Finite Group Databasebeta

← #11738949

Subgroups — #11738949

Order 896.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90316C211yesmaximal, characteristic
2#90210C211yesmaximal, characteristic
3C7×#485C211yesmaximal, characteristic
4#89815C211yesmaximal, characteristic
5#89768C211yesmaximal, characteristic
6#89650C211yesmaximal, characteristic, complemented
7#89649C211yesmaximal, characteristic, complemented
8#89628C211yesmaximal, characteristic, complemented
9#89626C211yesmaximal, characteristic, complemented
10#89569C211yesmaximal, characteristic, complemented
11#89568C211yesmaximal, characteristic, complemented
12#89566C211yesmaximal, characteristic, complemented
13#89564C211yesmaximal, characteristic, complemented
14#89494C211yesmaximal, characteristic
15#89492C211yesmaximal, characteristic
16C7×C22⋊Q8C2211yescharacteristic
17C7×#117C2211yescharacteristic
18Q8×Dic7C2211yescharacteristic
19#6507C2211yescharacteristic
20#6489C2211yescharacteristic
21C2×C4⋊Dic7C2211yescharacteristic
22#6487C2211yescharacteristic
23#6453—21no—
24#6453C2211yescharacteristic
25#6453C2211yescharacteristic
26#6453C2211yescharacteristic
27#6452—21no—
28#6451C2211yescharacteristic
29#6451C2211yescharacteristic
30#6450C2211yescharacteristic
31#6442—21no—
32#6442—21no—
33#6441—21no—
34#6441—21no—
35#6433—21no—
36#6432—21no—
37C7×#107C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×#103C2211yescharacteristic
40C7×#103C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6392C2211yescharacteristic
46#6392C2211yescharacteristic
47#6390C2211yescharacteristic
48#6390C2211yescharacteristic
49#6389C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3242—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×Dic7—21no—
63C2×Dic14—21no—
64C2×Dic14—21no—
65C2×Dic14C2311yescharacteristic
66C2×C7×C8C2311yescharacteristic
67C2×C7×C8C2311yescharacteristic
68C7×C4⋊C4—21no—
69C7×C4⋊C4C2311yescharacteristic
70C7×C4⋊C4C2311yescharacteristic
71C7×C4⋊C4C2311yescharacteristic
72C7×C22⋊C4—21no—
73C7×C22⋊C4—21no—
74C42×C7—21no—
75C22⋊Dic7—21no—
76C22⋊Dic7D411yescharacteristic
77C22⋊Dic7D411yescharacteristic
78C4⋊Dic7—21no—
79C4⋊Dic7—21no—
80C4⋊Dic7—21no—
81C4⋊Dic7—21no—
82C4⋊Dic7—21no—
83C4⋊Dic7D411yescharacteristic
84C4⋊Dic7D411yescharacteristic
85C4⋊Dic7C2311yescharacteristic
86C4⋊Dic7C2311yescharacteristic
87C4⋊Dic7C2311yescharacteristic
88Dic7⋊C4—21no—
89Dic7⋊C4—21no—
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7C2311yescharacteristic
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#544—71no—
104#541—71no—
105#501—71no—
106#499—71no—
107#491—71no—
108#487—71no—
109#485—71no—
110#485D711yescharacteristic
111#484—71no—
112#483—71no—
113#479—71no—
114#479—71no—
115#477—71no—
116#475—71no—
117#433—71no—
118C23×C7C2×D411yescharacteristic
119C7×Q8—21no—
120C7×Q8—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7C2×D411yescharacteristic
128C2×C4×C7C2411yescharacteristic
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—81no—
134C2×Dic7C2×D411yescharacteristic
135C2×Dic7C2×D411yescharacteristic
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138Dic14—21no—
139Dic14—21no—
140Dic14—81no—
141Dic14—81no—
142C7×C8—21no—
143C7×C8—21no—
144C7⋊C8—21no—
145C7⋊C8—21no—
146C4⋊Q8—71no—
147C4⋊Q8—71no—
148C4⋊Q8—71no—
149C4⋊Q8—141no—
150#126—141no—
151#126—141no—
