Finite Group Databasebeta

← #13371435

Subgroups — #13371435

Order 1792.

878 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739604C211yesmaximal, characteristic
2C22×#6381C211yesmaximal, characteristic
3#11731323C211yesmaximal, characteristic
4C7×#1823C211yesmaximal, characteristic
5#11727701C211yesmaximal, characteristic
6#11727678C211yesmaximal, characteristic
7#11727266C211yesmaximal, characteristic
8#11726613C212yesmaximal, complemented
9#11726569C212yesmaximal, complemented
10#11726541C212yesmaximal, complemented
11#11726539C212yesmaximal, complemented
12C22×C7×C4⋊C4C2211yescharacteristic
13C2×#6489—21no—
14C22×C4⋊Dic7C2211yescharacteristic
15C23×C7⋊C8C2211yescharacteristic
16C2×#6440—21no—
17C2×C7×C22⋊C8C2211yescharacteristic
18C7×#380C2211yescharacteristic
19#89984C2211yescharacteristic
20#89983—21no—
21C4×C22⋊Dic7C2211yescharacteristic
22C2×C8×Dic7C2212yes—
23#89757C2212yes—
24#89753—22no—
25C4⋊C4×Dic7C2212yes—
26C4⋊C4×Dic7C2212yes—
27#89745C2212yes—
28C2×#6381C418yescomplemented
29C2×#6381C2212yes—
30C2×#6381C2211yescharacteristic
31C2×#6381C2211yescharacteristic
32C2×#6381C2211yescharacteristic
33C2×#6381C2211yescharacteristic
34#89715—21no—
35#89426—22no—
36C7×#340C2212yes—
37C7×#340C2212yes—
38#89346C2212yes—
39#89346C2212yes—
40#89345C2212yes—
41#89324C2212yes—
42#22611—71nomaximal, Sylow
43C23×C4×C7—21no—
44C23×C4×C7C2311yescharacteristic
45C23×Dic7—21no—
46C22×C7×C8C2312yes—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4C2×C414yes—
50C2×C7×C4⋊C4C2312yes—
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C7×C4⋊C4C2311yescharacteristic
53C2×C7×C22⋊C4—21no—
54C2×C22⋊Dic7—21no—
55C2×C22⋊Dic7D411yescharacteristic
56C2×C22⋊Dic7D411yescharacteristic
57#6489—44no—
58C2×C4⋊Dic7—21no—
59C2×C4⋊Dic7—22no—
60C2×C4⋊Dic7—22no—
61C2×C4⋊Dic7C2×C414yes—
62C2×C4⋊Dic7C2312yes—
63C2×C4⋊Dic7C2311yescharacteristic
64C2×C4⋊Dic7C2311yescharacteristic
65C2×Dic7⋊C4—22no—
66C2×C4×Dic7—22no—
67C2×C4×Dic7—22no—
68C2×C4×Dic7C2312yes—
69C22×C7⋊C8—24no—
70C22×C7⋊C8C2×C414yes—
71C22×C7⋊C8C2311yescharacteristic
72C22×C7⋊C8C2311yescharacteristic
73#6440—44no—
74C7×C22⋊C8—22no—
75C7×#95—21no—
76C7⋊#95—21no—
77C7⋊#95—22no—
78C7⋊#95—22no—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95D412yes—
82C8×Dic7—24no—
83#6381—24no—
84#6381C2×C414yes—
85#6381C2×C414yes—
86#6381C2×C414yes—
87#6381C2×C414yes—
88C22×#107—71no—
89#2344—71no—
90#2344—71no—
91#1942—72no—
92#1918—72no—
93#1918—72no—
94#1823—71no—
95#1823D711yescharacteristic
96#1808—71no—
97C4×#107—72no—
98C8×C22⋊C4—71no—
99C24×C7C2×D411yescharacteristic
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—24no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—42no—
107C22×C4×C7—42no—
108C22×C4×C7C22×C412yes—
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7#4112yes—
111C22×C4×C7C2411yescharacteristic
112C22×Dic7—24no—
113C22×Dic7—21no—
114C22×Dic7—21no—
115C22×Dic7—22no—
116C22×Dic7—42no—
117C22×Dic7—42no—
118C22×Dic7C2×D412yes—
119C22×Dic7C2×D412yes—
120C2×C7×C8—22no—
121C2×C7×C8—22no—
