Finite Group Databasebeta

← #31908303

Subgroups — #31908303

Order 18176.

280 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 71 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#22599085C211yesmaximal, characteristic
2#22590152C211yesmaximal, characteristic
3#22586333C212yesmaximal
4#22586327C212yesmaximal
5C71×#1841C211yesmaximal, characteristic
6#17986556—21no—
7#17986397C411yescharacteristic, complemented
8#17986379C411yescharacteristic, complemented
9#17986350—21no—
10#17986257—21no—
11#17986257C2212yes—
12#17986107—21no—
13#17986107C412yescomplemented
14#17985820—21no—
15#17985781C2212yes—
16#17985777—21no—
17C4×M16×C71C2211yescharacteristic
18#344×C71—21no—
19#339×C71—21no—
20#338×C71C2211yescharacteristic
21H(3,ℤ/4)×C71C2211yescharacteristic
22C2×Dic284—21no—
23C2×Dic284C2×C412yes—
24#14556128—21no—
25#14556128—21no—
26#14556120—41no—
27#14556107—22no—
28#14556107—41no—
29C4×Dic142—22no—
30C4×Dic142—41no—
31#14556087—41no—
32#14556068—22no—
33#14556060—21no—
34#14556060C2×C412yes—
35#14556053—22no—
36#14556053—22no—
37C2×M16×C71—21no—
38C2×M16×C71D411yescharacteristic
39C2×M16×C71C2311yescharacteristic
40#117×C71—21no—
41C2×C42×C71D411yescharacteristic
42#97×C71C2×C411yescharacteristic
43#97×C71D411yescharacteristic
44C4×C8×C71C2×C411yescharacteristic
45C4×C8×C71D411yescharacteristic
46C2×Dic142—22no—
47C2×Dic142—41no—
48Dic284—42no—
49Dic284—42no—
50C4⋊Dic71—22no—
51C4⋊Dic71—41no—
52C4⋊Dic71—41no—
53C4⋊Dic71—41no—
54Dic71⋊C4—42no—
55Dic71⋊C4—42no—
56C22⋊Dic71—42no—
57C4×Dic71—42no—
58C71⋊C16—42no—
59C22×C4×C71—21no—
60C22×C4×C71C2×D411yescharacteristic
61M16×C71—41no—
62M16×C71—41no—
63M16×C71—41no—
64M16×C71—41no—
65C2×C8×C71—41no—
66C2×C8×C71C22⋊C411yescharacteristic
67C2×C8×C71C22⋊C411yescharacteristic
68C2×C8×C71C22×C411yescharacteristic
69C2×C8×C71C2×D411yescharacteristic
70C4⋊C4×C71—41no—
71C22⋊C4×C71—41no—
72C42×C71—21no—
73C42×C71—41no—
74C42×C71C22⋊C411yescharacteristic
75C42×C71C22⋊C411yescharacteristic
76C23×C71AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—42no—
82C2×C4×C71—81no—
83C2×C4×C71C2×C22⋊C411yescharacteristic
84C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
85C2×Dic71—42no—
86C2×Dic71—42no—
87Dic142—42no—
88Dic142—42no—
89Dic142—82no—
90C8×C71—21no—
91C8×C71—21no—
92C8×C71—21no—
93C8×C71—21no—
94C8×C71—81no—
95C22×C71—21no—
96C22×C71—41no—
97C22×C71#41811yescharacteristic
98C4×C71—22no—
99C4×C71—21no—
100C4×C71—41no—
101C4×C71—161no—
102C4×C71#41011yescharacteristic
103C4×C71#42111yescharacteristic
104Dic71—82no—
105Dic71—82no—
106#11838—711nomaximal, Sylow
107C2×C71—21no—
108C2×C71—41no—
109C2×C71#151411yescharacteristic
110#3185—711no—
111#2165—711no—
112#1841D7111yescharacteristic
113#1342—712no—
114#1340—712no—
115C71#1183811yescharacteristic, complemented, Sylow
116#533—1421no—
117#498—711no—
118#494—711no—
119#487—1421no—
120#470—712no—
121#470—1421no—
122#441—712no—
123#441—1421no—
124C4×M16C2×D7111yescharacteristic
125#364—1421no—
126#349—712no—
127#347—1421no—
128#344—21no—
129#339—21no—
130#338C2×D7111yescharacteristic
131H(3,ℤ/4)C2×D7111yescharacteristic
132C2×Q16—712no—
133C2×Q16—1421no—
134C2×M16—21no—
135C2×M16C22⋊D7111yescharacteristic
136C2×M16C22×D7111yescharacteristic
137C4⋊Q8—1421no—
138C4⋊Q8—1421no—
139#125—2841no—
140C22⋊Q8—2841no—
141C22⋊Q8—1422no—
142C4×Q8—2841no—
143C4×Q8—1422no—
144#117—21no—
145#117—2841no—
146C2×C42C22⋊D7111yescharacteristic
147M32—1422no—
148#106—712no—
149#106—1421no—
150#103—1422no—
151#103—1422no—
152#97C4×D7111yescharacteristic
153#97D28411yescharacteristic
154C4×C8C4×D7111yescharacteristic
155C4×C8D28411yescharacteristic
156C2×Q8—2841no—
157C2×Q8—1422no—
158C22×C4—21no—
159C22×C4C2×C22⋊D7111yescharacteristic
160Q16—2842no—
161Q16—2842no—
162M16—41no—
163M16—41no—
164M16—41no—
165M16—41no—
166C2×C8—41no—
167C2×C8(C2×D71)⋊C411yescharacteristic
168C2×C8(C2×D71)⋊C411yescharacteristic
169C2×C8C2×C4×D7111yescharacteristic
170C2×C8C2×D28411yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—2842no—
173C4⋊C4—2841no—
174C4⋊C4—2841no—
175C4⋊C4—2841no—
176C4⋊C4—2842no—
177C4⋊C4—1422no—
178C22⋊C4—41no—
179C22⋊C4—2842no—
180C42—21no—
181C42—41no—
182C42—2842no—
183C42(C2×D71)⋊C411yescharacteristic
184C42(C2×D71)⋊C411yescharacteristic
185C16—2842no—
186C23#1455615311yescharacteristic
187Q8—2842no—
188Q8—2842no—
189Q8—5682no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—2842no—
197C2×C4—2842no—
198C2×C4C2×(C2×D71)⋊C411yescharacteristic
199C2×C4#1455615311yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—81no—
205C22—21no—
206C22—41no—
207C22#1798600311yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—161no—
212C4—5682no—
213C4—5682no—
214C4#1798597211yescharacteristic
215C4#1798601711yescharacteristic
216C2—21no—
217C2—41no—
218C2#2258688611yescharacteristic
subgroupG/H
centerC2#22586886
commutatorC2×C8×C71C22×C4
FrattiniC2×M16C22×D71
FittingC71×#1841C2
socleC2×C71#1514
prime radicalC23×C71AGL(1,ℤ/8)
cosocleC2×M16×C71C23
hypercenter#1841D71
hypocenterC71#11838
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #1841

Complements

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

NH ≅ G/Ntype
7. #17986397C4semidirect
8. #17986379C4semidirect
13. #17986107C4semidirect
115. C71#11838semidirect

Maximal subgroups

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

subgroupindex
#225990852
#225901522
#225863332
#225863272
C71×#18412
#1183871