Finite Group Databasebeta

← #11739063

Subgroups — #11739063

Order 896.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90417C211yesmaximal, characteristic, complemented
2#90417C211yesmaximal, characteristic, complemented
3C7×#487C211yesmaximal, characteristic, complemented
4#89859C211yesmaximal, characteristic, complemented
5#89855C211yesmaximal, characteristic, complemented
6#89584C212yesmaximal, complemented
7#89583C212yesmaximal, complemented
8#89581C212yesmaximal, complemented
9#89575C212yesmaximal, complemented
10C4×D56C211yesmaximal, characteristic, complemented
11#89460C211yesmaximal, characteristic, complemented
12C2×#1161—21no—
13C2×#1161C2212yes—
14C7×C4⋊Q8C2211yescharacteristic
15C7×C4⋊Q8C2211yescharacteristic
16#6509—21no—
17#6509C2212yes—
18C2×#1143C2212yes—
19C2×D56C2211yescharacteristic
20C2×#1131C2212yes—
21#6458—21no—
22#6458C2212yes—
23#6457—22no—
24#6457—22no—
25D7×C4⋊C4—21no—
26D7×C4⋊C4C2212yes—
27C4×D28C2211yescharacteristic
28C4×D28C2211yescharacteristic
29C4×D28C2211yescharacteristic
30C7×#103C2212yes—
31C7×#103C2212yes—
32C4×C7×C8C2211yescharacteristic
33#6409C2212yes—
34#6395C2211yescharacteristic
35#6395C2211yescharacteristic
36(C2×D7)⋊C8C2212yes—
37#6392C2211yescharacteristic
38#6383C2212yes—
39#6381C2212yes—
40C4⋊(C7⋊C8)C2211yescharacteristic
41#3330—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2312yes—
44#1161—24no—
45#1161—42no—
46#1161—44no—
47C2×D28—22no—
48C2×D28—22no—
49C2×D28D411yescharacteristic
50C2×D28C2311yescharacteristic
51C2×D28C2311yescharacteristic
52C2×C4×D7—22no—
53C2×C4×D7—21no—
54C2×C4×D7—21no—
55C2×C4×D7—22no—
56C2×C4×D7C2312yes—
57C2×C7×C8C2311yescharacteristic
58C2×C7×C8C2311yescharacteristic
59C7×C4⋊C4—21no—
60C7×C4⋊C4—21no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4C2312yes—
63C42×C7C2311yescharacteristic
64#1143—42no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4D412yes—
70C4⋊Dic7D411yescharacteristic
71C4⋊Dic7C2311yescharacteristic
72C4⋊Dic7C2311yescharacteristic
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75C2×C7⋊C8C2312yes—
76D56—22no—
77#1131—42no—
78#547—71no—
79#547—71no—
80#487D711yescharacteristic, complemented
81#484—72no—
82#474—71no—
83#468—72no—
84#461—71no—
85#449—72no—
86C4×D8—71no—
87#436—71no—
88#417—72no—
89C22×D7—21no—
90C22×D7—21no—
91C22×D7C2×D412yes—
92C7×Q8—22no—
93C7×Q8—22no—
94C7×Q8—42no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2411yescharacteristic
100C2×Dic7—21no—
101C2×Dic7—21no—
102C2×Dic7C2×D412yes—
103D28—22no—
104D28—21no—
105D28—21no—
106D28—21no—
107D28—44no—
108D28—44no—
109D28#4112yescomplemented
110C4×D7—22no—
111C4×D7—22no—
112C4×D7—44no—
113C4×D7—44no—
114C4×D7—42no—
115C4×D7—42no—
116C7×C8—21no—
117C7×C8—21no—
118C7⋊C8—22no—
119C2×#41—72no—
120C2×#41—141no—
121C2×SD16—72no—
122C2×D8—71no—
123C2×M16—72no—
124C4⋊Q8C2×D711yescharacteristic
125C4⋊Q8C2×D711yescharacteristic
126#123—142no—
127#123—142no—
128C22⋊Q8—72no—
129C22⋊Q8—141no—
130#121—72no—
131#121—141no—
132C4×D4—71no—
133C4×D4—71no—
