Finite Group Databasebeta

← #13371422

Subgroups — #13371422

Order 1792.

850 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#11731324C211yesmaximal, characteristic
3#11731323C211yesmaximal, characteristic
4C7×#1807C211yesmaximal, characteristic
5C8×C22⋊Dic7C211yesmaximal, characteristic
6#11728313C212yesmaximal, complemented
7Dic7×#106C212yesmaximal, complemented
8#11727256C211yesmaximal, characteristic
9#11727255C211yesmaximal, characteristic
10#11726572C212yesmaximal, complemented
11#11726570C212yesmaximal, complemented
12C23×C7×C8C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C2×#6440—21no—
16C2×C7×#106C2212yes—
17C7×#380C2211yescharacteristic
18C7×#380C2211yescharacteristic
19C4×C22⋊Dic7C2211yescharacteristic
20C2×C22⋊(C7⋊C8)C2211yescharacteristic
21C2×#6391C418yescomplemented
22C2×#6391C2211yescharacteristic
23C2×#6391C2211yescharacteristic
24C2×#6391C2211yescharacteristic
25C2×#6391C2211yescharacteristic
26C2×C8×Dic7C2212yes—
27C4⋊C4×Dic7C2212yes—
28C4⋊C4×Dic7C2212yes—
29#89745C2212yes—
30#89745C2212yes—
31#89715—21no—
32#89714—21no—
33C22⋊C4×Dic7—21no—
34#89426—22no—
35#89425—22no—
36C7×#340C2212yes—
37#89345C2212yes—
38#89324C2212yes—
39#89324C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#22609—71nomaximal, Sylow
43C23×C4×C7C2311yescharacteristic
44C23×Dic7—21no—
45C23×Dic7—21no—
46C22×C7×C8—24no—
47C22×C7×C8C2×C414yes—
48C22×C7×C8C2311yescharacteristic
49C22×C7×C8C2311yescharacteristic
50C2×C7×C4⋊C4C2312yes—
51C2×C7×C4⋊C4C2312yes—
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C22⋊Dic7D411yescharacteristic
55C2×C22⋊Dic7D411yescharacteristic
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⋊Dic7C2311yescharacteristic
63C2×C4⋊Dic7C2311yescharacteristic
64C2×C4⋊Dic7C2311yescharacteristic
65C2×C4⋊Dic7C2311yescharacteristic
66C2×Dic7⋊C4—22no—
67C2×Dic7⋊C4—22no—
68C2×C4×Dic7—22no—
69C2×C4×Dic7—22no—
70C2×C4×Dic7C2312yes—
71C22×C7⋊C8C2312yes—
72#6440—44no—
73C7×#106—24no—
74C7×#95—21no—
75C7×#95—21no—
76C7⋊#95—22no—
77C7⋊#95—22no—
78C7⋊#95—22no—
79C7⋊#95—22no—
80C7⋊#95D412yes—
81C22⋊(C7⋊C8)—22no—
82#6391C2×C414yes—
83#6391C2×C414yes—
84#6391C2×C414yes—
85#6391C2×C414yes—
86C8×Dic7—24no—
87C22×#106—71no—
88#2345—71no—
89#2344—71no—
90#1941—72no—
91#1920—72no—
92#1919—72no—
93#1824—71no—
94#1823—71no—
95#1807D711yescharacteristic
96C4×#106—72no—
97C8×C22⋊C4—71no—
98C24×C7C2×D411yescharacteristic
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7C22×C412yes—
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—42no—
114C22×Dic7—42no—
115C22×Dic7—82no—
116C22×Dic7C2×D412yes—
117C22×Dic7C2×D412yes—
118C2×C7×C8—22no—
119C2×C7×C8—28no—
120C2×C7×C8C22×C412yes—
121C2×C7×C8C22×C412yes—
122C2×C7×C8#4114yes—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—22no—
127C7×C22⋊C4—42no—
128C7×C22⋊C4—42no—
129C22⋊Dic7—22no—
130C22⋊Dic7SD1614yes—
131C4⋊Dic7—24no—
132C4⋊Dic7—24no—
