Finite Group Databasebeta

← #13248842

Subgroups — #13248842

Order 1792.

768 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89765C211yesmaximal, characteristic
2#11731450C211yesmaximal, characteristic
3C7×#1823C211yesmaximal, characteristic, complemented
4#11729307C211yesmaximal, characteristic, complemented
5C2×#89263C211yesmaximal, characteristic
6#11725481C211yesmaximal, characteristic, complemented
7#11725430C211yesmaximal, characteristic, complemented
8C22×C7×C4⋊C4C2211yescharacteristic
9C22×(C2×D7)⋊C4—21no—
10C22×(C2×D7)⋊C4C2211yescharacteristic
11C2×D7×C22⋊C4—21no—
12C2×C7×C22⋊C8C2211yescharacteristic
13C7×#380C2211yescharacteristic
14#89984C2211yescharacteristic
15C2×C7⋊#95—21no—
16C2×C7⋊#95C2211yescharacteristic
17C2×C22⋊(C7⋊C8)C2211yescharacteristic
18#89765—22no—
19#89765C414yescomplemented
20#89761—22no—
21#89714—21no—
22#89440—22no—
23C7×#340—22no—
24#89324—22no—
25#89263—22no—
26#9025—71nomaximal, Sylow
27C24×D7D411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7C2311yescharacteristic
30C23×Dic7—21no—
31C23×Dic7D411yescharacteristic
32C22×C4×D7—22no—
33C22×C7×C8—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4C2×C412yescomplemented
40C2×C7×C4⋊C4D412yes—
41C2×C7×C22⋊C4—21no—
42C2×C22⋊Dic7—21no—
43C2×(C2×D7)⋊C4—21no—
44C2×(C2×D7)⋊C4—21no—
45C2×(C2×D7)⋊C4—21no—
46C2×(C2×D7)⋊C4—21no—
47C2×(C2×D7)⋊C4—22no—
48C2×(C2×D7)⋊C4—24no—
49C2×(C2×D7)⋊C4—42no—
50C2×(C2×D7)⋊C4—42no—
51C2×(C2×D7)⋊C4C2×C412yes—
52C2×(C2×D7)⋊C4D412yes—
53C2×C4⋊Dic7D412yes—
54C2×Dic7⋊C4—22no—
55C22×C7⋊C8—21no—
56D7×C22⋊C4—42no—
57D7×C22⋊C4—42no—
58C7×C22⋊C8—21no—
59C7×C22⋊C8—21no—
60C7×#95—21no—
61C7⋊#95—22no—
62C7⋊#95—21no—
63C7⋊#95—21no—
64C7⋊#95—24no—
65C7⋊#95C2×C412yescomplemented
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)—21no—
68C2×#397—71no—
69#2315—71no—
70#1823—71no—
71#1823D711yescharacteristic, complemented
72C2×#323—71no—
73#1294—71no—
74#1294—71no—
75C24×C7C2×D411yescharacteristic
76C23×D7—21no—
77C23×D7—21no—
78C23×D7—21no—
79C23×D7—21no—
80C23×D7—42no—
81C23×D7—42no—
82C23×D7C22⋊C412yes—
83C23×D7D812yes—
84C23×D7SD1612yes—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—42no—
88C22×C4×C7—41no—
89C22×C4×C7—41no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7C22×C411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7#4111yescharacteristic
99C22×C4×C7#4111yescharacteristic
100C22×Dic7—21no—
101C22×Dic7—21no—
102C22×Dic7—22no—
103C22×Dic7—22no—
104C22×Dic7—42no—
105C22×Dic7—42no—
106C22×Dic7—42no—
107C22×Dic7—41no—
108C22×Dic7—41no—
109C22×Dic7C22⋊C412yes—
110C2×C4×D7—44no—
111C2×C4×D7—82no—
112C2×C4×D7—82no—
113C2×C7×C8—42no—
114C2×C7×C8—41no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—42no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—41no—
122C22⋊Dic7—41no—
123C22⋊Dic7—41no—
124(C2×D7)⋊C4—22no—
125(C2×D7)⋊C4—22no—
126(C2×D7)⋊C4—22no—
127(C2×D7)⋊C4—22no—
128(C2×D7)⋊C4—44no—
129(C2×D7)⋊C4—48no—
130C4⋊Dic7—42no—
131Dic7⋊C4—82no—
132C2×C7⋊C8—42no—
133C2×C7⋊C8—41no—
134C22×C4⋊C4C2×D711yescharacteristic
135C22×C22⋊C4—71no—
136C22×C22⋊C4—141no—
137C22×C22⋊C4—141no—
138C2×C22⋊C8—71no—
