Finite Group Databasebeta

← #13364181

Subgroups — #13364181

Order 1792.

862 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic7⋊C8C211yesmaximal, characteristic
2#11731309C211yesmaximal, characteristic
3#11731308C211yesmaximal, characteristic
4C7×#1807C211yesmaximal, characteristic
5#11729725C211yesmaximal, characteristic
6#11728318C212yesmaximal, complemented
7#11728317C212yesmaximal, complemented
8#11727266C211yesmaximal, characteristic
9#11727265C211yesmaximal, characteristic
10#11726552C212yesmaximal, complemented
11#11726549C212yesmaximal, complemented
12C23×C7×C8C2211yescharacteristic
13C22×C4×Dic7C2211yescharacteristic
14C23×C7⋊C8C2211yescharacteristic
15C2×#6440—21no—
16C2×C7×#106C2212yes—
17C7×#380C2211yescharacteristic
18C7×#380C2211yescharacteristic
19#89984C2211yescharacteristic
20#89984C2211yescharacteristic
21C2×#6392C2212yes—
22C2×Dic7⋊C8C418yescomplemented
23C2×Dic7⋊C8C2211yescharacteristic
24C2×Dic7⋊C8C2211yescharacteristic
25C2×Dic7⋊C8C2211yescharacteristic
26C2×Dic7⋊C8C2211yescharacteristic
27#89749C2212yes—
28#89748C2212yes—
29C4⋊C4×Dic7C2212yes—
30C4⋊C4×Dic7C2212yes—
31C2×#6382C2212yes—
32C2×#6381C2212yes—
33#89715—21no—
34#89714—21no—
35C22⋊C4×Dic7—21no—
36#89423—21no—
37#89422—21no—
38#89421—21no—
39#89420—21no—
40C7×#340C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21962—71nomaximal, Sylow
45C23×C4×C7C2311yescharacteristic
46C23×Dic7D411yescharacteristic
47C23×Dic7D411yescharacteristic
48C22×C7×C8—42no—
49C22×C7×C8C2×C414yes—
50C22×C7×C8C2311yescharacteristic
51C22×C7×C8C2311yescharacteristic
52C2×C7×C4⋊C4C2312yes—
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—21no—
56C2×C22⋊Dic7—21no—
57C2×C22⋊Dic7—21no—
58C2×C4⋊Dic7C2312yes—
59C2×C4⋊Dic7C2312yes—
60C2×Dic7⋊C4—22no—
61C2×Dic7⋊C4—22no—
62C2×C4×Dic7—22no—
63C2×C4×Dic7—22no—
64C2×C4×Dic7C2×C414yes—
65C2×C4×Dic7D411yescharacteristic
66C2×C4×Dic7D411yescharacteristic
67C2×C4×Dic7Q814yes—
68C2×C4×Dic7C2311yescharacteristic
69C2×C4×Dic7C2311yescharacteristic
70C22×C7⋊C8—42no—
71C22×C7⋊C8C2×C414yes—
72C22×C7⋊C8C2311yescharacteristic
73C22×C7⋊C8C2311yescharacteristic
74#6440—44no—
75C7×#106—42no—
76C7×#95—21no—
77C7×#95—21no—
78C7⋊#95—21no—
79C7⋊#95—21no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82C7⋊#95—22no—
83C7⋊#95—22no—
84#6392—42no—
85Dic7⋊C8C2×C414yes—
86Dic7⋊C8C2×C414yes—
87Dic7⋊C8C2×C414yes—
88Dic7⋊C8C2×C414yes—
89#6382—42no—
90#6381—42no—
91C22×C4⋊C8—71no—
92#2285—71no—
93#2283—71no—
94#1949—72no—
95#1949—72no—
96#1948—72no—
97#1947—72no—
98#1808—71no—
99#1808—71no—
100#1807—71no—
101#1807D711yescharacteristic
102C24×C7C2×D411yescharacteristic
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7C22×C412yes—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×Q812yes—
110C22×C4×C7C2411yescharacteristic
111C22×Dic7—22no—
112C22×Dic7—22no—
113C22×Dic7—24no—
114C22×Dic7—22no—
115C22×Dic7—22no—
116C22×Dic7C4⋊C414yes—
117C22×Dic7C4⋊C414yes—
118C22×Dic7C2×D411yescharacteristic
119C22×Dic7C2×D411yescharacteristic
120C22×Dic7C2×D411yescharacteristic
121C22×Dic7C2×D411yescharacteristic
