Finite Group Databasebeta

← #11736281

Subgroups — #11736281

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
2C2×#6512C211yesmaximal, characteristic, complemented
3C22×#1132C211yesmaximal, characteristic, complemented
4#90216C211yesmaximal, characteristic, complemented
5C7×#416C211yesmaximal, characteristic, complemented
6#89882C211yesmaximal, characteristic, complemented
7#89776C211yesmaximal, characteristic, complemented
8#89578C214yesmaximal, complemented
9#89553C214yesmaximal, complemented
10C2×C7×#41C2211yescharacteristic
11#6554—24no—
12#6554—24no—
13C2×Q8×D7C2212yescomplemented
14C2×#1159—22no—
15C2×#1159C2212yescomplemented
16C2×#1157—21no—
17C2×#1157C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C22×Dic14—21no—
20C22×Dic14C2211yescharacteristic
21C22×C7×C8C2211yescharacteristic
22C7×#117C2211yescharacteristic
23#6512—24no—
24C2×#1143C2212yes—
25C2×#1141C2212yes—
26C2×#1136C2211yescharacteristic
27C2×#1132—24no—
28C2×#1132C2214yes—
29#6458C2214yes—
30#6445—22no—
31#6438—21no—
32#6437—21no—
33C7×#103C2212yescomplemented
34C7×#102C2212yescomplemented
35(C2×D7)⋊C8C2214yescomplemented
36#6384C2214yescomplemented
37#3010—71nomaximal, Sylow
38C23×D7—21no—
39C7×#41—22no—
40C7×#41D414yescomplemented
41C2×C7×Q8C2311yescharacteristic
42C2×C7×D4—21no—
43C2×C7×D4C2311yescharacteristic
44C22×C4×C7—21no—
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—22no—
47C2×C22⋊D7D412yes—
48C22×Dic7—21no—
49C22×Dic7—22no—
50Q8×D7—24no—
51Q8×D7—24no—
52#1159—216no—
53#1159—24no—
54#1159—24no—
55#1157—28no—
56#1157—22no—
57#1157—22no—
58C2×D28—22no—
59C2×D28—22no—
60C2×D28—21no—
61C2×D28—24no—
62C2×D28D411yescharacteristic
63C2×D28C2312yes—
64C2×C4×D7—22no—
65C2×C4×D7C2312yes—
66C2×Dic14—28no—
67C2×Dic14—21no—
68C2×Dic14—22no—
69C2×Dic14—22no—
70C2×Dic14D411yescharacteristic
71C2×Dic14D414yescomplemented
72C2×Dic14C2312yes—
73C2×C7×C8—22no—
74C2×C7×C8C2312yes—
75C7×C4⋊C4C2312yes—
76C7×C22⋊C4—21no—
77C42×C7—21no—
78#1143—24no—
79#1141—24no—
80(C2×D7)⋊C4—24no—
81#1136—22no—
82C2×C7⋊C8C2312yes—
83#1132—28no—
84C2×2−1+4—71no—
85C2×AGL(1,ℤ/8)—71no—
86C22×SD16—71no—
87#535—71no—
88#449—74no—
89#448—74no—
90#419—71no—
91#416D711yescharacteristic, complemented
92#408—71no—
93C23×C7—21no—
94C23×C7C2×D411yescharacteristic
95C22×D7—22no—
96C22×D7—42no—
97C22×D7—82no—
98C22×D7C2×D412yes—
99C7×Q8—21no—
100C7×Q8C2×D412yes—
101C7×D4—24no—
102C7×D4—21no—
103C7×D4C2×D412yes—
104C2×C4×C7—21no—
105C2×C4×C7—24no—
106C2×C4×C7—22no—
107C2×C4×C7C2×D412yes—
108C2×C4×C7C2×D411yescharacteristic
109C2×C4×C7C2×D412yes—
110C2×C4×C7C2411yescharacteristic
111C22⋊D7—28no—
112C22⋊D7—24no—
113C2×Dic7—28no—
114C2×Dic7—24no—
115C2×Dic7—22no—
116C2×Dic7C2×D412yes—
117D28—22no—
118D28—22no—
119D28—22no—
120D28—42no—
121D28—84no—
122C4×D7—28no—
123C4×D7—22no—
124C4×D7—22no—
125Dic14—28no—
126Dic14—22no—
127Dic14—24no—
128Dic14—22no—
129Dic14C2×D414yes—
130C7×C8—22no—
131C7⋊C8—22no—
1322−1+4—144no—
1332−1+4—144no—
134C2×#41—71no—
135C2×#41—72no—
136C2×#41—141no—
137C2×#41—142no—
138C2×#41C2×D711yescharacteristic
139C22×Q8—72no—
140C22×Q8—71no—
141C22×Q8—141no—
142C22×D4—71no—
143AGL(1,ℤ/8)—144no—
144C2×SD16—72no—
145C2×SD16—74no—
146C2×SD16—144no—
147C2×D8—72no—
148C2×M16—71no—
149C22×C8C2×D711yescharacteristic
150#127—141no—
