Finite Group Databasebeta

← #11739413

Subgroups — #11739413

Order 896.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90412C211yesmaximal, characteristic, complemented
2#90381C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic
5#89835C211yesmaximal, characteristic, complemented
6#89663C211yesmaximal, characteristic
7#89661C211yesmaximal, characteristic
8#89650C211yesmaximal, characteristic, complemented
9#89649C211yesmaximal, characteristic, complemented
10#89643C211yesmaximal, characteristic
11#89628C211yesmaximal, characteristic, complemented
12#89626C211yesmaximal, characteristic, complemented
13#89624C211yesmaximal, characteristic
14#89482C211yesmaximal, characteristic
15#89473C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#125C2211yescharacteristic
18Q8×Dic7—21no—
19#6507—21no—
20C2×#1132C2211yescharacteristic
21#6460—21no—
22#6459—21no—
23#6459—21no—
24#6456C2211yescharacteristic
25#6456C2211yescharacteristic
26D7×C4⋊C4—21no—
27#6453C2211yescharacteristic
28#6453C2211yescharacteristic
29#6453C2211yescharacteristic
30#6452—21no—
31#6452—21no—
32#6451C2211yescharacteristic
33#6450C2211yescharacteristic
34#6450C2211yescharacteristic
35C4×D28C2211yescharacteristic
36C4×Dic14C2211yescharacteristic
37C4×Dic14C2211yescharacteristic
38C7×#107C2211yescharacteristic
39C7×#107C2211yescharacteristic
40C7×#106C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#97C2211yescharacteristic
43#6395C2211yescharacteristic
44#6392C2211yescharacteristic
45#6390C2211yescharacteristic
46#6389C2211yescharacteristic
47Dic7⋊C8C2211yescharacteristic
48Dic7⋊C8C2211yescharacteristic
49C8×Dic7C2211yescharacteristic
50#6384C2211yescharacteristic
51#6384C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3388—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×D28C2311yescharacteristic
62C2×C4×D7—21no—
63C2×C4×D7—21no—
64C2×C4×D7—21no—
65C2×Dic14—21no—
66C2×Dic14D411yescharacteristic
67C2×Dic14C2311yescharacteristic
68C2×C7×C8C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C42×C7C2311yescharacteristic
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81C4⋊Dic7—21no—
82C4⋊Dic7—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7D411yescharacteristic
85C4⋊Dic7C2311yescharacteristic
86C4⋊Dic7C2311yescharacteristic
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×Dic7C2311yescharacteristic
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#1132Q814yescomplemented
104#557—71no—
105#554—71no—
106#501—71no—
107#501D711yescharacteristic, complemented
108#499—71no—
109#479—71no—
110#479—71no—
111#478—71no—
112#478—71no—
113#476—71no—
114#475—71no—
115#446—71no—
116#440—71no—
117#440—71no—
118C4×SD16—71no—
119C22×D7—21no—
120C7×Q8—42no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7C2×D411yescharacteristic
126C2×C4×C7C2×D411yescharacteristic
127C2×C4×C7C2411yescharacteristic
128C2×Dic7—21no—
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7C2×D411yescharacteristic
132C2×Dic7C2×D411yescharacteristic
133C2×Dic7C2×D411yescharacteristic
134C2×Dic7C2×D411yescharacteristic
135D28—21no—
136D28C2×Q812yes—
137C4×D7—42no—
138C4×D7—81no—
139C4×D7—81no—
140Dic14—21no—
141Dic14—41no—
142Dic14—81no—
143Dic14C2×Q812yes—
144C7×C8—21no—
145C7×C8C2×Q812yes—
146C7⋊C8—21no—
147C7⋊C8—21no—
148C2×SD16—71no—
149C4⋊Q8—71no—
150C4⋊Q8—141no—
151C4⋊Q8C2×D711yescharacteristic