122C2×C7×C8—22no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—22no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4C22×C412yes—
129C7×C4⋊C4C22×C412yes—
130C7×C22⋊C4—42no—
131C22⋊Dic7—22no—
132C22⋊Dic7—42no—
133C22⋊Dic7D814yes—
134C4⋊Dic7—24no—
135C4⋊Dic7—21no—
136C4⋊Dic7—21no—
137C4⋊Dic7—22no—
138C4⋊Dic7—22no—
139C4⋊Dic7—44no—
140C4⋊Dic7C22×C412yes—
141C4⋊Dic7C22×C412yes—
142Dic7⋊C4—44no—
143C4×Dic7—22no—
144C4×Dic7—22no—
145C4×Dic7—44no—
146C4×Dic7—44no—
147C2×C7⋊C8—22no—
148C2×C7⋊C8—28no—
149C2×C7⋊C8C22×C412yes—
150C2×C7⋊C8C22×C412yes—
151C2×C7⋊C8#4114yes—
152C23×C8—71no—
153C2×#117—141no—
154C2×#117—141no—
155C22×C4⋊C4—71no—
156C22×C4⋊C4C2×D711yescharacteristic
157C2×#107—78no—
158C2×#107—71no—
159C2×#107—71no—
160C2×#107—71no—
161C2×#107—71no—
162C2×#107—72no—
163C2×C22⋊C8C2×D711yescharacteristic
164C2×C4×C8—72no—
165#389—72no—
166#389—72no—
167#387—142no—
168#387—142no—
169#381—141no—
170#381—141no—
171#380—71no—
172#380C2×D711yescharacteristic
173C4×C4⋊C4—72no—
174C4×C4⋊C4—72no—
175C4×C22⋊C4—71no—
176#340—72no—
177#340—72no—
178#340—72no—
179#340C2×D712yes—
180#340C2×D712yes—
181#336—72no—
182C23×C7—22no—
183C23×C7—22no—
184C23×C7C4×D412yes—
185C23×C7C2×D812yes—
186C23×C7#13512yes—
187C23×C7C22×D411yescharacteristic
188C2×C4×C7—22no—
189C2×C4×C7—22no—
190C2×C4×C7—22no—
191C2×C4×C7—24no—
192C2×C4×C7—22no—
193C2×C4×C7—22no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—48no—
199C2×C4×C7—42no—
200C2×C4×C7—42no—
201C2×C4×C7—42no—
202C2×C4×C7C4×D412yes—
203C2×C4×C7C23×C411yescharacteristic
204C2×C4×C7C2×#4111yescharacteristic
205C2×C4×C7C2×#4112yes—
206C2×C4×C72−1+412yes—
207C2×Dic7—22no—
208C2×Dic7—22no—
209C2×Dic7—22no—
210C2×Dic7—22no—
211C2×Dic7—22no—
212C2×Dic7—22no—
213C2×Dic7—22no—
214C2×Dic7—41no—
215C2×Dic7—41no—
216C2×Dic7—41no—
217C2×Dic7—41no—
218C2×Dic7—48no—
219C2×Dic7—42no—
220C2×Dic7—42no—
221C2×Dic7—42no—
222C2×Dic7C2×D814yes—
223C7×C8—42no—
224C7⋊C8—22no—
225C7⋊C8#11714yescomplemented
226C23×C4—21no—
227C23×C4—141no—
228C23×C4C22×D711yescharacteristic
229C22×C8—71no—
230C22×C8—74no—
231C22×C8—71no—
232C22×C8—144no—
233C22×C8C22×D712yes—
234#117—284no—
235#117—284no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—74no—
239C2×C4⋊C4—71no—
240C2×C4⋊C4—71no—
241C2×C4⋊C4—72no—
242C2×C4⋊C4—141no—
243C2×C4⋊C4—142no—
244C2×C4⋊C4—142no—
245C2×C4⋊C4—142no—
246C2×C4⋊C4C4×D714yes—
247C2×C4⋊C4C22×D712yes—
248C2×C4⋊C4C22×D711yescharacteristic
249C2×C4⋊C4C22×D711yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—71no—
252C2×C22⋊C4—71no—
253C2×C22⋊C4—141no—
254C2×C42—72no—
255C2×C42—142no—
256C2×C42—142no—
257#107—74no—
258#107—74no—
259#107—74no—
260#107—74no—
261#107—144no—
262C22⋊C8—22no—
263C4×C8—144no—
264#95—21no—
265#95—72no—
266#95—142no—
267#95—142no—
268#95—142no—
269#95—142no—
270#95—141no—
271C22×C7—24no—
272C22×C7—24no—
273C22×C7—22no—
274C22×C7—22no—
275C22×C7C4×D814yes—
276C22×C7C2×C4×D411yescharacteristic
277C22×C7#54712yes—
278C22×C7C22×D811yescharacteristic
279C22×C7C2×#13511yescharacteristic