134C4×D4—71no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—141no—
137#107—71no—
138#107—72no—
139C4⋊C8—71no—
140#103—72no—
141#103C2×D712yes—
142#103C2×D712yes—
143#102—71no—
144#102—71no—
145#102—72no—
146C22⋊C8—72no—
147C4×C8C2×D711yescharacteristic
148C22×C7C22×D411yescharacteristic
149C2×D7—22no—
150C2×D7—22no—
151C2×D7—22no—
152C2×D7—22no—
153C2×D7—41no—
154C2×D7—41no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—41no—
158C4×C7—21no—
159C4×C7—42no—
160C4×C7—42no—
161C4×C7AGL(1,ℤ/8)12yes—
162C4×C7C2×#4111yescharacteristic
163C4×C7H(5,𝔽2)11yescharacteristic
164Dic7—22no—
165Dic7—41no—
166Dic7—41no—
167#41—144no—
168#41—282no—
169#41—284no—
170C2×Q8—21no—
171C2×Q8C22×D712yes—
172C2×D4—71no—
173C2×D4—71no—
174C2×D4—71no—
175C2×D4—142no—
176C2×D4—142no—
177C22×C4—72no—
178C22×C4—142no—
179C22×C4—142no—
180C22×C4—141no—
181C22×C4—141no—
182SD16—282no—
183D8—142no—
184M16—282no—
185C2×C8—72no—
186C2×C8C22×D711yescharacteristic
187C2×C8C22×D711yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—71no—
192C4⋊C4—71no—
193C4⋊C4—71no—
194C4⋊C4—142no—
195C4⋊C4—142no—
196C4⋊C4C22×D712yes—
197C22⋊C4—72no—
198C22⋊C4—142no—
199C22⋊C4—142no—
200C22⋊C4—141no—
201C22⋊C4—141no—
202C42C22×D711yescharacteristic
203C2×C7#54611yescharacteristic
204C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C7#57811yescharacteristic
206D7—42no—
207D7—42no—
208D7—81no—
209C23—72no—
210C23—141no—
211C23—141no—
212Q8—22no—
213Q8—22no—
214Q8—42no—
215D4—72no—
216D4—142no—
217D4—141no—
218D4—141no—
219D4—141no—
220D4—284no—
221D4—284no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—72no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—141no—
228C2×C4—141no—
229C2×C4—284no—
230C2×C4—284no—
231C2×C4—282no—
232C2×C4—282no—
233C2×C4D4×D711yescharacteristic
234C2×C4D4×D711yescharacteristic
235C2×C4C23×D711yescharacteristic
236C8—21no—
237C8—21no—
238C8—142no—
239C7#333011yescharacteristic, complemented, Sylow
240C22—142no—
241C22—142no—
242C22—142no—
243C22—142no—
244C22—281no—
245C22—281no—
246C22—281no—
247C22—281no—
248C22—281no—
249C22C2×D4×D711yescharacteristic
250C4—21no—
251C4—42no—
252C4—42no—
253C4—142no—
254C4—281no—
255C4—281no—
256C4#647712yes—
257C4D7×#4111yescharacteristic
258C4#655311yescharacteristic
259C2—282no—
260C2—282no—
261C2—561no—
262C2#9036711yescharacteristic
263C2C2×#647711yescharacteristic
264C2#9045711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socleC22×C7C22×D4
hypercenter#487D7
hypocenterC7#3330
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #90417C2semidirect
2. #90417C2semidirect
3. C7×#487C2semidirect
4. #89859C2semidirect
5. #89855C2semidirect
6. #89584C2semidirect
7. #89583C2semidirect
8. #89581C2semidirect
9. #89575C2semidirect
10. C4×D56C2semidirect
11. #89460C2semidirect
80. #487D7semidirect
109. D28#41semidirect
239. C7#3330semidirect

Maximal subgroups

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

subgroupindex
#904172
C7×#4872
#898592
#898552
#895842
#895832
#895812
#895752
C4×D562
#894602
#33307