Finite Group Databasebeta

← #31908649

Subgroups — #31908649

Order 18176.

274 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 79 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#22590152C212yesmaximal
2#22586375C214yesmaximal, complemented
3C71×#3139C211yesmaximal, characteristic
4#17985820—22no—
5#17985781—22no—
6#17985777C2214yes—
7#533×C71C411yescharacteristic
8C2×C4⋊Q8×C71C411yescharacteristic
9#491×C71C412yes—
10#491×C71C2212yes—
11#487×C71—22no—
12#419×C71—22no—
13C4×M16×C71C2211yescharacteristic
14#14556068—24no—
15C22×Q8×C71—21no—
16C2×M16×C71D412yes—
17C4⋊Q8×C71—21no—
18C4⋊Q8×C71—21no—
19C4⋊Q8×C71—41no—
20C4⋊Q8×C71C2×C412yes—
21#125×C71C2×C412yes—
22C22⋊Q8×C71—41no—
23C4×Q8×C71—41no—
24#117×C71—21no—
25C2×C4⋊C4×C71—41no—
26C2×C42×C71C2×C411yescharacteristic
27#103×C71—24no—
28#103×C71—24no—
29#97×C71C2×C411yescharacteristic
30#97×C71C2311yescharacteristic
31C4×C8×C71D412yes—
32C71⋊C16—44no—
33C2×Q8×C71—22no—
34C2×Q8×C71—41no—
35C2×Q8×C71—44no—
36C2×Q8×C71—41no—
37C22×C4×C71—21no—
38C22×C4×C71—41no—
39C22×C4×C71C22⋊C411yescharacteristic
40M16×C71—42no—
41C2×C8×C71C22⋊C412yes—
42C2×C8×C71C2×D412yes—
43C4⋊C4×C71—22no—
44C4⋊C4×C71—42no—
45C4⋊C4×C71—42no—
46C4⋊C4×C71—42no—
47C4⋊C4×C71—41no—
48C22⋊C4×C71—41no—
49C42×C71—21no—
50C42×C71—41no—
51C42×C71C22⋊C411yescharacteristic
52C42×C71C22×C411yescharacteristic
53C23×C71#10011yescharacteristic
54Q8×C71—44no—
55Q8×C71—82no—
56Q8×C71—82no—
57C2×C4×C71—21no—
58C2×C4×C71—22no—
59C2×C4×C71—21no—
60C2×C4×C71—42no—
61C2×C4×C71—42no—
62C2×C4×C71—42no—
63C2×C4×C71—82no—
64C2×C4×C71#10011yescharacteristic
65C2×C4×C71C2×C22⋊C411yescharacteristic
66C8×C71—22no—
67C8×C71—22no—
68C22×C71—21no—
69C22×C71—41no—
70C22×C71C2×#10011yescharacteristic
71C4×C71—22no—
72C4×C71—21no—
73C4×C71—41no—
74C4×C71—82no—
75C4×C71—82no—
76C4×C71#42112yes—
77#11916—711nomaximal, Sylow
78C2×C71—21no—
79C2×C71—41no—
80C2×C71#153311yescharacteristic
81#3139D7111yescharacteristic
82#2165—712no—
83#1356—714no—
84C71#1191611yescharacteristic, complemented, Sylow
85#533Dic7111yescharacteristic
86C2×C4⋊Q8Dic7111yescharacteristic
87#491Dic7112yes—
88#491C2×D7112yes—
89#487—22no—
90#419—22no—
91C4×M16C2×D7111yescharacteristic
92#364—1422no—
93#349—1422no—
94#347—714no—
95C22×Q8—21no—
96C2×M16C22⋊D7112yes—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—41no—
100C4⋊Q8C2×Dic7112yes—
101#125C2×Dic7112yes—
102C22⋊Q8—41no—
103C4×Q8—41no—
104#117—21no—
105C2×C4⋊C4—41no—
106C2×C42C2×Dic7111yescharacteristic
107M32—1424no—
108#103—24no—
109#103—24no—
110#97C2×Dic7111yescharacteristic
111#97C22×D7111yescharacteristic
112C4×C8C22⋊D7112yes—
113C2×Q8—22no—
114C2×Q8—41no—
115C2×Q8—44no—
116C2×Q8—41no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4C22⋊Dic7111yescharacteristic
120M16—42no—
121C2×C8C22⋊Dic7112yes—
122C2×C8C2×C22⋊D7112yes—
123C4⋊C4—22no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—41no—
128C22⋊C4—41no—
129C42—21no—
130C42—41no—
131C42C22⋊Dic7111yescharacteristic
132C42C22×Dic7111yescharacteristic
133C16—2844no—
134C23#1455604511yescharacteristic
135Q8—44no—
136Q8—82no—
137Q8—82no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—82no—
145C2×C4#1455604511yescharacteristic
146C2×C4C2×C22⋊Dic7111yescharacteristic
147C8—22no—
148C8—22no—
149C22—21no—
150C22—41no—
151C22C2×#1455604511yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—82no—
156C4—82no—
157C4#1798601612yes—
158C2—21no—
159C2—41no—
160C2#2258696411yescharacteristic
subgroupG/H
centerC2#22586964
commutatorC42×C71C22×C4
Frattini#97C22×D71
FittingC71×#3139C2
socleC2×C71#1533
prime radicalC23×C71#100
cosocle#97×C71C23
hypercenter#3139D71
hypocenterC71#11916
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #22586375C2semidirect
84. C71#11916semidirect

Maximal subgroups

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

subgroupindex
#225901522
#225863752
C71×#31392
#1191671