Finite Group Databasebeta

← #31890978

Subgroups — #31890978

Order 18176.

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

normal subgroups only

This group has 398 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#22590943C211yesmaximal, characteristic
2#22588420C211yesmaximal, characteristic
3#22586249C214yesmaximal
4C71×#1821C211yesmaximal, characteristic
5#17985963C2212yes—
6#17985885—22no—
7#17985885—22no—
8#17985862C2212yes—
9#17985859C412yes—
10C4×C22⋊Dic71C412yes—
11#17985768—22no—
12C22×C4⋊C4×C71C2211yescharacteristic
13#407×C71C2212yes—
14#343×C71—22no—
15C2×#11765849—22no—
16C2×C4⋊Dic71—22no—
17C2×C22⋊Dic71C2×C414yes—
18C2×C4×Dic71—22no—
19C22⋊(C71⋊C8)—24no—
20C71⋊#95—22no—
21C71⋊#95—22no—
22C71⋊#95—22no—
23C71⋊#95—22no—
24C23×C4×C71C2×C411yescharacteristic
25C23×C4×C71C2×C411yescharacteristic
26C23×C4×C71C2311yescharacteristic
27C2×M16×C71—22no—
28C2×C4⋊C4×C71—24no—
29C2×C4⋊C4×C71D412yes—
30C2×C4⋊C4×C71D412yes—
31C22⋊C8×C71—24no—
32C22×Dic71—24no—
33#11765849—44no—
34C2×C71⋊C8—42no—
35C4⋊Dic71—82no—
36C22⋊Dic71—44no—
37C4×Dic71—82no—
38C24×C71C22×C411yescharacteristic
39C22×C4×C71—22no—
40C22×C4×C71—22no—
41C22×C4×C71—42no—
42C22×C4×C71—81no—
43C22×C4×C71—81no—
44C22×C4×C71C22⋊C412yes—
45C22×C4×C71C22⋊C412yes—
46C22×C4×C71C2×D411yescharacteristic
47C22×C4×C71C2×D411yescharacteristic
48C22×C4×C71#4114yes—
49M16×C71—44no—
50C2×C8×C71—42no—
51C4⋊C4×C71—24no—
52C4⋊C4×C71—24no—
53C23×C71—22no—
54C23×C71—41no—
55C23×C71—41no—
56C23×C71C2×C22⋊C411yescharacteristic
57C23×C71#11712yes—
58C2×C4×C71—22no—
59C2×C4×C71—22no—
60C2×C4×C71—22no—
61C2×C4×C71—22no—
62C2×C4×C71—44no—
63C2×C4×C71—42no—
64C2×C4×C71—42no—
65C2×C4×C71—82no—
66C2×C4×C71—82no—
67C2×C4×C71#12312yes—
68C2×C4×C71#12312yes—
69C2×Dic71—44no—
70C2×Dic71—44no—
71C2×Dic71—44no—
72C8×C71—82no—
73C71⋊C8—82no—
74C22×C71—21no—
75C22×C71—21no—
76C22×C71—22no—
77C22×C71—22no—
78C22×C71—22no—
79C22×C71—44no—
80C22×C71#38111yescharacteristic
81C22×C71#41012yes—
82C4×C71—22no—
83C4×C71—22no—
84C4×C71—42no—
85C4×C71—42no—
86Dic71—84no—
87#8755—711nomaximal, Sylow
88C2×C71—22no—
89C2×C71—42no—
90C2×C71#182011yescharacteristic
91C2×C71#182911yescharacteristic
92C2×C71#183011yescharacteristic
93#2313—711no—
94#1821—711no—
95#1821D7111yescharacteristic
96#1306—714no—
97C71#875511yescharacteristic, complemented, Sylow
98C22×C4⋊C4C2×D7111yescharacteristic
99#407—712no—
100#407C2×D7112yes—
101#387—1422no—
102#387—1422no—
103#381—712no—
104#380—712no—
105C4×C22⋊C4—712no—
106#343—22no—
107#343—1422no—
108C23×C4C4×D7111yescharacteristic
109C23×C4C4×D7111yescharacteristic
110C23×C4C22×D7111yescharacteristic
111C2×M16—22no—
112C2×M16—1422no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—1422no—
115C2×C4⋊C4D28412yes—
116C2×C4⋊C4C22⋊D7112yes—
117C2×C22⋊C4—714no—
118C2×C42—1422no—
119C22⋊C8—24no—
120C22⋊C8—1424no—
121#95—1422no—
122#95—1422no—
123#95—1422no—
124#95—1422no—
125C24C2×C4×D7111yescharacteristic
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—81no—
130C22×C4—81no—
131C22×C4—1424no—
132C22×C4(C2×D71)⋊C412yes—
133C22×C4(C2×D71)⋊C412yes—
134C22×C4C2×C22⋊D7111yescharacteristic
135C22×C4C2×D28411yescharacteristic
136C22×C4#1176586414yes—
137M16—44no—
138M16—2844no—
139C2×C8—42no—
140C2×C8—2842no—
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—5682no—
144C22⋊C4—2844no—
145C42—5682no—
146C23—22no—
147C23—41no—
148C23—41no—
149C23C2×(C2×D71)⋊C411yescharacteristic
150C23#1455608412yes—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—2844no—
161C2×C4—2844no—
162C2×C4—2844no—
163C2×C4#1455610912yes—
164C2×C4#1455611212yes—
165C8—82no—
166C8—5682no—
167C22—21no—
168C22—21no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—44no—
173C22#1798586011yescharacteristic
174C22#1798597212yes—
175C4—22no—
176C4—22no—
177C4—42no—
178C4—42no—
179C4—5684no—
180C2—22no—
181C2—42no—
182C2#2258841511yescharacteristic
183C2#2258844911yescharacteristic
184C2#2258845211yescharacteristic
subgroupG/H
centerC22#17985860
commutator, prime radicalC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1821C2
socleC22×C71#381
cosocleC23×C4×C71C23
hypercenter#1821D71
hypocenterC71#8755
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
97. C71#8755semidirect

Maximal subgroups

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

subgroupindex
#225909432
#225884202
#225862492
C71×#18212
#875571