Finite Group Databasebeta

← #11736997

Subgroups — #11736997

Order 896.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#6472C212yesmaximal, complemented
2#90242C211yesmaximal, characteristic
3D4×Dic14C211yesmaximal, characteristic, complemented
4C7×#436C211yesmaximal, characteristic, complemented
5#89909C212yesmaximal, complemented
6#89621C211yesmaximal, characteristic, complemented
7#89613C211yesmaximal, characteristic, complemented
8#89511C212yesmaximal, complemented
9#89505C212yesmaximal, complemented
10#89471C211yesmaximal, characteristic, complemented
11C4×#1132C211yesmaximal, characteristic, complemented
12C2×#1157C2212yes—
13C22×Dic14C2212yes—
14C2×C7×M16C2212yes—
15C4×C7×D4C2211yescharacteristic
16#6501—21no—
17D4×Dic7—21no—
18#6487C2212yes—
19#6487C2212yes—
20#6472—28no—
21C2×Dic28C2212yes—
22C2×Dic28C2212yes—
23C2×Dic28C2212yes—
24C2×#1132C2212yes—
25C2×#1132C2211yescharacteristic
26D7×C4⋊C4—21no—
27#6451—21no—
28#6446—22no—
29#6441—22no—
30C4×D28C2211yescharacteristic
31#6432C2211yescharacteristic
32#6432C2211yescharacteristic
33C4×Dic14C2211yescharacteristic
34C7×C4⋊C8C2211yescharacteristic
35C7×C22⋊C8C2212yes—
36C4×C7×C8C2211yescharacteristic
37#6395C2211yescharacteristic
38#6391C2211yescharacteristic
39#6390C2212yes—
40#6390C2212yes—
41#6390C2211yescharacteristic
42#3276—71nomaximal, Sylow
43C2×C7×D4D411yescharacteristic
44C22×C4×C7C2312yes—
45C2×C22⋊D7—22no—
46C22×Dic7—22no—
47#1157—24no—
48#1157—42no—
49C2×D28C2311yescharacteristic
50C2×C4×D7—21no—
51C2×C4×D7—22no—
52C2×Dic14—24no—
53C2×Dic14—22no—
54C2×Dic14—42no—
55C2×Dic14C2312yes—
56C2×Dic14C2312yes—
57C2×Dic14C2311yescharacteristic
58C7×M16—24no—
59C2×C7×C8C2311yescharacteristic
60C2×C7×C8C2312yes—
61C2×C7×C8C2311yescharacteristic
62C7×C4⋊C4D411yescharacteristic
63C7×C22⋊C4D412yes—
64C42×C7C2311yescharacteristic
65C22⋊Dic7—22no—
66C22⋊Dic7—22no—
67(C2×D7)⋊C4—21no—
68C4⋊Dic7—21no—
69C4⋊Dic7—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7—21no—
72C4⋊Dic7C2311yescharacteristic
73C4⋊Dic7C2311yescharacteristic
74Dic7⋊C4—21no—
75Dic7⋊C4—22no—
76Dic7⋊C4—22no—
77C4×Dic7—21no—
78Dic28—24no—
79Dic28—24no—
80Dic28—42no—
81#1132—24no—
82#1132D414yescomplemented
83C2×#137—72no—
84D4×Q8—71no—
85#547—71no—
86#494—71no—
87#470—72no—
88#462—71no—
89#461—71no—
90#452—72no—
91#451—72no—
92C4×SD16—71no—
93#436D711yescharacteristic, complemented
94C23×C7C2×D412yes—
95C22×D7—21no—
96C7×D4—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7C2×D412yes—
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2411yescharacteristic
103C22⋊D7—44no—
104C2×Dic7—21no—
105C2×Dic7—21no—
106C2×Dic7—21no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—44no—
110D28—21no—
111D28C2×D412yes—
112C4×D7—44no—
113C4×D7—42no—
114Dic14—21no—
115Dic14—22no—
116Dic14—22no—
117Dic14—22no—
118Dic14—22no—
119Dic14—44no—
120Dic14C2×D412yes—
121C7×C8—22no—
122C7×C8—21no—
123C7×C8C2×D412yes—
124C2×#41—72no—
125C22×Q8—72no—
126#137—148no—
127C2×Q16—72no—
128C2×Q16—72no—
129C2×Q16—72no—
130C2×SD16—71no—
131C2×SD16—72no—
132C2×M16C2×D712yes—
133C4⋊Q8—71no—
134C4⋊Q8—71no—
135C4⋊Q8—141no—
136#123—142no—
137C22⋊Q8—72no—
138C22⋊Q8—72no—
