Finite Group Databasebeta

← #11738449

Subgroups — #11738449

Order 896.

338 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#90413C211yesmaximal, characteristic, complemented
2#90259C211yesmaximal, characteristic
3C7×#475C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic
5#89802C211yesmaximal, characteristic, complemented
6#89642C211yesmaximal, characteristic
7#89626C211yesmaximal, characteristic, complemented
8#89625C211yesmaximal, characteristic, complemented
9#89624C211yesmaximal, characteristic
10#89617C211yesmaximal, characteristic
11#89600C211yesmaximal, characteristic, complemented
12#89570C211yesmaximal, characteristic
13#89567C211yesmaximal, characteristic
14#89565C211yesmaximal, characteristic, complemented
15#89564C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18#6510C2211yescharacteristic
19#6507—21no—
20#6507—21no—
21#6507C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23C2×Dic28C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25#6461—21no—
26#6459—21no—
27#6456C2211yescharacteristic
28#6451—21no—
29#6451C2211yescharacteristic
30#6451C2211yescharacteristic
31#6450—21no—
32#6450C2211yescharacteristic
33#6450C2211yescharacteristic
34#6450C2211yescharacteristic
35#6439—21no—
36#6438C2211yescharacteristic
37#6435—21no—
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×#106C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45#6409C2211yescharacteristic
46#6395C2211yescharacteristic
47#6390C2211yescharacteristic
48#6389C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6377C2211yescharacteristic
59#3369—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×D28C2311yescharacteristic
63C2×C4×D7—21no—
64C2×Dic14—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⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C42×C7—21no—
77C42×C7C2311yescharacteristic
78#1144—22no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—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—
106Q82—71no—
107#556—71no—
108#489—71no—
109#489—71no—
110#487—71no—
111#476—71no—
112#475—71no—
113#475D711yescharacteristic, complemented
114#464—71no—
115#464—71no—
116#462—71no—
117#446—71no—
118#441—71no—
119#441—71no—
120C4×SD16—71no—
121C22×D7—21no—
122C7×Q8—21no—
123C7×Q8—21no—
124C7×Q8—81no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2411yescharacteristic
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7C2×D411yescharacteristic
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139D28—21no—
140D28—21no—
141C4×D7—42no—
142Dic14—21no—
143Dic14—22no—
144Dic14—21no—
145Dic14—21no—
146Dic14—81no—
147Dic14—81no—
148Dic14#4112yes—
149C7×C8—21no—
150C7×C8—21no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C2×Q16—71no—
154C2×Q16—71no—
155C2×SD16—71no—
156C4⋊Q8—71no—
157C4⋊Q8—71no—
158C4⋊Q8—71no—
159C4⋊Q8—141no—
160C4⋊Q8—141no—
161C4⋊Q8—141no—
162C4⋊Q8C2×D711yescharacteristic
163#126—141no—
164#126—141no—
165#124—71no—
166#124—71no—
167C22⋊Q8—141no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—141no—
174C4×Q8C2×D711yescharacteristic
175C4×D4—71no—
176#117—141no—
177#106—71no—
178#106C2×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#103C2×D711yescharacteristic
189#103C2×D711yescharacteristic
190#102—71no—
191#102—71no—
192#102—71no—
193#97—71no—
194#97—71no—
195C4×C8—71no—
196C22×C7C22×D411yescharacteristic
197C2×D7—41no—
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—22no—
209Dic7—21no—
210Dic7—21no—
211Dic7—21no—
212Dic7—41no—
213Dic7—41no—
214Dic7#13712yes—
215C2×Q8—21no—
216C2×Q8—71no—
217C2×Q8—71no—
218C2×Q8—71no—
219C2×Q8—141no—
220C2×Q8—141no—
221C2×Q8C22×D711yescharacteristic
222C2×D4—71no—
223C22×C4—141no—
224Q16—142no—
225Q16—142no—
226SD16—142no—
227C2×C8—71no—
228C2×C8—71no—
229C2×C8C22×D711yescharacteristic
230C2×C8C22×D711yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4C22×D711yescharacteristic
246C4⋊C4C22×D711yescharacteristic
247C4⋊C4C22×D711yescharacteristic
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C22⋊C4—141no—
252C22⋊C4—141no—
253C42—21no—
254C42—71no—
255C42—71no—
256C42—141no—
257C42—141no—
258C42—141no—
259C42—141no—
260C42C22×D711yescharacteristic
261C2×C7#54811yescharacteristic
262C2×C7C2×#13711yescharacteristic
263C2×C7#57711yescharacteristic
264D7—81no—
265C23—141no—
266Q8—21no—
267Q8—21no—
268Q8—72no—
269Q8—81no—
270Q8—142no—
271Q8—141no—
272Q8—141no—
273Q8—141no—
274Q8—561no—
275Q8—561no—
276D4—141no—
277D4—141no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—71no—
283C2×C4—71no—
284C2×C4—71no—
285C2×C4—71no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—282no—
290C2×C4D4×D711yescharacteristic
291C2×C4D4×D711yescharacteristic
292C2×C4C23×D711yescharacteristic
293C8—21no—
294C8—21no—
295C8—141no—
296C8—141no—
297C7#336911yescharacteristic, complemented, Sylow
298C22—281no—
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—142no—
313C4—281no—
314C4—281no—
315C4#655111yescharacteristic
316C4D7×#4111yescharacteristic
317C2—561no—
318C2#9032411yescharacteristic
319C2#9045311yescharacteristic
320C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#475D7
hypocenterC7#3369
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #90413C2semidirect
3. C7×#475C2semidirect
5. #89802C2semidirect
7. #89626C2semidirect
8. #89625C2semidirect
11. #89600C2semidirect
14. #89565C2semidirect
15. #89564C2semidirect
113. #475D7semidirect
297. C7#3369semidirect

Maximal subgroups

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

subgroupindex
#904132
#902592
C7×#4752
#898612
#898022
#896422
#896262
#896252
#896242
#896172
#896002
#895702
#895672
#895652
#895642
#33697