151#124—141no—
152#121—74no—
153#120—142no—
154#117C2×D711yescharacteristic
155#103—74no—
156#103C2×D712yescomplemented
157#102C2×D712yescomplemented
158C22⋊C8—74no—
159C22×C7—22no—
160C22×C7—22no—
161C22×C7—42no—
162C22×C7#12012yes—
163C22×C7C22×D411yescharacteristic
164C2×D7—22no—
165C2×D7—24no—
166C2×D7—42no—
167C2×D7—44no—
168C2×D7—84no—
169C4×C7—22no—
170C4×C7—42no—
171C4×C7#12012yes—
172C4×C7C22×D411yescharacteristic
173C4×C7C22×D411yescharacteristic
174Dic7—22no—
175Dic7—24no—
176C24—141no—
177#41—22no—
178#41—148no—
179#41—144no—
180#41—1416no—
181#41—144no—
182#41—142no—
183#41—142no—
184#41D2814yescomplemented
185C2×Q8—71no—
186C2×Q8—74no—
187C2×Q8—72no—
188C2×Q8—142no—
189C2×Q8—148no—
190C2×Q8—144no—
191C2×Q8—144no—
192C2×Q8—142no—
193C2×Q8—141no—
194C2×Q8C22×D711yescharacteristic
195C2×D4—21no—
196C2×D4—72no—
197C2×D4—71no—
198C2×D4—72no—
199C2×D4—144no—
200C2×D4—142no—
201C2×D4—142no—
202C2×D4—142no—
203C2×D4—141no—
204C2×D4C22×D711yescharacteristic
205C22×C4—21no—
206C22×C4—72no—
207C22×C4—142no—
208C22×C4—142no—
209C22×C4—141no—
210C22×C4C22×D711yescharacteristic
211SD16—148no—
212SD16—144no—
213D8—144no—
214M16—142no—
215C2×C8—22no—
216C2×C8—72no—
217C2×C8C22×D712yes—
218C4⋊C4C22×D712yes—
219C22⋊C4—21no—
220C22⋊C4—144no—
221C42—21no—
222C2×C7—22no—
223C2×C7—42no—
224C2×C7C2×#12011yescharacteristic
225C2×C7#57712yes—
226D7—42no—
227D7—82no—
228C23—21no—
229C23—72no—
230C23—142no—
231C23—282no—
232C23—562no—
233C23D4×D711yescharacteristic
234Q8—21no—
235Q8—74no—
236Q8—148no—
237Q8—142no—
238Q8—144no—
239Q8—142no—
240Q8C2×D2812yes—
241D4—24no—
242D4—21no—
243D4—148no—
244D4—144no—
245D4—142no—
246D4—142no—
247D4—142no—
248D4—282no—
249D4—564no—
250D4C2×D2812yes—
251C2×C4—21no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—72no—
255C2×C4—148no—
256C2×C4—148no—
257C2×C4—144no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—142no—
261C2×C4C2×D2812yes—
262C2×C4D4×D711yescharacteristic
263C2×C4D4×D712yes—
264C2×C4C23×D711yescharacteristic
265C8—22no—
266C8—142no—
267C7#301011yescharacteristic, complemented, Sylow
268C22—22no—
269C22—22no—
270C22—42no—
271C22—142no—
272C22—144no—
273C22—282no—
274C22—284no—
275C22—564no—
276C22#644512yes—
277C22C2×D4×D711yescharacteristic
278C4—22no—
279C4—42no—
280C4—142no—
281C4—144no—
282C4#644512yes—
283C4C22×D2811yescharacteristic
284C4C2×D4×D711yescharacteristic
285C2—22no—
286C2—42no—
287C2—282no—
288C2—562no—
289C2C2×#644511yescharacteristic
290C2#9046112yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#416C2
socleC22×C7C22×D4
hypercenter#416D7
hypocenterC7#3010
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#6554C2semidirect
2. C2×#6512C2semidirect
3. C22×#1132C2semidirect
4. #90216C2semidirect
5. C7×#416C2semidirect
6. #89882C2semidirect
7. #89776C2semidirect
8. #89578C2semidirect
9. #89553C2semidirect
13. C2×Q8×D7C22semidirect
15. C2×#1159C22semidirect
33. C7×#103C22semidirect
34. C7×#102C22semidirect
35. (C2×D7)⋊C8C22semidirect
36. #6384C22semidirect
40. C7×#41D4semidirect
71. C2×Dic14D4semidirect
91. #416D7semidirect
156. #103C2×D7semidirect
157. #102C2×D7semidirect
184. #41D28semidirect
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
C2×#65122
C22×#11322
#902162
C7×#4162
#898822
#897762
#895782
#895532
#30107