⋊C4×#64425C212yesmaximal, complemented
6#72246064—22no—
7#72246062C2212yescomplemented
8C22⋊C4×#3862C2212yescomplemented
9C2×D4×#64425C211yesmaximal, characteristic
10C2×#11924222C411yescharacteristic, complemented
11C2×#11924222C2211yescharacteristic
12C23×#64425C411yescharacteristic, complemented
13C23×#64425C2211yescharacteristic
14C2×C4⋊#64425C411yescharacteristic, complemented
15C2×D4×#3862C2211yescharacteristic
16C2×C4×#64425—22no—
17C2×C4×#64425C411yescharacteristic, complemented
18D4×#64425—22no—
19D4×#64425—22no—
20C2×D4×C7⋊D7C2×C311yescharacteristic
21C22⋊C4×C7⋊D7C2×C312yescomplemented
22C72×#99C2×C311yescharacteristic, complemented
23#13096150C2×C311yescharacteristic, complemented
24#13096124C2×C312yescomplemented
25#99×AGL(1,𝔽7)—72nomaximal
26C23×#3862C2×C411yescharacteristic, complemented
27C23×#3862C2311yescharacteristic
28C22×#64425—21no—
29C22×#64425—21no—
30C22×#64425—42no—
31C22×#64425—42no—
32C22×#64425C2×C411yescharacteristic
33C22×#64425C2×C411yescharacteristic
34C22×#64425C2×C411yescharacteristic, complemented
35C22×#64425D412yes—
36C22×#64425D412yes—
37D4×#3862—22no—
38C2×C4×#3862—22no—
39C2×C4×#3862C2×C411yescharacteristic, complemented
40#11924222—22no—
41#11924222—22no—
42C2×#10596875—22no—
43C2×#10596875C2×C411yescharacteristic, complemented
44C4⋊#64425—22no—
45C4×#64425—21no—
46C4×#64425—21no—
47C4×#64425—44no—
48C23×C7⋊D7C3×C411yescharacteristic, complemented
49C23×C7⋊D7C22×C311yescharacteristic
50C2×C72×D4C22×C311yescharacteristic
51C2×C22⋊(C7⋊D7)C3×C411yescharacteristic, complemented
52C2×C22⋊(C7⋊D7)C22×C311yescharacteristic
53D4×C7⋊D7—22no—
54D4×C7⋊D7—22no—
55C2×C4⋊(C7⋊D7)C3×C411yescharacteristic, complemented
56C2×C4×C7⋊D7—22no—
57C2×C4×C7⋊D7C3×C411yescharacteristic, complemented
58C72×C22⋊C4C22×C312yescomplemented
59C22⋊(C7⋊Dic7)C22×C312yescomplemented
60#11716070—22no—
61C2×D4×AGL(1,𝔽7)—72no—
62C22⋊C4×AGL(1,𝔽7)—74no—
63ΓL(1,𝔽8)×#99—142no—
64#6408⋊C3—72no—
65#6380⋊C3—74no—
66C22×#3862—21no—
67C22×#3862—41no—
68C22×#3862—41no—
69C22×#3862C22×C411yescharacteristic
70C22×#3862C2×D412yes—
71C2×#64425—22no—
72C2×#64425—24no—
73C2×#64425—21no—
74C2×#64425—41no—
75C2×#64425—42no—
76C2×#64425—42no—
77C2×#64425—42no—
78C2×#64425—41no—
79C2×#64425C22⋊C411yescharacteristic
80C2×#64425C22⋊C412yes—
81C2×#64425C22⋊C411yescharacteristic
82C4×#3862—21no—
83C4×#3862—42no—
84#10596875—21no—
85#10596875—42no—
86D7×#99—72no—
87D7×#99—212no—
