Finite Group Databasebeta

← #13362286

Subgroups — #13362286

Order 1792.

1110 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89963C211yesmaximal, characteristic
2C2×#89956C211yesmaximal, characteristic, complemented
3C2×#89641C211yesmaximal, characteristic
4C2×#89560C212yesmaximal, complemented
5#11733929C212yesmaximal, complemented
6C7×#1971C211yesmaximal, characteristic, complemented
7#11728308C211yesmaximal, characteristic, complemented
8#11727981C212yesmaximal, complemented
9#11727579C211yesmaximal, characteristic
10#11727558C211yesmaximal, characteristic, complemented
11#11727430C212yesmaximal, complemented
12C22×C7×Q16C2211yescharacteristic
13C2×#6510C2212yes—
14C2×Q8×Dic7C2212yes—
15C22×#1143C2212yes—
16C22×D56C2211yescharacteristic
17C2×#6456C2212yes—
18C2×C7×#106C2211yescharacteristic
19C2×C7×#103C2212yes—
20C2×C7×#97C2211yescharacteristic
21C7×#391C2212yes—
22C2×#6409C2212yes—
23#89963C418yescomplemented
24#89956—24no—
25C2×#6395C2212yes—
26C2×#6391C2211yescharacteristic
27C2×#6389C2211yescharacteristic
28C2×#6389C2211yescharacteristic
29C2×C8×Dic7C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31#89748C2212yes—
32C2×#6383C2212yes—
33C2×#6382C2212yes—
34#89711C2212yes—
35#89704C2211yescharacteristic
36C2×C4×C7⋊C8C2211yescharacteristic
37#89641—24no—
38#89560—28no—
39#89441—24no—
40#89440—24no—
41#21694—71nomaximal, Sylow
42C22×C7×Q8C2312yes—
43C22×D28C2312yes—
44C22×C4×D7—22no—
45C2×C7×Q16—22no—
46C2×C7×Q16—22no—
47C2×C7×Q16C2×C414yescomplemented
48C22×C7×C8C2311yescharacteristic
49C22×C7×C8C2311yescharacteristic
50C2×C7×C4⋊C4C2312yes—
51C2×C42×C7C2311yescharacteristic
52#6510—24no—
53#6510C2×C418yescomplemented
54Q8×Dic7—24no—
55Q8×Dic7—24no—
56C2×#1143—24no—
57C2×#1143—24no—
58C2×#1143C2×C418yescomplemented
59C2×(C2×D7)⋊C4—24no—
60C2×(C2×D7)⋊C4—22no—
61C2×(C2×D7)⋊C4—24no—
62C2×C4⋊Dic7—22no—
63C2×C4⋊Dic7C2312yes—
64C2×Dic7⋊C4—24no—
65C2×C4×Dic7—22no—
66C2×C4×Dic7C2311yescharacteristic
67C2×C4×Dic7C2311yescharacteristic
68C22×C7⋊C8C2311yescharacteristic
69C22×C7⋊C8C2311yescharacteristic
70C2×D56—22no—
71C2×D56—22no—
72C2×D56C2×C414yes—
73#6456—24no—
74#6456—24no—
75C7×#106—22no—
76C7×#103—24no—
77C7×#97—22no—
78C7×#95—24no—
79#6409—24no—
80C7⋊#95D414yes—
81#6395—24no—
82#6391—22no—
83#6389—22no—
84#6389D414yes—
85C8×Dic7C2×C414yescomplemented
86C8×Dic7D414yes—
87#6383—24no—
88#6382—24no—
89C4×C7⋊C8—22no—
90C2×#495—71no—
91C2×#442—71no—
92C2×#441—71no—
93C2×C4×SD16—72no—
94#2326—72no—
95#1972—71no—
96#1971D711yescharacteristic, complemented
97#1968—72no—
98#1959—72no—
99#1844—71no—
100#1726—71no—
101C23×D7—22no—
102C2×C7×Q8—22no—
103C2×C7×Q8—22no—
104C2×C7×Q8—22no—
105C2×C7×Q8—22no—
106C2×C7×Q8—22no—
107C2×C7×Q8C22×C414yes—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7C2×D412yes—
111C22×C4×C7C2411yescharacteristic
112C22×Dic7—22no—
113C22×Dic7C2×D412yes—
114C22×Dic7C2×D412yes—
115C2×D28—22no—
116C2×D28—22no—
117C2×D28—22no—
118C2×D28—22no—
119C2×D28—22no—
120C2×D28C22×C414yes—
121C2×C4×D7—44no—
122C2×C4×D7—84no—
123C7×Q16—28no—
124C2×C7×C8—21no—
125C2×C7×C8—21no—
126C2×C7×C8C22×C412yes—
127C2×C7×C8C2×D412yes—
