Finite Group Databasebeta

← #31885039

Subgroups — #31885039

Order 18176.

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

normal subgroups only

This group has 196 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#22589491C211yesmaximal, characteristic
2#22586274C211yesmaximal, characteristic
3C71×#1394C211yesmaximal, characteristic
4C2×#14556120C411yescharacteristic
5C2×#14556052—21no—
6#17985761—21no—
7#17985754C411yescharacteristic
8#17985725—21no—
9C2×#101×C71—21no—
10#391×C71C2211yescharacteristic
11C22×C71⋊C8—21no—
12#14556120—42no—
13C2×C4⋊Dic71—21no—
14C2×Dic71⋊C4—41no—
15#14556052—42no—
16C22⋊(C71⋊C8)—41no—
17C22×Q8×C71D411yescharacteristic
18C2×M16×C71—41no—
19C2×C4⋊C4×C71D411yescharacteristic
20C2×C42×C71C2×C411yescharacteristic
21#101×C71—22no—
22#101×C71—41no—
23#95×C71—41no—
24C22×Dic71—41no—
25C2×C71⋊C8—41no—
26C2×C71⋊C8—41no—
27C2×C71⋊C8—41no—
28C2×C71⋊C8—81no—
29C4⋊Dic71—22no—
30C4⋊Dic71—41no—
31Dic71⋊C4—42no—
32Dic71⋊C4—42no—
33C2×Q8×C71—21no—
34C2×Q8×C71—41no—
35C2×Q8×C71—41no—
36C2×Q8×C71SD1611yescharacteristic
37C2×Q8×C71Q1611yescharacteristic
38C22×C4×C71—21no—
39C22×C4×C71—21no—
40C22×C4×C71—21no—
41C22×C4×C71C22⋊C411yescharacteristic
42M16×C71—42no—
43M16×C71—81no—
44C2×C8×C71—81no—
45C4⋊C4×C71—41no—
46C42×C71—41no—
47C23×C71#9911yescharacteristic
48Q8×C71—41no—
49Q8×C71—41no—
50Q8×C71—81no—
51C2×C4×C71—21no—
52C2×C4×C71—21no—
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—41no—
61C2×C4×C71—81no—
62C2×C4×C71#10311yescharacteristic
63C2×C4×C71#10411yescharacteristic
64C2×Dic71—42no—
65C2×Dic71—81no—
66C2×Dic71—81no—
67C8×C71—82no—
68C71⋊C8—81no—
69C71⋊C8—161no—
70C22×C71—21no—
71C22×C71—21no—
72C22×C71#32811yescharacteristic
73C22×C71AΣL(1,𝔽16)11yescharacteristic
74C22×C71#35411yescharacteristic
75C4×C71—21no—
76C4×C71—21no—
77C4×C71—41no—
78C4×C71—41no—
79C4×C71—81no—
80C4×C71—81no—
81Dic71—82no—
82#7429—711nomaximal, Sylow
83C2×C71—21no—
84C2×C71—21no—
85C2×C71#135111yescharacteristic
86C2×C71#141311yescharacteristic
87C2×C71#141411yescharacteristic
88#2026—711no—
89#1394D7111yescharacteristic
90#1319—711no—
91C71#742911yescharacteristic, complemented, Sylow
92C2×#125—711no—
93C2×#103—1421no—
94C2×#101—21no—
95#391C2×D7111yescharacteristic
96#340—1421no—
97#336—711no—
98#324—1421no—
99C22×Q8C22⋊D7111yescharacteristic
100C2×M16—41no—
101C22×C8—1421no—
102#125—2842no—
103C2×C4⋊C4—2841no—
104C2×C4⋊C4—1421no—
105C2×C4⋊C4D28411yescharacteristic
106C2×C42C4×D7111yescharacteristic
107#103—2842no—
108#101—22no—
109#101—41no—
110C22⋊C8—2841no—
111#95—41no—
112C2×Q8—21no—
113C2×Q8—41no—
114C2×Q8—41no—
115C2×Q8#1176585311yescharacteristic
116C2×Q8#1176585511yescharacteristic
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—2841no—
121C22×C4(C2×D71)⋊C411yescharacteristic
122M16—42no—
123M16—81no—
124C2×C8—81no—
125C2×C8—2841no—
126C2×C8—2841no—
127C2×C8—2841no—
128C2×C8—5681no—
129C4⋊C4—41no—
130C4⋊C4—2841no—
131C4⋊C4—2842no—
132C4⋊C4—2842no—
133C4⋊C4—1422no—
134C42—41no—
135C23#1455604211yescharacteristic
136Q8—41no—
137Q8—41no—
138Q8—81no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—81no—
150C2×C4—2842no—
151C2×C4—5681no—
152C2×C4—5681no—
153C2×C4#1455605111yescharacteristic
154C2×C4#1455605611yescharacteristic
155C8—82no—
156C8—5681no—
157C8—11361no—
158C22—21no—
159C22—21no—
160C22#1798573511yescharacteristic
161C22#1798578511yescharacteristic
162C22#1798579311yescharacteristic
163C4—21no—
164C4—21no—
165C4—41no—
166C4—41no—
167C4—81no—
168C4—81no—
169C4—5682no—
170C2—21no—
171C2—21no—
172C2#2258636011yescharacteristic
173C2#2258651211yescharacteristic
174C2#2258651611yescharacteristic
subgroupG/H
centerC22#17985793
commutatorC2×C42×C71C2×C4
Frattini#391C2×D71
FittingC71×#1394C2
socleC22×C71#354
prime radicalC23×C71#99
cosocle#391×C71C22
hypercenter#1394D71
hypocenterC71#7429
LCSG ▷ C2×C42×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1394

Complements

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

NH ≅ G/Ntype
91. C71#7429semidirect

Maximal subgroups

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

subgroupindex
#225894912
#225862742
C71×#13942
#742971