Finite Group Databasebeta

← #11736682

Subgroups — #11736682

Order 896.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90224C211yesmaximal, characteristic
2#90202C211yesmaximal, characteristic
3C7×#428C211yesmaximal, characteristic, complemented
4#89909C211yesmaximal, characteristic, complemented
5#89900C211yesmaximal, characteristic
6#89768C211yesmaximal, characteristic, complemented
7#89743C211yesmaximal, characteristic, complemented
8#89660C211yesmaximal, characteristic, complemented
9#89653C211yesmaximal, characteristic, complemented
10#89647C211yesmaximal, characteristic, complemented
11#89645C211yesmaximal, characteristic, complemented
12#89640C211yesmaximal, characteristic, complemented
13#89634C211yesmaximal, characteristic, complemented
14#89627C211yesmaximal, characteristic, complemented
15#89626C211yesmaximal, characteristic, complemented
16C2×C7×M16C2211yescharacteristic
17C7×#117C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C4×C22⋊D7—21no—
20C4×C22⋊D7C2211yescharacteristic
21#6487—21no—
22#6487C2211yescharacteristic
23#6487C2211yescharacteristic
24C2×Dic28C2211yescharacteristic
25C2×#1131C2211yescharacteristic
26#6460C2211yescharacteristic
27#6460C2211yescharacteristic
28#6459—21no—
29#6455C2211yescharacteristic
30D7×C4⋊C4C2211yescharacteristic
31#6452—21no—
32#6451C2211yescharacteristic
33#6451C2211yescharacteristic
34#6450C2211yescharacteristic
35#6450C2211yescharacteristic
36#6448—21no—
37#6442—21no—
38#6439—21no—
39C4×Dic14—21no—
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45(C2×D7)⋊C8C2211yescharacteristic
46#6391C2211yescharacteristic
47#6390C2211yescharacteristic
48#6390C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50C8×Dic7C2211yescharacteristic
51#6384C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3356—71nomaximal, Sylow
60C22×C4×C7—21no—
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7D411yescharacteristic
63C2×C4×D7—21no—
64C2×C4×D7C2311yescharacteristic
65C2×Dic14—21no—
66C2×Dic14—21no—
67C2×Dic14C2311yescharacteristic
68C2×Dic14C2311yescharacteristic
69C7×M16—22no—
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C7×C4⋊C4—21no—
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C22⋊C4—21no—
78C42×C7—21no—
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7D411yescharacteristic
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4D411yescharacteristic
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104Dic28—22no—
105#1131—22no—
106#556—71no—
107#554—71no—
108#498—71no—
109#485—71no—
110#485—71no—
111#483—71no—
112#483—71no—
113#477—71no—
114#475—71no—
115#470—71no—
116#470—71no—
117C4×Q16—71no—
118#436—71no—
119#428—71no—
120#428D711yescharacteristic, complemented
121C23×C7C2×D411yescharacteristic
122C22×D7C2×D411yescharacteristic
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—81no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2411yescharacteristic
132C22⋊D7—22no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C4×D7—21no—
141C4×D7—21no—
142C4×D7—81no—
143Dic14—21no—
144Dic14—21no—
145Dic14—21no—
146Dic14—21no—
147Dic14—42no—
148Dic14—81no—
149C7×C8—21no—
150C7×C8#4112yes—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C2×Q16—71no—
154C2×M16—71no—
155C2×M16C2×D711yescharacteristic
156C4⋊Q8—71no—
157C4⋊Q8—71no—
158#126—141no—
159#126—141no—
160#125—141no—
161#124—141no—
162C22⋊Q8—71no—
163C22⋊Q8—71no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166C22⋊Q8—141no—
167C22⋊Q8—141no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—141no—
171C4×D4—71no—
172C4×D4—141no—
173#117—71no—
174#117C2×D711yescharacteristic
175C2×C4⋊C4—71no—
176C2×C4⋊C4C2×D711yescharacteristic
177#107—71no—
178#107C2×D711yescharacteristic
179#107C2×D711yescharacteristic
180#106—71no—
181#106—71no—
182#106—71no—
183#106—71no—
184#106C2×D711yescharacteristic
185#106C2×D711yescharacteristic
186C4⋊C8—71no—
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193C22⋊C8—71no—
194C22⋊C8—71no—
195C4×C8—71no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7C22×D411yescharacteristic
200C2×D7—21no—
201C2×D7—21no—
202C2×D7—21no—
203C4×C7—21no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7C2×#4111yescharacteristic
209C4×C72−1+411yescharacteristic
210Dic7—21no—
211Dic7—21no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216Dic7#13712yes—
217C2×Q8—71no—
218C2×Q8—71no—
219C2×Q8—141no—
220C2×Q8—141no—
221C2×D4—71no—
222C22×C4—21no—
223C22×C4—71no—
224C22×C4—141no—
225C22×C4C22×D711yescharacteristic
226Q16—142no—
227M16—22no—
228M16—142no—
229C2×C8—71no—
230C2×C8—71no—
231C2×C8C22×D711yescharacteristic
232C2×C8C22×D711yescharacteristic
233C4⋊C4—21no—
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4C22×D711yescharacteristic
247C4⋊C4C22×D711yescharacteristic
248C4⋊C4C22×D711yescharacteristic
249C4⋊C4C22×D711yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—71no—
252C22⋊C4—71no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C42—21no—
259C42—71no—
260C42—141no—
261C42—141no—
262C42—141no—
263C2×C7—41no—
264C2×C7#54711yescharacteristic
265C2×C7C2×#13711yescharacteristic
266C2×C7#57711yescharacteristic
267D7—41no—
268C23—71no—
269C23D4×D711yescharacteristic
270Q8—141no—
271Q8—141no—
272Q8—141no—
273Q8—141no—
274Q8—282no—
275Q8—561no—
276D4—142no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—71no—
284C2×C4—71no—
285C2×C4—71no—
286C2×C4—81no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—561no—
294C2×C4D4×D711yescharacteristic
295C2×C4C23×D711yescharacteristic
296C8—21no—
297C8—141no—
298C8—141no—
299C8#115712yes—
300C7#335611yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22—21no—
304C22—141no—
305C22—141no—
306C22—141no—
307C22C2×D4×D711yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—72no—
314C4—141no—
315C4—141no—
316C4—281no—
317C4—281no—
318C4—281no—
319C4—281no—
320C4C2×#115711yescharacteristic
321C4#655111yescharacteristic
322C2—41no—
323C2—281no—
324C2#9019811yescharacteristic
325C2#9046611yescharacteristic
326C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radicalC2×C22⋊D7D4
hypercenter#428D7
hypocenterC7#3356
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. C7×#428C2semidirect
4. #89909C2semidirect
6. #89768C2semidirect
7. #89743C2semidirect
8. #89660C2semidirect
9. #89653C2semidirect
10. #89647C2semidirect
11. #89645C2semidirect
12. #89640C2semidirect
13. #89634C2semidirect
14. #89627C2semidirect
15. #89626C2semidirect
120. #428D7semidirect
300. C7#3356semidirect

Maximal subgroups

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

subgroupindex
#902242
#902022
C7×#4282
#899092
#899002
#897682
#897432
#896602
#896532
#896472
#896452
#896402
#896342
#896272
#896262
#33567