133C4⋊Dic7—21no—
134C4⋊Dic7—21no—
135C4⋊Dic7—21no—
136C4⋊Dic7—21no—
137C4⋊Dic7C22×C412yes—
138C4⋊Dic7C22×C412yes—
139C4⋊Dic7C22×C412yes—
140C4⋊Dic7C22×C412yes—
141Dic7⋊C4—44no—
142Dic7⋊C4—82no—
143C4×Dic7—22no—
144C4×Dic7—22no—
145C4×Dic7—44no—
146C4×Dic7—82no—
147C2×C7⋊C8—22no—
148C2×C7⋊C8—22no—
149C2×C7⋊C8—22no—
150C23×C8C2×D711yescharacteristic
151C2×#117—141no—
152C22×C4⋊C4—71no—
153C22×C4⋊C4—71no—
154C2×#106—78no—
155C2×#106—71no—
156C2×#106—71no—
157C2×#106—71no—
158C2×#106—71no—
159C2×#106C2×D712yes—
160C2×C22⋊C8—71no—
161C2×C4×C8—72no—
162#389—72no—
163#389—72no—
164#387—142no—
165#387—142no—
166#381—141no—
167#380—141no—
168#380C2×D711yescharacteristic
169#380C2×D711yescharacteristic
170C4×C4⋊C4—72no—
171C4×C4⋊C4—72no—
172C4×C22⋊C4—71no—
173C4×C22⋊C4—141no—
174#340—72no—
175#340—72no—
176#340—72no—
177#340—72no—
178#340C2×D712yes—
179#336—72no—
180C23×C7—22no—
181C23×C7—22no—
182C23×C7C4×D412yes—
183C23×C7C2×SD1612yes—
184C23×C7#13512yes—
185C23×C7C22×D411yescharacteristic
186C2×C4×C7—22no—
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—24no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
193C2×C4×C7—42no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7C4×D412yes—
197C2×C4×C7C23×C411yescharacteristic
198C2×C4×C7C2×#4112yes—
199C2×C4×C7C2×#4111yescharacteristic
200C2×C4×C72−1+412yes—
201C2×Dic7—22no—
202C2×Dic7—22no—
203C2×Dic7—22no—
204C2×Dic7—22no—
205C2×Dic7—22no—
206C2×Dic7—22no—
207C2×Dic7—22no—
208C2×Dic7—22no—
209C2×Dic7—22no—
210C2×Dic7—41no—
211C2×Dic7—41no—
212C2×Dic7—41no—
213C2×Dic7—41no—
214C2×Dic7—42no—
215C2×Dic7—42no—
216C2×Dic7—48no—
217C2×Dic7—84no—
218C2×Dic7C2×SD1614yes—
219C7×C8—22no—
220C7×C8#11714yescomplemented
221C7⋊C8—42no—
222C23×C4—141no—
223C23×C4—141no—
224C23×C4C22×D711yescharacteristic
225C22×C8—24no—
226C22×C8—72no—
227C22×C8C4×D714yes—
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—141no—
238C2×C4⋊C4—141no—
239C2×C4⋊C4—142no—
240C2×C4⋊C4—142no—
241C2×C4⋊C4—142no—
242C2×C4⋊C4—142no—
243C2×C4⋊C4C22×D712yes—
244C2×C4⋊C4C22×D712yes—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—71no—
248C2×C22⋊C4—71no—
249C2×C42—72no—
250C2×C42—142no—
251C2×C42—142no—
252#106—24no—
253#106—74no—
254#106—74no—
255#106—74no—
256#106—74no—
257C22⋊C8—142no—
258C4×C8—144no—
259#95—21no—
260#95—21no—
261#95—72no—
262#95—142no—
263#95—142no—
264#95—142no—
265#95—142no—
266C22×C7—24no—
267C22×C7—24no—
268C22×C7—22no—
269C22×C7—22no—
270C22×C7C4×SD1614yes—
271C22×C7C2×C4×D411yescharacteristic
272C22×C7#54712yes—
273C22×C7C22×SD1611yescharacteristic
274C22×C7C2×#13511yescharacteristic
275C22×C7#57712yes—
276C4×C7—22no—
277C4×C7—82no—
278C4×C7—82no—
279C4×C7C2×#11711yescharacteristic
280C4×C7#51911yescharacteristic
281C4×C7#55212yes—
282Dic7—24no—
283Dic7—42no—
284Dic7—42no—
285Dic7—42no—
286Dic7—42no—
287Dic7—42no—
288C24D4×D711yescharacteristic
