Finite Group Databasebeta

← C7×#2127 #11730596

Subgroups — C7×#2127 #11730596

Order 896.

286 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 84 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×#535C211yesmaximal, characteristic
2C2×C7×#99C212yesmaximal, complemented
3C7×#354C212yesmaximal, complemented
4C7×#353C212yesmaximal, complemented
5C2×C7×#41C411yescharacteristic, complemented
6C22×C7×D4C2211yescharacteristic
7C7×#127C411yescharacteristic, complemented
8C7×#127C2211yescharacteristic
9C7×#124C411yescharacteristic, complemented
10C7×#124C2211yescharacteristic
11C7×#121—22no—
12C7×#120—22no—
13C7×#117C411yescharacteristic, complemented
14C2×C7×C22⋊C4—22no—
15C7×#99—22no—
16C7×#99C2214yes—
17#2127C711yesmaximal, characteristic, direct product factor, Sylow
18C24×C7D412yes—
19C7×#41—42no—
20C2×C7×Q8C2×C411yescharacteristic, complemented
21C2×C7×D4—21no—
22C2×C7×D4—22no—
23C2×C7×D4—22no—
24C2×C7×D4—42no—
25C2×C7×D4C2×C411yescharacteristic
26C2×C7×D4C2×C411yescharacteristic, complemented
27C2×C7×D4D412yes—
28C2×C7×D4C2311yescharacteristic
29C22×C4×C7—21no—
30C22×C4×C7—42no—
31C22×C4×C7C2×C411yescharacteristic
32C7×C4⋊C4—21no—
33C7×C22⋊C4—22no—
34C7×C22⋊C4—21no—
35C7×C22⋊C4—22no—
36C7×C22⋊C4—24no—
37C7×C22⋊C4—42no—
38C42×C7C2×C411yescharacteristic, complemented
39C42×C7C2×C411yescharacteristic, complemented
40#535C2×C711yescharacteristic
41C2×#99C2×C712yescomplemented
42#354C2×C712yescomplemented
43#353C2×C712yescomplemented
44C23×C7—22no—
45C23×C7—22no—
46C23×C7—21no—
47C23×C7—82no—
48C23×C7C22⋊C412yes—
49C23×C7C22⋊C411yescharacteristic
50C23×C7C2×D412yes—
51C7×Q8—41no—
52C7×D4—41no—
53C7×D4—42no—
54C7×D4—41no—
55C7×D4—41no—
56C7×D4—42no—
57C7×D4—82no—
58C7×D4—82no—
59C2×C4×C7—22no—
60C2×C4×C7—21no—
61C2×C4×C7—21no—
62C2×C4×C7—21no—
63C2×C4×C7—42no—
64C2×C4×C7—44no—
65C2×C4×C7—84no—
66C2×C4×C7C22⋊C411yescharacteristic
67C2×C4×C7C22×C411yescharacteristic
68C2×#41C4×C711yescharacteristic, complemented
69C22×D4C22×C711yescharacteristic
70#127C4×C711yescharacteristic, complemented
71#127C22×C711yescharacteristic
72#124C4×C711yescharacteristic, complemented
73#124C22×C711yescharacteristic
74#121—22no—
75#120—22no—
76#117C4×C711yescharacteristic, complemented
77C2×C22⋊C4—22no—
78#99—22no—
79#99C22×C714yes—
80C22×C7—22no—
81C22×C7—22no—
82C22×C7—41no—
83C22×C7—42no—
84C22×C7—42no—
85C22×C7—44no—
86C22×C7—42no—
87C22×C7—41no—
88C22×C7—82no—
89C22×C7#9912yes—
90C22×C7C2×C22⋊C411yescharacteristic
91C4×C7—21no—
92C4×C7—21no—
93C4×C7—41no—
94C4×C7—41no—
95C4×C7—41no—
96C4×C7—84no—
97C24C7×D412yes—
98#41—42no—
99C2×Q8C2×C4×C711yescharacteristic, complemented
100C2×D4—21no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—42no—
104C2×D4C2×C4×C711yescharacteristic
105C2×D4C2×C4×C711yescharacteristic, complemented
106C2×D4C7×D412yes—
107C2×D4C23×C711yescharacteristic
108C22×C4—21no—
109C22×C4—42no—
110C22×C4C2×C4×C711yescharacteristic
111C4⋊C4—21no—
112C22⋊C4—21no—
113C22⋊C4—22no—
114C22⋊C4—22no—
115C22⋊C4—24no—
116C22⋊C4—42no—
117C42C2×C4×C711yescharacteristic, complemented
118C42C2×C4×C711yescharacteristic, complemented
119C2×C7—21no—
120C2×C7—22no—
121C2×C7—42no—
122C2×C7—42no—
123C2×C7—81no—
124C2×C7C2×#9911yescharacteristic
125C23—22no—
126C23—22no—
127C23—21no—
128C23—82no—
129C23C7×C22⋊C412yes—
130C23C7×C22⋊C411yescharacteristic
131C23C2×C7×D412yes—
132Q8—41no—
133D4—41no—
134D4—42no—
135D4—41no—
136D4—41no—
137D4—42no—
138D4—82no—
139D4—82no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4—84no—
147C2×C4C7×C22⋊C411yescharacteristic
148C2×C4C22×C4×C711yescharacteristic
149C7#212711yescharacteristic, direct product factor, Sylow
150C22—22no—
151C22—22no—
152C22—41no—
153C22—42no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—41no—
158C22—82no—
159C22C7×#9912yes—
160C22C2×C7×C22⋊C411yescharacteristic
161C4—21no—
162C4—21no—
163C4—41no—
164C4—41no—
165C4—41no—
166C4—84no—
167C2—21no—
168C2—22no—
169C2—42no—
170C2—42no—
171C2—81no—
172C2C2×C7×#9911yescharacteristic
subgroupG/H
center, socleC2×C7C2×#99
commutatorC2×C4C22×C4×C7
Frattini, cosocleC2×D4C23×C7
prime radicalC7×#535C2
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#99C2semidirect
3. C7×#354C2semidirect
4. C7×#353C2semidirect
5. C2×C7×#41C4semidirect
7. C7×#127C4semidirect
9. C7×#124C4semidirect
13. C7×#117C4semidirect
17. #2127C7direct (N × H)
20. C2×C7×Q8C2×C4semidirect
26. C2×C7×D4C2×C4semidirect
38. C42×C7C2×C4semidirect
39. C42×C7C2×C4semidirect
41. C2×#99C2×C7semidirect
42. #354C2×C7semidirect
43. #353C2×C7semidirect
68. C2×#41C4×C7semidirect
70. #127C4×C7semidirect
72. #124C4×C7semidirect
76. #117C4×C7semidirect
99. C2×Q8C2×C4×C7semidirect
105. C2×D4C2×C4×C7semidirect
117. C42C2×C4×C7semidirect
118. C42C2×C4×C7semidirect
149. C7#2127direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5352
C2×C7×#992
C7×#3542
C7×#3532
#21277