Finite Group Databasebeta

← #13371470

Subgroups — #13371470

Order 1792.

838 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6391C211yesmaximal, characteristic
2#11730902C211yesmaximal, characteristic
3#11730901C211yesmaximal, characteristic
4C7×#1752C211yesmaximal, characteristic
5#11729725C211yesmaximal, characteristic
6#11727679C211yesmaximal, characteristic
7#11727678C211yesmaximal, characteristic
8#11727574C212yesmaximal, complemented
9#11727568C212yesmaximal, complemented
10#11726377C212yesmaximal, complemented
11#11726376C212yesmaximal, complemented
12C23×C7×C8C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C2×#6440—21no—
16C2×C7×C22⋊C8C2211yescharacteristic
17C2×C7×#97C2212yes—
18C4×C7×C22⋊C4C2211yescharacteristic
19#89984C2211yescharacteristic
20#89984C2211yescharacteristic
21C2×#6392C2212yes—
22C2×#6391C418yescomplemented
23C2×#6391C2211yescharacteristic
24C2×#6391C2211yescharacteristic
25C2×#6391C2211yescharacteristic
26C2×#6391C2211yescharacteristic
27#89715—21no—
28#89714—21no—
29C22⋊C4×Dic7—21no—
30C4×C4⋊Dic7C2212yes—
31C4×C4⋊Dic7C2212yes—
32#89700C2212yes—
33#89700C2212yes—
34#89426—22no—
35#89425—22no—
36C7×#336C2212yes—
37#89346C2212yes—
38#89346C2212yes—
39#89346C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#22615—71nomaximal, Sylow
43C23×C4×C7C2311yescharacteristic
44C23×Dic7—21no—
45C23×Dic7—21no—
46C22×C7×C8—24no—
47C22×C7×C8C2×C414yes—
48C22×C7×C8C2312yes—
49C22×C7×C8C2311yescharacteristic
50C22×C7×C8C2311yescharacteristic
51C2×C7×C22⋊C4D411yescharacteristic
52C2×C7×C22⋊C4D411yescharacteristic
53C2×C42×C7C2312yes—
54C2×C22⋊Dic7—21no—
55C2×C22⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—21no—
58C2×C4⋊Dic7—22no—
59C2×C4⋊Dic7—22no—
60C2×C4⋊Dic7C2×C414yes—
61C2×C4⋊Dic7C2×C414yes—
62C2×C4⋊Dic7C2312yes—
63C2×C4⋊Dic7C2312yes—
64C2×C4⋊Dic7C2311yescharacteristic
65C2×C4⋊Dic7C2311yescharacteristic
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2311yescharacteristic
68C2×Dic7⋊C4—22no—
69C2×Dic7⋊C4—22no—
70C2×C4×Dic7—22no—
71C2×C4×Dic7—22no—
72#6440—44no—
73C7×C22⋊C8—22no—
74C7×#97—24no—
75C7×#95D412yes—
76C7⋊#95—21no—
77C7⋊#95—22no—
78C7⋊#95—22no—
79C7⋊#95—21no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82#6392—24no—
83#6391C2×C414yes—
84#6391C2×C414yes—
85#6391C2×C414yes—
86#6391C2×C414yes—
87C22×#106—71no—
88#2345—71no—
89#2344—71no—
90#1944—72no—
91#1920—72no—
92#1919—72no—
93#1824—71no—
94#1823—71no—
95#1808—71no—
96#1776—72no—
97#1752D711yescharacteristic
98C24×C7C2×D411yescharacteristic
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7C22×C412yes—
102C22×C4×C7C2×D412yes—
103C22×C4×C7C2×D412yes—
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7#4112yes—
106C22×C4×C7C2411yescharacteristic
107C22×Dic7—24no—
108C22×Dic7—21no—
109C22×Dic7—21no—
110C22×Dic7—21no—
111C22×Dic7—21no—
112C22×Dic7—24no—
113C22×Dic7—22no—
114C22×Dic7—22no—
115C22×Dic7—42no—
116C22×Dic7—42no—
117C22×Dic7—82no—
118C2×C7×C8—22no—
119C2×C7×C8—28no—
120C2×C7×C8—22no—
121C2×C7×C8—22no—
122C2×C7×C8—22no—
123C2×C7×C8C22×C412yes—
124C2×C7×C8C22×C412yes—