122C2×C7×C8—22no—
123C2×C7×C8—22no—
124C2×C7×C8—44no—
125C2×C7×C8C22×C412yes—
126C2×C7×C8C22×C412yes—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4—22no—
130C7×C4⋊C4—22no—
131C7×C22⋊C4—42no—
132C7×C22⋊C4—42no—
133C22⋊Dic7—42no—
134C22⋊Dic7—42no—
135C4⋊Dic7—22no—
136C4⋊Dic7—22no—
137C4⋊Dic7—22no—
138C4⋊Dic7—22no—
139Dic7⋊C4—44no—
140Dic7⋊C4—44no—
141C4×Dic7—44no—
142C4×Dic7—44no—
143C4×Dic7C4⋊C414yes—
144C4×Dic7C4⋊C414yes—
145C4×Dic7C22×C412yes—
146C4×Dic7C22×C412yes—
147C4×Dic7C2×Q812yes—
148C4×Dic7C2×Q812yes—
149C2×C7⋊C8—22no—
150C2×C7⋊C8—22no—
151C2×C7⋊C8—44no—
152C2×C7⋊C8C22×C412yes—
153C2×C7⋊C8C22×C412yes—
154C23×C8—71no—
155C23×C8C2×D711yescharacteristic
156C2×#117—141no—
157C22×C42—71no—
158C2×#107—72no—
159C2×#107—72no—
160C2×#106—72no—
161C2×#106C2×D712yes—
162C2×C4⋊C8—78no—
163C2×C4⋊C8—71no—
164C2×C4⋊C8—71no—
165C2×C4⋊C8—71no—
166C2×C4⋊C8—71no—
167#384—72no—
168#383—141no—
169#382—72no—
170#382—141no—
171#381—141no—
172#380—71no—
173#380—71no—
174#380—141no—
175#380C2×D711yescharacteristic
176#380C2×D711yescharacteristic
177C4×C4⋊C4—72no—
178C4×C4⋊C4—72no—
179C4×C4⋊C4—141no—
180C4×C22⋊C4—141no—
181#376—141no—
182#340—72no—
183#340—72no—
184#340—72no—
185#340C2×D712yes—
186C23×C7—22no—
187C23×C7—22no—
188C23×C7C2×C4⋊C412yes—
189C23×C7AGL(1,ℤ/8)12yes—
190C23×C7#13712yes—
191C23×C7C22×D411yescharacteristic
192C2×C4×C7—22no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—24no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—42no—
199C2×C4×C7—42no—
200C2×C4×C7—42no—
201C2×C4×C7—42no—
202C2×C4×C7C2×C4⋊C412yes—
203C2×C4×C7C23×C411yescharacteristic
204C2×C4×C7C22×Q811yescharacteristic
205C2×C4×C7H(5,𝔽2)12yes—
206C2×C4×C72−1+412yes—
207C2×Dic7—22no—
208C2×Dic7—21no—
209C2×Dic7—21no—
210C2×Dic7—21no—
211C2×Dic7—21no—
212C2×Dic7—28no—
213C2×Dic7—28no—
214C2×Dic7—42no—
215C2×Dic7—42no—
216C2×Dic7—42no—
217C2×Dic7—42no—
218C2×Dic7—42no—
219C2×Dic7—42no—
220C2×Dic7C2×C4⋊C412yes—
221C2×Dic7C2×C4⋊C412yes—
222C2×Dic7C2×C4⋊C412yes—
223C2×Dic7C2×C4⋊C412yes—
224C2×Dic7#13514yes—
225C7×C8—22no—
226C7×C8—22no—
227C7⋊C8—22no—
228C7⋊C8—22no—
229C23×C4—71no—
230C23×C4—71no—
231C23×C4C22×D711yescharacteristic
232C22×C8—42no—
233C22×C8—71no—
234C22×C8—71no—
235C22×C8—74no—
236C22×C8—282no—
237C22×C8C4×D714yes—
238C22×C8C22×D711yescharacteristic
239C22×C8C22×D711yescharacteristic
240#117—284no—
241C2×C4⋊C4—72no—
242C2×C4⋊C4—72no—
243C2×C4⋊C4—142no—
244C2×C4⋊C4—142no—
245C2×C4⋊C4C22×D712yes—
246C2×C4⋊C4C22×D712yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—141no—
250C2×C22⋊C4—141no—
251C2×C42—74no—
252C2×C42—71no—
253C2×C42—71no—
254C2×C42—74no—
255C2×C42—71no—
256C2×C42—71no—
257C2×C42—142no—
258C2×C42—142no—
259#107—282no—
260#107—282no—
261#106—42no—
262#106—282no—
263C4⋊C8—74no—
264C4⋊C8—74no—
265C4⋊C8—74no—
266C4⋊C8—74no—
267#95—21no—
268#95—21no—
269#95—142no—
270#95—142no—
271#95—142no—
272#95—142no—
273#95—141no—
274#95—141no—
275C22×C7—22no—
276C22×C7—22no—
277C22×C7—24no—
278C22×C7—24no—
279C22×C7#42814yes—
280C22×C7C22×C4⋊C411yescharacteristic
281C22×C7#53712yes—
282C22×C7C2×#13512yes—