88C23×C72C2×C3×C411yescharacteristic, complemented
89C23×C72C23×C311yescharacteristic
90C22×C7⋊D7—21no—
91C22×C7⋊D7—21no—
92C22×C7⋊D7—42no—
93C22×C7⋊D7—42no—
94C22×C7⋊D7C2×C3×C411yescharacteristic
95C22×C7⋊D7C2×C3×C411yescharacteristic
96C22×C7⋊D7C2×C3×C411yescharacteristic, complemented
97C22×C7⋊D7C3×D412yes—
98C22×C7⋊D7C3×D412yes—
99C72×D4—22no—
100C2×C4×C72—22no—
101C2×C4×C72C2×C3×C411yescharacteristic, complemented
102C22⋊(C7⋊D7)—22no—
103C22⋊(C7⋊D7)—22no—
104C2×C7⋊Dic7—22no—
105C2×C7⋊Dic7C2×C3×C411yescharacteristic, complemented
106C4⋊(C7⋊D7)—22no—
107C4×C7⋊D7—21no—
108C4×C7⋊D7—21no—
109C4×C7⋊D7—44no—
110C23×AGL(1,𝔽7)—72no—
111C23×AGL(1,𝔽7)—72no—
112C2×D4×ΓL(1,𝔽8)—142no—
113C2×(C22⋊D7)⋊C3—72no—
114C2×(C22⋊D7)⋊C3—72no—
115D4×AGL(1,𝔽7)—144no—
116D4×AGL(1,𝔽7)—144no—
117C2×C4⋊AGL(1,𝔽7)—72no—
118C2×C4×AGL(1,𝔽7)—72no—
119C2×C4×AGL(1,𝔽7)—144no—
120C22⋊C4×ΓL(1,𝔽8)—144no—
121(C22⋊Dic7)⋊C3—74no—
122(C2×AGL(1,𝔽7))⋊C4—144no—
123C2×#3862—22no—
124C2×#3862—21no—
125C2×#3862—41no—
126C2×#3862C2×C22⋊C411yescharacteristic
127#64425—21no—
128#64425—22no—
129#64425—41no—
130#64425#9912yesdirect product factor
131C2×D4×D7—72no—
132C2×D4×D7—212no—
133D7×C22⋊C4—74no—
134D7×C22⋊C4—214no—
135C7×#99—32no—
136C7×#99AGL(1,𝔽7)12yescomplemented
137#6408—72no—
138#6408—212no—
139#6380—74no—
140#6380—214no—
141C22×C72—21no—
142C22×C72—41no—
143C22×C72—41no—
144C22×C72C22×C3×C411yescharacteristic
145C22×C72C2×C3×D412yes—
146C2×C7⋊D7—24no—
147C2×C7⋊D7—22no—
148C2×C7⋊D7—21no—
149C2×C7⋊D7—41no—
150C2×C7⋊D7—42no—
151C2×C7⋊D7—42no—
152C2×C7⋊D7—42no—
153C2×C7⋊D7—41no—
154C2×C7⋊D7C3×C22⋊C411yescharacteristic
155C2×C7⋊D7C3×C22⋊C412yes—
156C2×C7⋊D7C3×C22⋊C411yescharacteristic
157C4×C72—21no—
158C4×C72—42no—
159C7⋊Dic7—21no—
160C7⋊Dic7—42no—
161C2×C3×#99—491noHall
162C23×ΓL(1,𝔽8)—142no—
163C23×ΓL(1,𝔽8)—142no—
164C22×AGL(1,𝔽7)—72no—
165C22×AGL(1,𝔽7)—74no—
166C22×AGL(1,𝔽7)—72no—
167C22×AGL(1,𝔽7)—74no—
168C22×AGL(1,𝔽7)—72no—
169C22×AGL(1,𝔽7)—142no—
170C22×AGL(1,𝔽7)—142no—
171C22×AGL(1,𝔽7)—284no—
172C22×AGL(1,𝔽7)—284no—
173D4×ΓL(1,𝔽8)—284no—
174C2×C4×ΓL(1,𝔽8)—142no—
175C2×C4×ΓL(1,𝔽8)—284no—
176(C22⋊D7)⋊C3—144no—
177(C22⋊D7)⋊C3—144no—
178C2×ΓL(1,𝔽8)⋊C4—72no—
179C2×ΓL(1,𝔽8)⋊C4—144no—
180C4⋊AGL(1,𝔽7)—144no—
181C4×AGL(1,𝔽7)—142no—
182C4×AGL(1,𝔽7)—142no—
183C4×AGL(1,𝔽7)—288no—
184#3862C2×#9911yescharacteristic, complemented, Hall
185C23×D7—72no—
186C23×D7—72no—
187C23×D7—212no—
188C23×D7—212no—
