Finite Group Databasebeta

← #31892401

Subgroups — #31892401

Order 18176.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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#22598685C211yesmaximal, characteristic
2#22587002C211yesmaximal, characteristic
3#22586812C212yesmaximal, complemented
4#22586238C212yesmaximal, complemented
5C71×#2980C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7#17986367—21no—
8#17986363—21no—
9#17986358C414yes—
10#17986052C2212yes—
11#17985995—22no—
12C2×#14556039C2211yescharacteristic
13#17985748—21no—
14#17985743—21no—
15#17985729—24no—
16#532×C71C2211yescharacteristic
17#435×C71—24no—
18#432×C71C2212yes—
19#408×C71—22no—
20C22×C71⋊C8—21no—
21#14556128C2×C412yes—
22#14556120—21no—
23#14556107—22no—
24C4×Dic142—22no—
25#14556087—21no—
26C4⋊(C71⋊C8)—22no—
27#14556052—24no—
28#14556049—24no—
29C22⋊(C71⋊C8)—22no—
30#14556039—21no—
31#14556039C2×C412yes—
32C4×C71⋊C8—22no—
33C2×#41×C71D412yes—
34C2×M16×C71—22no—
35C22×C8×C71—22no—
36C4⋊Q8×C71D411yescharacteristic
37#127×C71D411yescharacteristic
38#124×C71C2×C412yes—
39#121×C71—22no—
40C4×D4×C71—22no—
41C2×C42×C71C2311yescharacteristic
42C4⋊C8×C71D414yes—
43C22⋊C8×C71—24no—
44C22⋊C8×C71—22no—
45#97×C71—22no—
46C2×Dic142—22no—
47C2×C71⋊C8—22no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—42no—
50C4⋊Dic71—22no—
51C4⋊Dic71—22no—
52Dic71⋊C4—42no—
53C22⋊Dic71—41no—
54C4×Dic71—41no—
55#41×C71—42no—
56#41×C71—42no—
57C2×Q8×C71C22⋊C412yes—
58C2×D4×C71—22no—
59C2×D4×C71—22no—
60C2×D4×C71C22⋊C412yes—
61C22×C4×C71—22no—
62C22×C4×C71C2×D411yescharacteristic
63C22×C4×C71#4112yes—
64M16×C71—82no—
65C2×C8×C71—24no—
66C2×C8×C71—82no—
67C4⋊C4×C71—22no—
68C22⋊C4×C71—24no—
69C42×C71C22×C411yescharacteristic
70C42×C71C2×D412yes—
71C42×C71C2×D411yescharacteristic
72C23×C71—22no—
73C23×C71#12311yescharacteristic
74Q8×C71—42no—
75D4×C71—42no—
76D4×C71—82no—
77D4×C71—82no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—42no—
81C2×C4×C71—42no—
82C2×C4×C71—42no—
83C2×C4×C71C2×C22⋊C411yescharacteristic
84C2×C4×C71C4×D412yes—
85C2×C4×C71#12011yescharacteristic
86C2×C4×C71#12112yes—
87C2×Dic71—42no—
88Dic142—44no—
89C8×C71—44no—
90C71⋊C8—42no—
91C71⋊C8—82no—
92C22×C71—22no—
93C22×C71—21no—
94C22×C71—44no—
95C22×C71—42no—
96C22×C71#38611yescharacteristic
97C4×C71—22no—
98C4×C71—22no—
99C4×C71—21no—
100C4×C71—42no—
101C4×C71#45512yes—
102Dic71—82no—
103#9005—711nomaximal, Sylow
104C2×C71—41no—
105C2×C71—82no—
106C2×C71#188112yes—
107C2×C71#188511yescharacteristic
108#3135—711no—
109#2980D7111yescharacteristic
110#1543—711no—
111#1497—712no—
112#1302—712no—
113C71#900511yescharacteristic, complemented, Sylow
114#533—711no—
115#532C2×D7111yescharacteristic
116#491—1421no—
117#490—1421no—
118#489—714no—
119#435—24no—
120#432C2×D7112yes—
121#431—712no—
122#416—1422no—
123#408—22no—
124C2×#97—711no—
125#333—1421no—
126#331—1421no—
127Q8⋊C8—1424no—
128C2×#41C22⋊D7112yes—
129C2×M16—22no—
130C22×C8—22no—
131C22×C8—1421no—
132C4⋊Q8—712no—
133C4⋊Q8D28411yescharacteristic
134#127D28411yescharacteristic
135#125—1421no—
136#124C4×D7112yes—
137C22⋊Q8—1422no—
138#121—22no—
139C4×Q8—1422no—
140C4×D4—22no—
141#117—1421no—
142C2×C42C22×D7111yescharacteristic
143C4⋊C8—1422no—
144C4⋊C8C22⋊D7114yes—
145#103—1424no—
146#102—1424no—
147C22⋊C8—24no—
148C22⋊C8—22no—
149C22⋊C8—1422no—
150#97—22no—
151#97—712no—
152#97—1421no—
153C4×C8—1422no—
154#41—42no—
155#41—42no—
156C2×Q8—1422no—
157C2×Q8(C2×D71)⋊C412yes—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4(C2×D71)⋊C412yes—
161C22×C4—22no—
162C22×C4C2×C22⋊D7111yescharacteristic
163C22×C4#1176586512yes—
164M16—82no—
165C2×C8—24no—
166C2×C8—82no—
167C2×C8—2842no—
168C2×C8—2842no—
169C2×C8—1422no—
170C4⋊C4—22no—
171C4⋊C4—2842no—
172C4⋊C4—1422no—
173C4⋊C4—1422no—
174C22⋊C4—24no—
175C22⋊C4—2841no—
176C42—2841no—
177C42C2×C4×D7111yescharacteristic
178C42C2×C22⋊D7112yes—
179C42C2×D28411yescharacteristic
180C23—22no—
181C23#1455611111yescharacteristic
182Q8—42no—
183Q8—2844no—
184D4—42no—
185D4—82no—
186D4—82no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—2842no—
193C2×C4C2×(C2×D71)⋊C411yescharacteristic
194C2×C4C4×C22⋊D7112yes—
195C2×C4#1455609711yescharacteristic
196C2×C4#1455610212yes—
197C8—44no—
198C8—2842no—
199C8—5682no—
200C22—22no—
201C22—21no—
202C22—44no—
203C22—42no—
204C22#1798587811yescharacteristic
205C4—22no—
206C4—22no—
207C4—21no—
208C4—42no—
209C4—5682no—
210C4#1798617712yes—
211C2—41no—
212C2—82no—
213C2#2258870212yes—
214C2#2258872811yescharacteristic
subgroupG/H
centerC2×C4#14556097
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2980C2
socleC22×C71#386
prime radical#532×C71C22
cosocleC2×C42×C71C23
hypercenter#2980D71
hypocenterC71#9005
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
3. #22586812C2semidirect
4. #22586238C2semidirect
113. C71#9005semidirect

Maximal subgroups

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

subgroupindex
#225986852
#225870022
#225868122
#225862382
C71×#29802
#900571