Finite Group Databasebeta

← #13411930

Subgroups — #13411930

Order 1792.

734 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2552C211yesmaximal, characteristic, complemented
2#11731083C211yesmaximal, characteristic
3#11729497C212yesmaximal
4#11729492C212yesmaximal, complemented
5#11727384C212yesmaximal
6#11727357C212yesmaximal, complemented
7#11727338C211yesmaximal, characteristic
8#11727328C211yesmaximal, characteristic, complemented
9#11727238C211yesmaximal, characteristic, complemented
10C4×#6383C211yesmaximal, characteristic
11#11727220C211yesmaximal, characteristic, complemented
12C2×C7×#125C2212yes—
13C2×#6455—21no—
14C2×C4×D28C2211yescharacteristic
15C7×#398—22no—
16C7×#389—21no—
17C7×#389C2212yes—
18C7×#384—21no—
19C7×#384C2211yescharacteristic
20C4×C7×C4⋊C4C2211yescharacteristic
21C4×C7×C4⋊C4C2211yescharacteristic
22#89759—21no—
23C2×#6383C2211yescharacteristic
24C2×#6383C2211yescharacteristic
25C2×#6383C2212yes—
26C2×#6381C2212yes—
27#89711C2212yes—
28#89707C2211yescharacteristic
29C4×C4⋊Dic7C2212yes—
30C2×C4⋊(C7⋊C8)C2211yescharacteristic
31C2×C4⋊(C7⋊C8)C2212yes—
32C2×C4×C7⋊C8C2211yescharacteristic
33#89442—22no—
34#89324C2212yes—
35#89324C2211yescharacteristic
36#89324C2212yes—
37#89324C2212yes—
38#89324C2212yes—
39#89324C2211yescharacteristic
40#89319C2212yes—
41#26025—71nomaximal, Sylow
42C22×D28C2311yescharacteristic
43C22×C4×D7—21no—
44C7×#125—24no—
45C7×#125—42no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—22no—
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4D411yescharacteristic
52C2×C7×C4⋊C4D411yescharacteristic
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C7×C4⋊C4C2311yescharacteristic
56C2×C42×C7—21no—
57C2×C42×C7C2312yes—
58C2×C42×C7C2311yescharacteristic
59C2×(C2×D7)⋊C4—21no—
60C2×(C2×D7)⋊C4—22no—
61C2×(C2×D7)⋊C4—22no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7C2312yes—
65C2×C4⋊Dic7C2311yescharacteristic
66C2×C4×Dic7—22no—
67C22×C7⋊C8C2311yescharacteristic
68C22×C7⋊C8C2311yescharacteristic
69C22×C7⋊C8C2312yes—
70#6455—44no—
71C4×D28—22no—
72C4×D28Q814yes—
73C7×#95—21no—
74C7×#95—22no—
75C7×#95D412yes—
76C7⋊#95—22no—
77#6383—22no—
78#6383—22no—
79#6383—42no—
80#6381—24no—
81C4⋊(C7⋊C8)—22no—
82C4⋊(C7⋊C8)—42no—
83C4×C7⋊C8—22no—
84#2552D711yescharacteristic, complemented
85#2354—71no—
86#2067—72no—
87#2025—72no—
88#1989—72no—
89#1947—71no—
90#1939—71no—
91#1921—72no—
92#1834—71no—
93#1802—71no—
94C4×#102—71no—
95C23×D7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—22no—
99C22×C4×C7C2×D412yes—
100C22×C4×C7C2×D412yes—
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2411yescharacteristic
104C22×Dic7—21no—
105C22×Dic7—22no—
106C2×D28—21no—
107C2×D28—21no—
108C2×D28—21no—
109C2×D28—21no—
110C2×D28—21no—
111C2×D28C2×Q812yes—
112C2×C4×D7—24no—
113C2×C4×D7—42no—
114C2×C4×D7—42no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—44no—
122C7×C4⋊C4—44no—
123C7×C4⋊C4—42no—
124C7×C4⋊C4—81no—
125C7×C4⋊C4—82no—
126C7×C4⋊C4#4112yes—
127C7×C4⋊C4#4112yes—
128C42×C7—22no—
129C42×C7—22no—
130C42×C7—42no—
131C42×C7C2×Q812yes—
132C42×C7#4112yes—
133(C2×D7)⋊C4—24no—
134(C2×D7)⋊C4—44no—
135(C2×D7)⋊C4—82no—
136C4⋊Dic7—21no—
137C4⋊Dic7—22no—
138C4⋊Dic7—22no—
139C4⋊Dic7—42no—
140C4⋊Dic7—81no—
