Finite Group Databasebeta

← #11741412

Subgroups — #11741412

Order 896.

480 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 145 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×C7×#137C211yesmaximal, characteristic
2#41×Dic7C211yesmaximal, characteristic
3C2×#6514C211yesmaximal, characteristic
4C2×#6507C211yesmaximal, characteristic, complemented
5#89997C211yesmaximal, characteristic, complemented
6#89954C214yesmaximal, complemented
7#89934C214yesmaximal, complemented
8#89892C211yesmaximal, characteristic, complemented
9#89890C211yesmaximal, characteristic, complemented
10C2×C7×#41C2211yescharacteristic
11C22×C7×Q8C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C7×#137—24no—
14C2×C7×Q16C2212yes—
15C2×C7×SD16C2212yes—
16C2×C7×M16C2211yescharacteristic
17#6514—24no—
18Q8×Dic7C2212yes—
19#6507—21no—
20#6507—21no—
21#6507C2214yes—
22C2×#1144C2212yes—
23D4×Dic7—22no—
24D4×Dic7C2212yes—
25C2×#1142C2212yes—
26#6489—21no—
27#6489C2211yescharacteristic
28C2×Dic7⋊C4—22no—
29C2×C4×Dic7—21no—
30C2×C4×Dic7C2211yescharacteristic
31C2×#1136C2211yescharacteristic
32#6409C2214yes—
33#6390C2214yes—
34Dic7⋊C8C2214yes—
35#3039—71nomaximal, Sylow
36C7×#41—22no—
37C7×#41D414yes—
38C2×C7×Q8—21no—
39C2×C7×Q8—22no—
40C2×C7×Q8—22no—
41C2×C7×Q8C2312yes—
42C2×C7×Q8C2311yescharacteristic
43C2×C7×D4—21no—
44C2×C7×D4C2311yescharacteristic
45C22×C4×C7—21no—
46C22×C4×C7—21no—
47C22×C4×C7C2311yescharacteristic
48C22×Dic7—21no—
49C22×Dic7—22no—
50C22×Dic7D412yes—
51C2×Dic14—22no—
52C2×Dic14—22no—
53C2×Dic14—21no—
54C2×Dic14C2312yes—
55C7×Q16—24no—
56C7×SD16—24no—
57C7×M16—22no—
58C2×C7×C8C2312yes—
59C22⋊Dic7—21no—
60C22⋊Dic7—22no—
61#1144—24no—
62#1142—24no—
63C4⋊Dic7—22no—
64C4⋊Dic7C2312yes—
65Dic7⋊C4—24no—
66Dic7⋊C4—24no—
67C4×Dic7—21no—
68C4×Dic7—22no—
69C4×Dic7D411yescharacteristic
70C4×Dic7D411yescharacteristic
71C4×Dic7C2312yes—
72#1136—22no—
73C2×C7⋊C8C2312yes—
74C2×#137—71no—
75C2×#137D711yescharacteristic
76C2×C4⋊Q8—71no—
77C4×#41—71no—
78#462—74no—
79#461—74no—
80#423—71no—
81#419—71no—
82#419—71no—
83C23×C7—21no—
84C23×C7C2×D411yescharacteristic
85C7×Q8—21no—
86C7×Q8—22no—
87C7×Q8—22no—
88C7×Q8—42no—
89C7×Q8C2×D412yes—
90C7×D4—24no—
91C7×D4—21no—
92C7×D4C2×D412yes—
93C2×C4×C7—21no—
94C2×C4×C7—24no—
95C2×C4×C7—22no—
96C2×C4×C7—42no—
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2×D412yes—
99C2×C4×C7#4112yes—
100C2×C4×C7C2411yescharacteristic
101C2×Dic7—24no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—42no—
105C2×Dic7—44no—
106C2×Dic7C2×D412yes—
107C2×Dic7C2×D412yes—
108Dic14—22no—
109Dic14—22no—
110Dic14—42no—
111C7×C8—22no—
112C7⋊C8—22no—
113C2×#41C2×D711yescharacteristic
114C22×Q8—71no—
115C22×Q8C2×D711yescharacteristic
116#137—24no—
117#137—144no—
118C2×Q16—72no—
119C2×Q16C2×D712yes—
120C2×SD16—72no—
121C2×SD16C2×D712yes—
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—71no—
131#117—141no—
132C2×C4⋊C4—142no—
133C2×C42—71no—
134C2×C42—141no—
135C4⋊C8—74no—
136#103—74no—
137#103—74no—
138C22×C7—22no—
139C22×C7—22no—
140C22×C7—42no—
141C22×C7#12112yes—
142C22×C7C22×D411yescharacteristic
143C4×C7—22no—
144C4×C7—42no—
145C4×C7#12112yes—
146C4×C7C22×D411yescharacteristic
147C4×C7C2×#4111yescharacteristic
148Dic7—22no—
149Dic7—42no—
150Dic7—42no—
151Dic7#13714yescomplemented
152#41—22no—
153#41C22⋊D714yes—
154C2×Q8—21no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—72no—
158C2×Q8—141no—
159C2×Q8—142no—
160C2×Q8—142no—
161C2×Q8C22×D712yes—
162C2×Q8C22×D711yescharacteristic
163C2×D4—21no—
164C2×D4C22×D711yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—72no—
168C22×C4—142no—
169C22×C4—141no—
170C22×C4C22×D711yescharacteristic
171Q16—24no—
172Q16—144no—
173SD16—24no—
174SD16—144no—
175M16—22no—
176M16—142no—
177C2×C8—72no—
178C2×C8C22×D712yes—
179C4⋊C4—72no—
180C4⋊C4—144no—
181C4⋊C4—144no—
182C4⋊C4—142no—
183C22⋊C4—142no—
184C22⋊C4—141no—
185C42—71no—
186C42—71no—
187C42—72no—
188C42—142no—
189C42—141no—
190C2×C7—22no—
191C2×C7—42no—
192C2×C7C2×#12111yescharacteristic
193C2×C7C2×#13712yes—
194C23—21no—
195C23D4×D711yescharacteristic
196Q8—21no—
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—142no—
201Q8—142no—
202Q8—282no—
203Q8C2×C22⋊D712yes—
204D4—24no—
205D4—21no—
206D4C2×C22⋊D712yes—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—72no—
212C2×C4—72no—
213C2×C4—144no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—284no—
217C2×C4—282no—
218C2×C4D4×D711yescharacteristic
219C2×C4#115912yes—
220C2×C4C2×C22⋊D712yes—
221C2×C4C23×D711yescharacteristic
222C8—22no—
223C8—142no—
224C7#303911yescharacteristic, complemented, Sylow
225C22—22no—
226C22—22no—
227C22—42no—
228C22#650212yes—
229C22C2×D4×D711yescharacteristic
230C4—22no—
231C4—42no—
232C4—74no—
233C4—142no—
234C4—282no—
235C4—282no—
236C4#650212yes—
237C4C2×#115911yescharacteristic
238C4C22×C22⋊D711yescharacteristic
239C2—22no—
240C2—42no—
241C2D7×#13712yes—
242C2C2×#650211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C7×#137C2
socleC22×C7C22×D4
prime radicalC2×C7×#41C22
hypercenterC2×#137D7
hypocenterC7#3039
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
4. C2×#6507C2semidirect
5. #89997C2semidirect
6. #89954C2semidirect
7. #89934C2semidirect
8. #89892C2semidirect
9. #89890C2semidirect
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×C7×#1372
#41×Dic72
C2×#65142
C2×#65072
#899972
#899542
#899342
#898922
#898902
#30397