Finite Group Databasebeta

← #31890964

Subgroups — #31890964

Order 18176.

538 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 99 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#22588414C212yesmaximal
2C71⋊#1304C214yesmaximal
3C71×#2312C211yesmaximal, characteristic
4#17985855C2214yes—
5#17985766—24no—
6C22×C22⋊C4×C71C2211yescharacteristic
7#388×C71—24no—
8#380×C71C412yescomplemented
9#379×C71C2212yes—
10C4×C22⋊C4×C71C412yescomplemented
11C2×C22⋊Dic71—24no—
12C2×C22⋊Dic71—24no—
13C2×C22⋊Dic71—24no—
14C25×C71C2311yescharacteristic
15C23×C4×C71C2×C411yescharacteristic
16C23×C4×C71C2×C411yescharacteristic
17C2×C4⋊C4×C71—22no—
18C2×C22⋊C4×C71—24no—
19C2×C22⋊C4×C71—22no—
20C2×C22⋊C4×C71—22no—
21C2×C22⋊C4×C71C2×C414yes—
22C2×C22⋊C4×C71D414yes—
23C2×C42×C71—22no—
24#95×C71—22no—
25#95×C71—22no—
26C22×Dic71—44no—
27C22⋊Dic71—48no—
28C22⋊Dic71—84no—
29C22⋊Dic71—84no—
30C24×C71—42no—
31C24×C71—81no—
32C24×C71—81no—
33C24×C71C22×C411yescharacteristic
34C24×C71C2×D412yes—
35C24×C71#4114yes—
36C22×C4×C71—22no—
37C22×C4×C71—22no—
38C22×C4×C71—24no—
39C22×C4×C71—81no—
40C22×C4×C71—81no—
41C22×C4×C71C22⋊C412yes—
42C22×C4×C71C22⋊C412yes—
43C4⋊C4×C71—82no—
44C22⋊C4×C71—28no—
45C22⋊C4×C71—44no—
46C22⋊C4×C71—82no—
47C22⋊C4×C71—82no—
48C42×C71—82no—
49C23×C71—22no—
50C23×C71—24no—
51C23×C71—41no—
52C23×C71—41no—
53C23×C71—42no—
54C23×C71—42no—
55C23×C71—44no—
56C23×C71—44no—
57C23×C71—44no—
58C23×C71—88no—
59C23×C71C2×C22⋊C411yescharacteristic
60C23×C71#11712yes—
61C23×C71#12314yes—
62C2×C4×C71—22no—
63C2×C4×C71—22no—
64C2×C4×C71—42no—
65C2×C4×C71—42no—
66C2×C4×C71—44no—
67C2×C4×C71—44no—
68C2×C4×C71—44no—
69C2×C4×C71—82no—
70C2×C4×C71—82no—
71C2×Dic71—84no—
72C2×Dic71—84no—
73C2×Dic71—84no—
74C22×C71—21no—
75C22×C71—21no—
76C22×C71—24no—
77C22×C71—22no—
78C22×C71—24no—
79C22×C71—24no—
80C22×C71—22no—
81C22×C71—22no—
82C22×C71—44no—
83C22×C71—84no—
84C22×C71—88no—
85C22×C71#38111yescharacteristic
86C22×C71#41012yes—
87C4×C71—42no—
88C4×C71—42no—
89C4×C71—84no—
90Dic71—164no—
91#8750—711nomaximal, Sylow
92C2×C71—22no—
93C2×C71—44no—
94C2×C71—42no—
95C2×C71#182011yescharacteristic
96C2×C71#182811yescharacteristic
97C2×C71#182811yescharacteristic
98#2312D7111yescharacteristic
99#1819—712no—
100#1304—714no—
101C71#875011yescharacteristic, complemented, Sylow
102C22×C22⋊C4C2×D7111yescharacteristic
103#388—24no—
104#380Dic7112yescomplemented
105#379—714no—
106#379C2×D7112yes—
107C4×C22⋊C4Dic7112yescomplemented
108#342—1424no—
109C25C22×D7111yescharacteristic
110C23×C4C2×Dic7111yescharacteristic
111C23×C4C2×Dic7111yescharacteristic
112C2×C4⋊C4—22no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—1424no—
117C2×C22⋊C4—1424no—
118C2×C22⋊C4—1424no—
119C2×C22⋊C4C2×Dic7114yes—
120C2×C22⋊C4C22⋊D7114yes—
121C2×C42—22no—
122#95—22no—
123#95—22no—
124C24—42no—
125C24—81no—
126C24—81no—
127C24C22×Dic7111yescharacteristic
128C24C2×C22⋊D7112yes—
129C24#1176586514yes—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4—2844no—
136C22×C4C22⋊Dic7112yes—
137C22×C4C22⋊Dic7112yes—
138C4⋊C4—82no—
139C22⋊C4—28no—
140C22⋊C4—44no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C22⋊C4—2848no—
144C22⋊C4—5684no—
145C22⋊C4—5684no—
146C42—82no—
147C23—22no—
148C23—24no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—44no—
154C23—44no—
155C23—44no—
156C23—88no—
157C23C2×C22⋊Dic7111yescharacteristic
158C23#1455608712yes—
159C23#1455611114yes—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—5684no—
170C2×C4—5684no—
171C2×C4—5684no—
172C22—21no—
173C22—21no—
174C22—24no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—24no—
179C22—24no—
180C22—44no—
181C22—84no—
182C22—88no—
183C22#1798586211yescharacteristic
184C22#1798597312yes—
185C4—42no—
186C4—42no—
187C4—84no—
188C4—11364no—
189C2—22no—
190C2—44no—
191C2—42no—
192C2#2258841711yescharacteristic
193C2#2258844811yescharacteristic
194C2#2258844811yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC24×C71C22×C4
FrattiniC25C22×D71
FittingC71×#2312C2
socleC22×C71#381
prime radical, cosocleC25×C71C23
hypercenter#2312D71
hypocenterC71#8750
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C25 ◁ #2312

Complements

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

NH ≅ G/Ntype
8. #380×C71C4semidirect
10. C4×C22⋊C4×C71C4semidirect
101. C71#8750semidirect
104. #380Dic71semidirect
107. C4×C22⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884142
C71⋊#13042
C71×#23122
#875071