Finite Group Databasebeta

← #13364394

Subgroups — #13364394

Order 1792.

762 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89481C211yesmaximal, characteristic
2#11731803C211yesmaximal, characteristic
3#11731670C211yesmaximal, characteristic
4C7×#1847C211yesmaximal, characteristic, complemented
5#11728386C212yesmaximal
6#11728369C212yesmaximal, complemented
7#11728334C212yesmaximal
8#11728318C212yesmaximal, complemented
9#11727569C211yesmaximal, characteristic, complemented
10#11727331C211yesmaximal, characteristic, complemented
11#11727301C211yesmaximal, characteristic, complemented
12C2×#6455C2211yescharacteristic
13C2×D7×C4⋊C4C2211yescharacteristic
14C2×#6435C2211yescharacteristic
15C2×C7×#107C2212yes—
16C2×C7×#106C2212yes—
17C2×C7×#97C2211yescharacteristic
18C7×#384C2211yescharacteristic
19C7×#382C2211yescharacteristic
20C2×(C2×D7)⋊C8C2212yes—
21C2×Dic7⋊C8C2212yes—
22#89762—22no—
23#89759C2212yes—
24#89750—22no—
25#89749C2211yescharacteristic
26#89748C2211yescharacteristic
27C4⋊C4×Dic7C2212yes—
28C2×#6381C2212yes—
29C2×#6381C2212yes—
30#89708C2211yescharacteristic
31#89707C2211yescharacteristic
32C2×C4×C7⋊C8C2211yescharacteristic
33#89481C418yescomplemented
34#89440—22no—
35#89419—22no—
36#89346C2212yes—
37#89346C2212yes—
38#89324C2212yes—
39#89324C2212yes—
40#21991—71nomaximal, Sylow
41C22×C4×D7—21no—
42C22×C4×D7C2311yescharacteristic
43C22×C7×C8C2312yes—
44C2×C7×C4⋊C4—22no—
45C2×C7×C4⋊C4C2311yescharacteristic
46C2×C7×C4⋊C4C2311yescharacteristic
47C2×C7×C4⋊C4C2312yes—
48C2×C42×C7C2311yescharacteristic
49C2×(C2×D7)⋊C4—21no—
50C2×(C2×D7)⋊C4—22no—
51C2×(C2×D7)⋊C4D412yes—
52C2×C4⋊Dic7—22no—
53C2×C4⋊Dic7C2311yescharacteristic
54C2×C4⋊Dic7C2311yescharacteristic
55C2×C4⋊Dic7C2312yes—
56C2×Dic7⋊C4—21no—
57C2×Dic7⋊C4—22no—
58C2×Dic7⋊C4D412yes—
59C2×C4×Dic7—21no—
60C2×C4×Dic7—22no—
61C2×C4×Dic7C2311yescharacteristic
62C22×C7⋊C8C2312yes—
63#6455—42no—
64D7×C4⋊C4—42no—
65#6435C2×C414yes—
66#6435Q814yes—
67C7×#107—42no—
68C7×#106—42no—
69C7×#97C2×C414yescomplemented
70C7×#95—22no—
71C7⋊#95—22no—
72C7⋊#95—22no—
73C7⋊#95—22no—
74C7⋊#95—22no—
75(C2×D7)⋊C8C2×C418yes—
76Dic7⋊C8C2×C418yescomplemented
77#6381—42no—
78#6381—42no—
79C4×C7⋊C8C2×C414yescomplemented
80C2×#433—71no—
81#2317—71no—
82#2314—71no—
83#1949—72no—
84#1947—72no—
85#1854—72no—
86#1852—72no—
87#1847D711yescharacteristic, complemented
88#1845—71no—
89#1836—71no—
90#1835—71no—
91C23×D7C2×D411yescharacteristic
92C22×C4×C7—21no—
93C22×C4×C7—22no—
94C22×C4×C7—21no—
95C22×C4×C7C2×D412yes—
96C22×C4×C7C2411yescharacteristic
97C22×Dic7—22no—
98C22×Dic7—21no—
99C22×Dic7—21no—
100C22×Dic7C2×D412yes—
101C22×Dic7C2×D411yescharacteristic
102C2×C4×D7—22no—
103C2×C4×D7—22no—
104C2×C4×D7—42no—
105C2×C4×D7—82no—
106C2×C4×D7C22×C412yes—
107C2×C4×D7C2×Q812yes—
108C2×C7×C8—22no—
109C2×C7×C8—22no—
