Finite Group Databasebeta

← #31890971

Subgroups — #31890971

Order 18176.

754 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 107 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#22588261C212yesmaximal
2#22586248C214yesmaximal, complemented
3C71×#2308C211yesmaximal, characteristic
4C2×#14556045—24no—
5#17985963C2214yes—
6C23×D4×C71C2211yescharacteristic
7C22×C22⋊C4×C71C412yes—
8#386×C71—24no—
9#381×C71C2212yes—
10#379×C71C412yes—
11C2×#11765849—24no—
12#14556045—48no—
13C22⋊(C71⋊C8)—28no—
14C25×C71C2×C411yescharacteristic
15C25×C71C2×C411yescharacteristic
16C22×D4×C71—24no—
17C22×D4×C71—44no—
18C22×D4×C71D414yes—
19C23×C4×C71—22no—
20C23×C4×C71C2311yescharacteristic
21C2×C22⋊C4×C71—22no—
22C2×C22⋊C4×C71—24no—
23C2×C22⋊C4×C71—44no—
24C2×C22⋊C4×C71C2×C414yes—
25#95×C71—24no—
26#11765849—48no—
27C2×C71⋊C8—44no—
28C24×C71—24no—
29C24×C71—22no—
30C24×C71—22no—
31C24×C71—42no—
32C24×C71—82no—
33C24×C71—82no—
34C24×C71C22⋊C412yes—
35C24×C71C22⋊C412yes—
36C24×C71C22×C411yescharacteristic
37C2×D4×C71—28no—
38C2×D4×C71—28no—
39C2×D4×C71—44no—
40C2×D4×C71—416no—
41C22×C4×C71—24no—
42C22×C4×C71—42no—
43C22×C4×C71—44no—
44C22×C4×C71—84no—
45C22×C4×C71C2×D412yes—
46C22×C4×C71#4114yes—
47C22⋊C4×C71—44no—
48C22⋊C4×C71—44no—
49C22⋊C4×C71—84no—
50C23×C71—28no—
51C23×C71—22no—
52C23×C71—22no—
53C23×C71—22no—
54C23×C71—22no—
55C23×C71—24no—
56C23×C71—42no—
57C23×C71—44no—
58C23×C71—44no—
59C23×C71—44no—
60C23×C71—84no—
61C23×C71—84no—
62C23×C71—88no—
63C23×C71—88no—
64C23×C71#9914yes—
65C23×C71C2×C22⋊C411yescharacteristic
66C23×C71#11712yes—
67D4×C71—416no—
68C2×C4×C71—24no—
69C2×C4×C71—44no—
70C2×C4×C71—48no—
71C2×C4×C71—44no—
72C2×C4×C71—88no—
73C2×C4×C71#12314yes—
74C71⋊C8—84no—
75C22×C71—22no—
76C22×C71—22no—
77C22×C71—24no—
78C22×C71—24no—
79C22×C71—44no—
80C22×C71—44no—
81C22×C71—44no—
82C22×C71—44no—
83C22×C71—42no—
84C22×C71—44no—
85C22×C71—84no—
86C22×C71—88no—
87C22×C71—88no—
88C22×C71#38111yescharacteristic
89C22×C71C2×#9912yes—
90C4×C71—24no—
91C4×C71—84no—
92#8752—711nomaximal, Sylow
93C2×C71—22no—
94C2×C71—42no—
95C2×C71—44no—
96C2×C71—82no—
97C2×C71#181911yescharacteristic
98C2×C71#182912yes—
99#2308D7111yescharacteristic
100#1783—712no—
101#1306—714no—
102C71#875211yescharacteristic, complemented, Sylow
103C23×D4C2×D7111yescharacteristic
104C22×C22⋊C4Dic7112yes—
105C2×#100—1424no—
106#407—714no—
107#386—24no—
108#381C2×D7112yes—
109#379Dic7112yes—
110C25C2×Dic7111yescharacteristic
111C25C2×Dic7111yescharacteristic
112C22×D4—24no—
113C22×D4—44no—
114C22×D4C22⋊D7114yes—
115C23×C4—22no—
116C23×C4C22×D7111yescharacteristic
117C2×M16—1424no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—44no—
121C2×C22⋊C4C2×Dic7114yes—
122#100—2848no—
123C22⋊C8—1428no—
124#95—24no—
125C24—24no—
126C24—22no—
127C24—22no—
128C24—42no—
129C24—82no—
130C24—82no—
131C24C22⋊Dic7112yes—
132C24C22⋊Dic7112yes—
133C24C22×Dic7111yescharacteristic
134C2×D4—28no—
135C2×D4—28no—
136C2×D4—44no—
137C2×D4—416no—
138C22×C4—24no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4—84no—
142C22×C4C2×C22⋊D7112yes—
143C22×C4#1176586514yes—
144M16—2848no—
145C2×C8—2844no—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C22⋊C4—84no—
149C23—28no—
150C23—22no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—24no—
155C23—42no—
156C23—44no—
157C23—44no—
158C23—44no—
159C23—84no—
160C23—84no—
161C23—88no—
162C23—88no—
163C23#1455604314yes—
164C23C2×C22⋊Dic7111yescharacteristic
165C23#1455608712yes—
166D4—416no—
167C2×C4—24no—
168C2×C4—44no—
169C2×C4—48no—
170C2×C4—44no—
171C2×C4—88no—
172C2×C4#1455611114yes—
173C8—5684no—
174C22—22no—
175C22—24no—
176C22—22no—
177C22—24no—
178C22—44no—
179C22—44no—
180C22—44no—
181C22—44no—
182C22—44no—
183C22—42no—
184C22—84no—
185C22—88no—
186C22—88no—
187C22#1798586211yescharacteristic
188C22C2×#1455604312yes—
189C4—24no—
190C4—84no—
191C2—22no—
192C2—42no—
193C2—44no—
194C2—82no—
195C2#2258841411yescharacteristic
196C2#2258845112yes—
subgroupG/H
centerC22#17985862
commutatorC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2308C2
socleC22×C71#381
prime radicalC23×D4×C71C22
cosocleC23×C4×C71C23
hypercenter#2308D71
hypocenterC71#8752
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2308

Complements

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

NH ≅ G/Ntype
2. #22586248C2semidirect
102. C71#8752semidirect

Maximal subgroups

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

subgroupindex
#225882612
#225862482
C71×#23082
#875271