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—72no—
294C22×C4—72no—
295C22×C4—144no—
296C22×C4—141no—
297C22×C4—141no—
298C22×C4—141no—
299C22×C4—141no—
300C22×C4—144no—
301C22×C4—282no—
302C22×C4—282no—
303C22×C4—562no—
304C22×C4C2×C4×D712yes—
305C22×C4D4×D711yescharacteristic
306C22×C4#116112yes—
307C22×C4C23×D711yescharacteristic
308C2×C8—22no—
309C2×C8—28no—
310C2×C8—142no—
311C2×C8—142no—
312C2×C8—142no—
313C2×C8C2×C4×D712yes—
314C2×C8C2×C4×D712yes—
315C2×C8#115914yes—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—72no—
321C4⋊C4—72no—
322C4⋊C4—72no—
323C4⋊C4—72no—
324C4⋊C4—144no—
325C4⋊C4—144no—
326C4⋊C4—141no—
327C4⋊C4—141no—
328C4⋊C4—141no—
329C4⋊C4—141no—
330C4⋊C4—284no—
331C4⋊C4—562no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—74no—
335C22⋊C4—142no—
336C42—142no—
337C42—142no—
338C42—284no—
339C42—562no—
340C2×C7—24no—
341C2×C7#234411yescharacteristic
342C2×C7C2×C4×SD1611yescharacteristic
343C2×C7#294911yescharacteristic
344C2×C7#333512yes—
345C2×C7#333712yes—
346C23—22no—
347C23—22no—
348C23#645612yes—
349C23D7×SD1612yes—
350C23#647912yes—
351C23C2×D4×D711yescharacteristic
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—74no—
363C2×C4—142no—
364C2×C4—142no—
365C2×C4—142no—
366C2×C4—142no—
367C2×C4—142no—
368C2×C4—142no—
369C2×C4—142no—
370C2×C4—142no—
371C2×C4—142no—
372C2×C4—288no—
373C2×C4—281no—
374C2×C4—281no—
375C2×C4—281no—
376C2×C4—281no—
377C2×C4—282no—
378C2×C4—282no—
379C2×C4—564no—
380C2×C4#645612yes—
381C2×C4C22×C4×D711yescharacteristic
382C2×C4C2×#115912yes—
383C2×C4C2×#116111yescharacteristic
384C2×C4#655412yes—
385C8—22no—
386C8—282no—
387C8#644014yescomplemented
388C7#2260911yescharacteristic, complemented, Sylow
389C22—22no—
390C22—22no—
391C22—24no—
392C22—24no—
393C22#8962414yes—
394C22C2×#645611yescharacteristic
395C22#9030212yes—
396C22C2×D7×SD1611yescharacteristic
397C22C2×#647911yescharacteristic
398C22#9046112yes—
399C4—22no—
400C4—82no—
401C4—82no—
402C4—144no—
403C4—282no—
404C4—282no—
405C4—282no—
406C4—282no—
407C4—282no—
408C4C2×#644011yescharacteristic
409C4#9020711yescharacteristic
410C4#9033312yes—
411C2—24no—
412C2#1173132311yescharacteristic
413C2C2×#8962411yescharacteristic
414C2#1173661311yescharacteristic
415C2#1173809312yes—
416C2#1173809412yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1807C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1807D7
hypocenterC7#22609
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
6. #11728313C2semidirect
7. Dic7×#106C2semidirect
10. #11726572C2semidirect
11. #11726570C2semidirect
21. C2×#6391C4semidirect
220. C7×C8#117semidirect
387. C8#6440semidirect
388. C7#22609semidirect

Maximal subgroups

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

subgroupindex
C22×#63912
#117313242
#117313232
C7×#18072
C8×C22⋊Dic72
#117283132
Dic7×#1062
#117272562
#117272552
#117265722
#117265702
#226097