Finite Group Databasebeta

← #31928468

Subgroups — #31928468

Order 18176.

3274 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 3274 conjugacy classes of subgroups — too many to draw. Only its 411 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
1C2×#17985855C212yesmaximal, complemented
2#22586589C216yesmaximal, complemented
3#22586581C216yesmaximal, complemented
4C22×#120×C71C211yesmaximal, characteristic
5C22×C22⋊Dic71C22112yes—
6C22×C22⋊Dic71C2214yescomplemented
7#17985855—28no—
8C2×C71⋊#95C22112yescomplemented
9C26×C71C2211yescharacteristic
10C23×D4×C71C2213yes—
11C2×#120×C71—28no—
12C2×#120×C71C418yescomplemented
13C22×C22⋊C4×C71C2213yes—
14C23×Dic71C2318yes—
15C2×C22⋊Dic71—224no—
16C2×C22⋊Dic71—424no—
17C2×C22⋊Dic71—412no—
18C71⋊#95—412no—
19C25×C71—412no—
20C25×C71C2×C414yes—
21C25×C71D418yes—
22C25×C71C2311yescharacteristic
23C25×C71C2313yes—
24C22×D4×C71—212no—
25C22×D4×C71—412no—
26C22×D4×C71C2×C4112yes—
27C23×C4×C71C2313yes—
28#120×C71—232no—
29C2×C22⋊C4×C71—212no—
30C2×C22⋊C4×C71C2×C4112yes—
31C22×Dic71—28no—
32C22×Dic71—224no—
33C22×Dic71—224no—
34C22⋊Dic71—448no—
35C22⋊Dic71—416no—
36C24×C71—212no—
37C24×C71—224no—
38C24×C71—22no—
39C24×C71—212no—
40C24×C71—424no—
41C24×C71—448no—
42C24×C71—448no—
43C24×C71—416no—
44C24×C71C22⋊C4116yescomplemented
45C24×C71C22×C412yes—
46C24×C71C22×C416yes—
47C24×C71C2×D4112yes—
48C24×C71C2411yescharacteristic
49C2×D4×C71—224no—
50C2×D4×C71—424no—
51C2×D4×C71—448no—
52C22×C4×C71—26no—
53C22×C4×C71—212no—
54C22×C4×C71C22×C416yes—
55C22⋊C4×C71—224no—
56C23×C71—212no—
57C23×C71—224no—
58C23×C71—224no—
59C23×C71—224no—
60C23×C71—224no—
61C23×C71—22no—
62C23×C71—212no—
63C23×C71—448no—
64C23×C71—464no—
65C23×C71—4128no—
66C23×C71—436no—
67C23×C71—412no—
68C23×C71—412no—
69C23×C71—412no—
70C23×C71—44no—
71C23×C71C2×C22⋊C4112yes—
72C23×C71C23×C411yescharacteristic
73C23×C71C22×D412yes—
74C23×C71H(5,𝔽2)112yes—
75D4×C71—448no—
76C2×C4×C71—26no—
77C2×C4×C71—212no—
78C2×C4×C71—412no—
79C2×Dic71—424no—
80C2×Dic71—424no—
81C2×Dic71—48no—
82C22×C71—212no—
83C22×C71—224no—
84C22×C71—464no—
85C22×C71—424no—
86C22×C71—448no—
87C22×C71—46no—
88C22×C71—48no—
89C22×C71—88no—
90C22×C71C22×C22⋊C411yescharacteristic
91C22×C71#51816yes—
92C22×C71#534112yes—
93C22×C71#560112yes—
94C22×C71H(3,𝔽4)14yes—
95C4×C71—46no—
96Dic71—88no—
97#14657—711nomaximal, Sylow
98C2×C71—412no—
99C2×C71—82no—
100C2×C71#230813yes—
101C2×C71#237514yes—
102C2×C71#281218yes—
103C22×#120D7111yescharacteristic
104C2×#379—712no—
105#1440—716no—
106#1438—716no—
107C71#1465711yescharacteristic, complemented, Sylow
108C26C2×D7111yescharacteristic
109C23×D4C2×D7113yes—
110C2×#120—28no—
111C2×#120Dic7118yescomplemented
112C22×C22⋊C4—714no—
113C22×C22⋊C4—7112no—
114C22×C22⋊C4C2×D7113yes—
115#379—1428no—
116C2×#95—7112no—
117C25—412no—
118C25C2×Dic7114yes—
119C25C22⋊D7118yes—
120C25C22×D7111yescharacteristic
121C25C22×D7113yes—
122C22×D4—212no—
123C22×D4—412no—
124C22×D4C2×Dic71112yes—
125C23×C4—718no—
126C23×C4C22×D7113yes—
127#120—232no—
128C2×C22⋊C4—212no—
129C2×C22⋊C4—28424no—
130C2×C22⋊C4—28412no—
131C2×C22⋊C4—14224no—
132C2×C22⋊C4C2×Dic71112yes—
133#95—28412no—
134C24—224no—
135C24—212no—
136C24—22no—
137C24—212no—
138C24—424no—
139C24—448no—
140C24—448no—
141C24—416no—
142C24C22⋊Dic71116yescomplemented
143C24C22×Dic7112yes—
144C24C22×Dic7116yes—
145C24C2×C22⋊D71112yes—
146C24C23×D7111yescharacteristic
147C2×D4—224no—
148C2×D4—424no—
149C2×D4—448no—
150C22×C4—26no—
151C22×C4—212no—
152C22×C4—1428no—
153C22×C4—14224no—
154C22×C4—14224no—
155C22×C4C22×Dic7116yes—
156C22⋊C4—224no—
157C22⋊C4—28448no—
158C22⋊C4—28416no—
159C23—212no—
160C23—224no—
161C23—224no—
162C23—224no—
163C23—224no—
164C23—22no—
165C23—212no—
166C23—448no—
167C23—4128no—
168C23—464no—
169C23—436no—
170C23—412no—
171C23—412no—
172C23—44no—
173C23—412no—
174C23C2×C22⋊Dic71112yes—
175C23C23×Dic7111yescharacteristic
176C23C22×C22⋊D7112yes—
177C23#14556173112yes—
178D4—448no—
179C2×C4—26no—
180C2×C4—212no—
181C2×C4—412no—
182C2×C4—28424no—
183C2×C4—2848no—
184C2×C4—28424no—
185C22—212no—
186C22—224no—
187C22—464no—
188C22—448no—
189C22—424no—
190C22—46no—
191C22—48no—
192C22—88no—
193C22C22×C22⋊Dic7111yescharacteristic
194C22#1798647416yes—
195C22#17986558112yes—
196C22#17986738112yes—
197C22#1798674214yes—
198C4—46no—
199C4—5688no—
200C2—412no—
201C2—82no—
202C2#2259091813yes—
203C2#2259148114yes—
204C2#2259633518yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
Fitting, prime radicalC22×#120×C71C2
socle, cosocleC24×C71C24
hypercenterC22×#120D71
hypocenterC71#14657
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22×#120

Complements

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

NH ≅ G/Ntype
1. C2×#17985855C2semidirect
2. #22586589C2semidirect
3. #22586581C2semidirect
6. C22×C22⋊Dic71C22semidirect
8. C2×C71⋊#95C22semidirect
12. C2×#120×C71C4semidirect
44. C24×C71C22⋊C4semidirect
107. C71#14657semidirect
111. C2×#120Dic71semidirect
142. C24C22⋊Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179858552
#225865892
#225865812
C22×#120×C712
#1465771