128C2×C7×C8C2×D412yes—
129C2×C7×C8#4112yes—
130C7×C4⋊C4—22no—
131C7×C4⋊C4—22no—
132C42×C7—21no—
133C42×C7—21no—
134#1143—216no—
135(C2×D7)⋊C4—216no—
136(C2×D7)⋊C4—44no—
137(C2×D7)⋊C4—84no—
138C4⋊Dic7—22no—
139C4⋊Dic7—22no—
140C4⋊Dic7—44no—
141Dic7⋊C4—84no—
142C4×Dic7—44no—
143C4×Dic7C22×C412yes—
144C4×Dic7C2×D412yes—
145C4×Dic7C2×D412yes—
146C4×Dic7#4112yes—
147C2×C7⋊C8—21no—
148C2×C7⋊C8—21no—
149C2×C7⋊C8C22×C412yes—
150C2×C7⋊C8C2×D412yes—
151C2×C7⋊C8C2×D412yes—
152C2×C7⋊C8#4112yes—
153D56—28no—
154C22×Q16C2×D711yescharacteristic
155C22×SD16—72no—
156C22×D8—71no—
157C2×#124—72no—
158C2×C4×Q8—72no—
159C2×C4×D4—72no—
160#495—78no—
161#442—144no—
162#441—144no—
163C4×SD16—148no—
164C2×#106—71no—
165C2×#106—72no—
166C2×#106C2×D711yescharacteristic
167C2×#103—72no—
168C2×#103C2×D712yes—
169C2×#102—72no—
170C2×#102—72no—
171C2×#97—71no—
172C2×#97—71no—
173C2×#97C2×D711yescharacteristic
174C2×C4×C8—71no—
175C2×C4×C8—71no—
176C2×C4×C8—71no—
177#391C2×D712yes—
178#390—72no—
179#388—144no—
180#385—144no—
181#384—72no—
182#376—71no—
183C23×C7C22×D411yescharacteristic
184C22×D7—24no—
185C22×D7—42no—
186C22×D7—42no—
187C22×D7—44no—
188C22×D7—44no—
189C7×Q8—24no—
190C7×Q8—24no—
191C7×Q8—42no—
192C7×Q8—42no—
193C2×C4×C7—24no—
194C2×C4×C7—22no—
195C2×C4×C7—24no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—42no—
199C2×C4×C7—44no—
200C2×C4×C7C23×C411yescharacteristic
201C2×C4×C7C22×D411yescharacteristic
202C2×C4×C7C22×D411yescharacteristic
203C2×C4×C7C2×#4111yescharacteristic
204C2×C4×C7C2×#4111yescharacteristic
205C2×C4×C7C2×#4111yescharacteristic
206C2×Dic7—24no—
207C2×Dic7—24no—
208C2×Dic7—42no—
209C2×Dic7—44no—
210C2×Dic7C4×D414yes—
211C2×Dic7#12714yes—
212D28—24no—
213D28—24no—
214D28—42no—
215D28—42no—
216C4×D7—88no—
217C7×C8—22no—
218C7×C8C4×D414yes—
219C7⋊C8—22no—
220C7⋊C8C4×D414yes—
221C22×Q8C22×D712yes—
222C22×D4—72no—
223C23×C4—142no—
224C2×Q16—22no—
225C2×Q16—22no—
226C2×Q16C4×D714yescomplemented
227C2×SD16—78no—
228C2×SD16—144no—
229C2×SD16—144no—
230C2×D8—74no—
231C2×D8—142no—
232C2×D8—142no—
233C22×C8—71no—
234C22×C8—71no—
235C22×C8C22×D711yescharacteristic
236C22×C8C22×D711yescharacteristic
237#124—78no—
238#124—144no—
239C4×Q8—144no—
240C4×Q8—144no—
241C4×D4—144no—
242C4×D4—144no—
243C2×C4⋊C4—72no—
244C2×C4⋊C4—142no—
245C2×C4⋊C4—144no—
246C2×C4⋊C4C22×D712yes—
247C2×C22⋊C4—144no—
248C2×C22⋊C4—142no—
249C2×C22⋊C4—144no—
250C2×C42—71no—
251C2×C42—71no—
252C2×C42—142no—
253C2×C42C22×D711yescharacteristic
254#106—22no—
255#106—144no—
256#106—142no—
257#103—24no—
258#103—144no—
259#102—144no—
260#102—144no—
261#97—22no—
262#97—74no—
263#97—142no—
264C4×C8—74no—
265C4×C8—74no—
266C4×C8—142no—
267#95—24no—
268#95—74no—
269C22×C7C2×C4×D411yescharacteristic
270C22×C7C2×#12711yescharacteristic
271C22×C7#53211yescharacteristic
272C22×C7#57714yes—
273C2×D7—44no—
274C2×D7—48no—
275C2×D7—82no—
276C2×D7—82no—
277C2×D7—84no—
278C4×C7—44no—
279C4×C7—42no—
280C4×C7—82no—
281C4×C7C2×C4×D411yescharacteristic
282C4×C7C2×C4×D411yescharacteristic
283C4×C7#53211yescharacteristic
284C4×C7#53211yescharacteristic
285Dic7—24no—
286Dic7—24no—
