Finite Group Databasebeta

← #11736671

Subgroups — #11736671

Order 896.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90221C211yesmaximal, characteristic
2#90202C211yesmaximal, characteristic, complemented
3C7×#428C211yesmaximal, characteristic, complemented
4#89906C211yesmaximal, characteristic, complemented
5#89899C211yesmaximal, characteristic, complemented
6#89768C211yesmaximal, characteristic, complemented
7#89739C211yesmaximal, characteristic, complemented
8#89662C211yesmaximal, characteristic, complemented
9#89661C211yesmaximal, characteristic, complemented
10#89656C211yesmaximal, characteristic, complemented
11D7×#107C211yesmaximal, characteristic, complemented
12#89636C211yesmaximal, characteristic, complemented
13#89635C211yesmaximal, characteristic, complemented
14#89629C211yesmaximal, characteristic, complemented
15#89628C211yesmaximal, characteristic, complemented
16C2×C7×M16C2211yescharacteristic
17C7×#117C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19#6493C2211yescharacteristic
20C4×C22⋊D7—21no—
21C4×C22⋊D7C2211yescharacteristic
22#6487—21no—
23#6487C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25C2×C8×D7C2211yescharacteristic
26#6460C2211yescharacteristic
27#6459—21no—
28#6458C2211yescharacteristic
29#6456C2211yescharacteristic
30D7×C4⋊C4C2211yescharacteristic
31D7×C4⋊C4C2211yescharacteristic
32#6453C2211yescharacteristic
33#6452—21no—
34#6451C2211yescharacteristic
35#6450C2211yescharacteristic
36#6447—21no—
37#6446—21no—
38#6439—21no—
39C4×D28—21no—
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6395C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47#6392C2211yescharacteristic
48#6390C2211yescharacteristic
49#6389C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51#6384C2211yescharacteristic
52#6384C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3346—71nomaximal, Sylow
60C22×C4×C7—21no—
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—21no—
63C2×C22⋊D7D411yescharacteristic
64C2×D28—21no—
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7C2311yescharacteristic
70C2×Dic14—21no—
71C2×Dic14C2311yescharacteristic
72C7×M16—22no—
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C42×C7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7D411yescharacteristic
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4D411yescharacteristic
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104#1132—22no—
105C8×D7—22no—
106#554—71no—
107#553—71no—
108#501—71no—
109#485—71no—
110#484—71no—
111#481—71no—
112#480—71no—
113#479—71no—
114#474—71no—
115#468—71no—
116#465—71no—
117#440—71no—
118#435—71no—
119#428D711yescharacteristic, complemented
120C2×#107—71no—
121C23×C7C2×D411yescharacteristic
122C22×D7—21no—
123C22×D7C2×D411yescharacteristic
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—81no—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C22⋊D7—22no—
134C22⋊D7—81no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141D28—21no—
142D28—21no—
143D28—42no—
144C4×D7—21no—
145C4×D7—21no—
146C4×D7—81no—
147C4×D7—81no—
148C4×D7—81no—
149Dic14—21no—
150Dic14—21no—
151Dic14—81no—
152C7×C8—21no—
153C7×C8#4112yes—
154C7⋊C8—21no—
155C7⋊C8—21no—
156C2×SD16—71no—
157C2×M16C2×D711yescharacteristic
158C22×C8—71no—
159C4⋊Q8—71no—
160#126—141no—
161#125—71no—
162#125—141no—
163#123—141no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166C22⋊Q8—141no—
167C22⋊Q8—141no—
168#121—71no—
169#121—71no—
170#121—141no—
171C4×Q8—71no—
172C4×D4—71no—
173C4×D4—71no—
174C4×D4—141no—
175C4×D4—141no—
176#117C2×D711yescharacteristic
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4C2×D711yescharacteristic
180#107—71no—
181#107—71no—
182#107—71no—
183#107—71no—
184#107C2×D711yescharacteristic
185#107C2×D711yescharacteristic
186#106—71no—
187#106C2×D711yescharacteristic
188#106C2×D711yescharacteristic
189C4⋊C8—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193#102—71no—
194#102—71no—
195#102—71no—
196C22⋊C8—71no—
197C22⋊C8—71no—
198#97—71no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7C22×D411yescharacteristic
203C2×D7—21no—
204C2×D7—21no—
205C2×D7—41no—
206C2×D7—41no—
207C2×D7—41no—
208C2×D7#13712yes—
209C4×C7—21no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7C2×#4111yescharacteristic
215C4×C72−1+411yescharacteristic
216Dic7—21no—
217Dic7—21no—
218Dic7—21no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222C2×Q8—71no—
223C2×Q8—141no—
224C2×D4—71no—
225C2×D4—71no—
226C2×D4—141no—
227C2×D4—141no—
228C22×C4—21no—
229C22×C4—71no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4C22×D711yescharacteristic
234SD16—142no—
235M16—22no—
236C2×C8—71no—
237C2×C8—71no—
238C2×C8—142no—
239C2×C8C22×D711yescharacteristic
240C2×C8C22×D711yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—71no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—71no—
259C22⋊C4—71no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C42—21no—
266C42—71no—
267C42—141no—
268C2×C7—41no—
269C2×C7#54711yescharacteristic
270C2×C7C2×#13711yescharacteristic
271C2×C7#57611yescharacteristic
272D7—22no—
273D7—81no—
274C23—71no—
275C23—141no—
276C23D4×D711yescharacteristic
277Q8—141no—
278Q8—141no—
279Q8—561no—
280D4—142no—
281D4—141no—
282D4—141no—
283D4—282no—
284D4—561no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—71no—
292C2×C4—71no—
293C2×C4—71no—
294C2×C4—81no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—561no—
301C2×C4—561no—
302C2×C4—561no—
303C2×C4D4×D711yescharacteristic
304C2×C4C23×D711yescharacteristic
305C8—21no—
306C8—141no—
307C8—141no—
308C8#115712yes—
309C7#334611yescharacteristic, complemented, Sylow
310C22—21no—
311C22—21no—
312C22—21no—
313C22—72no—
314C22—141no—
315C22—141no—
316C22—281no—
317C22—281no—
318C22—281no—
319C22C2×D4×D711yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—141no—
326C4—141no—
327C4—141no—
328C4—281no—
329C4—281no—
330C4—281no—
331C4C2×#115711yescharacteristic
332C4#655111yescharacteristic
333C2—41no—
334C2—142no—
335C2—561no—
336C2#9019811yescharacteristic
337C2#9046511yescharacteristic
338C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radical#90221C2
hypercenter#428D7
hypocenterC7#3346
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
2. #90202C2semidirect
3. C7×#428C2semidirect
4. #89906C2semidirect
5. #89899C2semidirect
6. #89768C2semidirect
7. #89739C2semidirect
8. #89662C2semidirect
9. #89661C2semidirect
10. #89656C2semidirect
11. D7×#107C2semidirect
12. #89636C2semidirect
13. #89635C2semidirect
14. #89629C2semidirect
15. #89628C2semidirect
119. #428D7semidirect
309. C7#3346semidirect

Maximal subgroups

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

subgroupindex
#902212
#902022
C7×#4282
#899062
#898992
#897682
#897392
#896622
#896612
#896562
D7×#1072
#896362
#896352
#896292
#896282
#33467