Finite Group Databasebeta

← #11736765

Subgroups — #11736765

Order 896.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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#90161C211yesmaximal, characteristic
2C2×#6433C211yesmaximal, characteristic
3C7×#432C211yesmaximal, characteristic
4#89601C214yesmaximal, complemented
5#89494C212yesmaximal
6#89493C212yesmaximal
7#89475C212yesmaximal, complemented
8#89474C212yesmaximal, complemented
9C2×C42×C7C2211yescharacteristic
10#6489—21no—
11C2×C4⋊Dic7C2211yescharacteristic
12C2×C4⋊Dic7C2211yescharacteristic
13#6487—21no—
14#6487C2212yes—
15C2×Dic7⋊C4—21no—
16C2×Dic7⋊C4—21no—
17#6433—21no—
18#6433—21no—
19#6433—21no—
20#6433C2212yes—
21#6433C2212yes—
22#6432C2211yescharacteristic
23#6432C2211yescharacteristic
24C4×Dic14—21no—
25C4×Dic14C2212yes—
26C7×C4⋊C8C2212yes—
27C7×C22⋊C8C2212yes—
28C7×#97C2212yes—
29#6392C2214yes—
30#6391C2214yes—
31#6390C2214yes—
32#6390C2214yes—
33#3236—71nomaximal, Sylow
34C22×C4×C7D411yescharacteristic
35C22×C4×C7D411yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C22×Dic7—21no—
38C22×Dic7—21no—
39C2×Dic14—21no—
40C2×Dic14C2312yes—
41C2×C7×C8C2314yes—
42C42×C7D411yescharacteristic
43C42×C7D411yescharacteristic
44C42×C7C2311yescharacteristic
45C42×C7C2311yescharacteristic
46C22⋊Dic7—22no—
47C4⋊Dic7—21no—
48C4⋊Dic7—21no—
49C4⋊Dic7—21no—
50C4⋊Dic7—21no—
51C4⋊Dic7—21no—
52C4⋊Dic7C2312yes—
53C4⋊Dic7C2312yes—
54C4⋊Dic7C2312yes—
55Dic7⋊C4—22no—
56Dic7⋊C4—22no—
57Dic7⋊C4—22no—
58Dic7⋊C4—22no—
59Dic7⋊C4—22no—
60Dic7⋊C4—22no—
61C4×Dic7—22no—
62#533—71no—
63C2×#125—71no—
64#501—72no—
65#491—72no—
66#484—72no—
67#483—72no—
68#479—74no—
69#432D711yescharacteristic
70C23×C7C2×D411yescharacteristic
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—41no—
74C2×C4×C7C2×D411yescharacteristic
75C2×C4×C7C2×D411yescharacteristic
76C2×C4×C7C2×D411yescharacteristic
77C2×C4×C7C2×D411yescharacteristic
78C2×C4×C7C2×D411yescharacteristic
79C2×C4×C7C2411yescharacteristic
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—42no—
85C2×Dic7—42no—
86Dic14—24no—
87Dic14—42no—
88C7×C8—24no—
89C4⋊Q8—71no—
90C4⋊Q8—71no—
91#125—72no—
92#125—72no—
93#125—141no—
94#125—141no—
95#125—141no—
96C22⋊Q8—72no—
97C22⋊Q8—141no—
98C4×Q8—72no—
99C4×Q8—141no—
100#117—141no—
101C2×C4⋊C4—71no—
102C2×C4⋊C4—71no—
103C2×C4⋊C4—141no—
104C2×C4⋊C4—141no—
105C2×C42C2×D711yescharacteristic
106#107—74no—
107#106—74no—
108C4⋊C8C2×D712yes—
109#103—74no—
110#103—74no—
111C22⋊C8C2×D712yes—
112#97C2×D712yes—
113C22×C7—21no—
114C22×C7—21no—
115C22×C7#13712yes—
116C22×C7C22×D411yescharacteristic
117C4×C7—21no—
118C4×C7—21no—
119C4×C7—21no—
120C4×C7—21no—
121C4×C7—21no—
122C4×C72−1+412yes—
123Dic7—42no—
124Dic7—42no—
125Dic7—42no—
126Dic7—42no—
127C2×Q8—72no—
128C2×Q8—141no—
129C22×C4—141no—
130C22×C4—141no—
131C22×C4D2811yescharacteristic
132C22×C4D2811yescharacteristic
133C22×C4C22×D711yescharacteristic
134C2×C8C22×D714yes—
135C4⋊C4—72no—
136C4⋊C4—72no—
137C4⋊C4—72no—
138C4⋊C4—142no—
139C4⋊C4—142no—
140C4⋊C4—142no—
141C4⋊C4—142no—
142C4⋊C4—142no—
143C4⋊C4—142no—
144C4⋊C4—141no—
145C4⋊C4—141no—
146C4⋊C4—141no—
147C4⋊C4—141no—
148C4⋊C4—141no—
149C22⋊C4—142no—
150C42—142no—
151C42D2811yescharacteristic
152C42D2811yescharacteristic
153C42C22×D711yescharacteristic
154C42C22×D711yescharacteristic
155C2×C7—22no—
156C2×C7#53611yescharacteristic
157C2×C7C2×#13711yescharacteristic
158C2×C7#57511yescharacteristic
159C23C2×D2811yescharacteristic
160Q8—144no—
161Q8—282no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—142no—
166C2×C4—142no—
167C2×C4—142no—
168C2×C4—142no—
169C2×C4—282no—
170C2×C4—282no—
171C2×C4C2×D2811yescharacteristic
172C2×C4C2×D2811yescharacteristic
173C2×C4C2×D2811yescharacteristic
174C2×C4C2×D2811yescharacteristic
175C2×C4C2×D2811yescharacteristic
176C2×C4C23×D711yescharacteristic
177C8—24no—
178C7#323611yescharacteristic, complemented, Sylow
179C22—21no—
180C22—21no—
181C22#647212yes—
182C22C22×D2811yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—282no—
189C4—282no—
190C4—282no—
191C4—282no—
192C4#655412yes—
193C2—22no—
194C2#9021711yescharacteristic
195C2C2×#647211yescharacteristic
196C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#432C2
socleC22×C7C22×D4
prime radicalC23×C7C2×D4
hypercenter#432D7
hypocenterC7#3236
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
4. #89601C2semidirect
7. #89475C2semidirect
8. #89474C2semidirect
178. C7#3236semidirect

Maximal subgroups

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

subgroupindex
#901612
C2×#64332
C7×#4322
#896012
#894942
#894932
#894752
#894742
#32367