125C2×C7×C8#4114yes—
126C7×C22⋊C4—22no—
127C7×C22⋊C4—22no—
128C42×C7—22no—
129C42×C7—22no—
130C22⋊Dic7—42no—
131C22⋊Dic7—42no—
132C4⋊Dic7—24no—
133C4⋊Dic7—24no—
134C4⋊Dic7—21no—
135C4⋊Dic7—21no—
136C4⋊Dic7—21no—
137C4⋊Dic7—21no—
138C4⋊Dic7—22no—
139C4⋊Dic7—22no—
140C4⋊Dic7—22no—
141C4⋊Dic7—22no—
142C4⋊Dic7C22×C412yes—
143C4⋊Dic7C22×C412yes—
144C4⋊Dic7C22×C412yes—
145C4⋊Dic7C22×C412yes—
146Dic7⋊C4—44no—
147Dic7⋊C4—82no—
148C4×Dic7—44no—
149C4×Dic7—82no—
150C23×C8C2×D711yescharacteristic
151C2×#117—141no—
152C22×C4⋊C4—71no—
153C22×C4⋊C4—71no—
154C2×#107—72no—
155C2×#106—78no—
156C2×#106—71no—
157C2×#106—71no—
158C2×#106—71no—
159C2×#106—71no—
160C2×C22⋊C8C2×D711yescharacteristic
161C2×#97C2×D712yes—
162#389—72no—
163#389—72no—
164#387—142no—
165#387—142no—
166#381—141no—
167#380—71no—
168#380—71no—
169#380—141no—
170C4×C4⋊C4—72no—
171C4×C4⋊C4—72no—
172C4×C22⋊C4—141no—
173C4×C22⋊C4C2×D711yescharacteristic
174#340—72no—
175#340—72no—
176#340—72no—
177#340—72no—
178#340—72no—
179#336C2×D712yes—
180C23×C7—22no—
181C23×C7—22no—
182C23×C7C4×D412yes—
183C23×C7AGL(1,ℤ/8)12yes—
184C23×C7#13712yes—
185C23×C7C22×D411yescharacteristic
186C2×C4×C7—22no—
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—24no—
190C2×C4×C7—22no—
191C2×C4×C7—22no—
192C2×C4×C7—22no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—22no—
196C2×C4×C7C4×D412yes—
197C2×C4×C7C23×C411yescharacteristic
198C2×C4×C7C2×#4111yescharacteristic
199C2×C4×C7C2×#4112yes—
200C2×C4×C72−1+412yes—
201C2×Dic7—22no—
202C2×Dic7—22no—
203C2×Dic7—22no—
204C2×Dic7—22no—
205C2×Dic7—41no—
206C2×Dic7—41no—
207C2×Dic7—41no—
208C2×Dic7—41no—
209C2×Dic7—42no—
210C2×Dic7—42no—
211C2×Dic7—48no—
212C2×Dic7—42no—
213C2×Dic7—42no—
214C2×Dic7—42no—
215C2×Dic7—42no—
216C2×Dic7—42no—
217C2×Dic7—42no—
218C2×Dic7—84no—
219C7×C8—22no—
220C7×C8—42no—
221C7×C8#11714yescomplemented
222C23×C4—141no—
223C23×C4—141no—
224C23×C4C22×D711yescharacteristic
225C22×C8—24no—
226C22×C8C4×D714yes—
227C22×C8C22×D712yes—
228C22×C8C22×D711yescharacteristic
229C22×C8C22×D711yescharacteristic
230#117—284no—
231C2×C4⋊C4—74no—
232C2×C4⋊C4—74no—
233C2×C4⋊C4—71no—
234C2×C4⋊C4—71no—
235C2×C4⋊C4—71no—
236C2×C4⋊C4—71no—
237C2×C4⋊C4—72no—
238C2×C4⋊C4—72no—
239C2×C4⋊C4—141no—
240C2×C4⋊C4—141no—
241C2×C4⋊C4—142no—
242C2×C4⋊C4—142no—
243C2×C4⋊C4—142no—
244C2×C4⋊C4—142no—
245C2×C22⋊C4—141no—
246C2×C22⋊C4—141no—
247C2×C22⋊C4D2811yescharacteristic
248C2×C22⋊C4D2811yescharacteristic
249C2×C42—142no—
250C2×C42—142no—
251C2×C42C22×D712yes—
252#107—144no—
253#106—74no—
254#106—74no—
255#106—74no—
256#106—74no—
257C22⋊C8—22no—
258#97—24no—
259#95—142no—
260#95—142no—
261#95—142no—
262#95—141no—
263#95—141no—
264#95—142no—
265#95D2812yes—
266C22×C7—24no—
267C22×C7—24no—
268C22×C7—22no—
269C22×C7—22no—
270C22×C7#44014yes—
271C22×C7C2×C4×D411yescharacteristic
272C22×C7#54712yes—
273C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C7C2×#13711yescharacteristic
275C22×C7#57712yes—
276C4×C7—22no—
277C4×C7—42no—
278C4×C7—42no—
279C4×C7C2×#11711yescharacteristic
280C4×C7#51911yescharacteristic
281C4×C7#55212yes—