141C4⋊Dic7C2×Q812yes—
142C4×Dic7—44no—
143C2×C7⋊C8—21no—
144C2×C7⋊C8—21no—
145C2×C7⋊C8—22no—
146C2×C7⋊C8—22no—
147C2×C7⋊C8—21no—
148C2×C7⋊C8—21no—
149C2×C7⋊C8—21no—
150C2×C7⋊C8—22no—
151C2×C7⋊C8—21no—
152C2×#125C2×D712yes—
153C2×C4×D4—71no—
154C2×#117—141no—
155C2×#107—72no—
156C2×C4⋊C8—71no—
157C2×C4⋊C8—72no—
158C2×#102—71no—
159C2×#102—71no—
160C2×#102—72no—
161C2×C4×C8—71no—
162#398—22no—
163#390—72no—
164#389—21no—
165#389C2×D712yes—
166#388—142no—
167#384—21no—
168#384—71no—
169#384C2×D711yescharacteristic
170#380—141no—
171C4×C4⋊C4—72no—
172C4×C4⋊C4C2×D711yescharacteristic
173C4×C4⋊C4C2×D711yescharacteristic
174#340—72no—
175#340—72no—
176#340—72no—
177#340—71no—
178#340—72no—
179#340—71no—
180#336—72no—
181C23×C7C22×D411yescharacteristic
182C22×D7—22no—
183C22×D7—41no—
184C22×D7—41no—
185C22×D7—41no—
186C22×D7—41no—
187C22×D7—41no—
188C22×D7—41no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—22no—
192C2×C4×C7—22no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—22no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7—22no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7—42no—
206C2×C4×C7C22⋊Q812yes—
207C2×C4×C7C22⋊Q812yes—
208C2×C4×C7#13512yes—
209C2×C4×C7AGL(1,ℤ/8)12yes—
210C2×C4×C7C22×Q811yescharacteristic
211C2×C4×C7C2×#4111yescharacteristic
212C2×C4×C7C2×#4111yescharacteristic
213C2×C4×C7C2×#4111yescharacteristic
214C2×C4×C7H(5,𝔽2)11yescharacteristic
215C2×C4×C72−1+411yescharacteristic
216C2×Dic7—22no—
217C2×Dic7—41no—
218C2×Dic7—41no—
219C2×Dic7—42no—
220C2×Dic7—42no—
221C2×Dic7—42no—
222D28—22no—
223D28—22no—
224D28—41no—
225D28—41no—
226C4×D7—48no—
227C7⋊C8—41no—
228C7⋊C8—42no—
229C7⋊C8—41no—
230C22×D4—71no—
231C23×C4—141no—
232C22×C8—71no—
233C22×C8—72no—
234C22×C8—71no—
235#125—24no—
236#125—42no—
237C4×D4—74no—
238C4×D4—142no—
239#117—284no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—71no—
246C2×C4⋊C4—72no—
247C2×C4⋊C4—141no—
248C2×C4⋊C4—141no—
249C2×C4⋊C4C22⋊D711yescharacteristic
250C2×C4⋊C4C22⋊D711yescharacteristic
251C2×C4⋊C4C22×D712yes—
252C2×C4⋊C4C22×D711yescharacteristic
253C2×C4⋊C4C22×D711yescharacteristic
254C2×C22⋊C4—141no—
255C2×C22⋊C4—142no—
256C2×C22⋊C4—142no—
257C2×C42—21no—
258C2×C42—142no—
259C2×C42C22×D712yes—
260C2×C42C22×D711yescharacteristic
261#107—144no—
262C4⋊C8—142no—
263C4⋊C8—282no—
264#102—142no—
265#102—142no—
266#102—282no—
267C4×C8—142no—
268#95—21no—
269#95—22no—
270#95—142no—
271#95C22⋊D712yes—
272C22×C7C2×C22⋊Q811yescharacteristic
273C22×C7#54611yescharacteristic
274C22×C7#54811yescharacteristic
275C22×C7C2×#13511yescharacteristic
276C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C7#57611yescharacteristic
278C22×C7#57811yescharacteristic
279C2×D7—42no—
280C2×D7—42no—
281C2×D7—42no—
282C2×D7—81no—
283C2×D7—81no—
284C2×D7—81no—
285C2×D7—81no—
286C4×C7—22no—
287C4×C7—42no—
288C4×C7—42no—
289C4×C7—42no—
290C4×C7—42no—
291C4×C7—82no—
292C4×C7#47814yes—
293C4×C7#53311yescharacteristic
294C4×C7#54411yescharacteristic
295C4×C7#55511yescharacteristic
296C4×C7#55611yescharacteristic
297Dic7—42no—
298Dic7—82no—
299C24—141no—
300C2×D4—72no—
301C2×D4—141no—
302C2×D4—141no—
303C2×D4—141no—
304C2×D4—141no—
305C2×D4—141no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—141no—
