Finite Group Databasebeta

← #31890951

Subgroups — #31890951

Order 18176.

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

normal subgroups only

This group has 814 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#22588255C212yesmaximal
2C71⋊#1304C214yesmaximal, complemented
3C71×#2308C211yesmaximal, characteristic
4C2×#14556043—24no—
5#17985855C2214yes—
6C23×D4×C71C2211yescharacteristic
7C22×C22⋊C4×C71C412yescomplemented
8#386×C71—24no—
9#381×C71C412yescomplemented
10#379×C71C2212yes—
11C2×C22⋊Dic71—24no—
12C2×C22⋊Dic71—28no—
13#14556043—48no—
14C25×C71C2×C411yescharacteristic
15C25×C71C2311yescharacteristic
16C22×D4×C71—24no—
17C22×D4×C71—44no—
18C22×D4×C71D414yes—
19C23×C4×C71—22no—
20C23×C4×C71C2×C411yescharacteristic
21C2×C22⋊C4×C71—22no—
22C2×C22⋊C4×C71—24no—
23C2×C22⋊C4×C71—44no—
24C2×C22⋊C4×C71C2×C414yes—
25#95×C71—24no—
26C22×Dic71—44no—
27C22⋊Dic71—48no—
28C22⋊Dic71—88no—
29C24×C71—24no—
30C24×C71—22no—
31C24×C71—42no—
32C24×C71—82no—
33C24×C71—82no—
34C24×C71C22⋊C412yes—
35C24×C71C22×C411yescharacteristic
36C24×C71C2×D412yes—
37C24×C71#4114yes—
38C2×D4×C71—28no—
39C2×D4×C71—28no—
40C2×D4×C71—44no—
41C2×D4×C71—416no—
42C22×C4×C71—22no—
43C22×C4×C71—24no—
44C22×C4×C71—42no—
45C22×C4×C71—44no—
46C22×C4×C71—84no—
47C22×C4×C71C22⋊C412yes—
48C22⋊C4×C71—44no—
49C22⋊C4×C71—44no—
50C22⋊C4×C71—84no—
51C23×C71—28no—
52C23×C71—22no—
53C23×C71—22no—
54C23×C71—22no—
55C23×C71—24no—
56C23×C71—42no—
57C23×C71—44no—
58C23×C71—44no—
59C23×C71—48no—
60C23×C71—44no—
61C23×C71—84no—
62C23×C71—88no—
63C23×C71—88no—
64C23×C71#9914yes—
65C23×C71C2×C22⋊C411yescharacteristic
66C23×C71#11712yes—
67C23×C71#12314yes—
68D4×C71—416no—
69C2×C4×C71—22no—
70C2×C4×C71—24no—
71C2×C4×C71—44no—
72C2×C4×C71—48no—
73C2×C4×C71—44no—
74C2×C4×C71—88no—
75C2×Dic71—84no—
76C2×Dic71—88no—
77C22×C71—22no—
78C22×C71—24no—
79C22×C71—22no—
80C22×C71—24no—
81C22×C71—28no—
82C22×C71—24no—
83C22×C71—44no—
84C22×C71—44no—
85C22×C71—44no—
86C22×C71—44no—
87C22×C71—84no—
88C22×C71—88no—
89C22×C71—88no—
90C22×C71#38111yescharacteristic
91C22×C71C2×#9912yes—
92C4×C71—42no—
93C4×C71—84no—
94Dic71—164no—
95#8746—711nomaximal, Sylow
96C2×C71—22no—
97C2×C71—44no—
98C2×C71—42no—
99C2×C71—44no—
100C2×C71#181911yescharacteristic
101C2×C71#182812yes—
102#2308D7111yescharacteristic
103#1781—712no—
104#1304—714no—
105C71#874611yescharacteristic, complemented, Sylow
106C23×D4C2×D7111yescharacteristic
107C22×C22⋊C4Dic7112yescomplemented
108C2×#99—1424no—
109#386—24no—
110#381Dic7112yescomplemented
111#379—714no—
112#379C2×D7112yes—
113C25C2×Dic7111yescharacteristic
114C25C22×D7111yescharacteristic
115C22×D4—24no—
116C22×D4—44no—
117C22×D4C22⋊D7114yes—
118C23×C4—22no—
119C23×C4C2×Dic7111yescharacteristic
120C2×C22⋊C4—22no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—44no—
123C2×C22⋊C4—1424no—
124C2×C22⋊C4—1428no—
125C2×C22⋊C4C2×Dic7114yes—
126#99—2848no—
127#95—24no—
128C24—24no—
129C24—22no—
130C24—42no—
131C24—82no—
132C24—82no—
133C24C22⋊Dic7112yes—
134C24C22×Dic7111yescharacteristic
135C24C2×C22⋊D7112yes—
136C24#1176586514yes—
137C2×D4—28no—
138C2×D4—28no—
139C2×D4—44no—
140C2×D4—416no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—84no—
146C22×C4—2844no—
147C22×C4C22⋊Dic7112yes—
148C22⋊C4—44no—
149C22⋊C4—44no—
150C22⋊C4—84no—
151C22⋊C4—2848no—
152C22⋊C4—5688no—
153C23—28no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—24no—
158C23—42no—
159C23—44no—
160C23—48no—
161C23—44no—
162C23—44no—
163C23—84no—
164C23—88no—
165C23—88no—
166C23#1455604314yes—
167C23C2×C22⋊Dic7111yescharacteristic
168C23#1455608712yes—
169C23#1455611114yes—
170D4—416no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—44no—
174C2×C4—48no—
175C2×C4—44no—
176C2×C4—88no—
177C2×C4—5684no—
178C2×C4—5688no—
179C22—22no—
180C22—28no—
181C22—24no—
182C22—22no—
183C22—24no—
184C22—24no—
185C22—44no—
186C22—44no—
187C22—44no—
188C22—44no—
189C22—84no—
190C22—88no—
191C22—88no—
192C22#1798586211yescharacteristic
193C22C2×#1455604312yes—
194C4—42no—
195C4—84no—
196C4—11364no—
197C2—22no—
198C2—44no—
199C2—42no—
200C2—44no—
201C2#2258841411yescharacteristic
202C2#2258844812yes—
subgroupG/H
centerC22#17985862
commutatorC24×C71C22×C4
FrattiniC25C22×D71
FittingC71×#2308C2
socleC22×C71#381
prime radicalC23×D4×C71C22
cosocleC25×C71C23
hypercenter#2308D71
hypocenterC71#8746
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C25 ◁ #2308

Complements

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

NH ≅ G/Ntype
2. C71⋊#1304C2semidirect
7. C22×C22⋊C4×C71C4semidirect
9. #381×C71C4semidirect
105. C71#8746semidirect
107. C22×C22⋊C4Dic71semidirect
110. #381Dic71semidirect

Maximal subgroups

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

subgroupindex
#225882552
C71⋊#13042
C71×#23082
#874671