Finite Group Databasebeta

← #11738964

Subgroups — #11738964

Order 896.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90319C211yesmaximal, characteristic
2#90214C211yesmaximal, characteristic
3C7×#485C211yesmaximal, characteristic, complemented
4#89815C211yesmaximal, characteristic, complemented
5#89768C211yesmaximal, characteristic, complemented
6#89660C211yesmaximal, characteristic, complemented
7#89652C211yesmaximal, characteristic, complemented
8#89634C211yesmaximal, characteristic, complemented
9#89633C211yesmaximal, characteristic, complemented
10#89586C211yesmaximal, characteristic, complemented
11#89581C211yesmaximal, characteristic, complemented
12#89574C211yesmaximal, characteristic, complemented
13D7×#103C211yesmaximal, characteristic, complemented
14#89499C211yesmaximal, characteristic
15#89494C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C7×#117C2211yescharacteristic
20#6509C2211yescharacteristic
21C2×C4⋊Dic7C2211yescharacteristic
22#6487C2211yescharacteristic
23C2×#1131C2211yescharacteristic
24C2×C8×D7C2211yescharacteristic
25#6460C2211yescharacteristic
26#6457—21no—
27#6455C2211yescharacteristic
28#6455C2211yescharacteristic
29#6455C2211yescharacteristic
30D7×C4⋊C4C2211yescharacteristic
31D7×C4⋊C4C2211yescharacteristic
32#6451—21no—
33#6448—21no—
34#6446—21no—
35(C22⋊D7)⋊C4—21no—
36(C22⋊D7)⋊C4—21no—
37C4×D28—21no—
38C4×Dic14—21no—
39C7×#107C2211yescharacteristic
40C7×#106C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46C22⋊(C7⋊C8)C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3097—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7D411yescharacteristic
63C2×C22⋊D7D411yescharacteristic
64C22×Dic7—21no—
65Q8×D7—22no—
66Q8×D7—22no—
67#1157—22no—
68#1157—22no—
69C2×D28D411yescharacteristic
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7C2311yescharacteristic
74C2×C4×D7C2311yescharacteristic
75C2×Dic14—21no—
76C2×Dic14D411yescharacteristic
77C2×Dic14C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C42×C7—21no—
87C22⋊Dic7—21no—
88(C2×D7)⋊C4—21no—
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⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—22no—
108C8×D7—22no—
109#536—71no—
110#520—71no—
111#485—71no—
112#485—71no—
113#485—71no—
114#485D711yescharacteristic, complemented
115#484—71no—
116#484—71no—
117#483—71no—
118#483—71no—
119#428—71no—
120#427—71no—
121#419—71no—
122C2×#103—71no—
123#408—71no—
124C23×C7C2×D411yescharacteristic
125C22×D7C2×D411yescharacteristic
126C22×D7C2×D411yescharacteristic
127C7×Q8—21no—
128C7×Q8—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C22⋊D7—22no—
138C22⋊D7—22no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—81no—
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146D28—22no—
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—42no—
150C4×D7—81no—
151C4×D7—81no—
152C4×D7#4112yes—
153C4×D7#4112yes—
154Dic14—22no—
155Dic14—21no—
156Dic14—21no—
157Dic14—42no—
158C7×C8—21no—
159C7×C8—21no—
160C7⋊C8—21no—
161C7⋊C8—21no—
162C2×#41—71no—
163C22×Q8—71no—
164C2×M16—71no—
165C22×C8—71no—
166C4⋊Q8—141no—
167#124—141no—
168#123—141no—
169#123—141no—
170C22⋊Q8—71no—
171C22⋊Q8—71no—
172C22⋊Q8—71no—
173C22⋊Q8C2×D711yescharacteristic
174C4×Q8—141no—
175C4×D4—141no—
176C4×D4—141no—
177C4×D4—141no—
178#117—71no—
179#117—71no—
180#117—71no—
181#117C2×D711yescharacteristic
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185#107—71no—
186#107—71no—
187#107—71no—
188#107C2×D711yescharacteristic
189#106—71no—
190#106—71no—
191#106—71no—
192#106C2×D711yescharacteristic
193#103—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103C2×D711yescharacteristic
200#103C2×D711yescharacteristic
201C22⋊C8—71no—
202C22⋊C8—71no—
203C22⋊C8—71no—
204C22⋊C8C2×D711yescharacteristic
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7C22×D411yescharacteristic
209C2×D7—21no—
210C2×D7—21no—
211C2×D7—21no—
212C2×D7—21no—
213C2×D7—21no—
214C2×D7#12312yes—
215C4×C7—21no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7C2×#4111yescharacteristic
221C4×C7C2×#4111yescharacteristic
222Dic7—21no—
223Dic7—41no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227Dic7#12312yes—
228#41—142no—
229#41—142no—
230C2×Q8—71no—
231C2×Q8—71no—
232C2×Q8—141no—
233C2×Q8—142no—
234C2×Q8—142no—
235C2×Q8C22×D711yescharacteristic
236C2×D4—71no—
237C2×D4—71no—
238C2×D4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4—141no—
244C22×C4—141no—
245C22×C4C22×D711yescharacteristic
246M16—142no—
247C2×C8—71no—
248C2×C8—71no—
249C2×C8—142no—
250C2×C8C22×D711yescharacteristic
251C2×C8C22×D711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4C22×D711yescharacteristic
263C4⋊C4C22×D711yescharacteristic
264C4⋊C4C22×D711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C42—21no—
274C42—141no—
275C42—141no—
276C42—141no—
277C2×C7—41no—
278C2×C7C2×#12311yescharacteristic
279C2×C7#57711yescharacteristic
280C2×C7#57811yescharacteristic
281D7—22no—
282D7—41no—
283C23—71no—
284C23—71no—
285C23D4×D711yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—142no—
289Q8—141no—
290Q8—141no—
291Q8—282no—
292D4—142no—
293D4—142no—
294D4—142no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—71no—
302C2×C4—72no—
303C2×C4—72no—
304C2×C4—71no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—282no—
312C2×C4—561no—
313C2×C4—561no—
314C2×C4—561no—
315C2×C4D4×D711yescharacteristic
316C2×C4C23×D711yescharacteristic
317C8—21no—
318C8—21no—
319C8—141no—
320C8—141no—
321C7#309711yescharacteristic, complemented, Sylow
322C22—21no—
323C22—21no—
324C22—21no—
325C22—72no—
326C22—141no—
327C22—141no—
328C22—141no—
329C22—141no—
330C22—141no—
331C22C2×D4×D711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—72no—
338C4—141no—
339C4—281no—
340C4—281no—
341C4—281no—
342C4—281no—
343C4D7×#4111yescharacteristic
344C4D7×#4111yescharacteristic
345C2—41no—
346C2—142no—
347C2—281no—
348C2D7×#12311yescharacteristic
349C2#9046211yescharacteristic
350C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#485C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenter#485D7
hypocenterC7#3097
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. C7×#485C2semidirect
4. #89815C2semidirect
5. #89768C2semidirect
6. #89660C2semidirect
7. #89652C2semidirect
8. #89634C2semidirect
9. #89633C2semidirect
10. #89586C2semidirect
11. #89581C2semidirect
12. #89574C2semidirect
13. D7×#103C2semidirect
15. #89494C2semidirect
114. #485D7semidirect
321. C7#3097semidirect

Maximal subgroups

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

subgroupindex
#903192
#902142
C7×#4852
#898152
#897682
#896602
#896522
#896342
#896332
#895862
#895812
#895742
D7×#1032
#894992
#894942
#30977