Finite Group Databasebeta

← #11736419

Subgroups — #11736419

Order 896.

480 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 137 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×#6507C211yesmaximal, characteristic, complemented
2C2×#6505C211yesmaximal, characteristic
3C2×#6472C211yesmaximal, characteristic
4#90213C211yesmaximal, characteristic
5C7×#419C211yesmaximal, characteristic, complemented
6#89890C211yesmaximal, characteristic, complemented
7#89776C211yesmaximal, characteristic, complemented
8#89590C214yesmaximal, complemented
9#89565C214yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C2×#1157C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×M16C2211yescharacteristic
14C7×#117C2211yescharacteristic
15#6507—21no—
16#6507—21no—
17#6507C2214yes—
18C2×#1144C2212yes—
19#6505—24no—
20C2×#1143C2212yes—
21C2×Dic7⋊C4—22no—
22C2×C4×Dic7C2211yescharacteristic
23C2×#1136C2211yescharacteristic
24#6472—24no—
25C2×Dic28C2212yes—
26C2×#1132C2212yes—
27#6456C2212yes—
28#6450C2212yes—
29(C22⋊D7)⋊C4—22no—
30#6435—21no—
31C42×D7—21no—
32C7×#103C2214yes—
33Dic7⋊C8C2214yes—
34#6384C2214yes—
35#3039—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8—22no—
38C2×C7×Q8—22no—
39C2×C7×Q8C2312yes—
40C22×C4×C7—21no—
41C22×C4×C7C2311yescharacteristic
42C2×C22⋊D7—21no—
43C22×Dic7—21no—
44C22×Dic7D412yes—
45#1157—22no—
46#1157D414yes—
47C2×D28C2311yescharacteristic
48C2×C4×D7—21no—
49C2×C4×D7—22no—
50C2×Dic14—22no—
51C2×Dic14—22no—
52C2×Dic14—21no—
53C2×Dic14C2312yes—
54C2×Dic14C2311yescharacteristic
55C7×M16—22no—
56C2×C7×C8C2312yes—
57C7×C4⋊C4C2312yes—
58C7×C22⋊C4—21no—
59C42×C7—21no—
60#1144—24no—
61#1143—24no—
62(C2×D7)⋊C4—22no—
63Dic7⋊C4—22no—
64Dic7⋊C4—24no—
65Dic7⋊C4—24no—
66C4×Dic7—22no—
67C4×Dic7D411yescharacteristic
68C4×Dic7D411yescharacteristic
69C4×Dic7C2312yes—
70#1136—22no—
71C2×C7⋊C8C2312yes—
72Dic28—24no—
73#1132—24no—
74C2×#137—71no—
75C2×#137—71no—
76C2×C4⋊Q8—71no—
77C4×#41—71no—
78#462—74no—
79#461—74no—
80#423—71no—
81#419—71no—
82#419D711yescharacteristic, complemented
83C23×C7C2×D411yescharacteristic
84C22×D7—21no—
85C7×Q8—22no—
86C7×Q8—22no—
87C7×Q8—42no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—42no—
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2×D412yes—
93C2×C4×C7#4112yes—
94C2×C4×C7C2411yescharacteristic
95C22⋊D7—24no—
96C2×Dic7—21no—
97C2×Dic7—24no—
98C2×Dic7—22no—
99C2×Dic7—42no—
100C2×Dic7C2×D412yes—
101C2×Dic7C2×D412yes—
102D28—21no—
103D28C2×D412yes—
104C4×D7—24no—
105C4×D7—44no—
106Dic14—21no—
107Dic14—22no—
108Dic14—22no—
109Dic14—42no—
110Dic14C2×D412yes—
111C7×C8—22no—
112C7⋊C8—22no—
113C2×#41—71no—
114C22×Q8—71no—
115C22×Q8C2×D711yescharacteristic
116#137—144no—
117#137—144no—
118C2×Q16—72no—
119C2×Q16—72no—
120C2×SD16—72no—
