Finite Group Databasebeta

← #11738472

Subgroups — #11738472

Order 896.

430 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 117 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#90268C211yesmaximal, characteristic, complemented
3C7×#475C211yesmaximal, characteristic, complemented
4#89860C211yesmaximal, characteristic, complemented
5C4×#1143C211yesmaximal, characteristic, complemented
6#89637C212yesmaximal, complemented
7#89635C212yesmaximal, complemented
8#89615C211yesmaximal, characteristic, complemented
9#89610C211yesmaximal, characteristic, complemented
10#89583C212yesmaximal, complemented
11#89575C212yesmaximal, complemented
12C2×#1161—21no—
13C2×#1161C2212yes—
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×Q8C2211yescharacteristic
16#6509—21no—
17C2×#1143C2211yescharacteristic
18C2×D56C2212yes—
19C2×#1131C2212yes—
20#6458—22no—
21#6458C2212yes—
22#6458C2212yes—
23#6457—22no—
24D7×C4⋊C4C2212yes—
25#6437—21no—
26#6437C2211yescharacteristic
27C4×D28—21no—
28C4×D28C2211yescharacteristic
29C4×D28C2211yescharacteristic
30C7×#106C2212yes—
31C7×C4⋊C8C2211yescharacteristic
32C7×#103C2212yes—
33#6409C2211yescharacteristic
34#6395C2212yes—
35(C2×D7)⋊C8C2212yes—
36#6383C2212yes—
37#6383C2212yes—
38#6383C2211yescharacteristic
39#6382C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3332—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2311yescharacteristic
44#1161—24no—
45#1161—42no—
46#1161—44no—
47C2×D28—22no—
48C2×D28—22no—
49C2×D28—21no—
50C2×D28—22no—
51C2×D28—21no—
52C2×D28D411yescharacteristic
53C2×D28C2311yescharacteristic
54C2×D28C2312yes—
55C2×C4×D7—22no—
56C2×C4×D7—21no—
57C2×C4×D7—22no—
58C2×C4×D7C2312yes—
59C2×C7×C8C2312yes—
60C7×C4⋊C4—21no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4C2312yes—
63C7×C4⋊C4C2311yescharacteristic
64C42×C7—21no—
65C42×C7C2311yescharacteristic
66#1143—22no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4D412yes—
71C4⋊Dic7D411yescharacteristic
72C4⋊Dic7C2311yescharacteristic
73Dic7⋊C4—22no—
74C2×C7⋊C8C2311yescharacteristic
75C2×C7⋊C8C2311yescharacteristic
76D56—42no—
77#1131—42no—
78#550—71no—
79#547—71no—
80#486—71no—
81#481—72no—
82#475D711yescharacteristic, complemented
83#469—72no—
84#459—71no—
85#449—72no—
86C4×SD16—71no—
87#436—71no—
88#417—72no—
89C22×D7—21no—
90C22×D7—22no—
91C22×D7C2×D412yes—
92C7×Q8—21no—
93C7×Q8—42no—
94C7×Q8#4112yescomplemented
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2411yescharacteristic
101C2×Dic7—21no—
102C2×Dic7C2×D412yes—
103D28—22no—
104D28—21no—
105D28—22no—
106D28—21no—
107D28—22no—
108D28—44no—
109D28—44no—
110D28—42no—
111D28—81no—
112D28—82no—
113C4×D7—22no—
114C4×D7—22no—
115C4×D7—44no—
116C4×D7—44no—
117C4×D7—42no—
118C7×C8—22no—
119C7⋊C8—21no—
120C7⋊C8—21no—
121C2×#41—72no—
122C2×#41—141no—
123C2×SD16—71no—
124C2×D8—72no—
125C2×M16—72no—
126C4⋊Q8C2×D711yescharacteristic
127#127—71no—
128#127—141no—
129#123—142no—
130C22⋊Q8—141no—
131#121—72no—
132#121—72no—
133#121—142no—
134C4×Q8C2×D711yescharacteristic
135C4×D4—71no—
136C4×D4—71no—