287Dic7—82no—
288C24—142no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8C2×C4×D714yes—
295C2×D4—74no—
296C2×D4—142no—
297C2×D4—142no—
298C2×D4—142no—
299C2×D4—142no—
300C2×D4—142no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—72no—
304C22×C4—72no—
305C22×C4—142no—
306C22×C4—284no—
307C22×C4—564no—
308C22×C4D4×D712yes—
309C22×C4C23×D711yescharacteristic
310Q16—28no—
311SD16—1416no—
312D8—148no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—72no—
316C2×C8—72no—
317C2×C8—72no—
318C2×C8—72no—
319C2×C8—141no—
320C2×C8—141no—
321C2×C8C2×C4×D712yes—
322C2×C8D4×D712yes—
323C2×C8D4×D712yes—
324C2×C8#115912yes—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—142no—
328C4⋊C4—142no—
329C4⋊C4—284no—
330C4⋊C4—564no—
331C22⋊C4—1416no—
332C22⋊C4—284no—
333C22⋊C4—564no—
334C42—21no—
335C42—21no—
336C42—72no—
337C42—72no—
338C42—72no—
339C42—72no—
340C42—284no—
341C2×C7C4×#12711yescharacteristic
342C2×C7#295212yes—
343C2×C7#315412yes—
344C2×C7#317312yes—
345D7—84no—
346C23—144no—
347C23—282no—
348C23—284no—
349C23—284no—
350C23—282no—
351C23C2×D4×D711yescharacteristic
352Q8—24no—
353Q8—24no—
354Q8—42no—
355Q8—42no—
356D4—144no—
357D4—144no—
358D4—282no—
359D4—282no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—74no—
368C2×C4—74no—
369C2×C4—144no—
370C2×C4—144no—
371C2×C4—282no—
372C2×C4—284no—
373C2×C4—568no—
374C2×C4C22×C4×D711yescharacteristic
375C2×C4C2×D4×D711yescharacteristic
376C2×C4C2×D4×D711yescharacteristic
377C2×C4C2×#115911yescharacteristic
378C2×C4D7×#4111yescharacteristic
379C2×C4D7×#4111yescharacteristic
380C8—22no—
381C8—74no—
382C8—142no—
383C8(C22⋊D7)⋊C414yes—
384C7#2169411yescharacteristic, complemented, Sylow
385C22—288no—
386C22—284no—
387C22—562no—
388C22—562no—
389C22—564no—
390C22C4×D4×D711yescharacteristic
391C22#9028711yescharacteristic
392C22D7×#12711yescharacteristic
393C22#9046914yes—
394C4—42no—
395C4—44no—
396C4—82no—
397C4—144no—
398C4—144no—
399C4—562no—
400C4C2×(C22⋊D7)⋊C411yescharacteristic
401C4C4×D4×D711yescharacteristic
402C4#9028711yescharacteristic
403C4#9035911yescharacteristic
404C2—564no—
405C2#1173373811yescharacteristic
406C2#1173727712yes—
407C2#1173833412yes—
408C2#1173931712yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1971C2
socleC23×C7C22×D4
prime radicalC22×D56C22
cosocleC22×C4×C7C24
hypercenter#1971D7
hypocenterC7#21694
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1971

Complements

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

NH ≅ G/Ntype
2. C2×#89956C2semidirect
4. C2×#89560C2semidirect
5. #11733929C2semidirect
6. C7×#1971C2semidirect
7. #11728308C2semidirect
8. #11727981C2semidirect
10. #11727558C2semidirect
11. #11727430C2semidirect
23. #89963C4semidirect
47. C2×C7×Q16C2×C4semidirect
53. #6510C2×C4semidirect
58. C2×#1143C2×C4semidirect
85. C8×Dic7C2×C4semidirect
96. #1971D7semidirect
226. C2×Q16C4×D7semidirect
384. C7#21694semidirect

Maximal subgroups

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

subgroupindex
C2×#899632
C2×#899562
C2×#896412
C2×#895602
#117339292
C7×#19712
#117283082
#117279812
#117275792
#117275582
#117274302
#216947