121C2×SD16—72no—
122C2×M16—71no—
123C2×M16C2×D711yescharacteristic
124C4⋊Q8—74no—
125C4⋊Q8—141no—
126C4⋊Q8—141no—
127C4×Q8—72no—
128C4×D4—72no—
129C4×D4—142no—
130#117—141no—
131#117C2×D711yescharacteristic
132C2×C4⋊C4—142no—
133C2×C42—71no—
134C2×C42—141no—
135C4⋊C8—74no—
136#103—74no—
137#103C2×D714yes—
138C22×C7—22no—
139C22×C7#12112yes—
140C22×C7C22×D411yescharacteristic
141C2×D7—22no—
142C2×D7—42no—
143C4×C7—42no—
144C4×C7—42no—
145C4×C7#12112yes—
146C4×C7C22×D411yescharacteristic
147C4×C7C2×#4111yescharacteristic
148Dic7—22no—
149Dic7—22no—
150Dic7—42no—
151Dic7#13714yescomplemented
152#41—74no—
153#41—142no—
154C2×Q8—21no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—71no—
158C2×Q8—72no—
159C2×Q8—141no—
160C2×Q8—142no—
161C2×Q8—142no—
162C2×Q8C22×D712yes—
163C2×D4—71no—
164C2×D4—141no—
165C22×C4—21no—
166C22×C4—72no—
167C22×C4—142no—
168C22×C4—141no—
169C22×C4—141no—
170C22×C4C22×D711yescharacteristic
171Q16—144no—
172Q16—144no—
173SD16—144no—
174SD16—144no—
175M16—22no—
176M16—142no—
177C2×C8—72no—
178C2×C8C22×D712yes—
179C4⋊C4—144no—
180C4⋊C4—144no—
181C4⋊C4—142no—
182C4⋊C4C22×D712yes—
183C22⋊C4—21no—
184C22⋊C4—142no—
185C42—21no—
186C42—71no—
187C42—71no—
188C42—72no—
189C42—142no—
190C2×C7—22no—
191C2×C7C2×#12111yescharacteristic
192C2×C7C2×#13712yes—
193D7—42no—
194C23—141no—
195C23D4×D711yescharacteristic
196Q8—22no—
197Q8—22no—
198Q8—42no—
199Q8—72no—
200Q8—141no—
201Q8—142no—
202Q8—142no—
203Q8—282no—
204D4—72no—
205D4—144no—
206D4—141no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—72no—
211C2×C4—72no—
212C2×C4—144no—
213C2×C4—141no—
214C2×C4—144no—
215C2×C4—142no—
216C2×C4—284no—
217C2×C4—282no—
218C2×C4#115712yes—
219C2×C4D4×D711yescharacteristic
220C2×C4D4×D712yes—
221C2×C4C23×D711yescharacteristic
222C8—22no—
223C8—142no—
224C7#303911yescharacteristic, complemented, Sylow
225C22—22no—
226C22—142no—
227C22—282no—
228C22#644712yes—
229C22C2×D4×D711yescharacteristic
230C4—42no—
231C4—42no—
232C4—74no—
233C4—142no—
234C4—142no—
235C4—282no—
236C4#644712yes—
237C4C2×#115711yescharacteristic
238C4C2×D4×D711yescharacteristic
239C2—22no—
240C2—282no—
241C2C2×#644711yescharacteristic
242C2D7×#13712yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#419C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenter#419D7
hypocenterC7#3039
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#6507C2semidirect
5. C7×#419C2semidirect
6. #89890C2semidirect
7. #89776C2semidirect
8. #89590C2semidirect
9. #89565C2semidirect
82. #419D7semidirect
151. Dic7#137semidirect
224. C7#3039semidirect

Maximal subgroups

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

subgroupindex
C2×#65072
C2×#65052
C2×#64722
#902132
C7×#4192
#898902
#897762
#895902
#895652
#30397