139C22⋊Q8—142no—
140#121—141no—
141C4×Q8—71no—
142C4×D4—71no—
143C4×D4—141no—
144C4×D4C2×D711yescharacteristic
145C2×C4⋊C4—141no—
146#106—71no—
147C4⋊C8C2×D711yescharacteristic
148#103—71no—
149#103—72no—
150#103—72no—
151#102—71no—
152C22⋊C8C2×D712yes—
153C4×C8C2×D711yescharacteristic
154C22×C7—22no—
155C22×C7—22no—
156C22×C7—22no—
157C22×C7C22×D411yescharacteristic
158C2×D7—22no—
159C2×D7—41no—
160C2×D7—41no—
161C4×C7—22no—
162C4×C7—21no—
163C4×C7#13712yescomplemented
164C4×C7C22×D411yescharacteristic
165C4×C7H(5,𝔽2)11yescharacteristic
166Dic7—22no—
167Dic7—42no—
168Dic7—42no—
169Dic7—41no—
170Dic7—41no—
171#41—144no—
172#41—282no—
173C2×Q8—71no—
174C2×Q8—72no—
175C2×Q8—72no—
176C2×Q8—144no—
177C2×Q8—142no—
178C2×Q8—282no—
179C2×D4—71no—
180C2×D4—142no—
181C2×D4D2811yescharacteristic
182C22×C4—142no—
183C22×C4—141no—
184C22×C4—142no—
185C22×C4C22×D712yes—
186Q16—144no—
187Q16—144no—
188Q16—282no—
189SD16—74no—
190SD16—144no—
191M16—24no—
192C2×C8C22×D711yescharacteristic
193C2×C8C22×D712yes—
194C2×C8C22×D711yescharacteristic
195C4⋊C4—71no—
196C4⋊C4—71no—
197C4⋊C4—142no—
198C4⋊C4—141no—
199C4⋊C4—142no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4D2811yescharacteristic
205C22⋊C4—142no—
206C22⋊C4—142no—
207C22⋊C4—141no—
208C22⋊C4D2812yes—
209C42—141no—
210C42C22×D711yescharacteristic
211C2×C7—42no—
212C2×C7D4211yescharacteristic
213C2×C7C2×#13711yescharacteristic
214C2×C7#57811yescharacteristic
215D7—42no—
216C23—141no—
217C23C2×D2812yes—
218Q8—72no—
219Q8—141no—
220Q8—142no—
221Q8—142no—
222Q8—142no—
223Q8—142no—
224Q8—284no—
225D4—22no—
226D4—72no—
227D4—141no—
228D4—284no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—141no—
232C2×C4—141no—
233C2×C4—141no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—284no—
237C2×C4—284no—
238C2×C4—282no—
239C2×C4C2×D2811yescharacteristic
240C2×C4C2×D2812yes—
241C2×C4C2×D2811yescharacteristic
242C2×C4C23×D711yescharacteristic
243C8—22no—
244C8—21no—
245C8D4×D712yes—
246C7#327611yescharacteristic, complemented, Sylow
247C22—22no—
248C22—22no—
249C22—22no—
250C22—142no—
251C22—281no—
252C22—281no—
253C22C22×D2811yescharacteristic
254C4—21no—
255C4—22no—
256C4—142no—
257C4—281no—
258C4—281no—
259C4—282no—
260C4—282no—
261C4#647212yescomplemented
262C4C2×D4×D711yescharacteristic
263C4#655311yescharacteristic
264C2—42no—
265C2—282no—
266C2D4×D2811yescharacteristic
267C2C2×#647211yescharacteristic
268C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#436C2
socleC22×C7C22×D4
prime radical#90242C2
hypercenter#436D7
hypocenterC7#3276
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#6472C2semidirect
3. D4×Dic14C2semidirect
4. C7×#436C2semidirect
5. #89909C2semidirect
6. #89621C2semidirect
7. #89613C2semidirect
8. #89511C2semidirect
9. #89505C2semidirect
10. #89471C2semidirect
11. C4×#1132C2semidirect
82. #1132D4semidirect
93. #436D7semidirect
163. C4×C7#137semidirect
246. C7#3276semidirect
261. C4#6472semidirect

Maximal subgroups

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

subgroupindex
C2×#64722
#902422
D4×Dic142
C7×#4362
#899092
#896212
#896132
#895112
#895052
#894712
C4×#11322
#32767