Finite Group Databasebeta

← #11738545

Subgroups — #11738545

Order 896.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90418C211yesmaximal, characteristic, complemented
2#90259C211yesmaximal, characteristic, complemented
3C7×#477C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89798C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89661C211yesmaximal, characteristic, complemented
8#89647C211yesmaximal, characteristic, complemented
9#89644C211yesmaximal, characteristic
10#89616C211yesmaximal, characteristic
11#89600C211yesmaximal, characteristic, complemented
12#89591C211yesmaximal, characteristic
13#89590C211yesmaximal, characteristic, complemented
14#89567C211yesmaximal, characteristic, complemented
15#89564C211yesmaximal, characteristic, complemented
16C2×Q8×D7—21no—
17C7×C4⋊Q8C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19#6510C2211yescharacteristic
20#6509—21no—
21#6507C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23C2×D56C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25#6461—21no—
26#6459—21no—
27#6459—21no—
28#6459—21no—
29#6456C2211yescharacteristic
30#6456C2211yescharacteristic
31#6456C2211yescharacteristic
32#6451C2211yescharacteristic
33#6451C2211yescharacteristic
34#6450C2211yescharacteristic
35#6439—21no—
36#6438C2211yescharacteristic
37C4×D28C2211yescharacteristic
38#6435—21no—
39C4×Dic14C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45#6409C2211yescharacteristic
46#6395C2211yescharacteristic
47#6390C2211yescharacteristic
48#6389C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3367—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62Q8×D7—44no—
63C2×D28C2311yescharacteristic
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×Dic14—21no—
70C2×Dic14—21no—
71C2×Dic14D411yescharacteristic
72C2×Dic14C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C42×C7—21no—
81C42×C7C2311yescharacteristic
82#1143—22no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91C4⋊Dic7D411yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105D56—22no—
106#1132—22no—
107#556—71no—
108D4×Q8—71no—
109#489—71no—
110#489—71no—
111#487—71no—
112#477—71no—
113#477D711yescharacteristic, complemented
114#474—71no—
115#463—71no—
116#463—71no—
117#461—71no—
118#446—71no—
119#440—71no—
120#440—71no—
121C4×D8—71no—
122C22×D7—21no—
123C22×D7—21no—
124C7×Q8—21no—
125C7×Q8—21no—
126C7×Q8—42no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140D28—21no—
141D28—21no—
142D28—21no—
143D28#4112yes—
144C4×D7—42no—
145C4×D7—42no—
146C4×D7—42no—
147C4×D7—42no—
148Dic14—22no—
149Dic14—21no—
150Dic14—21no—
151Dic14—42no—
152Dic14—42no—
153C7×C8—21no—
154C7×C8—21no—
155C7⋊C8—21no—
156C7⋊C8—21no—
157C22×Q8—141no—
158C2×SD16—71no—
159C2×SD16—71no—
160C2×D8—71no—
161C4⋊Q8—71no—
162C4⋊Q8—71no—
163C4⋊Q8—71no—
164C4⋊Q8C2×D711yescharacteristic
165#126—141no—
166#126—141no—
167#124—71no—
168#124—71no—
169C22⋊Q8—141no—
170C22⋊Q8—141no—
171C22⋊Q8—141no—
172C22⋊Q8—141no—
173C4×Q8—71no—
174C4×Q8—71no—
175C4×Q8C2×D711yescharacteristic
176C4×D4—71no—
177C4×D4—71no—
178C4×D4—71no—
179C4×D4—71no—
180#117—141no—
181#107—71no—
182#107C2×D711yescharacteristic
183#107C2×D711yescharacteristic
184C4⋊C8—71no—
185C4⋊C8—71no—
186C4⋊C8C2×D711yescharacteristic
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#103C2×D711yescharacteristic
192#103C2×D711yescharacteristic
193#102—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197#97—71no—
198#97—71no—
199C4×C8—71no—
200C22×C7C22×D411yescharacteristic
201C2×D7—22no—
202C2×D7—41no—
203C2×D7—41no—
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7C2×#4111yescharacteristic
214C4×C72−1+411yescharacteristic
215Dic7—21no—
216Dic7—21no—
217Dic7—21no—
218Dic7—41no—
219Dic7—41no—
220Dic7AGL(1,ℤ/8)12yes—
221C2×Q8—21no—
222C2×Q8—71no—
223C2×Q8—71no—
224C2×Q8—141no—
225C2×Q8—141no—
226C2×Q8—284no—
227C2×Q8C22×D711yescharacteristic
228C2×D4—71no—
229C2×D4—71no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4—141no—
234SD16—142no—
235SD16—142no—
236D8—142no—
237C2×C8—71no—
238C2×C8—71no—
239C2×C8C22×D711yescharacteristic
240C2×C8C22×D711yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—71no—
244C4⋊C4—71no—
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4C22×D711yescharacteristic
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C42—21no—
264C42—71no—
265C42—71no—
266C42—141no—
267C42C22×D711yescharacteristic
268C2×C7#54811yescharacteristic
269C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
270C2×C7#57811yescharacteristic
271D7—42no—
272D7—81no—
273C23—141no—
274C23—141no—
275Q8—21no—
276Q8—21no—
277Q8—42no—
278Q8—142no—
279Q8—141no—
280Q8—141no—
281Q8—282no—
282Q8—282no—
283D4—72no—
284D4—141no—
285D4—141no—
286D4—141no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—71no—
292C2×C4—71no—
293C2×C4—71no—
294C2×C4—71no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—282no—
301C2×C4D4×D711yescharacteristic
302C2×C4D4×D711yescharacteristic
303C2×C4C23×D711yescharacteristic
304C8—21no—
305C8—21no—
306C8—141no—
307C8—141no—
308C7#336711yescharacteristic, complemented, Sylow
309C22—142no—
310C22—281no—
311C22—281no—
312C22—281no—
313C22—281no—
314C22—281no—
315C22C2×D4×D711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—72no—
323C4—141no—
324C4—141no—
325C4—141no—
326C4—281no—
327C4—281no—
328C4#655111yescharacteristic
329C4D7×#4111yescharacteristic
330C2—282no—
331C2—561no—
332C2#9032411yescharacteristic
333C2#9045711yescharacteristic
334C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#477C2
socleC22×C7C22×D4
prime radicalC2×D56C22
hypercenter#477D7
hypocenterC7#3367
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #90418C2semidirect
2. #90259C2semidirect
3. C7×#477C2semidirect
4. #89861C2semidirect
5. #89798C2semidirect
6. #89662C2semidirect
7. #89661C2semidirect
8. #89647C2semidirect
11. #89600C2semidirect
13. #89590C2semidirect
14. #89567C2semidirect
15. #89564C2semidirect
113. #477D7semidirect
308. C7#3367semidirect

Maximal subgroups

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

subgroupindex
#904182
#902592
C7×#4772
#898612
#897982
#896622
#896612
#896472
#896442
#896162
#896002
#895912
#895902
#895672
#895642
#33677