139C2×C22⋊C8C2×D711yescharacteristic
140#397—74no—
141#397—142no—
142#385—142no—
143#385—142no—
144#380—71no—
145#380—141no—
146#380C2×D711yescharacteristic
147C2×#95—71no—
148C2×#95—141no—
149#340—22no—
150#340—142no—
151#323—142no—
152C23×C7—22no—
153C23×C7—22no—
154C23×C7C2×C22⋊C411yescharacteristic
155C23×C7#12011yescharacteristic
156C23×C7#12311yescharacteristic
157C23×C7C2×D811yescharacteristic
158C23×C7C2×SD1611yescharacteristic
159C23×C7AGL(1,ℤ/8)11yescharacteristic
160C23×C7#13711yescharacteristic
161C22×D7—22no—
162C22×D7—22no—
163C22×D7—22no—
164C22×D7—22no—
165C22×D7—22no—
166C22×D7—22no—
167C22×D7—42no—
168C22×D7—42no—
169C22×D7—42no—
170C22×D7—42no—
171C22×D7—44no—
172C22×D7—44no—
173C22×D7—82no—
174C22×D7—82no—
175C22×D7—81no—
176C22×D7—81no—
177C22×D7#10214yes—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7—81no—
193C2×C4×C7—81no—
194C2×C4×C7C4×D411yescharacteristic
195C2×C4×C7C4×D411yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×Dic7—22no—
199C2×Dic7—42no—
200C2×Dic7—42no—
201C2×Dic7—42no—
202C2×Dic7—42no—
203C2×Dic7—42no—
204C2×Dic7—42no—
205C2×Dic7—42no—
206C2×Dic7—42no—
207C2×Dic7—81no—
208C2×Dic7—82no—
209C2×Dic7—82no—
210C4×D7—88no—
211C7×C8—81no—
212C7⋊C8—81no—
213C25—71no—
214C23×C4—21no—
215C23×C4—71no—
216C23×C4—142no—
217C23×C4—141no—
218C23×C4C22×D711yescharacteristic
219C22×C8—21no—
220C22×C8—141no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—72no—
227C2×C4⋊C4—142no—
228C2×C4⋊C4C4×D712yescomplemented
229C2×C4⋊C4C22⋊D712yes—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—72no—
232C2×C22⋊C4—72no—
233C2×C22⋊C4—141no—
234C2×C22⋊C4—141no—
235C2×C22⋊C4—141no—
236C2×C22⋊C4—141no—
237C2×C22⋊C4—144no—
238C2×C22⋊C4—142no—
239C2×C22⋊C4—141no—
240C2×C22⋊C4—282no—
241C2×C22⋊C4—282no—
242C2×C22⋊C4—282no—
243C2×C22⋊C4—282no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—141no—
247C22⋊C8—141no—
248#95—21no—
249#95—72no—
250#95—144no—
251#95—142no—
252#95—141no—
253#95—141no—
254C22×C7—22no—
255C22×C7—22no—
256C22×C7—22no—
257C22×C7—22no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7#38611yescharacteristic
263C22×C7C2×#10211yescharacteristic
264C22×C7#41711yescharacteristic
265C22×C7#44711yescharacteristic
266C22×C7#44811yescharacteristic
267C22×C7#45512yes—
268C22×C7#45812yes—
269C22×C7#48011yescharacteristic
270C22×C7#48111yescharacteristic
271C2×D7—24no—
272C2×D7—44no—
273C2×D7—44no—
274C2×D7—42no—
275C2×D7—42no—
276C2×D7—44no—
277C2×D7—82no—
278C2×D7—82no—
279C2×D7—82no—
280C4×C7—21no—
281C4×C7—21no—
282C4×C7—21no—
283C4×C7—21no—
284C4×C7—82no—
285C4×C7—82no—
286Dic7—42no—
287Dic7—82no—
288Dic7—82no—
289C24—72no—
290C24—72no—
291C24—72no—
292C24—141no—
293C24—141no—
294C24—141no—
295C24—141no—
296C24—282no—
297C24—282no—
298C24D4×D711yescharacteristic
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—72no—
309C22×C4—141no—
310C22×C4—141no—
311C22×C4—142no—
312C22×C4—142no—
313C22×C4—284no—
314C22×C4—282no—
315C22×C4—282no—
316C22×C4—282no—
317C22×C4—281no—
318C22×C4—281no—
319C22×C4—562no—
320C22×C4—562no—
321C22×C4C2×C4×D711yescharacteristic
322C22×C4#115711yescharacteristic