110C2×C7×C8C22×C414yes—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—22no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—82no—
116C42×C7C22×C412yes—
117C42×C7C2×Q812yes—
118(C2×D7)⋊C4—42no—
119(C2×D7)⋊C4—82no—
120(C2×D7)⋊C4C4⋊C418yes—
121C4⋊Dic7—22no—
122C4⋊Dic7—22no—
123C4⋊Dic7—24no—
124C4⋊Dic7—82no—
125Dic7⋊C4—42no—
126Dic7⋊C4—82no—
127Dic7⋊C4C4⋊C418yes—
128C4×Dic7—42no—
129C4×Dic7—82no—
130C4×Dic7C22×C412yes—
131C4×Dic7C2×Q812yes—
132C2×C7⋊C8—22no—
133C2×C7⋊C8—22no—
134C2×C7⋊C8C22×C414yes—
135C2×#117—71no—
136C2×#117—71no—
137C22×C4⋊C4—71no—
138#433—78no—
139C2×#107—72no—
140C2×#107—72no—
141C2×#107C2×D712yes—
142C2×#106C2×D712yes—
143C2×C4⋊C8—72no—
144C2×C22⋊C8—72no—
145C2×#97C2×D711yescharacteristic
146C2×C4×C8—71no—
147#389—142no—
148#388—142no—
149#387—142no—
150#385—142no—
151#384—71no—
152#384—71no—
153#384C2×D711yescharacteristic
154#382—71no—
155#382—71no—
156#382C2×D711yescharacteristic
157#380—72no—
158C4×C4⋊C4—72no—
159#340—72no—
160#340—72no—
161#340—72no—
162#340—72no—
163C23×C7C22×D411yescharacteristic
164C22×D7—21no—
165C22×D7—21no—
166C22×D7—22no—
167C22×D7—22no—
168C22×D7C2×C4⋊C412yes—
169C2×C4×C7—22no—
170C2×C4×C7—22no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7C2×C4⋊C414yes—
179C2×C4×C7C23×C411yescharacteristic
180C2×C4×C7C22×Q811yescharacteristic
181C2×C4×C7H(5,𝔽2)12yes—
182C2×C4×C72−1+412yes—
183C2×Dic7—21no—
184C2×Dic7—21no—
185C2×Dic7—22no—
186C2×Dic7—22no—
187C2×Dic7—41no—
188C2×Dic7—41no—
189C2×Dic7—42no—
190C2×Dic7—42no—
191C2×Dic7—42no—
192C2×Dic7—42no—
193C2×Dic7—42no—
194C2×Dic7C2×C4⋊C414yes—
195C2×Dic7C2×C4⋊C412yes—
196C4×D7—24no—
197C4×D7—24no—
198C4×D7—84no—
199C7×C8—24no—
200C7⋊C8—24no—
201C23×C4—71no—
202C23×C4—141no—
203C22×C8—72no—
204C22×C8C22×D712yes—
205#117—74no—
206#117—74no—
207#117—282no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—72no—
210C2×C4⋊C4—71no—
211C2×C4⋊C4—71no—
212C2×C4⋊C4—72no—
213C2×C4⋊C4—141no—
214C2×C4⋊C4—142no—
215C2×C4⋊C4—142no—
216C2×C4⋊C4—282no—
217C2×C4⋊C4C22×D711yescharacteristic
218C2×C4⋊C4C22×D711yescharacteristic
219C2×C4⋊C4C22×D712yes—
220C2×C22⋊C4—72no—
221C2×C22⋊C4—141no—
222C2×C22⋊C4—142no—
223C2×C42—71no—
224C2×C42—141no—
225C2×C42—142no—
226C2×C42C22×D711yescharacteristic
227#107—42no—
228#107—282no—
229#107—282no—
230#106—42no—
231C4⋊C8—78no—
232C22⋊C8—78no—
233#97C4×D714yescomplemented
234C4×C8—74no—
235#95—22no—
236#95—142no—
237#95—142no—
238#95—142no—
239#95—142no—
240C22×C7C22×C4⋊C411yescharacteristic
241C22×C7#53511yescharacteristic
242C22×C7#53611yescharacteristic
243C22×C7#57612yes—
244C22×C7#57812yes—
245C2×D7—22no—
246C2×D7—24no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—42no—
250C4×C7—24no—
251C4×C7—81no—
252C4×C7—82no—
253C4×C7—81no—
254C4×C7#52012yes—
255C4×C7#54412yes—
256Dic7—22no—
257Dic7—24no—
