Finite Group Databasebeta

← #11737255

Subgroups — #11737255

Order 896.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1161C211yesmaximal, characteristic, complemented
2#90248C211yesmaximal, characteristic, complemented
3C7×#440C211yesmaximal, characteristic, complemented
4#89944C211yesmaximal, characteristic
5#89938C211yesmaximal, characteristic, complemented
6#89802C211yesmaximal, characteristic, complemented
7#89786C211yesmaximal, characteristic, complemented
8#89663C211yesmaximal, characteristic, complemented
9#89660C211yesmaximal, characteristic, complemented
10#89582C211yesmaximal, characteristic, complemented
11#89568C211yesmaximal, characteristic, complemented
12#89559C211yesmaximal, characteristic, complemented
13#89556C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15#89477C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1144C2211yescharacteristic
22#6501C2211yescharacteristic
23#6499C2211yescharacteristic
24C2×#1141C2211yescharacteristic
25C4×C22⋊D7—21no—
26#6489—21no—
27C2×#1132C2211yescharacteristic
28#6460C2211yescharacteristic
29#6456—21no—
30#6456C2211yescharacteristic
31#6455—21no—
32#6455C2211yescharacteristic
33#6453C2211yescharacteristic
34#6446—21no—
35#6442—21no—
36C4×D28—21no—
37C4×D28C2211yescharacteristic
38C42×D7—21no—
39C42×D7C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×#97C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47#6395C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6392C2211yescharacteristic
51#6390C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58#6377C2211yescharacteristic
59#3111—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64#1161—22no—
65#1161—22no—
66C2×D28—21no—
67C2×D28C2311yescharacteristic
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7D411yescharacteristic
72C2×C4×D7D411yescharacteristic
73C2×C4×D7C2311yescharacteristic
74C2×Dic14C2311yescharacteristic
75C7×SD16—22no—
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C7×C4⋊C4—21no—
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C42×C7—21no—
83C42×C7C2311yescharacteristic
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87#1144—22no—
88#1141—22no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7D411yescharacteristic
104C4×Dic7D411yescharacteristic
105C4×Dic7C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108#1132—22no—
109#529—71no—
110C4×#41—71no—
111#485—71no—
112#482—71no—
113#479—71no—
114#478—71no—
115#464—71no—
116#463—71no—
117#452—71no—
118#449—71no—
119#442—71no—
120#441—71no—
121#440—71no—
122#440D711yescharacteristic, complemented
123#432—71no—
124C23×C7—21no—
125C22×D7—21no—
126C22×D7C2×D411yescharacteristic
127C7×Q8—21no—
128C7×Q8#4112yes—
129C7×D4—21no—
130C7×D4—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—42no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C22⋊D7—42no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145D28—24no—
146D28—21no—
147D28#4112yes—
148C4×D7—24no—
149C4×D7—21no—
150C4×D7—21no—
151C4×D7—41no—
152C4×D7—41no—
153C4×D7—42no—
154C4×D7—42no—
155Dic14—21no—
156Dic14—21no—
157C7×C8—21no—
158C7×C8—21no—
159C7⋊C8—21no—
160C7⋊C8—21no—
161C2×#41—71no—
162C2×Q16—71no—
163C2×SD16—71no—
164C2×SD16C2×D711yescharacteristic
165C2×D8—71no—
166#126—141no—
167#125—71no—
168#124—71no—
169#123—141no—
170C22⋊Q8—71no—
171#121—71no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×Q8C2×D711yescharacteristic
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—141no—
178C4×D4—141no—
179C4×D4—141no—
180C4×D4C2×D711yescharacteristic
181#117—71no—
182#117—141no—
183#117—141no—
184C2×C42—71no—
185C2×C42—141no—
186#107—71no—
187#107C2×D711yescharacteristic
188#106—71no—
189#106—71no—
190C4⋊C8—71no—
191C4⋊C8—71no—
192#103—71no—
193#103—71no—
194#103—71no—
195#103C2×D711yescharacteristic
196#102—71no—
197#102—71no—
198#102—71no—
199#102C2×D711yescharacteristic
200C22⋊C8—71no—
201C22⋊C8—71no—
202#97—71no—
203#97C2×D711yescharacteristic
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7C22×D411yescharacteristic
208C2×D7—21no—
209C2×D7—22no—
210C2×D7—21no—
211C2×D7—21no—
212C2×D7—41no—
213C2×D7—41no—
214C4×C7—22no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7C2×#4111yescharacteristic
220C4×C7C2×#4111yescharacteristic
221Dic7—21no—
222Dic7—21no—
223Dic7—21no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227#41—142no—
228#41—142no—
229C2×Q8—71no—
230C2×Q8C22×D711yescharacteristic
231C2×D4—71no—
232C2×D4—141no—
233C2×D4—141no—
234C2×D4C22×D711yescharacteristic
235C22×C4—21no—
236C22×C4—71no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4—141no—
242Q16—142no—
243SD16—22no—
244SD16—142no—
245D8—142no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8C22×D711yescharacteristic
249C2×C8C22×D711yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—71no—
252C4⋊C4—71no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4C22×D711yescharacteristic
259C4⋊C4C22×D711yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C42—21no—
269C42—71no—
270C42—71no—
271C42—71no—
272C42—141no—
273C42—141no—
274C42—141no—
275C42C22×D711yescharacteristic
276C2×C7—81no—
277C2×C7#52511yescharacteristic
278C2×C7#57511yescharacteristic
279C2×C7#57511yescharacteristic
280D7—41no—
281D7—42no—
282C23—21no—
283C23—71no—
284C23—141no—
285Q8—21no—
286Q8—141no—
287Q8—141no—
288Q8#115712yes—
289D4—21no—
290D4—21no—
291D4—72no—
292D4—144no—
293D4—141no—
294D4—282no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—71no—
300C2×C4—71no—
301C2×C4—71no—
302C2×C4—144no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—281no—
309C2×C4—281no—
310C2×C4—282no—
311C2×C4—282no—
312C2×C4D4×D711yescharacteristic
313C2×C4D4×D711yescharacteristic
314C2×C4C23×D711yescharacteristic
315C8—21no—
316C8—21no—
317C8—141no—
318C8—141no—
319C7#311111yescharacteristic, complemented, Sylow
320C22—41no—
321C22—41no—
322C22—41no—
323C22—141no—
324C22—141no—
325C22—142no—
326C22—141no—
327C22—281no—
328C22—281no—
329C22C2×D4×D711yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—141no—
336C4—141no—
337C4—141no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4C2×#115711yescharacteristic
342C4D7×#4111yescharacteristic
343C2—81no—
344C2—281no—
345C2—282no—
346C2#9023811yescharacteristic
347C2#9046411yescharacteristic
348C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
prime radical#89944C2
hypercenter#440D7
hypocenterC7#3111
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C4×#1161C2semidirect
2. #90248C2semidirect
3. C7×#440C2semidirect
5. #89938C2semidirect
6. #89802C2semidirect
7. #89786C2semidirect
8. #89663C2semidirect
9. #89660C2semidirect
10. #89582C2semidirect
11. #89568C2semidirect
12. #89559C2semidirect
13. #89556C2semidirect
14. #89482C2semidirect
15. #89477C2semidirect
122. #440D7semidirect
319. C7#3111semidirect

Maximal subgroups

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

subgroupindex
C4×#11612
#902482
C7×#4402
#899442
#899382
#898022
#897862
#896632
#896602
#895822
#895682
#895592
#895562
#894822
#894772
#31117