Finite Group Databasebeta

← #31894989

Subgroups — #31894989

Order 18176.

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

normal subgroups only

This group has 534 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#22598714C211yesmaximal, characteristic
2#22587054C211yesmaximal, characteristic
3#22586890C212yesmaximal, complemented
4#22586204C212yesmaximal, complemented
5C71×#2380C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7#17986363—22no—
8#17986345C414yescomplemented
9#17986056C2212yes—
10#17986004—22no—
11C4×#11765849C2211yescharacteristic
12#17985743—22no—
13#17985735—24no—
14C2×#127×C71C2211yescharacteristic
15#386×C71—22no—
16#386×C71—24no—
17#383×C71C2212yes—
18C2×#11765849—21no—
19#14556128C2×C412yescomplemented
20#14556120—21no—
21#14556107—22no—
22C4×Dic142—22no—
23#14556087—21no—
24C4⋊(C71⋊C8)—22no—
25#14556049—28no—
26C22⋊(C71⋊C8)—22no—
27#14556039—21no—
28#14556039—22no—
29C4×C71⋊C8C2×C412yescomplemented
30C22×D4×C71—22no—
31C22×D4×C71D412yes—
32C23×C4×C71—22no—
33#127×C71—22no—
34#127×C71C2×C412yes—
35#127×C71D412yes—
36C2×C22⋊C4×C71—24no—
37C2×C22⋊C4×C71—22no—
38C2×C42×C71C2311yescharacteristic
39#95×C71—22no—
40#95×C71—22no—
41#95×C71D414yes—
42C2×Dic142—22no—
43#11765849—42no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8—42no—
46C4⋊Dic71—22no—
47C4⋊Dic71—22no—
48Dic71⋊C4—42no—
49C22⋊Dic71—41no—
50C4×Dic71—41no—
51C24×C71—22no—
52C2×D4×C71—24no—
53C2×D4×C71—24no—
54C2×D4×C71—22no—
55C2×D4×C71—42no—
56C2×D4×C71—42no—
57C2×D4×C71—82no—
58C2×D4×C71—82no—
59C2×D4×C71C22⋊C414yes—
60C22×C4×C71—24no—
61C22×C4×C71—84no—
62C22×C4×C71—82no—
63C22×C4×C71C2×D411yescharacteristic
64C22×C4×C71C2×D412yes—
65C22⋊C4×C71—84no—
66C22⋊C4×C71—82no—
67C42×C71C22×C411yescharacteristic
68C42×C71C2×D411yescharacteristic
69C42×C71#4112yes—
70C23×C71—24no—
71C23×C71—44no—
72C23×C71—44no—
73C23×C71—42no—
74C23×C71—42no—
75C23×C71#12011yescharacteristic
76D4×C71—44no—
77D4×C71—82no—
78D4×C71—84no—
79D4×C71—84no—
80C2×C4×C71—22no—
81C2×C4×C71—42no—
82C2×C4×C71—48no—
83C2×C4×C71—44no—
84C2×C4×C71—88no—
85C2×C4×C71C2×C22⋊C411yescharacteristic
86C2×C4×C71C4×D412yes—
87C2×C4×C71#12112yes—
88C2×C4×C71#12311yescharacteristic
89C2×Dic71—42no—
90Dic142—82no—
91C71⋊C8—42no—
92C71⋊C8—82no—
93C22×C71—22no—
94C22×C71—44no—
95C22×C71—48no—
96C22×C71—82no—
97C22×C71—82no—
98C22×C71—84no—
99C22×C71#38611yescharacteristic
100C22×C71#45512yes—
101C4×C71—21no—
102C4×C71—21no—
103C4×C71—22no—
104C4×C71—22no—
105C4×C71—84no—
106Dic71—82no—
107#9425—711nomaximal, Sylow
108C2×C71—22no—
109C2×C71—84no—
110C2×C71#188511yescharacteristic
111C2×C71#190412yes—
112#3138—711no—
113#2380D7111yescharacteristic
114#1555—711no—
115#1515—712no—
116#1288—712no—
117C71#942511yescharacteristic, complemented, Sylow
118#533—711no—
119C2×#127C2×D7111yescharacteristic
120#490—1422no—
121#486—714no—
122#432—712no—
123#418—1422no—
124C4×M16—711no—
125#386—22no—
126#386—24no—
127#383C2×D7112yes—
128#331—1422no—
129#328—1424no—
130C22×D4—22no—
131C22×D4C22⋊D7112yes—
132C23×C4—22no—
133C2×M16—1421no—
134C4⋊Q8—712no—
135#127—22no—
136#127C4×D7112yes—
137#127D28412yes—
138#125—1421no—
139C22⋊Q8—1422no—
140C4×Q8—1422no—
141#117—1421no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—22no—
144C2×C42C22×D7111yescharacteristic
145C4⋊C8—1422no—
146#102—1428no—
147C22⋊C8—1422no—
148#97—1421no—
149#97—1422no—
150C4×C8—712no—
151#95—22no—
152#95—22no—
153#95C22⋊D7114yes—
154C24—22no—
155C2×Q8—1422no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4(C2×D71)⋊C414yes—
164C22×C4—24no—
165C22×C4—84no—
166C22×C4—82no—
167C22×C4C2×C22⋊D7111yescharacteristic
168C22×C4C2×C22⋊D7112yes—
169M16—2842no—
170C2×C8—2842no—
171C2×C8—1422no—
172C4⋊C4—2842no—
173C4⋊C4—1422no—
174C4⋊C4—1422no—
175C22⋊C4—84no—
176C22⋊C4—82no—
177C22⋊C4—2841no—
178C42—2841no—
179C42C2×C4×D7111yescharacteristic
180C42C2×D28411yescharacteristic
181C42#1176586512yes—
182C23—24no—
183C23—44no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23#1455609711yescharacteristic
188Q8—5682no—
189D4—44no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—48no—
196C2×C4—44no—
197C2×C4—88no—
198C2×C4—2842no—
199C2×C4C2×(C2×D71)⋊C411yescharacteristic
200C2×C4C4×C22⋊D7112yes—
201C2×C4#1455610212yes—
202C2×C4#1455611111yescharacteristic
203C8—2842no—
204C8—5682no—
205C22—22no—
206C22—44no—
207C22—48no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22#1798587811yescharacteristic
212C22#1798617712yes—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—84no—
218C4—5682no—
219C2—22no—
220C2—84no—
221C2#2258872811yescharacteristic
222C2#2258882712yes—
subgroupG/H
centerC22#17985878
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2380C2
socleC22×C71#386
prime radicalC2×#127×C71C22
cosocleC2×C42×C71C23
hypercenter#2380D71
hypocenterC71#9425
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2380

Complements

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

NH ≅ G/Ntype
3. #22586890C2semidirect
4. #22586204C2semidirect
8. #17986345C4semidirect
19. #14556128C2×C4semidirect
29. C4×C71⋊C8C2×C4semidirect
117. C71#9425semidirect

Maximal subgroups

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

subgroupindex
#225987142
#225870542
#225868902
#225862042
C71×#23802
#942571