189C2×C7×D4—32no—
190C2×C7×D4C2×AGL(1,𝔽7)12yes—
191C2×C22⋊D7—72no—
192C2×C22⋊D7—72no—
193C2×C22⋊D7—212no—
194C2×C22⋊D7—212no—
195D4×D7—144no—
196D4×D7—144no—
197D4×D7—424no—
198D4×D7—424no—
199C2×D28—72no—
200C2×D28—212no—
201C2×C4×D7—72no—
202C2×C4×D7—144no—
203C2×C4×D7—212no—
204C2×C4×D7—424no—
205C7×C22⋊C4—34no—
206C7×C22⋊C4C2×AGL(1,𝔽7)14yescomplemented
207C22⋊Dic7—74no—
208C22⋊Dic7—214no—
209(C2×D7)⋊C4—144no—
210(C2×D7)⋊C4—424no—
211C2×C72—22no—
212C2×C72—21no—
213C2×C72—41no—
214C2×C72C2×C3×C22⋊C411yescharacteristic
215C7⋊D7—21no—
216C7⋊D7—22no—
217C7⋊D7—41no—
218C7⋊D7C3×#9912yescomplemented
219C22×C3×D4—491no—
220C2×C3×C22⋊C4—492no—
221C3×#99—491no—
222C3×#99—492no—
223C3×#99—981no—
224C22×ΓL(1,𝔽8)—142no—
225C22×ΓL(1,𝔽8)—144no—
226C22×ΓL(1,𝔽8)—282no—
227C22×ΓL(1,𝔽8)—562no—
228C22×ΓL(1,𝔽8)—562no—
229C2×AGL(1,𝔽7)—72no—
230C2×AGL(1,𝔽7)—72no—
231C2×AGL(1,𝔽7)—74no—
232C2×AGL(1,𝔽7)—148no—
233C2×AGL(1,𝔽7)—144no—
234C2×AGL(1,𝔽7)—142no—
235C2×AGL(1,𝔽7)—284no—
236C2×AGL(1,𝔽7)—282no—
237C2×AGL(1,𝔽7)—284no—
238C2×AGL(1,𝔽7)—284no—
239C2×AGL(1,𝔽7)—282no—
240C4×ΓL(1,𝔽8)—282no—
241C4×ΓL(1,𝔽8)—564no—
242ΓL(1,𝔽8)⋊C4—142no—
243ΓL(1,𝔽8)⋊C4—284no—
244C2×#99—491noSylow
245C23×C7—32no—
246C23×C7—32no—
247C23×C7C4×AGL(1,𝔽7)12yescomplemented
248C23×C7C22×AGL(1,𝔽7)12yes—
249C22×D7—72no—
250C22×D7—74no—
251C22×D7—72no—
252C22×D7—74no—
253C22×D7—72no—
254C22×D7—142no—
255C22×D7—142no—
256C22×D7—212no—
257C22×D7—214no—
258C22×D7—212no—
259C22×D7—214no—
260C22×D7—212no—
261C22×D7—284no—
262C22×D7—284no—
263C22×D7—422no—
264C22×D7—422no—
265C22×D7—844no—
266C22×D7—844no—
267C7×D4—24no—
268C7×D4—64no—
269C2×C4×C7—24no—
270C2×C4×C7—32no—
271C2×C4×C7—64no—
272C2×C4×C7C4×AGL(1,𝔽7)12yescomplemented
273C22⋊D7—144no—
274C22⋊D7—144no—
275C22⋊D7—424no—
276C22⋊D7—424no—
277C2×Dic7—72no—
278C2×Dic7—144no—
279C2×Dic7—212no—
280C2×Dic7—424no—
281D28—144no—
282D28—424no—
283C4×D7—142no—
284C4×D7—142no—
285C4×D7—288no—
286C4×D7—422no—
287C4×D7—422no—
288C4×D7—848no—
289C72C2×C3×#9911yescharacteristic, complemented, Sylow
290C24×C3—491no—
291C24×C3—491no—
292C2×C3×D4—491no—
293C2×C3×D4—491no—
294C2×C3×D4—491no—
295C2×C3×D4—982no—
296C2×C3×D4—982no—
297C2×C3×D4—981no—
298C22×C3×C4—491no—
299C22×C3×C4—982no—
300C3×C22⋊C4—492no—
301C3×C22⋊C4—982no—
302C3×C22⋊C4—982no—
303C2×ΓL(1,𝔽8)—142no—
304C2×ΓL(1,𝔽8)—284no—
305C2×ΓL(1,𝔽8)—282no—
306C2×ΓL(1,𝔽8)—562no—
307AGL(1,𝔽7)—74no—
308AGL(1,𝔽7)—144no—