258Dic7—81no—
259Dic7—81no—
260Dic7—82no—
261C24—71no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—72no—
266C22×C4—71no—
267C22×C4—72no—
268C22×C4—72no—
269C22×C4—142no—
270C22×C4—142no—
271C22×C4—142no—
272C22×C4—141no—
273C22×C4—141no—
274C22×C4—282no—
275C22×C4—562no—
276C22×C4D4×D712yes—
277C22×C4C23×D711yescharacteristic
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—74no—
281C2×C8—142no—
282C2×C8—142no—
283C2×C8C2×C4×D714yes—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—78no—
289C4⋊C4—82no—
290C4⋊C4—142no—
291C4⋊C4—142no—
292C4⋊C4—144no—
293C4⋊C4—282no—
294C4⋊C4—562no—
295C4⋊C4—562no—
296C22⋊C4—78no—
297C22⋊C4—282no—
298C22⋊C4—562no—
299C42—72no—
300C42—72no—
301C42—282no—
302C42—562no—
303C42C2×C4×D712yes—
304C42Q8×D712yes—
305C2×C7#231711yescharacteristic
306C2×C7#291312yes—
307C2×C7#322011yescharacteristic
308C2×C7#322311yescharacteristic
309C2×C7#324312yes—
310D7—42no—
311C23—72no—
312C23—141no—
313C23—141no—
314C23—142no—
315C23—142no—
316C23C2×D4×D711yescharacteristic
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—74no—
327C2×C4—72no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—142no—
331C2×C4—142no—
332C2×C4—144no—
333C2×C4—144no—
334C2×C4—281no—
335C2×C4—281no—
336C2×C4—282no—
337C2×C4—282no—
338C2×C4—282no—
339C2×C4—282no—
340C2×C4—282no—
341C2×C4—564no—
342C2×C4D7×C4⋊C414yes—
343C2×C4C22×C4×D711yescharacteristic
344C2×C4#654812yes—
345C2×C4C2×Q8×D711yescharacteristic
346C2×C4#655112yes—
347C8—24no—
348C8—144no—
349C7#2199111yescharacteristic, complemented, Sylow
350C22—144no—
351C22—142no—
352C22—282no—
353C22—281no—
354C22—281no—
355C22C2×D7×C4⋊C411yescharacteristic
356C22#9040611yescharacteristic
357C22#9041511yescharacteristic
358C22#9046512yes—
359C22#9047012yes—
360C4—24no—
361C4—81no—
362C4—82no—
363C4—81no—
364C4—144no—
365C4—142no—
366C4—562no—
367C4—561no—
368C4—561no—
369C4#9019512yes—
370C4#9042012yes—
371C2—282no—
372C2#1173180411yescharacteristic
373C2#1173665512yes—
374C2#1173929811yescharacteristic
375C2#1173932411yescharacteristic
376C2#1173943312yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1847C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1847D7
hypocenterC7#21991
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. C7×#1847C2semidirect
6. #11728369C2semidirect
8. #11728318C2semidirect
9. #11727569C2semidirect
10. #11727331C2semidirect
11. #11727301C2semidirect
33. #89481C4semidirect
69. C7×#97C2×C4semidirect
76. Dic7⋊C8C2×C4semidirect
79. C4×C7⋊C8C2×C4semidirect
87. #1847D7semidirect
233. #97C4×D7semidirect
349. C7#21991semidirect

Maximal subgroups

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

subgroupindex
C2×#894812
#117318032
#117316702
C7×#18472
#117283862
#117283692
#117283342
#117283182
#117275692
#117273312
#117273012
#219917