Finite Group Databasebeta

← #11739007

Subgroups — #11739007

Order 896.

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

normal subgroups only

This group has 400 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#90419C211yesmaximal, characteristic, complemented
2#90407C211yesmaximal, characteristic
3C7×#486C211yesmaximal, characteristic, complemented
4#89865C211yesmaximal, characteristic, complemented
5#89842C211yesmaximal, characteristic, complemented
6#89544C214yesmaximal, complemented
7#89540C214yesmaximal, complemented
8C4×C8×D7C211yesmaximal, characteristic, complemented
9#89453C211yesmaximal, characteristic, complemented
10C2×#1159C2212yes—
11C7×C4⋊Q8C2211yescharacteristic
12C7×#127C2211yescharacteristic
13#6509—21no—
14Q8×Dic7—21no—
15#6501—21no—
16#6501—21no—
17#6499C2212yes—
18D4×Dic7—21no—
19D4×Dic7—21no—
20C2×C8×D7C2212yes—
21#6460—21no—
22#6455C2212yes—
23#6453C2212yes—
24#6450—21no—
25C42×D7C2211yescharacteristic
26#6432C2211yescharacteristic
27#6432C2211yescharacteristic
28C7×#102C2214yes—
29C4×C7×C8C2211yescharacteristic
30#6406C2214yes—
31#6390C2214yes—
32C8×Dic7C2212yes—
33#6384C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3131—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×D4—21no—
38C2×C7×D4—21no—
39C2×C7×D4C2312yes—
40C2×C22⋊D7—22no—
41C22×Dic7—22no—
42#1159—28no—
43C2×C4×D7D411yescharacteristic
44C2×C4×D7D411yescharacteristic
45C2×C4×D7C2311yescharacteristic
46C2×Dic14—21no—
47C2×Dic14C2312yes—
48C2×C7×C8C2312yes—
49C7×C4⋊C4—21no—
50C7×C4⋊C4C2312yes—
51C42×C7C2311yescharacteristic
52C22⋊Dic7—22no—
53C22⋊Dic7—22no—
54(C2×D7)⋊C4—22no—
55C4⋊Dic7—21no—
56C4⋊Dic7—21no—
57C4⋊Dic7—21no—
58C4⋊Dic7—22no—
59C4⋊Dic7C2312yes—
60Dic7⋊C4—22no—
61C4×Dic7—22no—
62C4×Dic7D411yescharacteristic
63C4×Dic7D411yescharacteristic
64C4×Dic7C2311yescharacteristic
65C2×C7⋊C8C2312yes—
66C8×D7—24no—
67#533—71no—
68#532—71no—
69#488—74no—
70#487—71no—
71#487—71no—
72#486—71no—
73#486D711yescharacteristic, complemented
74#416—74no—
75C2×C4×C8—71no—
76C23×C7—22no—
77C22×D7C2×D411yescharacteristic
78C7×Q8—42no—
79C7×D4—24no—
80C7×D4—42no—
81C7×D4—42no—
82C2×C4×C7—22no—
83C2×C4×C7C2×D411yescharacteristic
84C2×C4×C7C2×D411yescharacteristic
85C2×C4×C7C2411yescharacteristic
86C22⋊D7—44no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—44no—
90C2×Dic7C2×D411yescharacteristic
91C2×Dic7C2×D411yescharacteristic
92C2×Dic7C2×D411yescharacteristic
93C4×D7—22no—
94C4×D7—22no—
95C4×D7#4114yes—
96Dic14—24no—
97Dic14—42no—
98C7×C8—22no—
99C7⋊C8—22no—
100C2×#41—72no—
101C22×C8—72no—
102C4⋊Q8—71no—
103C4⋊Q8—71no—
104C4⋊Q8C2×D711yescharacteristic
105#127C2×D711yescharacteristic
106#125—72no—
107#124—72no—
108C22⋊Q8—141no—
109C22⋊Q8—141no—
110#121—141no—
111#121—141no—
112C4×Q8—141no—
113C4×Q8—141no—
114C4×D4—141no—
115C4×D4—141no—
116#117—72no—
117C2×C42—71no—
118#103—74no—
119#103—74no—
120#102—74no—
121#102C2×D714yes—
122C4×C8—72no—
123C4×C8—71no—
124C4×C8C2×D711yescharacteristic
125C22×C7—42no—
126C22×C7—42no—
127C22×C7—42no—
128C22×C7C22×D411yescharacteristic
129C2×D7—21no—
130C2×D7—21no—
131C2×D7#12412yes—
132C4×C7—42no—
133C4×C7#13512yes—
134C4×C7#13512yes—
135C4×C7C2×#4112yes—
136Dic7—21no—
137Dic7—21no—
138Dic7—42no—
139Dic7—42no—
140Dic7#12412yes—
141#41—148no—
142C2×Q8—21no—
143C2×Q8—72no—
144C2×Q8—141no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—142no—
148C2×D4C22×D712yes—
149C22×C4—71no—
150C22×C4—71no—
151C22×C4—71no—
152C22×C4—142no—
153C2×C8—72no—
154C2×C8—144no—
155C2×C8C22×D712yes—
156C4⋊C4—21no—
157C4⋊C4—72no—
158C4⋊C4—142no—
159C4⋊C4—142no—
160C4⋊C4—141no—
161C4⋊C4—141no—
162C4⋊C4—141no—
163C4⋊C4C22×D712yes—
164C22⋊C4—142no—
165C22⋊C4—142no—
166C22⋊C4—142no—
167C42—71no—
168C42—71no—
169C42—71no—
170C42—142no—
171C42C22×D711yescharacteristic
172C2×C7—82no—
173C2×C7C2×#12411yescharacteristic
174C2×C7C2×#13511yescharacteristic
175C2×C7C2×#13511yescharacteristic
176D7—22no—
177C23—22no—
178C23—71no—
179Q8—42no—
180Q8—144no—
181Q8—282no—
182D4—24no—
183D4—42no—
184D4—42no—
185D4—284no—
186C2×C4—22no—
187C2×C4—71no—
188C2×C4—74no—
189C2×C4—71no—
190C2×C4—71no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—284no—
196C2×C4D4×D711yescharacteristic
197C2×C4D4×D711yescharacteristic
198C2×C4C23×D711yescharacteristic
199C8—22no—
200C8—142no—
201C7#313111yescharacteristic, complemented, Sylow
202C22—42no—
203C22—42no—
204C22—42no—
205C22—72no—
206C22—141no—
207C22—141no—
208C22C2×D4×D711yescharacteristic
209C4—42no—
210C4—72no—
211C4—141no—
212C4—141no—
213C4—282no—
214C4—282no—
215C4#647512yes—
216C4#647912yes—
217C4D7×#4112yes—
218C2—82no—
219C2—142no—
220C2D7×#12411yescharacteristic
221C2C2×#647511yescharacteristic
222C2C2×#647911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#486C2
socleC22×C7C22×D4
prime radical#90407C2
hypercenter#486D7
hypocenterC7#3131
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#486C2semidirect
4. #89865C2semidirect
5. #89842C2semidirect
6. #89544C2semidirect
7. #89540C2semidirect
8. C4×C8×D7C2semidirect
9. #89453C2semidirect
73. #486D7semidirect
201. C7#3131semidirect

Maximal subgroups

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

subgroupindex
#904192
#904072
C7×#4862
#898652
#898422
#895442
#895402
C4×C8×D72
#894532
#31317