Finite Group Databasebeta

← #11736280

Subgroups — #11736280

Order 896.

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

normal subgroups only

This group has 752 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×#6553C211yesmaximal, characteristic
2C2×#6514C211yesmaximal, characteristic, complemented
3C22×#1132C211yesmaximal, characteristic, complemented
4#90217C211yesmaximal, characteristic, complemented
5C7×#416C211yesmaximal, characteristic, complemented
6#89882C211yesmaximal, characteristic, complemented
7#89773C211yesmaximal, characteristic, complemented
8#89582C214yesmaximal, complemented
9#89548C214yesmaximal, complemented
10C2×C7×#41C2211yescharacteristic
11#6553—24no—
12#6553—24no—
13C2×#1161C2212yes—
14C2×D4×D7—22no—
15C2×D4×D7C2212yes—
16C2×#1157—21no—
17C2×#1157C2211yescharacteristic
18C22×D28—21no—
19C22×D28C2211yescharacteristic
20C22×Dic14C2211yescharacteristic, complemented
21C22×C7×C8C2211yescharacteristic, complemented
22C7×#117C2211yescharacteristic, complemented
23#6514—24no—
24C2×#1144C2212yescomplemented
25C2×#1142C2212yescomplemented
26C2×#1136C2211yescharacteristic, complemented
27C2×#1132—24no—
28C2×#1132C2214yescomplemented
29#6460C2214yescomplemented
30#6449—22no—
31#6438—21no—
32#6432—21no—
33C7×#103C2212yescomplemented
34C7×#102C2212yescomplemented
35(C2×D7)⋊C8C2214yescomplemented
36#6383C2214yescomplemented
37#3009—71nomaximal, Sylow
38C23×D7—21no—
39C23×D7—22no—
40C7×#41—22no—
41C7×#41D414yes—
42C2×C7×Q8C2311yescharacteristic
43C2×C7×D4—21no—
44C2×C7×D4C2311yescharacteristic
45C22×C4×C7—21no—
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—22no—
48C2×C22⋊D7D412yes—
49C22×Dic7—21no—
50#1161—24no—
51#1161—24no—
52D4×D7—216no—
53D4×D7—24no—
54D4×D7—24no—
55#1157—28no—
56#1157—22no—
57#1157—22no—
58C2×D28—28no—
59C2×D28—22no—
60C2×D28—21no—
61C2×D28—22no—
62C2×D28D411yescharacteristic
63C2×D28D414yes—
64C2×D28C2312yes—
65C2×C4×D7—22no—
66C2×C4×D7C2312yes—
67C2×Dic14—22no—
68C2×Dic14—22no—
69C2×Dic14—21no—
70C2×Dic14D411yescharacteristic
71C2×Dic14C2312yes—
72C2×C7×C8—22no—
73C2×C7×C8C2312yes—
74C7×C4⋊C4C2312yes—
75C7×C22⋊C4—21no—
76C42×C7—21no—
77#1144—24no—
78#1142—24no—
79(C2×D7)⋊C4—24no—
80C4⋊Dic7—24no—
81#1136—22no—
82C2×C7⋊C8C2312yes—
83#1132—28no—
84C2×H(5,𝔽2)—71no—
85C2×#137—71no—
86C22×SD16—71no—
87#536—71no—
88#452—74no—
89#450—74no—
90#418—71no—
91#416D711yescharacteristic, complemented
92#408—71no—
93C23×C7—21no—
94C23×C7C2×D411yescharacteristic
95C22×D7—22no—
96C22×D7—28no—
97C22×D7—24no—
98C22×D7—84no—
99C22×D7—82no—
100C22×D7C2×D412yes—
101C7×Q8—21no—
102C7×Q8C2×D412yes—
103C7×D4—24no—
104C7×D4—21no—
105C7×D4C2×D412yes—
106C2×C4×C7—21no—
107C2×C4×C7—24no—
108C2×C4×C7—22no—
109C2×C4×C7C2×D412yes—
110C2×C4×C7C2×D411yescharacteristic
111C2×C4×C7C2×D412yes—
112C2×C4×C7C2411yescharacteristic
113C22⋊D7—28no—
114C22⋊D7—24no—
115C2×Dic7—22no—
116C2×Dic7—42no—
117C2×Dic7C2×D412yes—
118D28—28no—
119D28—22no—
120D28—24no—
121D28—22no—
122D28C2×D414yes—
123C4×D7—28no—
124C4×D7—22no—
125C4×D7—22no—
126Dic14—22no—
127Dic14—22no—
128Dic14—22no—
129Dic14—42no—
130C7×C8—22no—
131C7⋊C8—22no—
132H(5,𝔽2)—144no—
133H(5,𝔽2)—144no—
134C2×#41—71no—
135C2×#41—72no—
136C2×#41—141no—
137C2×#41C2×D711yescharacteristic
138C22×Q8—71no—
139C22×D4—72no—
140C22×D4—71no—
141C22×D4—141no—
142C22×D4—142no—
143#137—144no—
144C2×Q16—72no—
145C2×SD16—72no—
146C2×SD16—74no—
147C2×SD16—144no—