310C22×C4—144no—
311C22×C4—142no—
312C22×C4—282no—
313C22×C4—282no—
314C22×C4C2×C22⋊D712yes—
315C22×C4C2×C22⋊D712yes—
316C22×C4C2×C22⋊D711yescharacteristic
317C22×C4C2×C22⋊D711yescharacteristic
318C22×C4C23×D711yescharacteristic
319C2×C8—141no—
320C2×C8—142no—
321C2×C8—141no—
322C2×C8—141no—
323C2×C8—141no—
324C2×C8—141no—
325C2×C8—142no—
326C2×C8—142no—
327C2×C8—141no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—72no—
338C4⋊C4—81no—
339C4⋊C4—82no—
340C4⋊C4—141no—
341C4⋊C4—142no—
342C4⋊C4—142no—
343C4⋊C4—282no—
344C4⋊C4—561no—
345C4⋊C4#115912yes—
346C4⋊C4#116112yes—
347C22⋊C4—144no—
348C22⋊C4—284no—
349C22⋊C4—562no—
350C42—22no—
351C42—22no—
352C42—42no—
353C42—284no—
354C42Q8×D712yes—
355C42#116112yes—
356C2×C7#243911yescharacteristic
357C2×C7C2×#47811yescharacteristic
358C2×C7#308311yescharacteristic
359C2×C7#330511yescharacteristic
360C2×C7#333011yescharacteristic
361C2×C7#336411yescharacteristic
362C2×C7#336611yescharacteristic
363D7—82no—
364C23—142no—
365C23—281no—
366C23—281no—
367C23—281no—
368C23—281no—
369C23—281no—
370C23—281no—
371C23C22×C22⋊D711yescharacteristic
372D4—142no—
373D4—142no—
374D4—281no—
375D4—281no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—142no—
394C2×C4—281no—
395C2×C4—281no—
396C2×C4—288no—
397C2×C4—282no—
398C2×C4—282no—
399C2×C4—282no—
400C2×C4#650912yes—
401C2×C4#650912yes—
402C2×C4#651212yes—
403C2×C4#651312yes—
404C2×C4C2×#115911yescharacteristic
405C2×C4#654811yescharacteristic
406C2×C4C2×Q8×D711yescharacteristic
407C2×C4C2×#116111yescharacteristic
408C2×C4C2×#116111yescharacteristic
409C2×C4#655111yescharacteristic
410C8—281no—
411C8—282no—
412C8—281no—
413C7#2602511yescharacteristic, complemented, Sylow
414C22—282no—
415C22—282no—
416C22—282no—
417C22—561no—
418C22—561no—
419C22—561no—
420C22—561no—
421C22#9049711yescharacteristic
422C22C2×#650911yescharacteristic
423C22#9050811yescharacteristic
424C22C2×#651211yescharacteristic
425C22C2×#651311yescharacteristic
426C22#9052611yescharacteristic
427C22#9052911yescharacteristic
428C4—22no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—82no—
434C4—282no—
435C4—562no—
436C4#8983814yes—
437C4#9041911yescharacteristic
438C4#9042011yescharacteristic
439C4#9042311yescharacteristic
440C4#9042511yescharacteristic
441C2—562no—
442C2C2×#8983811yescharacteristic
443C2#1174070111yescharacteristic
444C2#1174092011yescharacteristic
445C2#1174095911yescharacteristic
446C2#1174101611yescharacteristic
447C2#1174108311yescharacteristic
448C2#1174113411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2552C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2552D7
hypocenterC7#26025
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2552

Complements

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

NH ≅ G/Ntype
1. C7×#2552C2semidirect
4. #11729492C2semidirect
6. #11727357C2semidirect
8. #11727328C2semidirect
9. #11727238C2semidirect
11. #11727220C2semidirect
84. #2552D7semidirect
413. C7#26025semidirect

Maximal subgroups

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

subgroupindex
C7×#25522
#117310832
#117294972
#117294922
#117273842
#117273572
#117273382
#117273282
#117272382
C4×#63832
#117272202
#260257