Finite Group Databasebeta

← #11728136

Subgroups — #11728136

Order 896.

442 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#90519C211yesmaximal, characteristic
2C2×C42×D7C211yesmaximal, characteristic, complemented
3C2×C7×#104C211yesmaximal, characteristic, complemented
4C2×#6411C211yesmaximal, characteristic, complemented
5#89898C211yesmaximal, characteristic, complemented
6#89350C211yesmaximal, characteristic, complemented
7#89318C211yesmaximal, characteristic, complemented
8C2×C7×#41C2211yescharacteristic
9C22×C4×D7—21no—
10C22×C4×D7C2211yescharacteristic
11C2×C7×M16C2211yescharacteristic
12C2×C42×C7C2211yescharacteristic
13#6509C412yescomplemented
14#6501C412yescomplemented
15#6500—21no—
16#6498—21no—
17C4×C22⋊D7—22no—
18#6489C2211yescharacteristic
19C2×C4×Dic7—21no—
20C2×C4×Dic7C2211yescharacteristic
21C2×#1136C2211yescharacteristic
22C42×D7—22no—
23C42×D7—22no—
24C7×#104—22no—
25#6411—22no—
26(C2×D7)⋊C8—22no—
27#1896—71nomaximal, Sylow
28C23×D7D411yescharacteristic
29C7×#41—21no—
30C7×#41—21no—
31C7×#41—21no—
32C7×#41—21no—
33C2×C7×Q8C2×C411yescharacteristic, complemented
34C2×C7×D4—21no—
35C2×C7×D4C2×C411yescharacteristic, complemented
36C22×C4×C7—21no—
37C22×C4×C7—21no—
38C22×C4×C7C2311yescharacteristic
39C2×C22⋊D7—22no—
40C22×Dic7—21no—
41C22×Dic7D411yescharacteristic
42C2×C4×D7—24no—
43C2×C4×D7—28no—
44C2×C4×D7—21no—
45C2×C4×D7—21no—
46C2×C4×D7—21no—
47C2×C4×D7—21no—
48C2×C4×D7C2×C412yes—
49C2×C4×D7D412yes—
50C7×M16—21no—
51C7×M16—21no—
52C2×C7×C8—21no—
53C42×C7—21no—
54C42×C7D412yescomplemented
55C22⋊Dic7—21no—
56C22⋊Dic7—22no—
57(C2×D7)⋊C4—22no—
58C4⋊Dic7C2×C412yescomplemented
59Dic7⋊C4—22no—
60C4×Dic7—24no—
61C4×Dic7—21no—
62C4×Dic7—21no—
63C4×Dic7D412yescomplemented
64#1136—21no—
65#1136—21no—
66C2×C7⋊C8—21no—
67#525—71no—
68C22×C42—71no—
69C2×#104—71no—
70C2×#104D711yescharacteristic, complemented
71#407—71no—
72#339—71no—
73#339—71no—
74C23×C7—21no—
75C23×C7C2×D411yescharacteristic
76C22×D7—21no—
77C22×D7—21no—
78C22×D7—24no—
79C22×D7C22⋊C412yes—
80C7×Q8—21no—
81C7×Q8—21no—
82C7×D4—21no—
83C7×D4—21no—
84C7×D4—41no—
85C7×D4—41no—
86C2×C4×C7—24no—
87C2×C4×C7—22no—
88C2×C4×C7—21no—
89C2×C4×C7—41no—
90C2×C4×C7—41no—
91C2×C4×C7C22×C411yescharacteristic
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2×D411yescharacteristic
94C2×C4×C7C2×D411yescharacteristic
95C2×C4×C7#4111yescharacteristic
96C2×C4×C7#4111yescharacteristic
97C22⋊D7—44no—
98C2×Dic7—24no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7C22⋊C412yes—
104C4×D7—216no—
105C4×D7—22no—
106C4×D7—22no—
107C4×D7—22no—
108C4×D7—22no—
109C7×C8—41no—
110C7⋊C8—41no—
111C2×#41C2×D711yescharacteristic
112C23×C4—71no—
113C23×C4—141no—
114C2×M16—71no—
115C2×M16C2×D711yescharacteristic
116#123—141no—
117C22⋊Q8—72no—
118#121—72no—
119#120—141no—
120C4×D4—142no—
121#117—71no—
122C2×C42—71no—
123C2×C42—142no—
124C2×C42—141no—
125C2×C42—142no—
126C2×C42C2×D711yescharacteristic
127#104—22no—
128#104—142no—
129C22⋊C8—142no—
130C22×C7—22no—
131C22×C7—41no—
132C22×C7—42no—
133C22×C7C2×C22⋊C411yescharacteristic
134C22×C7#12011yescharacteristic
135C22×C7#12311yescharacteristic
136C2×D7—24no—
