Finite Group Databasebeta

← #11738951

Subgroups — #11738951

Order 896.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90334C211yesmaximal, characteristic
2#90210C211yesmaximal, characteristic, complemented
3C7×#485C211yesmaximal, characteristic, complemented
4#89816C211yesmaximal, characteristic, complemented
5#89775C211yesmaximal, characteristic
6#89662C211yesmaximal, characteristic, complemented
7#89646C211yesmaximal, characteristic, complemented
8#89643C211yesmaximal, characteristic, complemented
9#89628C211yesmaximal, characteristic, complemented
10#89572C211yesmaximal, characteristic, complemented
11#89568C211yesmaximal, characteristic, complemented
12#89567C211yesmaximal, characteristic, complemented
13#89566C211yesmaximal, characteristic, complemented
14#89508C211yesmaximal, characteristic
15#89492C211yesmaximal, characteristic, complemented
16C7×C22⋊Q8C2211yescharacteristic
17C7×#117C2211yescharacteristic
18#6510C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20#6493C2211yescharacteristic
21#6489C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23#6461—21no—
24#6460—21no—
25#6456C2211yescharacteristic
26#6453C2211yescharacteristic
27#6453C2211yescharacteristic
28#6451C2211yescharacteristic
29#6451C2211yescharacteristic
30#6451C2211yescharacteristic
31#6448—21no—
32#6446—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#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47#6392C2211yescharacteristic
48#6392C2211yescharacteristic
49#6389C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3243—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—21no—
63C22×Dic7—21no—
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×Dic14—21no—
67C2×Dic14—21no—
68C2×Dic14—21no—
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C22⋊C4—21no—
76C7×C22⋊C4—21no—
77C42×C7—21no—
78C22⋊Dic7D411yescharacteristic
79C22⋊Dic7D411yescharacteristic
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7D411yescharacteristic
89C4⋊Dic7D411yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98C4×Dic7—21no—
99C4×Dic7C2311yescharacteristic
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#544—71no—
104#542—71no—
105#501—71no—
106#500—71no—
107#489—71no—
108#488—71no—
109#485D711yescharacteristic, complemented
110#484—71no—
111#482—71no—
112#480—71no—
113#479—71no—
114#478—71no—
115#477—71no—
116#474—71no—
117#433—71no—
118C23×C7C2×D411yescharacteristic
119C22×D7—21no—
120C7×Q8—21no—
121C7×Q8—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7C2411yescharacteristic
130C22⋊D7—81no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—81no—
135C2×Dic7C2×D411yescharacteristic
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139D28—21no—
140D28—21no—
141C4×D7—81no—
142Dic14—81no—
143Dic14—81no—
144Dic14—81no—
145C7×C8—21no—
146C7×C8—21no—
147C7⋊C8—21no—
148C7⋊C8—21no—
149C4⋊Q8—71no—
150C4⋊Q8—71no—
151C4⋊Q8—71no—
152C4⋊Q8—141no—
153#126—141no—
154#125—71no—
155#125—71no—
156#125—141no—
157#124—71no—
158#124—141no—
159#123—141no—
160C22⋊Q8—141no—
161C22⋊Q8—141no—
162C22⋊Q8—141no—
163C22⋊Q8C2×D711yescharacteristic
164#121—71no—
165C4×Q8—71no—
166C4×D4—71no—
167#117—71no—
168#117C2×D711yescharacteristic
169C2×C4⋊C4—71no—
170#107—71no—
171#107—71no—
172#107—71no—
173#107—71no—
174#107—71no—
175#107C2×D711yescharacteristic
176#106—71no—
177#106C2×D711yescharacteristic
178C4⋊C8—71no—
179C4⋊C8—71no—
180#103—71no—
181#103—71no—
182#103C2×D711yescharacteristic
183#103C2×D711yescharacteristic
184#102—71no—
185#102—71no—
186#102—71no—
187#102—71no—
188C22⋊C8—71no—
189C22⋊C8C2×D711yescharacteristic
190#97—71no—
191C4×C8—71no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7C22×D411yescharacteristic
196C2×D7—41no—
197C2×D7—41no—
198C2×D7—41no—
199C4×C7—21no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C72−1+411yescharacteristic
205C4×C72−1+411yescharacteristic
206Dic7—21no—
207Dic7—21no—
208Dic7—21no—
209Dic7—21no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213C2×Q8—141no—
214C2×Q8—141no—
215C2×Q8—141no—
216C2×Q8C22×D711yescharacteristic
217C2×D4—71no—
218C2×D4—141no—
219C22×C4—141no—
220C22×C4—141no—
221C22×C4C22×D711yescharacteristic
222C2×C8—71no—
223C2×C8—71no—
224C2×C8C22×D711yescharacteristic
225C2×C8C22×D711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—71no—
228C4⋊C4—71no—
229C4⋊C4—71no—
230C4⋊C4—71no—
231C4⋊C4—71no—
232C4⋊C4—141no—
233C4⋊C4—141no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4C22×D711yescharacteristic
242C4⋊C4C22×D711yescharacteristic
243C4⋊C4C22×D711yescharacteristic
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—71no—
247C22⋊C4—71no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C22⋊C4—141no—
252C42—21no—
253C42—71no—
254C42—71no—
255C42—141no—
256C2×C7—41no—
257C2×C7#53611yescharacteristic
258C2×C7#57611yescharacteristic
259C2×C7#57811yescharacteristic
260D7—81no—
261C23—141no—
262C23D4×D711yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—561no—
266Q8—561no—
267Q8—561no—
268D4—141no—
269D4—141no—
270D4—561no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—71no—
278C2×C4—71no—
279C2×C4—71no—
280C2×C4—71no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—561no—
285C2×C4—561no—
286C2×C4D4×D711yescharacteristic
287C2×C4C23×D711yescharacteristic
288C8—21no—
289C8—21no—
290C8—141no—
291C8—141no—
292C7#324311yescharacteristic, complemented, Sylow
293C22—21no—
294C22—21no—
295C22—21no—
296C22—281no—
297C22—281no—
298C22—281no—
299C22C2×D4×D711yescharacteristic
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—141no—
306C4—141no—
307C4—141no—
308C4—141no—
309C4—281no—
310C4—281no—
311C4—281no—
312C4#655411yescharacteristic
313C4#655411yescharacteristic
314C2—41no—
315C2—561no—
316C2#9033911yescharacteristic
317C2#9046011yescharacteristic
318C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#485C2
socleC22×C7C22×D4
prime radical#6493C22
hypercenter#485D7
hypocenterC7#3243
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
2. #90210C2semidirect
3. C7×#485C2semidirect
4. #89816C2semidirect
6. #89662C2semidirect
7. #89646C2semidirect
8. #89643C2semidirect
9. #89628C2semidirect
10. #89572C2semidirect
11. #89568C2semidirect
12. #89567C2semidirect
13. #89566C2semidirect
15. #89492C2semidirect
109. #485D7semidirect
292. C7#3243semidirect

Maximal subgroups

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

subgroupindex
#903342
#902102
C7×#4852
#898162
#897752
#896622
#896462
#896432
#896282
#895722
#895682
#895672
#895662
#895082
#894922
#32437