Finite Group Databasebeta

← #31890936

Subgroups — #31890936

Order 18176.

418 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 91 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#22590838C211yesmaximal, characteristic
2#22588417C211yesmaximal, characteristic
3#22586194C214yesmaximal
4C71×#1822C211yesmaximal, characteristic
5#17985892—22no—
6#17985885—22no—
7#17985862C2212yes—
8C4×C4⋊Dic71C412yescomplemented
9C4×C22⋊Dic71C2212yes—
10#17985845C411yescharacteristic, complemented
11C42×Dic71C411yescharacteristic, complemented
12#17985766—22no—
13C2×#117×C71C2211yescharacteristic
14#407×C71C2212yes—
15#343×C71—22no—
16C2×C4⋊Dic71—22no—
17C2×C22⋊Dic71—22no—
18C2×C22⋊Dic71—22no—
19C2×C4×Dic71—22no—
20C2×C4×Dic71C2×C412yes—
21C2×C4×Dic71C2×C412yes—
22C71⋊#95—22no—
23C71⋊#95—22no—
24C71⋊#95—22no—
25C71⋊#95—24no—
26C23×C4×C71C2311yescharacteristic
27C2×M16×C71—22no—
28#117×C71—24no—
29C2×C4⋊C4×C71D412yes—
30C2×C22⋊C4×C71D412yes—
31C2×C42×C71C2×C411yescharacteristic
32C2×C42×C71C2×C411yescharacteristic
33C22⋊C8×C71—24no—
34C22×Dic71—22no—
35C22×Dic71—22no—
36C22×Dic71—42no—
37C4⋊Dic71—44no—
38C22⋊Dic71—44no—
39C22⋊Dic71—44no—
40C4×Dic71—44no—
41C4×Dic71—42no—
42C4×Dic71—42no—
43C24×C71C2×D411yescharacteristic
44C22×C4×C71—42no—
45C22×C4×C71C22⋊C412yes—
46C22×C4×C71C22⋊C412yes—
47C22×C4×C71C22×C411yescharacteristic
48C22×C4×C71C2×D411yescharacteristic
49C22×C4×C71#4114yes—
50M16×C71—44no—
51C2×C8×C71—42no—
52C4⋊C4×C71—24no—
53C22⋊C4×C71—24no—
54C42×C71—22no—
55C42×C71—22no—
56C23×C71—22no—
57C23×C71—42no—
58C23×C71C2×C22⋊C411yescharacteristic
59C23×C71#12312yes—
60C2×C4×C71—22no—
61C2×C4×C71—22no—
62C2×C4×C71—22no—
63C2×C4×C71—22no—
64C2×C4×C71—44no—
65C2×C4×C71—42no—
66C2×C4×C71—42no—
67C2×C4×C71#11712yes—
68C2×C4×C71#12312yes—
69C2×Dic71—24no—
70C2×Dic71—42no—
71C2×Dic71—42no—
72C2×Dic71—42no—
73C2×Dic71—42no—
74C2×Dic71—42no—
75C2×Dic71—82no—
76C2×Dic71—84no—
77C8×C71—82no—
78C22×C71—22no—
79C22×C71—22no—
80C22×C71—24no—
81C22×C71—82no—
82C22×C71#38111yescharacteristic
83C22×C71#41012yes—
84C4×C71—22no—
85C4×C71—22no—
86C4×C71—42no—
87C4×C71—42no—
88Dic71—44no—
89Dic71—82no—
90Dic71—162no—
91#8741—711nomaximal, Sylow
92C2×C71—22no—
93C2×C71—42no—
94C2×C71#182011yescharacteristic
95C2×C71#182712yes—
96#2298—711no—
97#1822D7111yescharacteristic
98#1820—711no—
99#1284—714no—
100C71#874111yescharacteristic, complemented, Sylow
101C2×#117C2×D7111yescharacteristic
102#407C2×D7112yes—
103#388—1422no—
104#387—1422no—
105#381—712no—
106C4×C4⋊C4—712no—
107C4×C22⋊C4—712no—
108#376—711no—
109C43—711no—
110#343—22no—
111#342—1422no—
112C23×C4C22×D7111yescharacteristic
113C2×M16—22no—
114#117—24no—
115C2×C4⋊C4—1422no—
116C2×C4⋊C4C22⋊D7112yes—
117C2×C22⋊C4—1422no—
118C2×C22⋊C4—1422no—
119C2×C22⋊C4D28412yes—
120C2×C42—712no—
121C2×C42—712no—
122C2×C42—1422no—
123C2×C42C4×D7111yescharacteristic
124C2×C42C4×D7111yescharacteristic
125C22⋊C8—24no—
126#95—1424no—
127#95—1422no—
128#95—1422no—
129#95—1422no—
130C24C2×D28411yescharacteristic
131C22×C4—42no—
132C22×C4—2842no—
133C22×C4—1422no—
134C22×C4—1422no—
135C22×C4(C2×D71)⋊C412yes—
136C22×C4(C2×D71)⋊C412yes—
137C22×C4C2×C4×D7111yescharacteristic
138C22×C4C2×C22⋊D7111yescharacteristic
139C22×C4#1176586414yes—
140M16—44no—
141C2×C8—42no—
142C4⋊C4—24no—
143C4⋊C4—2844no—
144C22⋊C4—24no—
145C22⋊C4—2844no—
146C22⋊C4—2844no—
147C42—22no—
148C42—22no—
149C42—2842no—
150C42—2842no—
151C42—2844no—
152C23—22no—
153C23—42no—
154C23C2×(C2×D71)⋊C411yescharacteristic
155C23#1455611212yes—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—44no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—2842no—
164C2×C4—2842no—
165C2×C4—2842no—
166C2×C4—2842no—
167C2×C4—2842no—
168C2×C4—5682no—
169C2×C4—5684no—
170C2×C4—1424no—
171C2×C4#1455608412yes—
172C2×C4#1455610912yes—
173C8—82no—
174C22—22no—
175C22—24no—
176C22—22no—
177C22—82no—
178C22#1798586011yescharacteristic
179C22#1798597212yes—
180C4—22no—
181C4—22no—
182C4—42no—
183C4—42no—
184C4—2844no—
185C4—5682no—
186C4—11362no—
187C2—22no—
188C2—42no—
189C2#2258841511yescharacteristic
190C2#2258844512yes—
subgroupG/H
centerC22#17985860
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1822C2
socleC22×C71#381
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#1822D71
hypocenterC71#8741
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
8. C4×C4⋊Dic71C4semidirect
10. #17985845C4semidirect
11. C42×Dic71C4semidirect
100. C71#8741semidirect

Maximal subgroups

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

subgroupindex
#225908382
#225884172
#225861942
C71×#18222
#874171