283C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C7C2×#13711yescharacteristic
285C4×C7—22no—
286C4×C7—82no—
287C4×C7—82no—
288C4×C7#51811yescharacteristic
289C4×C7#51911yescharacteristic
290C4×C7#54412yes—
291Dic7—22no—
292Dic7—22no—
293Dic7—22no—
294Dic7—82no—
295Dic7—82no—
296Dic7#42714yes—
297C24D4×D711yescharacteristic
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—71no—
303C22×C4—74no—
304C22×C4—74no—
305C22×C4—71no—
306C22×C4—71no—
307C22×C4—71no—
308C22×C4—144no—
309C22×C4—142no—
310C22×C4—142no—
311C22×C4—142no—
312C22×C4—142no—
313C22×C4C2×C4×D712yes—
314C22×C4D4×D711yescharacteristic
315C22×C4Q8×D712yes—
316C22×C4C23×D711yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—44no—
320C2×C8—72no—
321C2×C8—72no—
322C2×C8—142no—
323C2×C8—142no—
324C2×C8—284no—
325C2×C8C2×C4×D712yes—
326C2×C8C2×C4×D712yes—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—142no—
332C4⋊C4—142no—
333C4⋊C4—142no—
334C4⋊C4—142no—
335C4⋊C4—284no—
336C4⋊C4—284no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—282no—
340C22⋊C4—282no—
341C42—74no—
342C42—74no—
343C42—72no—
344C42—72no—
345C42—72no—
346C42—72no—
347C42—284no—
348C42—284no—
349C2×C7—24no—
350C2×C7#231611yescharacteristic
351C2×C7C2×#42711yescharacteristic
352C2×C7C2×#42811yescharacteristic
353C2×C7#324712yes—
354C2×C7#324812yes—
355C23—22no—
356C23—22no—
357C23D7×C4⋊C412yes—
358C23#647712yes—
359C23#647812yes—
360C23C2×D4×D711yescharacteristic
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—72no—
372C2×C4—74no—
373C2×C4—72no—
374C2×C4—72no—
375C2×C4—72no—
376C2×C4—148no—
377C2×C4—148no—
378C2×C4—141no—
379C2×C4—141no—
380C2×C4—141no—
381C2×C4—141no—
382C2×C4—142no—
383C2×C4—282no—
384C2×C4—282no—
385C2×C4—282no—
386C2×C4—282no—
387C2×C4—282no—
388C2×C4—282no—
389C2×C4D7×C4⋊C412yes—
390C2×C4C22×C4×D711yescharacteristic
391C2×C4#654812yes—
392C2×C4C2×Q8×D711yescharacteristic
393C2×C4#655412yes—
394C8—22no—
395C8—22no—
396C8—142no—
397C8—142no—
398C7#2196211yescharacteristic, complemented, Sylow
399C22—22no—
400C22—22no—
401C22—24no—
402C22—24no—
403C22#8963314yes—
404C22C2×D7×C4⋊C411yescharacteristic
405C22#9028812yes—
406C22C2×#647711yescharacteristic
407C22C2×#647811yescharacteristic
408C22D7×#13512yes—
409C4—22no—
410C4—74no—
411C4—82no—
412C4—82no—
413C4—142no—
414C4—142no—
415C4—142no—
416C4—562no—
417C4—562no—
418C4#9017711yescharacteristic
419C4#9020711yescharacteristic
420C4#9031412yes—
421C2—24no—
422C2#1173131411yescharacteristic
423C2C2×#8963311yescharacteristic
424C2D7×#42711yescharacteristic
425C2#1173808612yes—
426C2#1173808712yes—
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#21962
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. #11728318C2semidirect
7. #11728317C2semidirect
10. #11726552C2semidirect
11. #11726549C2semidirect
22. C2×Dic7⋊C8C4semidirect
398. C7#21962semidirect

Maximal subgroups

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

subgroupindex
C22×Dic7⋊C82
#117313092
#117313082
C7×#18072
#117297252
#117283182
#117283172
#117272662
#117272652
#117265522
#117265492
#219627