137C2×D7—24no—
138C2×D7—22no—
139C2×D7—22no—
140C2×D7#10414yescomplemented
141C4×C7—24no—
142C4×C7—41no—
143C4×C7C4×D411yescharacteristic
144C4×C7C4×D411yescharacteristic
145C4×C7#12111yescharacteristic
146C4×C7#12111yescharacteristic
147Dic7—24no—
148Dic7—22no—
149Dic7—42no—
150C24—71no—
151#41—21no—
152#41—21no—
153#41—21no—
154#41—21no—
155C2×Q8C4×D711yescharacteristic, complemented
156C2×D4—21no—
157C2×D4—142no—
158C2×D4C4×D711yescharacteristic, complemented
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—71no—
162C22×C4—72no—
163C22×C4—72no—
164C22×C4—144no—
165C22×C4—148no—
166C22×C4—141no—
167C22×C4—141no—
168C22×C4—141no—
169C22×C4—141no—
170C22×C4—141no—
171C22×C4C22×D711yescharacteristic
172M16—21no—
173M16—21no—
174M16—141no—
175M16—141no—
176C2×C8—21no—
177C2×C8—141no—
178C4⋊C4—72no—
179C4⋊C4—142no—
180C22⋊C4—142no—
181C22⋊C4—142no—
182C22⋊C4—141no—
183C42—21no—
184C42—72no—
185C42—144no—
186C42—141no—
187C42—141no—
188C42C22⋊D712yescomplemented
189C2×C7—21no—
190C2×C7—21no—
191C2×C7—81no—
192C2×C7#38611yescharacteristic
193C2×C7C2×#10412yes—
194D7—24no—
195C23—21no—
196C23—72no—
197C23—144no—
198C23—141no—
199C23—141no—
200C23D4×D711yescharacteristic
201Q8—21no—
202Q8—21no—
203D4—21no—
204D4—21no—
205D4—41no—
206D4—41no—
207D4—284no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—72no—
214C2×C4—1416no—
215C2×C4—144no—
216C2×C4—141no—
217C2×C4—142no—
218C2×C4—141no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4C2×C4×D711yescharacteristic
225C2×C4#115711yescharacteristic
226C2×C4D4×D711yescharacteristic
227C2×C4D4×D711yescharacteristic
228C2×C4#115911yescharacteristic
229C2×C4C2×C22⋊D711yescharacteristic
230C8—41no—
231C8—281no—
232C7#189611yescharacteristic, complemented, Sylow
233C22—22no—
234C22—41no—
235C22—42no—
236C22—74no—
237C22—144no—
238C22—144no—
239C22—142no—
240C22—142no—
241C22D7×C22⋊C411yescharacteristic
242C22#644611yescharacteristic
243C22#650011yescharacteristic
244C4—24no—
245C4—41no—
246C4—144no—
247C4—142no—
248C4—282no—
249C4(C22⋊D7)⋊C411yescharacteristic
250C4#644711yescharacteristic
251C4C4×C22⋊D711yescharacteristic
252C4#650111yescharacteristic
253C2—21no—
254C2—21no—
255C2—81no—
256C2—144no—
257C2D7×#10412yes—
258C2#8972811yescharacteristic
subgroupG/H
centerC2×C4D4×D7
commutatorC2×C4×C7C22×C4
FrattiniC22×C4C22×D7
FittingC2×C7×#104C2
socleC22×C7C2×C22⋊C4
prime radical#90519C2
cosocleC22×C4×C7C23
hypercenterC2×#104D7
hypocenterC7#1896
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×C42×D7C2semidirect
3. C2×C7×#104C2semidirect
4. C2×#6411C2semidirect
5. #89898C2semidirect
6. #89350C2semidirect
7. #89318C2semidirect
13. #6509C4semidirect
14. #6501C4semidirect
33. C2×C7×Q8C2×C4semidirect
35. C2×C7×D4C2×C4semidirect
54. C42×C7D4semidirect
58. C4⋊Dic7C2×C4semidirect
63. C4×Dic7D4semidirect
70. C2×#104D7semidirect
140. C2×D7#104semidirect
155. C2×Q8C4×D7semidirect
158. C2×D4C4×D7semidirect
188. C42C22⋊D7semidirect
232. C7#1896semidirect

Maximal subgroups

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

subgroupindex
#905192
C2×C42×D72
C2×C7×#1042
C2×#64112
#898982
#893502
#893182
#18967