152#125—71no—
153#125—71no—
154#125—71no—
155#125—141no—
156#125—141no—
157#125—141no—
158C22⋊Q8—71no—
159C22⋊Q8—141no—
160C22⋊Q8—141no—
161C22⋊Q8C2×D711yescharacteristic
162C4×Q8—71no—
163C4×Q8—71no—
164#117—71no—
165#117C2×D711yescharacteristic
166C2×C4⋊C4—71no—
167#107—71no—
168#107—71no—
169#107—71no—
170#107C2×D711yescharacteristic
171#106—71no—
172#106—71no—
173#106—71no—
174#106C2×D711yescharacteristic
175C4⋊C8—71no—
176C4⋊C8—71no—
177#103—71no—
178#103—71no—
179#103—71no—
180#103—71no—
181#103—71no—
182#103—71no—
183#103C2×D711yescharacteristic
184#103C2×D711yescharacteristic
185C22⋊C8—71no—
186C22⋊C8C2×D711yescharacteristic
187#97—71no—
188C4×C8—71no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7C22×D411yescharacteristic
193C4×C7—21no—
194C4×C7—41no—
195C4×C7—41no—
196C4×C7—41no—
197C4×C7—41no—
198C4×C72−1+411yescharacteristic
199C4×C72−1+411yescharacteristic
200Dic7—21no—
201Dic7—21no—
202Dic7—21no—
203Dic7—21no—
204Dic7—41no—
205Dic7—41no—
206Dic7—41no—
207Dic7—41no—
208C2×Q8—71no—
209C2×Q8—141no—
210C2×Q8—141no—
211C2×Q8C22×D711yescharacteristic
212C22×C4—141no—
213C22×C4C22×D711yescharacteristic
214C2×C8—71no—
215C2×C8—71no—
216C2×C8C22×D711yescharacteristic
217C2×C8C22×D711yescharacteristic
218C4⋊C4—21no—
219C4⋊C4—71no—
220C4⋊C4—71no—
221C4⋊C4—71no—
222C4⋊C4—71no—
223C4⋊C4—71no—
224C4⋊C4—141no—
225C4⋊C4—141no—
226C4⋊C4—141no—
227C4⋊C4—141no—
228C4⋊C4—141no—
229C4⋊C4—141no—
230C4⋊C4—141no—
231C4⋊C4—141no—
232C4⋊C4—141no—
233C4⋊C4—141no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4C22×D711yescharacteristic
239C4⋊C4C22×D711yescharacteristic
240C4⋊C4C22×D711yescharacteristic
241C22⋊C4—21no—
242C22⋊C4—21no—
243C22⋊C4—71no—
244C22⋊C4—71no—
245C22⋊C4—141no—
246C42—21no—
247C42—71no—
248C42—71no—
249C42—141no—
250C42—141no—
251C2×C7—41no—
252C2×C7#53611yescharacteristic
253C2×C7#57711yescharacteristic
254C2×C7#57811yescharacteristic
255C23D4×D711yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—141no—
259Q8—141no—
260Q8—561no—
261Q8—561no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—71no—
269C2×C4—71no—
270C2×C4—71no—
271C2×C4—71no—
272C2×C4—141no—
273C2×C4—141no—
274C2×C4—141no—
275C2×C4—141no—
276C2×C4—561no—
277C2×C4D4×D711yescharacteristic
278C2×C4C23×D711yescharacteristic
279C8—21no—
280C8—21no—
281C8—141no—
282C8—141no—
283C7#324211yescharacteristic, complemented, Sylow
284C22—21no—
285C22—21no—
286C22—21no—
287C22C2×D4×D711yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—141no—
294C4—141no—
295C4—141no—
296C4—141no—
297C4—281no—
298C4—281no—
299C4—281no—
300C4—281no—
301C4#655411yescharacteristic
302C4#655411yescharacteristic
303C2—41no—
304C2#9033911yescharacteristic
305C2#9046211yescharacteristic
306C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#485C2
socleC22×C7C22×D4
prime radicalC23×C7C2×D4
hypercenter#485D7
hypocenterC7#3242
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
6. #89650C2semidirect
7. #89649C2semidirect
8. #89628C2semidirect
9. #89626C2semidirect
10. #89569C2semidirect
11. #89568C2semidirect
12. #89566C2semidirect
13. #89564C2semidirect
283. C7#3242semidirect

Maximal subgroups

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

subgroupindex
#903162
#902102
C7×#4852
#898152
#897682
#896502
#896492
#896282
#896262
#895692
#895682
#895662
#895642
#894942
#894922
#32427