Finite Group Databasebeta

← #11738661

Subgroups — #11738661

Order 896.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90376C211yesmaximal, characteristic, complemented
2#90259C211yesmaximal, characteristic
3C7×#479C211yesmaximal, characteristic, complemented
4#89840C211yesmaximal, characteristic
5C4×#1144C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic
7#89645C211yesmaximal, characteristic, complemented
8#89629C211yesmaximal, characteristic, complemented
9#89624C211yesmaximal, characteristic
10#89617C211yesmaximal, characteristic
11#89602C211yesmaximal, characteristic, complemented
12#89572C211yesmaximal, characteristic
13#89570C211yesmaximal, characteristic
14#89565C211yesmaximal, characteristic, complemented
15#89564C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18#6510C2211yescharacteristic
19#6507C2211yescharacteristic
20C2×#1144C2211yescharacteristic
21C2×Dic28C2211yescharacteristic
22C2×#1132C2211yescharacteristic
23#6461—21no—
24#6459—21no—
25#6456C2211yescharacteristic
26#6453C2211yescharacteristic
27#6452—21no—
28#6452—21no—
29#6451—21no—
30#6451C2211yescharacteristic
31#6450—21no—
32#6450C2211yescharacteristic
33#6450C2211yescharacteristic
34#6450C2211yescharacteristic
35#6439—21no—
36#6438C2211yescharacteristic
37#6435—21no—
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45#6409C2211yescharacteristic
46#6395C2211yescharacteristic
47#6390C2211yescharacteristic
48Dic7⋊C8C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50C8×Dic7C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3364—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×D28C2311yescharacteristic
62C2×C4×D7—21no—
63C2×Dic14—21no—
64C2×Dic14D411yescharacteristic
65C2×Dic14C2311yescharacteristic
66C2×Dic14C2311yescharacteristic
67C2×C7×C8C2311yescharacteristic
68C2×C7×C8C2311yescharacteristic
69C7×C4⋊C4—21no—
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C42×C7—21no—
76C42×C7C2311yescharacteristic
77#1144—22no—
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7D411yescharacteristic
85C4⋊Dic7C2311yescharacteristic
86Dic7⋊C4—21no—
87Dic7⋊C4—21no—
88Dic7⋊C4—21no—
89Dic7⋊C4—21no—
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4D411yescharacteristic
95Dic7⋊C4D411yescharacteristic
96C4×Dic7—21no—
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7C2311yescharacteristic
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104Dic28—22no—
105#1132—22no—
106#557—71no—
107#556—71no—
108#488—71no—
109#488—71no—
110#487—71no—
111#479—71no—
112#479D711yescharacteristic, complemented
113#474—71no—
114#464—71no—
115#464—71no—
116#462—71no—
117C8×Q8—71no—
118C4×Q16—71no—
119C4×Q16—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—21no—
140C4×D7—42no—
141Dic14—21no—
142Dic14—22no—
143Dic14—21no—
144Dic14—21no—
145Dic14—42no—
146Dic14#4112yes—
147C7×C8—21no—
148C7×C8—21no—
149C7⋊C8—21no—
150C7⋊C8—21no—
151C2×Q16—71no—
152C2×Q16—71no—
153C2×SD16—71no—
154C4⋊Q8—71no—
155C4⋊Q8—71no—
156C4⋊Q8—141no—
157#126—141no—
158#126—141no—
159#125—71no—
160#125—141no—
161#125—141no—
162#125C2×D711yescharacteristic
163#124—71no—
164#124—71no—
165C22⋊Q8—141no—
166C4×Q8—71no—
167C4×Q8—71no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—141no—
172C4×Q8C2×D711yescharacteristic
173C4×D4—71no—
174#117—141no—
175#107—71no—
176#107C2×D711yescharacteristic
177#106C2×D711yescharacteristic
178C4⋊C8—71no—
179C4⋊C8—71no—
180C4⋊C8C2×D711yescharacteristic
181#103—71no—
182#103—71no—
183#103—71no—
184#103—71no—
185#103—71no—
186#103C2×D711yescharacteristic
187#103C2×D711yescharacteristic
188#102—71no—
189#102—71no—
190#102—71no—
191C4×C8—71no—
192C4×C8—71no—
193C4×C8—71no—
194C22×C7C22×D411yescharacteristic
195C2×D7—41no—
196C2×D7—41no—
197C2×D7—41no—
198C4×C7—21no—
199C4×C7—21no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7C2×#4111yescharacteristic
205C4×C72−1+411yescharacteristic
206Dic7—22no—
207Dic7—21no—
208Dic7—21no—
209Dic7—21no—
210Dic7—41no—
211Dic7—41no—
212Dic7#13512yes—
213C2×Q8—71no—
214C2×Q8—71no—
215C2×Q8—71no—
216C2×Q8—141no—
217C2×Q8C22×D711yescharacteristic
218C2×D4—71no—
219C22×C4—141no—
220Q16—142no—
221Q16—142no—
222SD16—142no—
223C2×C8—71no—
224C2×C8—71no—
225C2×C8C22×D711yescharacteristic
226C2×C8C22×D711yescharacteristic
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—71no—
231C4⋊C4—71no—
232C4⋊C4—71no—
233C4⋊C4—71no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4C22×D711yescharacteristic
244C4⋊C4C22×D711yescharacteristic
245C4⋊C4C22×D711yescharacteristic
246C22⋊C4—141no—
247C22⋊C4—141no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C42—21no—
252C42—71no—
253C42—71no—
254C42—141no—
255C42—141no—
256C42—141no—
257C42—141no—
258C42C22×D711yescharacteristic
259C2×C7#54811yescharacteristic
260C2×C7C2×#13511yescharacteristic
261C2×C7#57811yescharacteristic
262D7—81no—
263C23—141no—
264Q8—21no—
265Q8—21no—
266Q8—72no—
267Q8—142no—
268Q8—141no—
269Q8—141no—
270Q8—141no—
271Q8—282no—
272D4—141no—
273D4—141no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—71no—
279C2×C4—71no—
280C2×C4—71no—
281C2×C4—71no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—282no—
286C2×C4D4×D711yescharacteristic
287C2×C4D4×D711yescharacteristic
288C2×C4C23×D711yescharacteristic
289C8—21no—
290C8—21no—
291C8—141no—
292C8—141no—
293C7#336411yescharacteristic, complemented, Sylow
294C22—281no—
295C22—281no—
296C22—281no—
297C22C2×D4×D711yescharacteristic
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—72no—
305C4—141no—
306C4—141no—
307C4—141no—
308C4—142no—
309C4—281no—
310C4—281no—
311C4#655111yescharacteristic
312C4D7×#4111yescharacteristic
313C2—561no—
314C2#9032411yescharacteristic
315C2D7×#13511yescharacteristic
316C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#479D7
hypocenterC7#3364
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
1. #90376C2semidirect
3. C7×#479C2semidirect
5. C4×#1144C2semidirect
7. #89645C2semidirect
8. #89629C2semidirect
11. #89602C2semidirect
14. #89565C2semidirect
15. #89564C2semidirect
112. #479D7semidirect
293. C7#3364semidirect

Maximal subgroups

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

subgroupindex
#903762
#902592
C7×#4792
#898402
C4×#11442
#896622
#896452
#896292
#896242
#896172
#896022
#895722
#895702
#895652
#895642
#33647