280C22×C7#57812yes—
281C4×C7—22no—
282C4×C7—42no—
283C4×C7—42no—
284C4×C7—82no—
285C4×C7C2×#11711yescharacteristic
286C4×C7#51911yescharacteristic
287C4×C7#55212yes—
288Dic7—24no—
289Dic7—42no—
290Dic7—42no—
291Dic7—42no—
292Dic7—82no—
293C24D4×D711yescharacteristic
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—72no—
303C22×C4—72no—
304C22×C4—144no—
305C22×C4—141no—
306C22×C4—141no—
307C22×C4—142no—
308C22×C4—282no—
309C22×C4—282no—
310C22×C4C2×C4×D712yes—
311C22×C4D4×D711yescharacteristic
312C22×C4#115912yes—
313C22×C4C23×D711yescharacteristic
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—72no—
318C2×C8—74no—
319C2×C8—72no—
320C2×C8—148no—
321C2×C8—142no—
322C4⋊C4—24no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—72no—
328C4⋊C4—72no—
329C4⋊C4—144no—
330C4⋊C4—141no—
331C4⋊C4—141no—
332C4⋊C4—142no—
333C4⋊C4—142no—
334C4⋊C4—284no—
335C4⋊C4—284no—
336C4⋊C4C2×C4×D712yes—
337C4⋊C4C2×C4×D712yes—
338C22⋊C4—42no—
339C22⋊C4—74no—
340C22⋊C4—142no—
341C22⋊C4—282no—
342C42—142no—
343C42—142no—
344C42—284no—
345C42—284no—
346C2×C7—24no—
347C2×C7#234411yescharacteristic
348C2×C7C2×C4×D811yescharacteristic
349C2×C7#295011yescharacteristic
350C2×C7#333412yes—
351C2×C7#333612yes—
352C23—22no—
353C23—22no—
354C23(C22⋊D7)⋊C412yes—
355C23D7×D812yes—
356C23#647912yes—
357C23C2×D4×D711yescharacteristic
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—74no—
373C2×C4—142no—
374C2×C4—142no—
375C2×C4—142no—
376C2×C4—142no—
377C2×C4—142no—
378C2×C4—142no—
379C2×C4—142no—
380C2×C4—281no—
381C2×C4—281no—
382C2×C4—281no—
383C2×C4—281no—
384C2×C4—288no—
385C2×C4—282no—
386C2×C4—282no—
387C2×C4—282no—
388C2×C4(C22⋊D7)⋊C412yes—
389C2×C4C22×C4×D711yescharacteristic
390C2×C4C2×#115911yescharacteristic
391C2×C4D7×#4112yes—
392C2×C4#655412yes—
393C8—42no—
394C8—74no—
395C8—142no—
396C7#2261111yescharacteristic, complemented, Sylow
397C22—24no—
398C22—22no—
399C22—22no—
400C22—24no—
401C22#8952814yes—
402C22C2×(C22⋊D7)⋊C411yescharacteristic
403C22#9034612yes—
404C22C2×D7×D811yescharacteristic
405C22C2×#647911yescharacteristic
406C22#9047012yes—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—82no—
411C4—144no—
412C4—282no—
413C4—282no—
414C4—282no—
415C4—562no—
416C4#9020711yescharacteristic
417C4D7×#11711yescharacteristic
418C4#9034112yes—
419C2—24no—
420C2#1173150511yescharacteristic
421C2C2×#8952811yescharacteristic
422C2#1173639311yescharacteristic
423C2#1173873612yes—
424C2#1173888112yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1823C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1823D7
hypocenterC7#22611
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
8. #11726613C2semidirect
9. #11726569C2semidirect
10. #11726541C2semidirect
11. #11726539C2semidirect
28. C2×#6381C4semidirect
225. C7⋊C8#117semidirect
396. C7#22611semidirect

Maximal subgroups

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

subgroupindex
#117396042
C22×#63812
#117313232
C7×#18232
#117277012
#117276782
#117272662
#117266132
#117265692
#117265412
#117265392
#226117