137C4×D4—141no—
138C2×C4⋊C4—72no—
139#106—71no—
140#106C2×D712yes—
141C4⋊C8C2×D711yescharacteristic
142#103—71no—
143#103C2×D712yes—
144#102—72no—
145#102—72no—
146#102—71no—
147#102—72no—
148C22⋊C8—72no—
149C4×C8—71no—
150C22×C7C22×D411yescharacteristic
151C2×D7—22no—
152C2×D7—22no—
153C2×D7—22no—
154C2×D7—42no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—42no—
158C2×D7—42no—
159C2×D7—41no—
160C4×C7—22no—
161C4×C7—21no—
162C4×C7—42no—
163C4×C7—41no—
164C4×C7AGL(1,ℤ/8)12yes—
165C4×C7C2×#4111yescharacteristic
166C4×C7H(5,𝔽2)11yescharacteristic
167Dic7—22no—
168Dic7—41no—
169#41—144no—
170#41—282no—
171#41—284no—
172C2×Q8—21no—
173C2×Q8C22×D711yescharacteristic
174C2×D4—71no—
175C2×D4—72no—
176C2×D4—71no—
177C2×D4—142no—
178C2×D4—142no—
179C2×D4—142no—
180C2×D4—141no—
181C2×D4—141no—
182C22×C4—72no—
183C22×C4—142no—
184C22×C4—142no—
185C22×C4—141no—
186SD16—142no—
187D8—282no—
188M16—282no—
189C2×C8—71no—
190C2×C8—71no—
191C2×C8C22×D712yes—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—71no—
195C4⋊C4—71no—
196C4⋊C4—142no—
197C4⋊C4C22×D712yes—
198C4⋊C4C22×D711yescharacteristic
199C22⋊C4—72no—
200C22⋊C4—142no—
201C22⋊C4—142no—
202C22⋊C4—141no—
203C42—21no—
204C42C22×D711yescharacteristic
205C2×C7#54611yescharacteristic
206C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C7#57711yescharacteristic
208D7—42no—
209D7—82no—
210D7—81no—
211C23—72no—
212C23—141no—
213C23—142no—
214Q8—21no—
215Q8—42no—
216Q8#116112yescomplemented
217D4—142no—
218D4—141no—
219D4—142no—
220D4—141no—
221D4—142no—
222D4—284no—
223D4—284no—
224D4—282no—
225D4—561no—
226D4—562no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—72no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—141no—
234C2×C4—284no—
235C2×C4—284no—
236C2×C4—282no—
237C2×C4D4×D711yescharacteristic
238C2×C4D4×D711yescharacteristic
239C2×C4C23×D711yescharacteristic
240C8—22no—
241C8—141no—
242C8—141no—
243C7#333211yescharacteristic, complemented, Sylow
244C22—142no—
245C22—142no—
246C22—142no—
247C22—281no—
248C22—282no—
249C22—282no—
250C22—281no—
251C22—282no—
252C22—281no—
253C22C2×D4×D711yescharacteristic
254C4—21no—
255C4—22no—
256C4—42no—
257C4—41no—
258C4—142no—
259C4—281no—
260C4#647712yes—
261C4C2×#116111yescharacteristic
262C4#655311yescharacteristic
263C2—282no—
264C2—561no—
265C2—562no—
266C2#9032111yescharacteristic
267C2C2×#647711yescharacteristic
268C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
hypercenter#475D7
hypocenterC7#3332
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. #90417C2semidirect
2. #90268C2semidirect
3. C7×#475C2semidirect
4. #89860C2semidirect
5. C4×#1143C2semidirect
6. #89637C2semidirect
7. #89635C2semidirect
8. #89615C2semidirect
9. #89610C2semidirect
10. #89583C2semidirect
11. #89575C2semidirect
82. #475D7semidirect
94. C7×Q8#41semidirect
216. Q8#1161semidirect
243. C7#3332semidirect

Maximal subgroups

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

subgroupindex
#904172
#902682
C7×#4752
#898602
C4×#11432
#896372
#896352
#896152
#896102
#895832
#895752
#33327