Finite Group Databasebeta

← #31895067

Subgroups — #31895067

Order 18176.

452 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 111 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#22597442C211yesmaximal, characteristic, complemented
2#22587078C211yesmaximal, characteristic, complemented
3#22587008C212yesmaximal
4#22586230C212yesmaximal, complemented
5C71×#3138C211yesmaximal, characteristic
6#17986130—22no—
7Q8×C71⋊C8—22no—
8#17986060—22no—
9#17986052C2212yes—
10#17986023C2212yes—
11#17985745—22no—
12#17985743—22no—
13#17985743—22no—
14#533×C71C2211yescharacteristic
15C2×#127×C71C2211yescharacteristic
16#490×C71C414yes—
17#486×C71—22no—
18#418×C71—22no—
19C4×M16×C71C2211yescharacteristic
20C2×#11765849—22no—
21C4⋊(C71⋊C8)—24no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)D412yes—
25C4⋊(C71⋊C8)D412yes—
26C22⋊(C71⋊C8)—22no—
27#14556039—22no—
28C4×C71⋊C8—22no—
29C4×C71⋊C8—22no—
30C22×D4×C71—21no—
31C22×D4×C71—22no—
32C2×M16×C71—21no—
33C4⋊Q8×C71D412yes—
34#127×C71—21no—
35#127×C71—22no—
36#127×C71C2×C412yes—
37#125×C71C2×C412yes—
38C22⋊Q8×C71—22no—
39C4×Q8×C71—22no—
40#117×C71D412yes—
41C2×C42×C71C2311yescharacteristic
42#102×C71—28no—
43#97×C71C2×C412yes—
44C4×C8×C71—21no—
45#11765849—82no—
46C2×C71⋊C8—22no—
47C2×C71⋊C8—22no—
48C2×C71⋊C8—42no—
49C24×C71—41no—
50C2×Q8×C71—22no—
51C2×D4×C71—22no—
52C2×D4×C71—24no—
53C2×D4×C71—44no—
54C2×D4×C71—41no—
55C2×D4×C71—42no—
56C2×D4×C71—82no—
57C2×D4×C71—82no—
58C22×C4×C71C2×D411yescharacteristic
59C22×C4×C71#4112yes—
60M16×C71—42no—
61C2×C8×C71—22no—
62C4⋊C4×C71—24no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71C22⋊C414yes—
65C22⋊C4×C71—22no—
66C42×C71—22no—
67C42×C71C22×C411yescharacteristic
68C42×C71C2×D412yes—
69C42×C71C2×D411yescharacteristic
70C23×C71—42no—
71C23×C71—82no—
72C23×C71—84no—
73C23×C71#12311yescharacteristic
74Q8×C71—42no—
75Q8×C71—42no—
76D4×C71—44no—
77D4×C71—84no—
78D4×C71—82no—
79D4×C71—82no—
80D4×C71—82no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—24no—
85C2×C4×C71C2×C22⋊C411yescharacteristic
86C2×C4×C71C4×D412yes—
87C2×C4×C71#12011yescharacteristic
88C2×C4×C71#12112yes—
89C8×C71—42no—
90C71⋊C8—42no—
91C71⋊C8—42no—
92C71⋊C8—82no—
93C22×C71—21no—
94C22×C71—22no—
95C22×C71—82no—
96C22×C71—84no—
97C22×C71—164no—
98C22×C71#38611yescharacteristic
99C4×C71—21no—
100C4×C71—22no—
101C4×C71—44no—
102C4×C71#45412yes—
103C4×C71#47112yes—
104C4×C71#47212yes—
105#9438—711nomaximal, Sylow
106C2×C71—41no—
107C2×C71—162no—
108C2×C71#188811yescharacteristic
109C2×C71#190712yes—
110#3138D7111yescharacteristic
111#2986—711no—
112#1561—711no—
113#1545—712no—
114#1299—712no—
115C71#943811yescharacteristic, complemented, Sylow
116#533C2×D7111yescharacteristic
117C2×#127C2×D7111yescharacteristic
118#490Dic7114yes—
119#486—22no—
120#446—1422no—
121C8×Q8—1422no—
122#433—1422no—
123#431—712no—
124#423—712no—
125#418—22no—
126C4×M16C2×D7111yescharacteristic
127#332—1422no—
128#331—1422no—
129#331—1422no—
130C22×D4—21no—
131C22×D4—22no—
132C2×M16—21no—
133C2×M16—1422no—
134C4⋊Q8C22⋊D7112yes—
135#127—22no—
136#127—21no—
137#127C2×Dic7112yes—
138#125C2×Dic7112yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117C22⋊D7112yes—
142C2×C42C22×D7111yescharacteristic
143C4⋊C8—712no—
144C4⋊C8—712no—
145C4⋊C8—1424no—
146C4⋊C8—1422no—
147C4⋊C8—1422no—
148#102—28no—
149C22⋊C8—1422no—
150#97—1422no—
151#97C2×Dic7112yes—
152C4×C8—21no—
153C4×C8—1422no—
154C4×C8—1422no—
155C24—41no—
156C2×Q8—22no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—44no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—82no—
163C2×D4—82no—
164C22×C4C2×C22⋊D7111yescharacteristic
165C22×C4#1176586412yes—
166M16—42no—
167M16—5682no—
168C2×C8—22no—
169C2×C8—2842no—
170C2×C8—1422no—
171C2×C8—1422no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4C22⋊Dic7114yes—
175C22⋊C4—22no—
176C42—22no—
177C42C22×Dic7111yescharacteristic
178C42D4×D7112yes—
179C42C2×C22⋊D7111yescharacteristic
180C23—42no—
181C23—82no—
182C23—84no—
183C23#1455610911yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—44no—
187D4—84no—
188D4—82no—
189D4—82no—
190D4—82no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4C2×C22⋊Dic7111yescharacteristic
196C2×C4D4×Dic7112yes—
197C2×C4#1455609611yescharacteristic
198C2×C4#1455610012yes—
199C8—42no—
200C8—2842no—
201C8—2842no—
202C8—5682no—
203C22—21no—
204C22—22no—
205C22—82no—
206C22—84no—
207C22—164no—
208C22#1798587911yescharacteristic
209C4—21no—
210C4—22no—
211C4—44no—
212C4#1798617112yes—
213C4#1798626012yes—
214C4#1798626712yes—
215C2—41no—
216C2—162no—
217C2#2258874411yescharacteristic
218C2#2258885012yes—
subgroupG/H
centerC22#17985879
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3138C2
socleC22×C71#386
prime radicalC2×#127×C71C22
cosocleC2×C42×C71C23
hypercenter#3138D71
hypocenterC71#9438
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #22597442C2semidirect
2. #22587078C2semidirect
4. #22586230C2semidirect
115. C71#9438semidirect

Maximal subgroups

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

subgroupindex
#225974422
#225870782
#225870082
#225862302
C71×#31382
#943871