Finite Group Databasebeta

← #11739047

Subgroups — #11739047

Order 896.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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#90416C211yesmaximal, characteristic
2D7×C4⋊Q8C211yesmaximal, characteristic, complemented
3C7×#487C211yesmaximal, characteristic, complemented
4#89865C211yesmaximal, characteristic, complemented
5#89857C211yesmaximal, characteristic, complemented
6#89590C214yesmaximal, complemented
7#89578C214yesmaximal, complemented
8#89465C211yesmaximal, characteristic, complemented
9#89457C211yesmaximal, characteristic, complemented
10C2×#1161—21no—
11C2×Q8×D7C2212yes—
12C7×C4⋊Q8C2211yescharacteristic
13C7×C4⋊Q8C2211yescharacteristic
14#6510—21no—
15#6507C2212yes—
16C2×#1143C2214yes—
17C2×#1132C2214yes—
18#6458—21no—
19#6458C2212yes—
20#6456—21no—
21#6456C2212yes—
22D7×C4⋊C4—21no—
23D7×C4⋊C4—21no—
24#6451—21no—
25#6451—21no—
26#6437C2211yescharacteristic
27C42×D7C2211yescharacteristic
28#6432C2211yescharacteristic
29C7×#103C2214yes—
30C4×C7×C8C2211yescharacteristic
31(C2×D7)⋊C8C2212yes—
32Dic7⋊C8C2212yes—
33#6384C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3206—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8C2312yes—
38#1161—44no—
39Q8×D7—44no—
40C2×D28—22no—
41C2×D28—21no—
42C2×D28—21no—
43C2×D28C2312yes—
44C2×C4×D7—22no—
45C2×C4×D7—22no—
46C2×C4×D7D411yescharacteristic
47C2×C4×D7D411yescharacteristic
48C2×C4×D7C2311yescharacteristic
49C2×Dic14—22no—
50C2×Dic14C2312yes—
51C2×C7×C8C2312yes—
52C7×C4⋊C4—21no—
53C7×C4⋊C4—21no—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2312yes—
56C42×C7C2311yescharacteristic
57#1143—44no—
58(C2×D7)⋊C4—22no—
59(C2×D7)⋊C4—22no—
60C4⋊Dic7—21no—
61C4⋊Dic7—21no—
62Dic7⋊C4—22no—
63Dic7⋊C4—22no—
64C4×Dic7D411yescharacteristic
65C4×Dic7D411yescharacteristic
66C4×Dic7C2311yescharacteristic
67C2×C7⋊C8C2312yes—
68#1132—44no—
69#532—71no—
70C2×C4⋊Q8—71no—
71#492—71no—
72#492—71no—
73#487—71no—
74#487D711yescharacteristic, complemented
75#461—74no—
76#448—74no—
77#431—71no—
78C22×D7—22no—
79C22×D7C2×D411yescharacteristic
80C7×Q8—24no—
81C7×Q8—42no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7C2×D411yescharacteristic
85C2×C4×C7C2×D411yescharacteristic
86C2×C4×C7C2411yescharacteristic
87C2×Dic7—22no—
88C2×Dic7C2×D411yescharacteristic
89C2×Dic7C2×D411yescharacteristic
90C2×Dic7C2×D411yescharacteristic
91D28—24no—
92D28—44no—
93D28—42no—
94D28—42no—
95C4×D7—22no—
96C4×D7—22no—
97C4×D7—44no—
98C4×D7—44no—
99C4×D7SD1614yescomplemented
100Dic14—24no—
101Dic14—44no—
102C7×C8—22no—
103C7⋊C8—22no—
104C2×#41—141no—
105C22×Q8—72no—
106C2×SD16—74no—
107C2×SD16—74no—
108C4⋊Q8—72no—
109C4⋊Q8—71no—
110C4⋊Q8—141no—
111C4⋊Q8—141no—
112C4⋊Q8C2×D711yescharacteristic
113C4⋊Q8C2×D711yescharacteristic
114#127—71no—
115#124—141no—
116#121—72no—
117#121—141no—
118C4×D4—72no—
119C4×D4—141no—
120C2×C4⋊C4—141no—
121C2×C4⋊C4—141no—
122C2×C42—71no—
123C4⋊C8—72no—
124#103—74no—
125#103C2×D714yes—
126C22⋊C8—72no—
127C4×C8—71no—
128C4×C8C2×D711yescharacteristic
129C22×C7C22×D411yescharacteristic
130C2×D7—21no—
131C2×D7—21no—
132C2×D7—42no—
133C2×D7—42no—
134C2×D7—42no—
135C2×D7C2×SD1612yes—
136C4×C7—42no—
137C4×C7—42no—
138C4×C7C2×SD1612yes—
139C4×C7#13712yes—
140C4×C7H(5,𝔽2)12yes—
141Dic7—21no—
142Dic7—21no—
143Dic7—42no—
144Dic7C2×SD1612yes—
145#41—284no—
146C2×Q8—21no—
147C2×Q8—72no—
148C2×Q8—142no—
149C2×Q8—284no—
150C2×Q8C22×D712yes—
151C2×D4—72no—
152C2×D4—142no—
153C2×D4—141no—
154C2×D4—141no—
155C22×C4—71no—
156C22×C4—71no—
157C22×C4—71no—
158C22×C4—142no—
159C22×C4—142no—
160SD16—284no—
161SD16—284no—
162C2×C8—72no—
163C2×C8C22×D712yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—142no—
168C4⋊C4—142no—
169C4⋊C4—141no—
170C4⋊C4—141no—
171C4⋊C4C22×D712yes—
172C22⋊C4—142no—
173C22⋊C4—142no—
174C42—71no—
175C42—71no—
176C42—71no—
177C42C22×D711yescharacteristic
178C2×C7#53511yescharacteristic
179C2×C7C22×SD1611yescharacteristic
180C2×C7C2×#13711yescharacteristic
181D7—22no—
182D7—82no—
183C23—71no—
184C23—142no—
185Q8—24no—
186Q8—42no—
187Q8—144no—
188Q8—284no—
189D4—144no—
190D4—284no—
191D4—282no—
192D4—282no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—71no—
196C2×C4—74no—
197C2×C4—71no—
198C2×C4—71no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—284no—
203C2×C4—284no—
204C2×C4D4×D711yescharacteristic
205C2×C4D4×D711yescharacteristic
206C2×C4C23×D711yescharacteristic
207C8—22no—
208C8—142no—
209C7#320611yescharacteristic, complemented, Sylow
210C22—72no—
211C22—141no—
212C22—141no—
213C22—282no—
214C22—282no—
215C22—282no—
216C22C2×D4×D711yescharacteristic
217C4—42no—
218C4—42no—
219C4—72no—
220C4—141no—
221C4—141no—
222C4—282no—
223C4D7×SD1612yes—
224C4#648112yes—
225C4#655312yes—
226C2—142no—
227C2—562no—
228C2#9036511yescharacteristic
229C2C2×D7×SD1611yescharacteristic
230C2C2×#648111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#487D7
hypocenterC7#3206
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. D7×C4⋊Q8C2semidirect
3. C7×#487C2semidirect
4. #89865C2semidirect
5. #89857C2semidirect
6. #89590C2semidirect
7. #89578C2semidirect
8. #89465C2semidirect
9. #89457C2semidirect
74. #487D7semidirect
99. C4×D7SD16semidirect
209. C7#3206semidirect

Maximal subgroups

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

subgroupindex
#904162
D7×C4⋊Q82
C7×#4872
#898652
#898572
#895902
#895782
#894652
#894572
#32067