Finite Group Databasebeta

← #31885065

Subgroups — #31885065

Order 18176.

194 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 41 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#22589883C211yesmaximal, characteristic
2#22586467C211yesmaximal, characteristic
3C71×#1319C211yesmaximal, characteristic
4C2×#14556059—21no—
5C2×#14556052—21no—
6C2×#14556047—21no—
7#17985869C411yescharacteristic
8#17985754C411yescharacteristic
9#391×C71C2211yescharacteristic
10#324×C71—21no—
11C2×#11765849—41no—
12C22×C71⋊C8—21no—
13C2×C4⋊Dic71—21no—
14C2×C4⋊Dic71—41no—
15#14556059—42no—
16#14556052—42no—
17#14556047—22no—
18#14556047—41no—
19C22×Q8×C71D411yescharacteristic
20C2×C4⋊C4×C71D411yescharacteristic
21C2×C42×C71C2×C411yescharacteristic
22C22⋊C8×C71—41no—
23#95×C71—41no—
24C22×Dic71—41no—
25#11765849—42no—
26#11765849—81no—
27C2×C71⋊C8—22no—
28C2×C71⋊C8—41no—
29C2×C71⋊C8—41no—
30C2×C71⋊C8—81no—
31C4⋊Dic71—41no—
32C4⋊Dic71—41no—
33C4⋊Dic71—82no—
34C2×Q8×C71—21no—
35C2×Q8×C71—41no—
36C2×Q8×C71—41no—
37C2×Q8×C71SD1611yescharacteristic
38C2×Q8×C71Q1611yescharacteristic
39C22×C4×C71—21no—
40C22×C4×C71—21no—
41C22×C4×C71—21no—
42C22×C4×C71C22⋊C411yescharacteristic
43C2×C8×C71—81no—
44C4⋊C4×C71—41no—
45C42×C71—41no—
46C23×C71#9911yescharacteristic
47Q8×C71—41no—
48Q8×C71—41no—
49Q8×C71—81no—
50C2×C4×C71—21no—
51C2×C4×C71—21no—
52C2×C4×C71—22no—
53C2×C4×C71—21no—
54C2×C4×C71—21no—
55C2×C4×C71—41no—
56C2×C4×C71—41no—
57C2×C4×C71—41no—
58C2×C4×C71—41no—
59C2×C4×C71—41no—
60C2×C4×C71—81no—
61C2×C4×C71#10311yescharacteristic
62C2×C4×C71#10411yescharacteristic
63C2×Dic71—82no—
64C2×Dic71—81no—
65C8×C71—161no—
66C71⋊C8—42no—
67C71⋊C8—82no—
68C22×C71—21no—
69C22×C71—21no—
70C22×C71#32811yescharacteristic
71C22×C71#35211yescharacteristic
72C22×C71#35311yescharacteristic
73C4×C71—21no—
74C4×C71—21no—
75C4×C71—42no—
76C4×C71—41no—
77C4×C71—41no—
78C4×C71—81no—
79Dic71—161no—
80#7437—711nomaximal, Sylow
81C2×C71—21no—
82C2×C71—21no—
83C2×C71#135211yescharacteristic
84C2×C71#140711yescharacteristic
85C2×C71#141011yescharacteristic
86#2096—711no—
87#1394—711no—
88#1319D7111yescharacteristic
89C71#743711yescharacteristic, complemented, Sylow
90C2×#106—1421no—
91C2×#103—1421no—
92C2×#101—1421no—
93#391C2×D7111yescharacteristic
94#384—711no—
95#336—711no—
96#324—21no—
97C22×Q8D28411yescharacteristic
98C2×M16—2841no—
99C22×C8—1421no—
100C2×C4⋊C4—2841no—
101C2×C4⋊C4—1421no—
102C2×C4⋊C4C22⋊D7111yescharacteristic
103C2×C42C4×D7111yescharacteristic
104#106—2842no—
105#103—2842no—
106#101—2841no—
107#101—1422no—
108C22⋊C8—41no—
109#95—41no—
110C2×Q8—21no—
111C2×Q8—41no—
112C2×Q8—41no—
113C2×Q8#1176585411yescharacteristic
114C2×Q8Dic28411yescharacteristic
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—2841no—
119C22×C4(C2×D71)⋊C411yescharacteristic
120M16—2842no—
121M16—5681no—
122C2×C8—81no—
123C2×C8—2841no—
124C2×C8—2841no—
125C2×C8—5681no—
126C2×C8—1422no—
127C4⋊C4—41no—
128C4⋊C4—2841no—
129C4⋊C4—2841no—
130C4⋊C4—5682no—
131C42—41no—
132C23#1455604211yescharacteristic
133Q8—41no—
134Q8—41no—
135Q8—81no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—81no—
147C2×C4—5682no—
148C2×C4—5681no—
149C2×C4#1455605311yescharacteristic
150C2×C4#1455605411yescharacteristic
151C8—161no—
152C8—2842no—
153C8—5682no—
154C22—21no—
155C22—21no—
156C22#1798573711yescharacteristic
157C22#1798579011yescharacteristic
158C22#1798579111yescharacteristic
159C4—21no—
160C4—21no—
161C4—42no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—11361no—
166C2—21no—
167C2—21no—
168C2#2258636511yescharacteristic
169C2#2258649911yescharacteristic
170C2#2258650611yescharacteristic
subgroupG/H
centerC22#17985791
commutatorC2×C42×C71C2×C4
Frattini#391C2×D71
FittingC71×#1319C2
socleC22×C71#353
prime radicalC23×C71#99
cosocle#391×C71C22
hypercenter#1319D71
hypocenterC71#7437
LCSG ▷ C2×C42×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1319

Complements

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

NH ≅ G/Ntype
89. C71#7437semidirect

Maximal subgroups

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

subgroupindex
#225898832
#225864672
C71×#13192
#743771