323C22×C4D4×D711yescharacteristic
324C22×C4D4×D711yescharacteristic
325C22×C4#115911yescharacteristic
326C22×C4C2×C22⋊D711yescharacteristic
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—282no—
330C2×C8—281no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—282no—
337C4⋊C4—562no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—142no—
341C22⋊C4—142no—
342C22⋊C4—142no—
343C22⋊C4—142no—
344C22⋊C4—288no—
345C22⋊C4—284no—
346C22⋊C4—281no—
347C22⋊C4—281no—
348C2×C7—21no—
349C2×C7—21no—
350C2×C7—21no—
351C2×C7—21no—
352C2×C7#160712yes—
353C2×C7#160812yes—
354C2×C7#187411yescharacteristic
355C2×C7#188511yescharacteristic
356C2×C7#189011yescharacteristic
357D7—44no—
358C23—22no—
359C23—22no—
360C23—74no—
361C23—142no—
362C23—142no—
363C23—142no—
364C23—142no—
365C23—142no—
366C23—142no—
367C23—284no—
368C23—284no—
369C23—282no—
370C23—282no—
371C23—282no—
372C23—282no—
373C23—562no—
374C23—562no—
375C23—561no—
376C23—561no—
377C23D7×C22⋊C411yescharacteristic
378C23#644611yescharacteristic
379C23D7×D811yescharacteristic
380C23#647411yescharacteristic
381C23D7×SD1611yescharacteristic
382C23#647811yescharacteristic
383C23#650011yescharacteristic
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—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—142no—
401C2×C4—282no—
402C2×C4—282no—
403C2×C4—282no—
404C2×C4—282no—
405C2×C4—282no—
406C2×C4—282no—
407C2×C4—282no—
408C2×C4—282no—
409C2×C4—568no—
410C2×C4—561no—
411C2×C4—562no—
412C2×C4—562no—
413C2×C4(C22⋊D7)⋊C411yescharacteristic
414C2×C4#644711yescharacteristic
415C2×C4C4×C22⋊D711yescharacteristic
416C2×C4#650111yescharacteristic
417C8—81no—
418C8—561no—
419C7#902511yescharacteristic, complemented, Sylow
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—144no—
429C22—284no—
430C22—284no—
431C22—284no—
432C22—282no—
433C22—282no—
434C22—562no—
435C22—562no—
436C22—562no—
437C22D7×#10211yescharacteristic
438C22#8954311yescharacteristic
439C22#8954611yescharacteristic
440C22#8954911yescharacteristic
441C22#8972811yescharacteristic
442C22#8982512yes—
443C22#8992811yescharacteristic
444C22#8994211yescharacteristic
445C22#8997412yes—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—82no—
451C4—82no—
452C4—282no—
453C4—562no—
454C4—562no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—284no—
460C2#1172729211yescharacteristic
461C2#1172795411yescharacteristic
462C2#1172809711yescharacteristic
463C2#1172891812yes—
464C2#1172892612yes—
subgroupG/H
centerC23#6500
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1823C2
socleC23×C7#120
prime radicalC24×D7D4
cosocleC23×C4×C7C23
hypercenter#1823D7
hypocenterC7#9025
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. C7×#1823C2semidirect
4. #11729307C2semidirect
6. #11725481C2semidirect
7. #11725430C2semidirect
19. #89765C4semidirect
39. C2×C7×C4⋊C4C2×C4semidirect
65. C7⋊#95C2×C4semidirect
71. #1823D7semidirect
228. C2×C4⋊C4C4×D7semidirect
419. C7#9025semidirect

Maximal subgroups

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

subgroupindex
C2×#897652
#117314502
C7×#18232
#117293072
C2×#892632
#117254812
#117254302
#90257