Finite Group Databasebeta

← #11741305

Subgroups — #11741305

Order 896.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#6551C211yesmaximal, characteristic, complemented
2C22×C7×SD16C211yesmaximal, characteristic, complemented
3#90496C211yesmaximal, characteristic, complemented
4C2×#6495C211yesmaximal, characteristic, complemented
5#89997C211yesmaximal, characteristic, complemented
6#89944C214yesmaximal, complemented
7#89942C214yesmaximal, complemented
8#89885C211yesmaximal, characteristic, complemented
9#89882C211yesmaximal, characteristic, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C7×D4C2211yescharacteristic
12#6551—24no—
13#6551—24no—
14C2×#1161—21no—
15C2×#1161C2212yescomplemented
16C2×Q8×D7—21no—
17C2×Q8×D7C2212yescomplemented
18C2×#1157—22no—
19C2×#1157C2211yescharacteristic
20C2×#1157C2211yescharacteristic
21C2×C7×SD16—24no—
22C2×C7×SD16C2214yes—
23C22×C7×C8C2211yescharacteristic
24#6516—22no—
25#6503—21no—
26#6501C2214yes—
27#6499—21no—
28C2×#1142C2212yes—
29#6495—24no—
30C2×#1141C2212yes—
31#6489C2211yescharacteristic
32C2×#1136C2211yescharacteristic
33#6409C2214yescomplemented
34#6395C2212yescomplemented
35(C2×D7)⋊C8C2214yescomplemented
36#6390C2212yescomplemented
37#3010—71nomaximal, Sylow
38C24×C7—21no—
39C2×C7×Q8—21no—
40C2×C7×Q8—22no—
41C2×C7×Q8D414yescomplemented
42C2×C7×Q8C2312yes—
43C2×C7×D4—22no—
44C2×C7×D4—22no—
45C2×C7×D4—21no—
46C2×C7×D4C2312yes—
47C22×C4×C7—21no—
48C22×C4×C7C2311yescharacteristic
49C2×C22⋊D7—21no—
50C2×C22⋊D7—24no—
51C2×C22⋊D7D412yes—
52#1161—28no—
53#1161—24no—
54#1161—24no—
55Q8×D7—28no—
56Q8×D7—24no—
57Q8×D7—24no—
58#1157—216no—
59#1157—22no—
60#1157—22no—
61#1157—22no—
62#1157D414yescomplemented
63C2×D28—22no—
64C2×D28D411yescharacteristic
65C2×D28C2311yescharacteristic
66C2×C4×D7—22no—
67C2×C4×D7—21no—
68C2×C4×D7—22no—
69C2×C4×D7C2312yes—
70C2×Dic14—22no—
71C2×Dic14D411yescharacteristic
72C2×Dic14C2311yescharacteristic
73C7×SD16—28no—
74C2×C7×C8—22no—
75C2×C7×C8C2312yes—
76C22⋊Dic7—21no—
77C22⋊Dic7—24no—
78#1142—24no—
79#1141—24no—
80C4⋊Dic7C2312yes—
81C4×Dic7—21no—
82#1136—22no—
83C2×C7⋊C8C2312yes—
84C2×2−1+4—71no—
85C2×AGL(1,ℤ/8)—71no—
86C22×SD16D711yescharacteristic, complemented
87#535—71no—
88#449—74no—
89#448—74no—
90#419—71no—
91#416—71no—
92#408—71no—
93C23×C7—22no—
94C23×C7—42no—
95C23×C7—82no—
96C23×C7C2×D411yescharacteristic
97C22×D7—21no—
98C22×D7C2×D412yes—
99C7×Q8—24no—
100C7×Q8—22no—
101C7×Q8C2×D414yes—
102C7×D4—22no—
103C7×D4—22no—
104C7×D4—42no—
105C2×C4×C7—24no—
106C2×C4×C7—22no—
107C2×C4×C7C2×D411yescharacteristic
108C2×C4×C7C2×D412yes—
109C2×C4×C7C2×D412yes—
110C2×C4×C7C2411yescharacteristic
111C22⋊D7—24no—
112C22⋊D7—24no—
113C22⋊D7—84no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116C2×Dic7C2×D412yes—
117D28—28no—
118D28—22no—
119D28—21no—
120D28C2×D412yes—
121C4×D7—28no—
122C4×D7—28no—
123C4×D7—24no—
124C4×D7—22no—
125C4×D7—22no—
126Dic14—28no—
127Dic14—22no—
128Dic14—21no—
129Dic14C2×D412yes—
130C7×C8—22no—
131C7⋊C8—22no—
1322−1+4—144no—
1332−1+4—144no—
134C2×#41—71no—
135C2×#41—72no—
136C2×#41—71no—
137C2×#41—142no—
138C2×#41—141no—
139C22×Q8—72no—
140C22×Q8—141no—
141C22×Q8C2×D711yescharacteristic
142C22×D4C2×D711yescharacteristic
143AGL(1,ℤ/8)—144no—
144C2×SD16—24no—
145C2×SD16—72no—
146C2×SD16C2×D714yes—
147C2×D8—72no—