152#125—71no—
153#125—71no—
154#125—71no—
155#125—141no—
156#125—141no—
157#125C2×D711yescharacteristic
158C22⋊Q8—141no—
159C22⋊Q8—141no—
160C22⋊Q8—141no—
161C4×Q8—71no—
162C4×Q8—71no—
163C4×Q8—71no—
164C4×Q8—71no—
165C4×Q8—141no—
166C4×D4—71no—
167C4×D4—71no—
168C4×D4—71no—
169C2×C4⋊C4—141no—
170#107—71no—
171#107—71no—
172#107C2×D711yescharacteristic
173#107C2×D711yescharacteristic
174#106—71no—
175#106—71no—
176#106—71no—
177#106C2×D711yescharacteristic
178#106C2×D711yescharacteristic
179C4⋊C8—71no—
180C4⋊C8—71no—
181C4⋊C8—71no—
182#103—71no—
183#103—71no—
184#103—71no—
185#102—71no—
186#102—71no—
187#102—71no—
188#97—71no—
189#97C2×D711yescharacteristic
190C4×C8—71no—
191C22×C7C22×D411yescharacteristic
192C2×D7—22no—
193C2×D7—41no—
194C2×D7—41no—
195C4×C7—21no—
196C4×C7—21no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C7—41no—
200C4×C7—41no—
201C4×C7C22×Q811yescharacteristic
202C4×C72−1+411yescharacteristic
203Dic7—22no—
204Dic7—21no—
205Dic7—21no—
206Dic7—21no—
207Dic7—21no—
208Dic7—41no—
209Dic7—41no—
210C2×Q8—21no—
211C2×Q8—71no—
212C2×Q8—71no—
213C2×Q8—141no—
214C2×D4—71no—
215C22×C4—141no—
216C22×C4—141no—
217C22×C4—141no—
218SD16—74no—
219C2×C8—71no—
220C2×C8—71no—
221C2×C8C22×D711yescharacteristic
222C2×C8C22×D711yescharacteristic
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—71no—
227C4⋊C4—71no—
228C4⋊C4—71no—
229C4⋊C4—71no—
230C4⋊C4—71no—
231C4⋊C4—141no—
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
244C4⋊C4C22×D711yescharacteristic
245C22⋊C4—141no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C42—71no—
249C42—71no—
250C42—141no—
251C42—141no—
252C42—141no—
253C42C22×D711yescharacteristic
254C2×C7D4×Q811yescharacteristic
255C2×C7#57511yescharacteristic
256C2×C7#57711yescharacteristic
257D7—42no—
258C23—141no—
259Q8—42no—
260Q8—72no—
261Q8—141no—
262Q8—281no—
263Q8—561no—
264D4—72no—
265D4—141no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—71no—
271C2×C4—71no—
272C2×C4—71no—
273C2×C4—71no—
274C2×C4—141no—
275C2×C4—141no—
276C2×C4—141no—
277C2×C4—282no—
278C2×C4—561no—
279C2×C4—561no—
280C2×C4D4×D711yescharacteristic
281C2×C4D4×D711yescharacteristic
282C2×C4C23×D711yescharacteristic
283C8—21no—
284C8—141no—
285C8—141no—
286C8Q8×D712yes—
287C7#338811yescharacteristic, complemented, Sylow
288C22—142no—
289C22—281no—
290C22—281no—
291C22C2×D4×D711yescharacteristic
292C4—21no—
293C4—21no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—142no—
299C4—141no—
300C4—141no—
301C4—141no—
302C4—141no—
303C4—281no—
304C4—281no—
305C4C2×Q8×D711yescharacteristic
306C4#655111yescharacteristic
307C2—282no—
308C2#9041811yescharacteristic
309C2#9046411yescharacteristic
310C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#501C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#501D7
hypocenterC7#3388
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #90412C2semidirect
3. C7×#501C2semidirect
5. #89835C2semidirect
8. #89650C2semidirect
9. #89649C2semidirect
11. #89628C2semidirect
12. #89626C2semidirect
15. #89473C2semidirect
103. #1132Q8semidirect
107. #501D7semidirect
287. C7#3388semidirect

Maximal subgroups

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

subgroupindex
#904122
#903812
C7×#5012
#898562
#898352
#896632
#896612
#896502
#896492
#896432
#896282
#896262
#896242
#894822
#894732
#33887