309AGL(1,𝔽7)—142no—
310AGL(1,𝔽7)—282no—
311C22×D4—491no—
312C2×C22⋊C4—492no—
313#99—491no—
314#99—492no—
315#99#6442511yescharacteristic, direct product factor
316C22×C7—22no—
317C22×C7—34no—
318C22×C7—32no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7—62no—
322C22×C7—122no—
323C22×C7—122no—
324C22×C7C2×C4×AGL(1,𝔽7)12yes—
325C22×C7D4×AGL(1,𝔽7)14yes—
326C2×D7—72no—
327C2×D7—72no—
328C2×D7—74no—
329C2×D7—148no—
330C2×D7—144no—
331C2×D7—142no—
332C2×D7—212no—
333C2×D7—212no—
334C2×D7—214no—
335C2×D7—284no—
336C2×D7—284no—
337C2×D7—282no—
338C2×D7—284no—
339C2×D7—282no—
340C2×D7—428no—
341C2×D7—424no—
342C2×D7—422no—
343C2×D7—844no—
344C2×D7—842no—
345C2×D7—844no—
346C2×D7—844no—
347C2×D7—842no—
348C4×C7—22no—
349C4×C7—44no—
350C4×C7—62no—
351C4×C7—124no—
352Dic7—142no—
353Dic7—284no—
354Dic7—422no—
355Dic7—844no—
356C23×C3—491no—
357C23×C3—492no—
358C23×C3—492no—
359C23×C3—491no—
360C23×C3—491no—
361C23×C3—981no—
362C23×C3—981no—
363C23×C3—981no—
364C23×C3—981no—
365C23×C3—1962no—
366C23×C3—1962no—
367C3×D4—982no—
368C3×D4—982no—
369C3×D4—982no—
370C3×D4—1962no—
371C2×C3×C4—491no—
372C2×C3×C4—981no—
373C2×C3×C4—981no—
374C2×C3×C4—982no—
375C2×C3×C4—981no—
376C2×C3×C4—1962no—
377C2×C3×C4—1964no—
378ΓL(1,𝔽8)—142no—
379C24—491no—
380C24—491no—
381C2×D4—491no—
382C2×D4—491no—
383C2×D4—491no—
384C2×D4—982no—
385C2×D4—982no—
386C2×D4C2×#6442511yescharacteristic
387C22×C4—491no—
388C22×C4—982no—
389C22⋊C4—492no—
390C22⋊C4—982no—
391C22⋊C4C2×#6442512yescomplemented
392C2×C7—24no—
393C2×C7—22no—
394C2×C7—32no—
395C2×C7—42no—
396C2×C7—64no—
397C2×C7—62no—
398C2×C7—122no—
399C2×C7C22⋊C4×AGL(1,𝔽7)12yes—
400D7—74no—
401D7—142no—
402D7—144no—
403D7—214no—
404D7—282no—
405D7—424no—
406D7—422no—
407D7—842no—
408C22×C3—491no—
409C22×C3—491no—
410C22×C3—492no—
411C22×C3—984no—
412C22×C3—982no—
413C22×C3—981no—
414C22×C3—982no—
415C22×C3—981no—
416C22×C3—3921no—
417C22×C3—3921no—
418C22×C3—1961no—
419C22×C3—1962no—
420C22×C3—1962no—
421C22×C3—1961no—
422C22×C3—1962no—
423C22×C3—1961no—
424C3×C4—981no—
425C3×C4—3922no—
426C3×C4—1961no—
427C3×C4—1962no—
428C23—492no—
429C23—491no—
430C23—492no—
431C23—491no—
432C23—491no—
433C23—981no—
434C23—981no—
435C23—1962no—
436C23—1962no—
437C23C4×#6442511yescharacteristic, complemented
438C23C22×#6442511yescharacteristic
439D4—22no—
440D4—982no—
441D4—982no—
442D4—982no—
443C2×C4—22no—
444C2×C4—491no—
445C2×C4—981no—