148C2×M16—71no—
149C22×C8C2×D711yescharacteristic
150#127—141no—
151#124—141no—
152#121—74no—
153#120—142no—
154#117—71no—
155#103—74no—
156#103—72no—
157#102—72no—
158C22⋊C8—74no—
159C22×C7—22no—
160C22×C7—42no—
161C22×C7—44no—
162C22×C7—84no—
163C22×C7#12012yes—
164C22×C7C22×D411yescharacteristic
165C2×D7—22no—
166C2×D7—22no—
167C2×D7—24no—
168C2×D7—42no—
169C4×C7—24no—
170C4×C7#12012yes—
171C4×C7C22×D411yescharacteristic
172C4×C7C22×D411yescharacteristic
173Dic7—22no—
174Dic7—22no—
175Dic7—42no—
176C24—21no—
177#41—74no—
178#41—1416no—
179#41—144no—
180#41—148no—
181#41—144no—
182#41—142no—
183#41—142no—
184#41—142no—
185C2×Q8—21no—
186C2×Q8—22no—
187C2×Q8—71no—
188C2×Q8—71no—
189C2×Q8—142no—
190C2×Q8—144no—
191C2×Q8—148no—
192C2×Q8—144no—
193C2×Q8C22⋊D714yescomplemented
194C2×Q8C22×D712yes—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—72no—
199C2×D4—71no—
200C2×D4—71no—
201C2×D4—142no—
202C2×D4—141no—
203C2×D4—144no—
204C2×D4C22×D712yes—
205C22×C4—21no—
206C22×C4—72no—
207C22×C4—142no—
208C22×C4—142no—
209C22×C4—141no—
210C22×C4C22×D711yescharacteristic
211SD16—28no—
212SD16—144no—
213D8—144no—
214M16—142no—
215C2×C8—22no—
216C2×C8—72no—
217C2×C8C22×D712yes—
218C4⋊C4—72no—
219C22⋊C4—144no—
220C22⋊C4—141no—
221C42—141no—
222C2×C7—22no—
223C2×C7—82no—
224C2×C7C2×#12011yescharacteristic
225C2×C7#57712yes—
226D7—42no—
227D7—42no—
228C23—22no—
229C23—42no—
230C23—72no—
231C23—82no—
232C23—141no—
233C23D4×D711yescharacteristic
234Q8—24no—
235Q8—22no—
236Q8—72no—
237Q8—148no—
238Q8—142no—
239Q8—141no—
240Q8C2×C22⋊D714yes—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—72no—
245D4—148no—
246D4—144no—
247D4—144no—
248D4—142no—
249D4—141no—
250D4—564no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—72no—
254C2×C4—148no—
255C2×C4—148no—
256C2×C4—141no—
257C2×C4—144no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—142no—
261C2×C4D4×D711yescharacteristic
262C2×C4D4×D712yes—
263C2×C4C2×C22⋊D712yes—
264C2×C4C23×D711yescharacteristic
265C8—22no—
266C8—142no—
267C7#301011yescharacteristic, complemented, Sylow
268C22—22no—
269C22—44no—
270C22—42no—
271C22—84no—
272C22—144no—
273C22—142no—
274C22—142no—
275C22—282no—
276C22#650012yes—
277C22C2×D4×D711yescharacteristic
278C4—24no—
279C4—142no—
280C4—142no—
281C4—282no—
282C4#650012yes—
283C4C2×D4×D711yescharacteristic
284C4C22×C22⋊D711yescharacteristic
285C2—22no—
286C2—82no—
287C2—282no—
288C2—282no—
289C2#9045312yes—
290C2C2×#650011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC22×C7×SD16C2
socleC22×C7C22×D4
hypercenterC22×SD16D7
hypocenterC7#3010
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#6551C2semidirect
2. C22×C7×SD16C2semidirect
3. #90496C2semidirect
4. C2×#6495C2semidirect
5. #89997C2semidirect
6. #89944C2semidirect
7. #89942C2semidirect
8. #89885C2semidirect
9. #89882C2semidirect
15. C2×#1161C22semidirect
17. C2×Q8×D7C22semidirect
33. #6409C22semidirect
34. #6395C22semidirect
35. (C2×D7)⋊C8C22semidirect
36. #6390C22semidirect
41. C2×C7×Q8D4semidirect
62. #1157D4semidirect
86. C22×SD16D7semidirect
193. C2×Q8C22⋊D7semidirect
267. C7#3010semidirect

Maximal subgroups

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

subgroupindex
C2×#65512
C22×C7×SD162
#904962
C2×#64952
#899972
#899442
#899422
#898852
#898822
#30107