Finite Group Databasebeta

← #11737304

Subgroups — #11737304

Order 896.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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#90252C212yesmaximal, complemented
2C7×#442C211yesmaximal, characteristic, complemented
3#89931C211yesmaximal, characteristic
4#89927C211yesmaximal, characteristic, complemented
5C4×#1142C212yesmaximal, complemented
6#89640C211yesmaximal, characteristic, complemented
7#89627C211yesmaximal, characteristic, complemented
8#89550C212yesmaximal, complemented
9#89537C212yesmaximal, complemented
10#89488C211yesmaximal, characteristic, complemented
11#89480C211yesmaximal, characteristic, complemented
12C2×C7×D8C2211yescharacteristic
13C4×C7×D4C2212yes—
14#6501C2212yes—
15#6499C2212yes—
16C2×#1142C2212yes—
17#6492—22no—
18#6487—22no—
19C2×Dic28C2211yescharacteristic
20C2×#1131C2211yescharacteristic
21C2×C8×D7C2211yescharacteristic
22#6460C2212yes—
23#6451C2212yes—
24#6448—22no—
25#6442—22no—
26#6435C2211yescharacteristic
27C4×Dic14C2212yes—
28C7×#106C2211yescharacteristic
29C7×#102C2212yes—
30C7×#97C2211yescharacteristic
31#6406C2212yes—
32#6391C2211yescharacteristic
33#6390C2212yes—
34#6389C2211yescharacteristic
35C8×Dic7C2211yescharacteristic
36#6384C2212yes—
37#6382C2212yes—
38C4×C7⋊C8C2211yescharacteristic
39#3173—71nomaximal, Sylow
40C2×C7×D4C2312yes—
41C22×C4×C7—22no—
42C2×C22⋊D7—22no—
43C2×C4×D7C2311yescharacteristic
44C2×Dic14—22no—
45C2×Dic14C2312yes—
46C7×D8—22no—
47C2×C7×C8C2311yescharacteristic
48C2×C7×C8C2311yescharacteristic
49C7×C4⋊C4C2312yes—
50C7×C22⋊C4—22no—
51C42×C7C2311yescharacteristic
52C22⋊Dic7—22no—
53C22⋊Dic7—22no—
54C22⋊Dic7—22no—
55#1142—24no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4D412yes—
58C4⋊Dic7—22no—
59C4⋊Dic7C2312yes—
60Dic7⋊C4—22no—
61Dic7⋊C4—22no—
62Dic7⋊C4D412yes—
63C4×Dic7—22no—
64C4×Dic7C2311yescharacteristic
65C2×C7⋊C8C2311yescharacteristic
66C2×C7⋊C8C2311yescharacteristic
67Dic28—22no—
68#1131—22no—
69C8×D7—22no—
70#542—72no—
71#498—71no—
72#495—71no—
73#489—72no—
74#470—71no—
75#469—71no—
76#465—72no—
77#442D711yescharacteristic, complemented
78#441—71no—
79C4×SD16—72no—
80#405—71no—
81C23×C7—22no—
82C22×D7C2×D411yescharacteristic
83C7×D4—22no—
84C7×D4—22no—
85C2×C4×C7—22no—
86C2×C4×C7—82no—
87C2×C4×C7C2×D412yes—
88C2×C4×C7C2411yescharacteristic
89C22⋊D7—82no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7C2×D412yes—
93C2×Dic7C2×D411yescharacteristic
94C4×D7—21no—
95C4×D7—21no—
96Dic14—22no—
97Dic14—22no—
98Dic14—82no—
99C7×C8—21no—
100C7×C8#4112yes—
101C7⋊C8—21no—
102C7⋊C8#4112yes—
103C2×Q16—71no—
104C2×SD16—72no—
105C2×D8C2×D711yescharacteristic
106C2×M16—71no—
107C22×C8—71no—
108C4⋊Q8—72no—
109#126—142no—
110#124—72no—
111#124—142no—
112#123—142no—
113C22⋊Q8—72no—
114C22⋊Q8—142no—
115#121—72no—
116C4×Q8—72no—
117C4×D4C2×D712yes—
118#117—71no—
119#106—71no—
120#106—72no—
121#106C2×D711yescharacteristic
122#103—72no—
123#103—72no—
124#102—72no—
125#102C2×D712yes—
126#97—71no—
127#97C2×D711yescharacteristic
128C4×C8—71no—
129C4×C8—71no—
130C22×C7—42no—
131C22×C7—42no—
132C22×C7—42no—
133C22×C7C22×D411yescharacteristic
134C2×D7—21no—
135C2×D7—22no—
136C4×C7—22no—
137C4×C7—42no—
138C4×C7C2×#4111yescharacteristic
139C4×C7C2×#4111yescharacteristic
140Dic7—21no—
141Dic7—22no—
142Dic7—42no—
143Dic7—42no—
144C2×Q8—72no—
145C2×Q8—142no—
146C2×D4—142no—
147C2×D4C22×D712yes—
148C22×C4—22no—
149C22×C4—71no—
150Q16—142no—
151SD16—144no—
152D8—22no—
153M16—142no—
154C2×C8—71no—
155C2×C8—71no—
156C2×C8—142no—
157C2×C8C22×D711yescharacteristic
158C2×C8C22×D711yescharacteristic
159C4⋊C4—72no—
160C4⋊C4—72no—
161C4⋊C4—142no—
162C4⋊C4—142no—
163C4⋊C4—142no—
164C4⋊C4C22×D712yes—
165C22⋊C4—22no—
166C22⋊C4—72no—
167C22⋊C4—142no—
168C22⋊C4—142no—
169C22⋊C4—142no—
170C22⋊C4—142no—
171C42—71no—
172C42—142no—
173C42C22×D711yescharacteristic
174C2×C7—82no—
175C2×C7#53211yescharacteristic
176C2×C7#57712yes—
177D7—41no—
178C23—22no—
179C23—71no—
180Q8—142no—
181Q8—142no—
182Q8—562no—
183D4—22no—
184D4—22no—
185D4—562no—
186C2×C4—22no—
187C2×C4—72no—
188C2×C4—71no—
189C2×C4—82no—
190C2×C4—141no—
191C2×C4—141no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4D4×D712yes—
195C2×C4C23×D711yescharacteristic
196C8—21no—
197C8—72no—
198C8—141no—
199C8#115712yes—
200C7#317311yescharacteristic, complemented, Sylow
201C22—42no—
202C22—42no—
203C22—42no—
204C22—142no—
205C22—141no—
206C22C2×D4×D711yescharacteristic
207C4—22no—
208C4—42no—
209C4—142no—
210C4—141no—
211C4—282no—
212C4—282no—
213C4C2×#115711yescharacteristic
214C4D7×#4111yescharacteristic
215C2—82no—
216C2—281no—
217C2#9023911yescharacteristic
218C2#9046612yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#442C2
socleC22×C7C22×D4
prime radical#89931C2
hypercenter#442D7
hypocenterC7#3173
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #90252C2semidirect
2. C7×#442C2semidirect
4. #89927C2semidirect
5. C4×#1142C2semidirect
6. #89640C2semidirect
7. #89627C2semidirect
8. #89550C2semidirect
9. #89537C2semidirect
10. #89488C2semidirect
11. #89480C2semidirect
77. #442D7semidirect
200. C7#3173semidirect

Maximal subgroups

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

subgroupindex
#902522
C7×#4422
#899312
#899272
C4×#11422
#896402
#896272
#895502
#895372
#894882
#894802
#31737