446C2×C4—981no—
447C2×C4—982no—
448C2×C4—1964no—
449C2×C4C4×#6442511yescharacteristic, complemented
450C7—32no—
451C7#99×AGL(1,𝔽7)12yescomplemented
452C2×C3—492no—
453C2×C3—981no—
454C2×C3—982no—
455C2×C3—981no—
456C2×C3—3921no—
457C2×C3—1962no—
458C2×C3—1961no—
459C2×C3—1961no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—491no—
464C22—491no—
465C22—492no—
466C22—984no—
467C22—982no—
468C22—981no—
469C22—1962no—
470C22—1961no—
471C22—1961no—
472C22—1962no—
473C22—1962no—
474C22D4×#6442512yes—
475C22C2×C4×#6442511yescharacteristic
476C4—21no—
477C4—42no—
478C4—981no—
479C4—1962no—
480C3—981noSylow
481C2—22no—
482C2—21no—
483C2—41no—
484C2—492no—
485C2—981no—
486C2—982no—
487C2—1961no—
488C2C22⋊C4×#6442511yescharacteristic
subgroupG/H
centerC2C22⋊C4×#64425
commutatorC22×C72C22×C3×C4
FrattiniC23C22×#64425
FittingC72×#99C2×C3
socleC2×C72C2×C3×C22⋊C4
prime radicalC2×D4×#64425C2
cosocleC23×C72C23×C3
hypercenter#99#64425
hypocenterC72C2×C3×#99
LCSG ▷ C22×C72 ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #72246200C2semidirect
2. #72246199C2semidirect
3. #99×#3862C2semidirect
4. #99×C7⋊D7C3semidirect
5. C22⋊C4×#64425C2semidirect
7. #72246062C22semidirect
8. C22⋊C4×#3862C22semidirect
10. C2×#11924222C4semidirect
12. C23×#64425C4semidirect
14. C2×C4⋊#64425C4semidirect
17. C2×C4×#64425C4semidirect
21. C22⋊C4×C7⋊D7C2×C3semidirect
22. C72×#99C2×C3semidirect
23. #13096150C2×C3semidirect
24. #13096124C2×C3semidirect
26. C23×#3862C2×C4semidirect
34. C22×#64425C2×C4semidirect
39. C2×C4×#3862C2×C4semidirect
43. C2×#10596875C2×C4semidirect
48. C23×C7⋊D7C3×C4semidirect
51. C2×C22⋊(C7⋊D7)C3×C4semidirect
55. C2×C4⋊(C7⋊D7)C3×C4semidirect
57. C2×C4×C7⋊D7C3×C4semidirect
58. C72×C22⋊C4C22×C3semidirect
59. C22⋊(C7⋊Dic7)C22×C3semidirect
88. C23×C72C2×C3×C4semidirect
96. C22×C7⋊D7C2×C3×C4semidirect
101. C2×C4×C72C2×C3×C4semidirect
105. C2×C7⋊Dic7C2×C3×C4semidirect
130. #64425#99direct (N × H)
136. C7×#99AGL(1,𝔽7)semidirect
184. #3862C2×#99semidirect
206. C7×C22⋊C4C2×AGL(1,𝔽7)semidirect
218. C7⋊D7C3×#99semidirect
247. C23×C7C4×AGL(1,𝔽7)semidirect
272. C2×C4×C7C4×AGL(1,𝔽7)semidirect
289. C72C2×C3×#99semidirect
315. #99#64425direct (N × H)
391. C22⋊C4C2×#64425semidirect
437. C23C4×#64425semidirect
449. C2×C4C4×#64425semidirect
451. C7#99×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
#722462002
#722461992
#99×#38622
#99×C7⋊D73
C22⋊C4×#644252
C2×D4×#644252
#99×AGL(1,𝔽7)7