Finite Group Databasebeta

← #11740835

Subgroups — #11740835

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×#6554C211yesmaximal, characteristic, complemented
2C7×#535C211yesmaximal, characteristic, complemented
3C22×#1142C211yesmaximal, characteristic, complemented
4C2×#6495C211yesmaximal, characteristic, complemented
5#90006C211yesmaximal, characteristic, complemented
6#89812C214yesmaximal, complemented
7#89809C214yesmaximal, complemented
8#89776C211yesmaximal, characteristic, complemented
9#89774C211yesmaximal, characteristic, complemented
10C2×C7×#41C2211yescharacteristic
11C22×C7×D4C2211yescharacteristic
12#6554—24no—
13#6554—24no—
14C2×Q8×D7—21no—
15C2×#1159—21no—
16C2×#1159—22no—
17C2×#1157C2211yescharacteristic
18C2×#1157C2212yescomplemented
19C22×Dic14C2211yescharacteristic
20C22×Dic14C2212yescomplemented
21C7×#127—21no—
22C7×#124—21no—
23C7×#121C2214yes—
24C7×#120—22no—
25C7×#117C2211yescharacteristic
26C2×#1142—24no—
27C2×#1142C2212yes—
28C2×#1142C2214yes—
29#6495—24no—
30C2×#1141C2212yes—
31C2×#1136C2211yescharacteristic
32C22×C7⋊C8C2211yescharacteristic
33C22⋊(C7⋊C8)C2214yescomplemented
34#6384C2212yescomplemented
35#6384C2214yescomplemented
36#6383C2212yescomplemented
37#3010—71nomaximal, Sylow
38C24×C7—21no—
39C7×#41—22no—
40C7×#41—22no—
41C2×C7×Q8D411yescharacteristic
42C2×C7×D4—24no—
43C2×C7×D4—22no—
44C2×C7×D4—22no—
45C2×C7×D4—21no—
46C2×C7×D4D412yes—
47C2×C7×D4D411yescharacteristic
48C2×C7×D4C2312yes—
49C22×C4×C7C2312yes—
50C22×C4×C7C2311yescharacteristic
51C2×C22⋊D7—21no—
52C2×C22⋊D7—22no—
53C22×Dic7—21no—
54C22×Dic7—22no—
55Q8×D7—28no—
56#1159—216no—
57#1159—28no—
58#1157—22no—
59#1157—24no—
60#1157—24no—
61#1157D414yescomplemented
62C2×D28C2311yescharacteristic
63C2×C4×D7—21no—
64C2×C4×D7—22no—
65C2×Dic14—21no—
66C2×Dic14—22no—
67C2×Dic14—22no—
68C2×Dic14—24no—
69C2×Dic14—24no—
70C2×Dic14D414yescomplemented
71C2×Dic14C2312yes—
72C2×Dic14C2311yescharacteristic
73C7×C4⋊C4C2312yes—
74C7×C22⋊C4—21no—
75C7×C22⋊C4—24no—
76C42×C7—21no—
77#1142—24no—
78#1142—28no—
79#1141—24no—
80#1136—22no—
81C2×C7⋊C8—22no—
82C2×C7⋊C8C2312yes—
83C2×C7⋊C8C2312yes—
84C2×2−1+4—71no—
85C2×AGL(1,ℤ/8)—71no—
86C22×SD16—71no—
87#535D711yescharacteristic, complemented
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×D412yes—
97C23×C7C2×D411yescharacteristic
98C22×D7—21no—
99C7×Q8—22no—
100C7×D4—24no—
101C7×D4—22no—
102C7×D4—22no—
103C7×D4—22no—
104C7×D4—42no—
105C7×D4—84no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7C2×D412yes—
110C2×C4×C7C2×D412yes—
111C2×C4×C7C2×D412yes—
112C2×C4×C7C2×D411yescharacteristic
113C2×C4×C7C2411yescharacteristic
114C22⋊D7—28no—
115C22⋊D7—24no—
116C2×Dic7—28no—
117C2×Dic7—24no—
118C2×Dic7—22no—
119C2×Dic7—21no—
120D28—21no—
121D28C2×D412yes—
122C4×D7—28no—
123C4×D7—24no—
124Dic14—28no—
125Dic14—24no—
126Dic14—22no—
127Dic14—21no—
128Dic14C2×D412yes—
129Dic14C2×D414yes—
130C7⋊C8—22no—
131C7⋊C8—22no—
1322−1+4—144no—
1332−1+4—144no—
134C2×#41—71no—
135C2×#41—72no—
136C2×#41—142no—
137C2×#41—141no—
138C2×#41C2×D711yescharacteristic
139C22×Q8—71no—
140C22×Q8—72no—
141C22×Q8—141no—
142C22×D4C2×D711yescharacteristic
143AGL(1,ℤ/8)—144no—
144C2×SD16—74no—
145C2×SD16—72no—
146C2×SD16—144no—
147C2×D8—72no—
148C2×M16—71no—
