Finite Group Databasebeta

← #11740739

Subgroups — #11740739

Order 896.

402 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C7×#532C211yesmaximal, characteristic
2#90161C211yesmaximal, characteristic, complemented
3#89865C211yesmaximal, characteristic, complemented
4#89850C211yesmaximal, characteristic, complemented
5#89848C211yesmaximal, characteristic, complemented
6#89842C211yesmaximal, characteristic, complemented
7#89814C214yesmaximal, complemented
8#89790C214yesmaximal, complemented
9C4×#1136C211yesmaximal, characteristic, complemented
10C2×C7×#41—21no—
11C7×C4⋊Q8C2211yescharacteristic
12C7×#127C2211yescharacteristic
13C7×#124—21no—
14C7×#121—21no—
15C7×#121C2212yes—
16C4×C7×D4—21no—
17C4×C7×D4C2212yes—
18C2×C42×C7C2211yescharacteristic
19C2×#1142C2214yes—
20#6489—21no—
21#6487—21no—
22#6487C2212yes—
23C2×#1136C2212yes—
24#6433—21no—
25#6432C2211yescharacteristic
26#6432C2211yescharacteristic
27C4×Dic14—21no—
28C4×Dic14C2212yes—
29#6406C2214yes—
30#6384C2214yes—
31#6381C2214yes—
32#6377C2212yes—
33C4×C7⋊C8C2211yescharacteristic
34C4×C7⋊C8C2211yescharacteristic
35#3182—71nomaximal, Sylow
36C7×#41—44no—
37C2×C7×Q8—21no—
38C2×C7×D4—22no—
39C2×C7×D4—21no—
40C2×C7×D4—21no—
41C2×C7×D4C2312yes—
42C22×C4×C7—22no—
43C22×C4×C7D411yescharacteristic
44C22×C4×C7D411yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C2×Dic14—21no—
47C2×Dic14C2312yes—
48C7×C4⋊C4—21no—
49C7×C4⋊C4C2312yes—
50C7×C22⋊C4—22no—
51C7×C22⋊C4—22no—
52C42×C7D411yescharacteristic
53C42×C7D411yescharacteristic
54C42×C7C2311yescharacteristic
55C42×C7C2311yescharacteristic
56C22⋊Dic7—22no—
57#1142—28no—
58C4⋊Dic7—21no—
59C4⋊Dic7—21no—
60C4⋊Dic7—21no—
61C4⋊Dic7C2312yes—
62Dic7⋊C4—22no—
63Dic7⋊C4—22no—
64C4×Dic7—22no—
65#1136—24no—
66C2×C7⋊C8C2312yes—
67C2×C7⋊C8C2312yes—
68#533—71no—
69#532D711yescharacteristic
70#500—71no—
71#492—71no—
72#487—71no—
73#486—71no—
74#468—74no—
75#440—74no—
76C4×M16—71no—
77C23×C7—22no—
78C23×C7C2×D411yescharacteristic
79C7×Q8—42no—
80C7×D4—24no—
81C7×D4—44no—
82C7×D4—42no—
83C7×D4—42no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—44no—
89C2×C4×C7C2×D411yescharacteristic
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2×D411yescharacteristic
94C2×C4×C7C2411yescharacteristic
95C2×Dic7—22no—
96C2×Dic7—22no—
97Dic14—24no—
98Dic14—42no—
99C7⋊C8—22no—
100C7⋊C8#4114yescomplemented
101C2×#41—21no—
102C2×SD16—74no—
103C2×M16—72no—
104C4⋊Q8—71no—
105C4⋊Q8—71no—
106C4⋊Q8C2×D711yescharacteristic
107#127C2×D711yescharacteristic
108#125—141no—
109#124—21no—
110C22⋊Q8—72no—
111C22⋊Q8—141no—
112#121—21no—
113#121C2×D712yes—
114C4×Q8—72no—
115C4×Q8—141no—
116C4×D4—21no—
117C4×D4C2×D712yes—
118#117—141no—
119C2×C42C2×D711yescharacteristic
120#107—74no—
121#103—74no—
122#102—74no—
123#97—72no—
124C4×C8—71no—
125C4×C8—71no—
126C22×C7—21no—
127C22×C7—21no—
128C22×C7—21no—
129C22×C7—42no—
130C22×C7—42no—
131C22×C7—42no—
132C22×C7C22×D411yescharacteristic
133C4×C7—21no—
134C4×C7—21no—
135C4×C7—21no—
136C4×C7—42no—
137C4×C7AGL(1,ℤ/8)12yes—
138C4×C7#13712yes—
139C4×C7C2×#4112yes—
140Dic7—42no—
141Dic7—42no—
142#41—44no—
143C2×Q8—21no—
144C2×Q8—72no—
145C2×Q8—141no—
146C2×D4—22no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C22×D712yes—
150C22×C4—22no—
151C22×C4C22⋊D711yescharacteristic
152C22×C4C22⋊D711yescharacteristic
153C22×C4C22×D711yescharacteristic
154SD16—148no—
155M16—144no—
156C2×C8—72no—
157C2×C8—72no—
158C4⋊C4—21no—
159C4⋊C4—72no—
160C4⋊C4—142no—
161C4⋊C4—142no—
162C4⋊C4—141no—
163C4⋊C4—141no—
164C4⋊C4—141no—
165C4⋊C4C22×D712yes—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—142no—
169C42—142no—
170C42C22⋊D711yescharacteristic
171C42C22⋊D711yescharacteristic
172C42C22×D711yescharacteristic
173C42C22×D711yescharacteristic
174C2×C7—41no—
175C2×C7—82no—
176C2×C7#53211yescharacteristic
177C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
178C2×C7C2×#13711yescharacteristic
179C23—22no—
180C23C2×C22⋊D711yescharacteristic
181Q8—42no—
182Q8—144no—
183Q8—282no—
184D4—24no—
185D4—44no—
186D4—42no—
187D4—42no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4C2×C22⋊D711yescharacteristic
196C2×C4C2×C22⋊D711yescharacteristic
197C2×C4C2×C22⋊D711yescharacteristic
198C2×C4C2×C22⋊D711yescharacteristic
199C2×C4C2×C22⋊D711yescharacteristic
200C2×C4C23×D711yescharacteristic
201C8—74no—
202C8—142no—
203C7#318211yescharacteristic, complemented, Sylow
204C22—21no—
205C22—21no—
206C22—21no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C22×C22⋊D711yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—282no—
216C4—282no—
217C4#649512yes—
218C4#651412yes—
219C4D7×#4112yes—
220C2—41no—
221C2—82no—
222C2C2×#649511yescharacteristic
223C2C2×#651411yescharacteristic
224C2#9052311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
Fitting, prime radicalC7×#532C2
socleC22×C7C22×D4
hypercenter#532D7
hypocenterC7#3182
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
2. #90161C2semidirect
3. #89865C2semidirect
4. #89850C2semidirect
5. #89848C2semidirect
6. #89842C2semidirect
7. #89814C2semidirect
8. #89790C2semidirect
9. C4×#1136C2semidirect
100. C7⋊C8#41semidirect
203. C7#3182semidirect

Maximal subgroups

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

subgroupindex
C7×#5322
#901612
#898652
#898502
#898482
#898422
#898142
#897902
C4×#11362
#31827