Finite Group Databasebeta

← #31892414

Subgroups — #31892414

Order 18176.

398 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 95 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#22598688C211yesmaximal, characteristic
2#22597446C211yesmaximal, characteristic, complemented
3#22586810C212yesmaximal
4#22586240C212yesmaximal, complemented
5C71×#1544C211yesmaximal, characteristic, complemented
6#17986549C2211yescharacteristic
7#17986369C412yes—
8#17986365C412yes—
9#17986361—22no—
10#17986130—24no—
11#17986060—22no—
12#17986056C2212yes—
13#17985993—22no—
14C2×#14556039C2211yescharacteristic
15#17985728—24no—
16#533×C71C2211yescharacteristic
17#431×C71C2212yes—
18#332×C71—22no—
19C2×#11765865—21no—
20C22×C71⋊C8—21no—
21#14556128C2×C411yescharacteristic
22#14556125C2×C411yescharacteristic
23#14556116—21no—
24#14556102—22no—
25C4×D284—22no—
26C4⋊(C71⋊C8)—24no—
27C4⋊(C71⋊C8)D414yes—
28#14556051—24no—
29#14556048—24no—
30C22⋊(C71⋊C8)—22no—
31#14556039—22no—
32#14556039—22no—
33#14556039—21no—
34#14556039C2×C412yes—
35C4×C71⋊C8—22no—
36C4⋊Q8×C71D412yes—
37#125×C71C2×C412yes—
38C22⋊Q8×C71—22no—
39C4×Q8×C71—22no—
40#117×C71D412yes—
41C2×C42×C71C2311yescharacteristic
42C4⋊C8×C71—22no—
43C22⋊C8×C71—22no—
44C4×C8×C71—22no—
45#11765865—44no—
46C2×Dic142—21no—
47C2×D284—21no—
48C2×D284—22no—
49C2×C22⋊D71—41no—
50C2×C4×D71—41no—
51C2×C71⋊C8—22no—
52C2×C71⋊C8—24no—
53C2×C71⋊C8—42no—
54C4⋊Dic71—22no—
55(C2×D71)⋊C4—42no—
56C2×Q8×C71—22no—
57C22×C4×C71C2×D411yescharacteristic
58C22×C4×C71#4112yes—
59C2×C8×C71—42no—
60C4⋊C4×C71—24no—
61C4⋊C4×C71—22no—
62C4⋊C4×C71C22⋊C414yes—
63C22⋊C4×C71—22no—
64C42×C71—22no—
65C42×C71C22×C411yescharacteristic
66C42×C71C2×D412yes—
67C42×C71C2×D411yescharacteristic
68C23×C71#12311yescharacteristic
69C22×D71—41no—
70Q8×C71—82no—
71C2×C4×C71—22no—
72C2×C4×C71—24no—
73C2×C4×C71—42no—
74C2×C4×C71C2×C22⋊C411yescharacteristic
75C2×C4×C71C4×D412yes—
76C2×C4×C71#12011yescharacteristic
77C2×C4×C71#12112yes—
78C22⋊D71—82no—
79C2×Dic71—41no—
80D284—44no—
81D284—42no—
82C4×D71—82no—
83Dic142—42no—
84C8×C71—82no—
85C71⋊C8—42no—
86C71⋊C8—44no—
87C22×C71—22no—
88C22×C71—21no—
89C22×C71#38611yescharacteristic
90C2×D71—82no—
91C2×D71—81no—
92C4×C71—22no—
93C4×C71—22no—
94C4×C71—21no—
95C4×C71—44no—
96C4×C71#45412yes—
97Dic71—81no—
98#9008—711nomaximal, Sylow
99C2×C71—41no—
100C2×C71#188112yes—
101C2×C71#188811yescharacteristic
102D71—161no—
103#3135—711no—
104#2987—711no—
105#1544D7111yescharacteristic, complemented
106#1496—712no—
107#1302—712no—
108C71#900811yescharacteristic, complemented, Sylow
109#533C2×D7111yescharacteristic
110#532—711no—
111#491—712no—
112#490—712no—
113#489—1422no—
114#446—1424no—
115#433—1422no—
116#432—712no—
117#431C2×D7112yes—
118#416—1422no—
119C2×#97—711no—
120#332—22no—
121D4⋊C8—1424no—
122C2×#41—1421no—
123C22×C8—1421no—
124C4⋊Q8—711no—
125C4⋊Q8C22⋊D7112yes—
126#127—711no—
127#125C4×D7112yes—
128#124—1421no—
129C22⋊Q8—22no—
130#121—1422no—
131C4×Q8—22no—
132C4×D4—1422no—
133#117D28412yes—
134C2×C42C22×D7111yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—714no—
137C4⋊C8—1424no—
138#103—1424no—
139#102—1424no—
140C22⋊C8—22no—
141C22⋊C8—1422no—
142#97—712no—
143#97—1422no—
144#97—1422no—
145#97—1421no—
146C4×C8—22no—
147C4×C8—1422no—
148#41—2844no—
149C2×Q8—22no—
150C2×Q8—1421no—
151C2×D4—2841no—
152C2×D4—1421no—
153C2×D4—1422no—
154C22×C4—2841no—
155C22×C4C2×D28411yescharacteristic
156C22×C4#1176586412yes—
157C2×C8—42no—
158C2×C8—2842no—
159C2×C8—1422no—
160C2×C8—1424no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—1422no—
164C4⋊C4(C2×D71)⋊C414yes—
165C22⋊C4—22no—
166C22⋊C4—2842no—
167C42—22no—
168C42C2×C4×D7111yescharacteristic
169C42D4×D7112yes—
170C42C2×C22⋊D7111yescharacteristic
171C23—2841no—
172C23#1455611211yescharacteristic
173Q8—82no—
174Q8—2842no—
175D4—2842no—
176D4—2844no—
177D4—5682no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—2841no—
182C2×C4—5682no—
183C2×C4C2×(C2×D71)⋊C411yescharacteristic
184C2×C4(C22⋊D71)⋊C412yes—
185C2×C4#1455609811yescharacteristic
186C2×C4#1455610012yes—
187C8—82no—
188C8—2842no—
189C8—2844no—
190C22—22no—
191C22—21no—
192C22—5682no—
193C22—5681no—
194C22#1798587611yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—5681no—
200C4#1798617412yes—
201C2—41no—
202C2—11361no—
203C2#2258870012yes—
204C2#2258874111yescharacteristic
subgroupG/H
centerC2×C4#14556098
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1544C2
socleC22×C71#386
prime radical#17986549C22
cosocleC2×C42×C71C23
hypercenter#1544D71
hypocenterC71#9008
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
2. #22597446C2semidirect
4. #22586240C2semidirect
5. C71×#1544C2semidirect
105. #1544D71semidirect
108. C71#9008semidirect

Maximal subgroups

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

subgroupindex
#225986882
#225974462
#225868102
#225862402
C71×#15442
#900871