149C22×C8—71no—
150#127—21no—
151#124—21no—
152#121C2×D714yes—
153#120—22no—
154#117C2×D711yescharacteristic
155#103—72no—
156#103—74no—
157#102—72no—
158C22⋊C8—74no—
159C22×C7—22no—
160C22×C7—22no—
161C22×C7—24no—
162C22×C7—42no—
163C22×C7—44no—
164C22×C7—84no—
165C22×C7#12012yes—
166C22×C7C22×D411yescharacteristic
167C2×D7—22no—
168C2×D7—42no—
169C4×C7—22no—
170C4×C7—42no—
171C4×C7#12012yes—
172C4×C7C22×D411yescharacteristic
173C4×C7C22×D411yescharacteristic
174Dic7—22no—
175Dic7—24no—
176C24—21no—
177#41—22no—
178#41—22no—
179#41—74no—
180#41—144no—
181#41—144no—
182#41—1416no—
183#41—148no—
184#41—142no—
185C2×Q8—74no—
186C2×Q8—72no—
187C2×Q8—71no—
188C2×Q8—144no—
189C2×Q8—144no—
190C2×Q8—148no—
191C2×Q8—142no—
192C2×Q8—142no—
193C2×Q8—141no—
194C2×Q8C22⋊D711yescharacteristic
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—71no—
200C2×D4—142no—
201C2×D4—141no—
202C2×D4C22⋊D712yes—
203C2×D4C22⋊D711yescharacteristic
204C2×D4C22×D712yes—
205C22×C4—142no—
206C22×C4—141no—
207C22×C4—142no—
208C22×C4—141no—
209C22×C4C22×D712yes—
210C22×C4C22×D711yescharacteristic
211SD16—148no—
212SD16—144no—
213D8—144no—
214M16—142no—
215C2×C8—72no—
216C2×C8—72no—
217C2×C8—142no—
218C4⋊C4C22×D712yes—
219C22⋊C4—21no—
220C22⋊C4—24no—
221C42—21no—
222C2×C7—22no—
223C2×C7—42no—
224C2×C7—82no—
225C2×C7C2×#12011yescharacteristic
226C2×C7#57712yes—
227D7—42no—
228C23—22no—
229C23—42no—
230C23—82no—
231C23—141no—
232C23C2×C22⋊D712yes—
233C23C2×C22⋊D711yescharacteristic
234Q8—22no—
235Q8—74no—
236Q8—72no—
237Q8—148no—
238Q8—144no—
239Q8—142no—
240Q8—141no—
241D4—24no—
242D4—22no—
243D4—22no—
244D4—22no—
245D4—42no—
246D4—72no—
247D4—84no—
248D4—148no—
249D4—144no—
250D4—141no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—148no—
255C2×C4—148no—
256C2×C4—144no—
257C2×C4—142no—
258C2×C4—141no—
259C2×C4—144no—
260C2×C4D4×D712yes—
261C2×C4D4×D712yes—
262C2×C4C2×C22⋊D712yes—
263C2×C4C2×C22⋊D711yescharacteristic
264C2×C4C23×D711yescharacteristic
265C8—142no—
266C8—142no—
267C7#301011yescharacteristic, complemented, Sylow
268C22—22no—
269C22—22no—
270C22—24no—
271C22—42no—
272C22—44no—
273C22—84no—
274C22—142no—
275C22—282no—
276C22#650012yes—
277C22C22×C22⋊D711yescharacteristic
278C4—22no—
279C4—42no—
280C4—144no—
281C4—142no—
282C4#650012yes—
283C4C2×D4×D711yescharacteristic
284C4C2×D4×D711yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—282no—
289C2C2×#650011yescharacteristic
290C2#9052712yes—
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#535C2
socleC22×C7C22×D4
hypercenter#535D7
hypocenterC7#3010
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#6554C2semidirect
2. C7×#535C2semidirect
3. C22×#1142C2semidirect
4. C2×#6495C2semidirect
5. #90006C2semidirect
6. #89812C2semidirect
7. #89809C2semidirect
8. #89776C2semidirect
9. #89774C2semidirect
18. C2×#1157C22semidirect
20. C22×Dic14C22semidirect
33. C22⋊(C7⋊C8)C22semidirect
34. #6384C22semidirect
35. #6384C22semidirect
36. #6383C22semidirect
61. #1157D4semidirect
70. C2×Dic14D4semidirect
87. #535D7semidirect
267. C7#3010semidirect

Maximal subgroups

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

subgroupindex
C2×#65542
C7×#5352
C22×#11422
C2×#64952
#900062
#898122
#898092
#897762
#897742
#30107