Finite Group Databasebeta

← #11738664

Subgroups — #11738664

Order 896.

340 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90381C211yesmaximal, characteristic
2#90258C211yesmaximal, characteristic, complemented
3C7×#479C211yesmaximal, characteristic, complemented
4#89841C211yesmaximal, characteristic, complemented
5#89798C211yesmaximal, characteristic
6#89661C211yesmaximal, characteristic
7#89647C211yesmaximal, characteristic, complemented
8#89629C211yesmaximal, characteristic, complemented
9#89624C211yesmaximal, characteristic
10#89613C211yesmaximal, characteristic
11#89600C211yesmaximal, characteristic, complemented
12#89590C211yesmaximal, characteristic
13#89590C211yesmaximal, characteristic
14#89569C211yesmaximal, characteristic, complemented
15#89564C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18#6507C2211yescharacteristic
19#6507C2211yescharacteristic
20C2×#1143C2211yescharacteristic
21C2×#1132C2211yescharacteristic
22C2×#1132C2211yescharacteristic
23#6460—21no—
24#6459—21no—
25#6459—21no—
26#6456C2211yescharacteristic
27#6456C2211yescharacteristic
28D7×C4⋊C4—21no—
29#6453C2211yescharacteristic
30#6452—21no—
31#6452—21no—
32#6451C2211yescharacteristic
33#6450C2211yescharacteristic
34#6450C2211yescharacteristic
35C4×D28C2211yescharacteristic
36#6433—21no—
37#6432C2211yescharacteristic
38C4×Dic14—21no—
39C4×Dic14C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45#6409C2211yescharacteristic
46#6390C2211yescharacteristic
47#6390C2211yescharacteristic
48#6389C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3372—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×D28C2311yescharacteristic
62C2×C4×D7—21no—
63C2×C4×D7—21no—
64C2×C4×D7—21no—
65C2×Dic14—21no—
66C2×Dic14D411yescharacteristic
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#1143—22no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7D411yescharacteristic
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⋊C4D411yescharacteristic
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7C2311yescharacteristic
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104#1132—22no—
105#1132—22no—
106#557—71no—
107#554—71no—
108#491—71no—
109#491—71no—
110#487—71no—
111#479—71no—
112#479D711yescharacteristic, complemented
113#477—71no—
114#461—71no—
115#461—71no—
116#461—71no—
117#446—71no—
118#440—71no—
119#440—71no—
120C4×SD16—71no—
121C22×D7—21no—
122C7×Q8—21no—
123C7×Q8—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2411yescharacteristic
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7C2×D411yescharacteristic
135C2×Dic7C2×D411yescharacteristic
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138D28—21no—
139D28#4112yes—
140C4×D7—42no—
141C4×D7—81no—
142C4×D7—81no—
143Dic14—22no—
144Dic14—21no—
145Dic14—21no—
146Dic14—21no—
147Dic14—21no—
148Dic14—81no—
149C7×C8—21no—
150C7×C8—21no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C2×SD16—71no—
154C2×SD16—71no—
155C2×SD16—71no—
156C4⋊Q8—71no—
157C4⋊Q8—71no—
158C4⋊Q8—71no—
159C4⋊Q8—71no—
160#125—71no—
161#125—141no—
162#125—141no—
163#125—141no—
164#125C2×D711yescharacteristic
165C22⋊Q8—141no—
166C22⋊Q8—141no—
167C22⋊Q8—141no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—141no—
172C4×Q8C2×D711yescharacteristic
173C4×D4—71no—
174C4×D4—71no—
175C4×D4—71no—
176C2×C4⋊C4—141no—
177#107—71no—
178#107C2×D711yescharacteristic
179#106C2×D711yescharacteristic
180C4⋊C8—71no—
181C4⋊C8—71no—
182C4⋊C8C2×D711yescharacteristic
183#103—71no—
184#103—71no—
185#103—71no—
186#103—71no—
187#103—71no—
188#103—71no—
189#103—71no—
190#103C2×D711yescharacteristic
191#103C2×D711yescharacteristic
192#102—71no—
193#97—71no—
194#97—71no—
195C4×C8—71no—
196C22×C7C22×D411yescharacteristic
197C2×D7—22no—
198C2×D7—41no—
199C2×D7—41no—
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7C2×#4111yescharacteristic
207C4×C72−1+411yescharacteristic
208Dic7—21no—
209Dic7—21no—
210Dic7—21no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7#13712yes—
215C2×Q8—71no—
216C2×Q8—71no—
217C2×Q8—71no—
218C2×Q8—141no—
219C2×Q8C22×D711yescharacteristic
220C2×D4—71no—
221C22×C4—141no—
222C22×C4—141no—
223C22×C4—141no—
224SD16—142no—
225SD16—142no—
226SD16—142no—
227C2×C8—71no—
228C2×C8—71no—
229C2×C8C22×D711yescharacteristic
230C2×C8C22×D711yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—141no—
239C4⋊C4—141no—
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—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C42—21no—
256C42—71no—
257C42—71no—
258C42—141no—
259C42—141no—
260C42C22×D711yescharacteristic
261C2×C7#54811yescharacteristic
262C2×C7C2×#13711yescharacteristic
263C2×C7#57711yescharacteristic
264D7—42no—
265C23—141no—
266Q8—21no—
267Q8—21no—
268Q8—142no—
269Q8—141no—
270Q8—141no—
271Q8—141no—
272Q8—141no—
273Q8—561no—
274D4—72no—
275D4—141no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—71no—
281C2×C4—71no—
282C2×C4—71no—
283C2×C4—71no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—282no—
288C2×C4—561no—
289C2×C4—561no—
290C2×C4D4×D711yescharacteristic
291C2×C4D4×D711yescharacteristic
292C2×C4C23×D711yescharacteristic
293C8—21no—
294C8—21no—
295C8—141no—
296C8—141no—
297C7#337211yescharacteristic, complemented, Sylow
298C22—142no—
299C22—281no—
300C22—281no—
301C22C2×D4×D711yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—72no—
309C4—141no—
310C4—141no—
311C4—141no—
312C4—281no—
313C4—281no—
314C4—281no—
315C4#655111yescharacteristic
316C4D7×#4111yescharacteristic
317C2—282no—
318C2#9032411yescharacteristic
319C2#9046111yescharacteristic
320C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#479D7
hypocenterC7#3372
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
2. #90258C2semidirect
3. C7×#479C2semidirect
4. #89841C2semidirect
7. #89647C2semidirect
8. #89629C2semidirect
11. #89600C2semidirect
14. #89569C2semidirect
15. #89564C2semidirect
112. #479D7semidirect
297. C7#3372semidirect

Maximal subgroups

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

subgroupindex
#903812
#902582
C7×#4792
#898412
#897982
#896612
#896472
#896292
#896242
#896132
#896002
#895902
#895692
#895642
#33727