148C2×M16—71no—
149C22×C8C2×D711yescharacteristic, complemented
150C4⋊Q8—141no—
151#124—141no—
152#123—142no—
153C22⋊Q8—74no—
154#117C2×D711yescharacteristic, complemented
155#103C2×D712yescomplemented
156#102—74no—
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—24no—
167C2×D7—44no—
168C2×D7—84no—
169C2×D7—88no—
170C4×C7—22no—
171C4×C7—42no—
172C4×C7#12012yes—
173C4×C7C22×D411yescharacteristic
174C4×C7C22×D411yescharacteristic
175Dic7—22no—
176Dic7—42no—
177C24—142no—
178C24—141no—
179#41—22no—
180#41—148no—
181#41—144no—
182#41—144no—
183#41—142no—
184#41—142no—
185#41D2814yes—
186C2×Q8—71no—
187C2×Q8—72no—
188C2×Q8—141no—
189C2×Q8—142no—
190C2×Q8—142no—
191C2×Q8C22×D711yescharacteristic
192C2×D4—21no—
193C2×D4—72no—
194C2×D4—74no—
195C2×D4—71no—
196C2×D4—72no—
197C2×D4—142no—
198C2×D4—148no—
199C2×D4—142no—
200C2×D4—144no—
201C2×D4—1416no—
202C2×D4—144no—
203C2×D4—142no—
204C2×D4—141no—
205C2×D4C22×D711yescharacteristic
206C22×C4—21no—
207C22×C4—72no—
208C22×C4—142no—
209C22×C4—141no—
210C22×C4C22×D711yescharacteristic
211Q16—144no—
212SD16—148no—
213SD16—144no—
214M16—142no—
215C2×C8—22no—
216C2×C8—72no—
217C2×C8C22×D712yes—
218C4⋊C4—144no—
219C4⋊C4C22×D712yes—
220C22⋊C4—21no—
221C22⋊C4—144no—
222C42—21no—
223C2×C7—22no—
224C2×C7—42no—
225C2×C7C2×#12011yescharacteristic
226C2×C7#57712yes—
227D7—42no—
228D7—44no—
229C23—21no—
230C23—72no—
231C23—148no—
232C23—142no—
233C23—144no—
234C23—562no—
235C23—564no—
236C23D4×D711yescharacteristic
237Q8—21no—
238Q8—142no—
239Q8—142no—
240Q8—142no—
241Q8—282no—
242Q8C2×D2812yes—
243D4—24no—
244D4—21no—
245D4—74no—
246D4—148no—
247D4—148no—
248D4—144no—
249D4—144no—
250D4—142no—
251D4—142no—
252D4C2×D2812yes—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—72no—
257C2×C4—148no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—142no—
261C2×C4—282no—
262C2×C4C2×D2812yes—
263C2×C4D4×D711yescharacteristic
264C2×C4D4×D712yes—
265C2×C4C23×D711yescharacteristic
266C8—22no—
267C8—142no—
268C7#300911yescharacteristic, complemented, Sylow
269C22—22no—
270C22—22no—
271C22—42no—
272C22—144no—
273C22—142no—
274C22—144no—
275C22—284no—
276C22—564no—
277C22—568no—
278C22#644512yes—
279C22C2×D4×D711yescharacteristic
280C4—22no—
281C4—42no—
282C4—142no—
283C4—282no—
284C4#644512yes—
285C4C22×D2811yescharacteristic
286C4C2×D4×D711yescharacteristic
287C2—22no—
288C2—42no—
289C2—282no—
290C2—284no—
291C2C2×#644511yescharacteristic
292C2#9046112yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#416C2
socleC22×C7C22×D4
prime radicalC2×#6553C2
hypercenter#416D7
hypocenterC7#3009
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
2. C2×#6514C2semidirect
3. C22×#1132C2semidirect
4. #90217C2semidirect
5. C7×#416C2semidirect
6. #89882C2semidirect
7. #89773C2semidirect
8. #89582C2semidirect
9. #89548C2semidirect
20. C22×Dic14C22semidirect
21. C22×C7×C8C22semidirect
22. C7×#117C22semidirect
24. C2×#1144C22semidirect
25. C2×#1142C22semidirect
26. C2×#1136C22semidirect
28. C2×#1132C22semidirect
29. #6460C22semidirect
33. C7×#103C22semidirect
34. C7×#102C22semidirect
35. (C2×D7)⋊C8C22semidirect
36. #6383C22semidirect
91. #416D7semidirect
149. C22×C8C2×D7semidirect
154. #117C2×D7semidirect
155. #103C2×D7semidirect
157. #102C2×D7semidirect
268. C7#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#65532
C2×#65142
C22×#11322
#902172
C7×#4162
#898822
#897732
#895822
#895482
#30097