282Dic7—42no—
283Dic7—42no—
284Dic7—42no—
285Dic7—42no—
286Dic7—82no—
287Dic7—82no—
288C24C2×D2811yescharacteristic
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—144no—
292C22×C4—141no—
293C22×C4—141no—
294C22×C4—141no—
295C22×C4—141no—
296C22×C4—144no—
297C22×C4—142no—
298C22×C4—142no—
299C22×C4—282no—
300C22×C4—282no—
301C22×C4—562no—
302C22×C4C2×C4×D712yes—
303C22×C4C2×D2812yes—
304C22×C4C2×D2812yes—
305C22×C4C2×D2811yescharacteristic
306C22×C4#115712yes—
307C22×C4C23×D711yescharacteristic
308C2×C8—22no—
309C2×C8—28no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8C2×C4×D712yes—
314C2×C8C2×C4×D712yes—
315C2×C8#115914yes—
316C4⋊C4—72no—
317C4⋊C4—72no—
318C4⋊C4—72no—
319C4⋊C4—72no—
320C4⋊C4—144no—
321C4⋊C4—144no—
322C4⋊C4—141no—
323C4⋊C4—141no—
324C4⋊C4—141no—
325C4⋊C4—141no—
326C4⋊C4—142no—
327C4⋊C4—142no—
328C4⋊C4—142no—
329C4⋊C4—142no—
330C4⋊C4—284no—
331C4⋊C4—562no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—282no—
335C22⋊C4—282no—
336C42—22no—
337C42—22no—
338C42—284no—
339C42—562no—
340C2×C7—24no—
341C2×C7#234411yescharacteristic
342C2×C7C2×#44011yescharacteristic
343C2×C7#294911yescharacteristic
344C2×C7#334412yes—
345C2×C7#334512yes—
346C23—22no—
347C23—22no—
348C23C4×D2812yes—
349C23#647112yes—
350C23#647212yes—
351C23C22×D2811yescharacteristic
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—142no—
363C2×C4—142no—
364C2×C4—142no—
365C2×C4—142no—
366C2×C4—281no—
367C2×C4—281no—
368C2×C4—281no—
369C2×C4—281no—
370C2×C4—282no—
371C2×C4—282no—
372C2×C4—288no—
373C2×C4—282no—
374C2×C4—282no—
375C2×C4—282no—
376C2×C4—282no—
377C2×C4—282no—
378C2×C4—282no—
379C2×C4—564no—
380C2×C4C4×D2812yes—
381C2×C4C22×C4×D711yescharacteristic
382C2×C4C2×#115711yescharacteristic
383C2×C4C2×#115912yes—
384C2×C4#655412yes—
385C8—22no—
386C8—42no—
387C8#644014yescomplemented
388C7#2261511yescharacteristic, complemented, Sylow
389C22—24no—
390C22—24no—
391C22—22no—
392C22—22no—
393C22#8948214yes—
394C22C2×C4×D2811yescharacteristic
395C22#9024612yes—
396C22C2×#647111yescharacteristic
397C22C2×#647211yescharacteristic
398C22#9044212yes—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—282no—
403C4—282no—
404C4—282no—
405C4—282no—
406C4—562no—
407C4—562no—
408C4C2×#644011yescharacteristic
409C4#9020711yescharacteristic
410C4#9023212yes—
411C2—24no—
412C2#1173090111yescharacteristic
413C2C2×#8948211yescharacteristic
414C2#1173601711yescharacteristic
415C2#1173691512yes—
416C2#1173691612yes—
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1752C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1752D7
hypocenterC7#22615
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
8. #11727574C2semidirect
9. #11727568C2semidirect
10. #11726377C2semidirect
11. #11726376C2semidirect
22. C2×#6391C4semidirect
221. C7×C8#117semidirect
387. C8#6440semidirect
388. C7#22615semidirect

Maximal subgroups

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

subgroupindex
C22×#63912
#117309022
#117309012
C7×#17522
#117297252
#117276792
#117276782
#117275742
#117275682
#117263772
#117263762
#226157