Finite Group Databasebeta

← #11726913

Subgroups — #11726913

Order 896.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#90366C211yesmaximal, characteristic
2#89512C212yesmaximal
3C7×#354C211yesmaximal, characteristic, complemented
4#89336C211yesmaximal, characteristic, complemented
5#89268C212yesmaximal, complemented
6C2×Q8×D7C411yescharacteristic, complemented
7C2×#1159C2211yescharacteristic
8C7×#124C2211yescharacteristic
9#6507C411yescharacteristic, complemented
10#6499C2211yescharacteristic
11#6446—22no—
12#6441—22no—
13#6440—22no—
14#6435C411yescharacteristic, complemented
15#6432C411yescharacteristic, complemented
16C7×#99C2212yes—
17#6408C2212yes—
18#6380—22no—
19#2129—71nomaximal, Sylow
20C2×C7×Q8C2×C411yescharacteristic
21C2×C7×D4C2311yescharacteristic
22C2×C22⋊D7D412yes—
23C22×Dic7D412yes—
24Q8×D7—42no—
25#1159—42no—
26C2×C4×D7—21no—
27C2×C4×D7C2×C411yescharacteristic
28C2×Dic14—21no—
29C2×Dic14C2×C411yescharacteristic
30C2×Dic14C2×C411yescharacteristic
31C7×C22⋊C4—22no—
32C7×C22⋊C4—22no—
33C42×C7C2×C411yescharacteristic
34C22⋊Dic7—22no—
35C22⋊Dic7—22no—
36(C2×D7)⋊C4—21no—
37C4⋊Dic7—22no—
38Dic7⋊C4—21no—
39Dic7⋊C4—22no—
40Dic7⋊C4—42no—
41C4×Dic7—42no—
42C4×Dic7C2×C411yescharacteristic
43#536—71no—
44#410—72no—
45#354—71no—
46#354—72no—
47#354D711yescharacteristic, complemented
48C23×C7C2×D412yes—
49C22×D7C22⋊C411yescharacteristic
50C7×Q8—41no—
51C7×D4—41no—
52C2×C4×C7—21no—
53C2×C4×C7—21no—
54C2×C4×C7—42no—
55C2×C4×C7C22×C411yescharacteristic
56C22⋊D7—42no—
57C2×Dic7—22no—
58C2×Dic7—22no—
59C2×Dic7—21no—
60C2×Dic7—21no—
61C2×Dic7—42no—
62C2×Dic7C22⋊C412yes—
63C2×Dic7C22⋊C411yescharacteristic
64C4×D7—22no—
65C4×D7—42no—
66Dic14—41no—
67Dic14—42no—
68Dic14—41no—
69C2×#41—71no—
70C22×Q8—71no—
71C4⋊Q8—71no—
72C4⋊Q8—71no—
73#124—71no—
74#124C2×D711yescharacteristic
75#123—142no—
76C22⋊Q8—142no—
77#117—71no—
78#117—142no—
79#99—72no—
80#99—142no—
81#99C2×D712yes—
82C22×C7—22no—
83C22×C7—42no—
84C22×C7C2×C22⋊C411yescharacteristic
85C2×D7—21no—
86C2×D7—41no—
87C4×C7—21no—
88C4×C7—41no—
89C4×C7—41no—
90C4×C7—82no—
91Dic7—22no—
92Dic7—41no—
93Dic7—41no—
94Dic7—82no—
95Dic7#9912yes—
96#41—282no—
97C2×Q8—71no—
98C2×Q8—71no—
99C2×Q8—141no—
100C2×Q8—282no—
101C2×Q8C4×D711yescharacteristic
102C2×D4—72no—
103C2×D4C22×D711yescharacteristic
104C22×C4—72no—
105C22×C4—71no—
106C22×C4—141no—
107C4⋊C4—141no—
108C4⋊C4—142no—
109C4⋊C4—142no—
110C4⋊C4—282no—
111C22⋊C4—22no—
112C22⋊C4—22no—
113C22⋊C4—141no—
114C22⋊C4—142no—
115C22⋊C4—142no—
116C42—71no—
117C42—282no—
118C42C4×D711yescharacteristic
119C2×C7—21no—
120C2×C7—42no—
121C2×C7C2×#9911yescharacteristic
122D7—41no—
123C23—71no—
124C23D4×D712yes—
125Q8—41no—
126Q8—281no—
127Q8—281no—
128Q8—282no—
129D4—41no—
130D4—282no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—71no—
135C2×C4—72no—
136C2×C4—142no—
137C2×C4—142no—
138C2×C4—142no—
139C2×C4—141no—
140C2×C4—141no—
141C2×C4—282no—
142C2×C4—282no—
143C2×C4C2×C4×D711yescharacteristic
144C7#212911yescharacteristic, complemented, Sylow
145C22—22no—
146C22—42no—
147C22—141no—
148C22—281no—
149C22D7×C22⋊C411yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—72no—
154C4—82no—
155C4—142no—
156C4—281no—
157C4—281no—
158C4—562no—
159C2—21no—
160C2—42no—
161C2—281no—
162C2D7×#9911yescharacteristic
subgroupG/H
centerC2D7×#99
commutatorC2×C4×C7C22×C4
FrattiniC2×D4C22×D7
FittingC7×#354C2
socleC2×C7C2×#99
prime radicalC2×#1159C22
cosocleC2×C7×D4C23
hypercenter#354D7
hypocenterC7#2129
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. C7×#354C2semidirect
4. #89336C2semidirect
5. #89268C2semidirect
6. C2×Q8×D7C4semidirect
9. #6507C4semidirect
14. #6435C4semidirect
15. #6432C4semidirect
47. #354D7semidirect
144. C7#2129semidirect

Maximal subgroups

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

subgroupindex
#903662
#895122
C7×#3542
#893362
#892682
#21297