Finite Group Databasebeta

← #11738656

Subgroups — #11738656

Order 896.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90388C211yesmaximal, characteristic
2#90263C211yesmaximal, characteristic
3C7×#479C211yesmaximal, characteristic, complemented
4#89835C211yesmaximal, characteristic, complemented
5#89803C211yesmaximal, characteristic, complemented
6#89660C211yesmaximal, characteristic
7#89645C211yesmaximal, characteristic, complemented
8#89639C211yesmaximal, characteristic
9#89625C211yesmaximal, characteristic, complemented
10#89621C211yesmaximal, characteristic, complemented
11#89612C211yesmaximal, characteristic
12#89580C211yesmaximal, characteristic
13#89580C211yesmaximal, characteristic
14#89562C211yesmaximal, characteristic, complemented
15#89561C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C7×#125C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20C2×#1144C2211yescharacteristic
21C2×#1144C2211yescharacteristic
22C2×Dic28C2211yescharacteristic
23C2×Dic28C2211yescharacteristic
24#6461—21no—
25#6461—21no—
26#6460C2211yescharacteristic
27#6460C2211yescharacteristic
28#6455—21no—
29#6455C2211yescharacteristic
30#6453—21no—
31#6453—21no—
32#6450—21no—
33#6450C2211yescharacteristic
34#6450C2211yescharacteristic
35#6450C2211yescharacteristic
36#6435—21no—
37#6435C2211yescharacteristic
38#6432C2211yescharacteristic
39C4×Dic14—21no—
40C4×Dic14C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×C4⋊C8C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50#6389C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3127—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61Q8×D7—22no—
62Q8×D7—22no—
63C2×C4×D7—21no—
64C2×C4×D7C2311yescharacteristic
65C2×Dic14—21no—
66C2×Dic14C2311yescharacteristic
67C2×Dic14C2311yescharacteristic
68C2×Dic14C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C42×C7—21no—
78C42×C7C2311yescharacteristic
79#1144—22no—
80#1144—22no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4D411yescharacteristic
85(C2×D7)⋊C4D411yescharacteristic
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4D411yescharacteristic
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107Dic28—22no—
108Dic28—22no—
109#541—71no—
110#519—71no—
111#485—71no—
112#485—71no—
113#479—71no—
114#479D711yescharacteristic, complemented
115#462—71no—
116#462—71no—
117#451—71no—
118#451—71no—
119#441—71no—
120#441—71no—
121C4×Q16—71no—
122C4×Q16—71no—
123#433—71no—
124C22×D7C2×D411yescharacteristic
125C7×Q8—21no—
126C7×Q8#4112yes—
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×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C4×D7—24no—
142C4×D7—21no—
143C4×D7—21no—
144Dic14—24no—
145Dic14—21no—
146Dic14—21no—
147Dic14—21no—
148Dic14—21no—
149Dic14—21no—
150Dic14#4112yes—
151C7×C8—21no—
152C7×C8—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C22×Q8—71no—
156C2×Q16—71no—
157C2×Q16—71no—
158C2×Q16—71no—
159C2×Q16—71no—
160C4⋊Q8—71no—
161#126—141no—
162#126—141no—
163#125—141no—
164#125—141no—
165#125C2×D711yescharacteristic
166C22⋊Q8—71no—
167C22⋊Q8—71no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—141no—
174C4×Q8—141no—
175C4×Q8C2×D711yescharacteristic
176#117—71no—
177#117—71no—
178#117—141no—
179#117—141no—
180#107—71no—
181#107C2×D711yescharacteristic
182#106—71no—
183#106C2×D711yescharacteristic
184C4⋊C8—71no—
185C4⋊C8C2×D711yescharacteristic
186#103—71no—
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#103—71no—
192#103C2×D711yescharacteristic
193#103C2×D711yescharacteristic
194C22⋊C8—71no—
195C22⋊C8—71no—
196#97—71no—
197C4×C8—71no—
198C22×C7C22×D411yescharacteristic
199C2×D7—21no—
200C2×D7—21no—
201C2×D7—21no—
202C4×C7—22no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7C2×#4111yescharacteristic
209C4×C7C2×#4111yescharacteristic
210Dic7—21no—
211Dic7—22no—
212Dic7—21no—
213Dic7—21no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217C2×Q8—71no—
218C2×Q8—71no—
219C2×Q8—71no—
220C2×Q8—141no—
221C2×Q8—142no—
222C2×Q8—142no—
223C2×Q8C22×D711yescharacteristic
224C22×C4—71no—
225C22×C4—141no—
226Q16—142no—
227Q16—142no—
228Q16—142no—
229Q16—142no—
230C2×C8—71no—
231C2×C8—71no—
232C2×C8C22×D711yescharacteristic
233C2×C8C22×D711yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C4⋊C4C22×D711yescharacteristic
252C22⋊C4—71no—
253C22⋊C4—71no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C42—21no—
258C42—71no—
259C42—141no—
260C42—141no—
261C42—141no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7#52511yescharacteristic
267C2×C7#57811yescharacteristic
268C2×C7#57811yescharacteristic
269D7—41no—
270C23—71no—
271Q8—21no—
272Q8—72no—
273Q8—144no—
274Q8—141no—
275Q8—141no—
276Q8—141no—
277Q8—141no—
278Q8—141no—
279Q8#116112yes—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—71no—
285C2×C4—71no—
286C2×C4—71no—
287C2×C4—144no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4D4×D711yescharacteristic
295C2×C4D4×D711yescharacteristic
296C2×C4C23×D711yescharacteristic
297C8—21no—
298C8—21no—
299C8—141no—
300C8—141no—
301C7#312711yescharacteristic, complemented, Sylow
302C22—141no—
303C22—141no—
304C22—141no—
305C22C2×D4×D711yescharacteristic
306C4—21no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—141no—
313C4—141no—
314C4—141no—
315C4—142no—
316C4—281no—
317C4—281no—
318C4—281no—
319C4C2×#116111yescharacteristic
320C4D7×#4111yescharacteristic
321C2—281no—
322C2#9031811yescharacteristic
323C2#9046211yescharacteristic
324C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#479D7
hypocenterC7#3127
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
3. C7×#479C2semidirect
4. #89835C2semidirect
5. #89803C2semidirect
7. #89645C2semidirect
9. #89625C2semidirect
10. #89621C2semidirect
14. #89562C2semidirect
15. #89561C2semidirect
114. #479D7semidirect
301. C7#3127semidirect

Maximal subgroups

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

subgroupindex
#903882
#902632
C7×#4792
#898352
#898032
#896602
#896452
#896392
#896252
#896212
#896122
#895802
#895622
#895612
#31277