Finite Group Databasebeta

← C71×#7525 #31828397

Subgroups — C71×#7525 #31828397

Order 18176.

138 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 30 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
1C71×#1662C211yesmaximal, characteristic
2C71×#1413C211yesmaximal, characteristic, complemented
3C71×#1356C211yesmaximal, characteristic, complemented
4#562×C71C411yescharacteristic
5#491×C71C2211yescharacteristic
6#489×C71—21no—
7#356×C71—21no—
8#347×C71—21no—
9#330×C71C411yescharacteristic
10#329×C71—21no—
11C4⋊Q8×C71D411yescharacteristic
12#125×C71C2×C411yescharacteristic
13#124×C71—21no—
14#123×C71—41no—
15C22⋊Q8×C71—41no—
16#121×C71—41no—
17M32×C71—41no—
18#103×C71—41no—
19#103×C71—41no—
20#102×C71—41no—
21#101×C71—41no—
22#97×C71—21no—
23#97×C71D411yescharacteristic
24C2×Q8×C71—21no—
25C2×Q8×C71—21no—
26C2×D4×C71—21no—
27C2×D4×C71—41no—
28C22×C4×C71—41no—
29M16×C71—81no—
30C2×C8×C71—21no—
31C2×C8×C71—41no—
32C4⋊C4×C71—21no—
33C4⋊C4×C71—41no—
34C4⋊C4×C71—41no—
35C4⋊C4×C71—41no—
36C22⋊C4×C71—41no—
37C22⋊C4×C71—41no—
38C22⋊C4×C71—41no—
39C42×C71C22⋊C411yescharacteristic
40C16×C71—42no—
41C23×C71—41no—
42Q8×C71—81no—
43Q8×C71—81no—
44D4×C71—81no—
45D4×C71—161no—
46C2×C4×C71—21no—
47C2×C4×C71—41no—
48C2×C4×C71—41no—
49C2×C4×C71—41no—
50C2×C4×C71—161no—
51C2×C4×C71#9911yescharacteristic
52C8×C71—41no—
53C8×C71—81no—
54C8×C71—161no—
55C22×C71—81no—
56C22×C71—81no—
57C22×C71—81no—
58C22×C71AΣL(1,𝔽16)11yescharacteristic
59C4×C71—21no—
60C4×C71—41no—
61C4×C71—81no—
62C4×C71—81no—
63C4×C71—81no—
64#7525C7111yesmaximal, characteristic, direct product factor, Sylow
65C2×C71—21no—
66C2×C71—161no—
67C2×C71#140411yescharacteristic
68#1662C2×C7111yescharacteristic
69#1413C2×C7111yescharacteristic, complemented
70#1356C2×C7111yescharacteristic, complemented
71C71#752511yescharacteristic, direct product factor, Sylow
72#562C4×C7111yescharacteristic
73#491C22×C7111yescharacteristic
74#489—21no—
75#356—21no—
76#347—21no—
77#330C4×C7111yescharacteristic
78#329—21no—
79C4⋊Q8D4×C7111yescharacteristic
80#125C2×C4×C7111yescharacteristic
81#124—21no—
82#123—41no—
83C22⋊Q8—41no—
84#121—41no—
85M32—41no—
86#103—41no—
87#103—41no—
88#102—41no—
89#101—41no—
90#97—21no—
91#97D4×C7111yescharacteristic
92C2×Q8—21no—
93C2×Q8—21no—
94C2×D4—21no—
95C2×D4—41no—
96C22×C4—41no—
97M16—81no—
98C2×C8—21no—
99C2×C8—41no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C42C22⋊C4×C7111yescharacteristic
108C16—42no—
109C23—41no—
110Q8—81no—
111Q8—81no—
112D4—81no—
113D4—161no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—161no—
119C2×C4#99×C7111yescharacteristic
120C8—41no—
121C8—81no—
122C8—161no—
123C22—81no—
124C22—81no—
125C22—81no—
126C22AΣL(1,𝔽16)×C7111yescharacteristic
127C4—21no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—81no—
132C2—21no—
133C2—161no—
134C2C71×#140411yescharacteristic
subgroupG/H
center, socleC2×C71#1404
commutator#125C2×C4×C71
Frattini, cosocle#491C22×C71
prime radical#562×C71C4
LCSG ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C22×C71 ◁ C2×C4×C71 ◁ C42×C71 ◁ #491×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#1413C2semidirect
3. C71×#1356C2semidirect
64. #7525C71direct (N × H)
69. #1413C2×C71semidirect
70. #1356C2×C71semidirect
71. C71#7525direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#16622
C71×#14132
C71×#13562
#752571