Finite Group Databasebeta

← #31885079

Subgroups — #31885079

Order 18176.

194 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 43 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#22586284C211yesmaximal, characteristic
2#22586221C211yesmaximal, characteristic
3C71×#2061C211yesmaximal, characteristic
4C2×#14556039—21no—
5#17985725—21no—
6#544×C71C411yescharacteristic
7#428×C71—21no—
8#382×C71C2211yescharacteristic
9#343×C71C411yescharacteristic
10#339×C71—21no—
11C22×C71⋊C8—41no—
12C22⋊(C71⋊C8)—41no—
13#14556039—22no—
14#14556039—41no—
15C2×M16×C71—21no—
16C4⋊Q8×C71—41no—
17#125×C71—41no—
18C22⋊Q8×C71—41no—
19#117×C71—21no—
20C2×C4⋊C4×C71C2×C411yescharacteristic
21C2×C4⋊C4×C71D411yescharacteristic
22C2×C42×C71D411yescharacteristic
23#107×C71—41no—
24#106×C71—41no—
25#95×C71—41no—
26C2×C71⋊C8—42no—
27C2×C71⋊C8—42no—
28C2×C71⋊C8—42no—
29C2×C71⋊C8—81no—
30C2×Q8×C71—41no—
31C22×C4×C71—21no—
32C22×C4×C71—21no—
33C22×C4×C71—21no—
34C22×C4×C71C22⋊C411yescharacteristic
35M16×C71—22no—
36M16×C71—41no—
37C2×C8×C71—41no—
38C4⋊C4×C71—21no—
39C4⋊C4×C71—21no—
40C4⋊C4×C71—41no—
41C4⋊C4×C71—41no—
42C4⋊C4×C71—41no—
43C4⋊C4×C71—42no—
44C22⋊C4×C71—41no—
45C42×C71—21no—
46C42×C71—41no—
47C42×C71SD1611yescharacteristic
48C42×C71Q1611yescharacteristic
49C23×C71#9911yescharacteristic
50Q8×C71—161no—
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—81no—
63C2×C4×C71#10311yescharacteristic
64C2×C4×C71#10411yescharacteristic
65C8×C71—42no—
66C71⋊C8—44no—
67C71⋊C8—161no—
68C22×C71—21no—
69C22×C71—21no—
70C22×C71#32811yescharacteristic
71C22×C71AΣL(1,𝔽16)11yescharacteristic
72C22×C71#35611yescharacteristic
73C4×C71—21no—
74C4×C71—21no—
75C4×C71—41no—
76C4×C71—41no—
77C4×C71—81no—
78C4×C71—81no—
79C4×C71—81no—
80C4×C71—81no—
81#7442—711nomaximal, Sylow
82C2×C71—21no—
83C2×C71—21no—
84C2×C71#135411yescharacteristic
85C2×C71#140311yescharacteristic
86C2×C71#140411yescharacteristic
87#2061D7111yescharacteristic
88#1324—711no—
89#1295—711no—
90C71#744211yescharacteristic, complemented, Sylow
91#544Dic7111yescharacteristic
92#428—21no—
93C2×#97—1421no—
94#382C2×D7111yescharacteristic
95#343Dic7111yescharacteristic
96#339—21no—
97#324—1421no—
98C2×M16—21no—
99C22×C8—2841no—
100C4⋊Q8—41no—
101#125—41no—
102C22⋊Q8—41no—
103#117—21no—
104C2×C4⋊C4C2×Dic7111yescharacteristic
105C2×C4⋊C4C22⋊D7111yescharacteristic
106C2×C42C22⋊D7111yescharacteristic
107#107—41no—
108#106—41no—
109C22⋊C8—2841no—
110#97—2841no—
111#97—1422no—
112#95—41no—
113C2×Q8—41no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4C22⋊Dic7111yescharacteristic
118M16—22no—
119M16—41no—
120C2×C8—41no—
121C2×C8—2842no—
122C2×C8—2842no—
123C2×C8—2842no—
124C2×C8—5681no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—42no—
131C22⋊C4—41no—
132C42—21no—
133C42—41no—
134C42#1176585211yescharacteristic
135C42#1176585511yescharacteristic
136C23#1455604311yescharacteristic
137Q8—161no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4#1455605211yescharacteristic
151C2×C4#1455605511yescharacteristic
152C8—42no—
153C8—2844no—
154C8—11361no—
155C22—21no—
156C22—21no—
157C22#1798573611yescharacteristic
158C22#1798578611yescharacteristic
159C22#1798579811yescharacteristic
160C4—21no—
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—81no—
166C4—81no—
167C4—81no—
168C2—21no—
169C2—21no—
170C2#2258637011yescharacteristic
171C2#2258648811yescharacteristic
172C2#2258649111yescharacteristic
subgroupG/H
centerC22#17985786
commutatorC2×C4⋊C4×C71C2×C4
Frattini#382C2×D71
FittingC71×#2061C2
socleC22×C71AΣL(1,𝔽16)
prime radicalC23×C71#99
cosocle#382×C71C22
hypercenter#2061D71
hypocenterC71#7442
LCSG ▷ C2×C4⋊C4×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #382 ◁ #2061

Complements

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

NH ≅ G/Ntype
90. C71#7442semidirect

Maximal subgroups

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

subgroupindex
#225862842
#225862212
C71×#20612
#744271