Finite Group Databasebeta

← #31908304

Subgroups — #31908304

Order 18176.

280 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 79 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#22590152C211yesmaximal, characteristic
2#22588489C211yesmaximal, characteristic
3#22586334C212yesmaximal
4#22586328C212yesmaximal
5C71×#3185C211yesmaximal, characteristic
6#17985820—21no—
7#17985781C2212yes—
8#17985777—21no—
9#17985770—21no—
10#17985759—21no—
11C71⋊#338C2211yescharacteristic
12#17985756C2211yescharacteristic
13#533×C71—21no—
14#498×C71C411yescharacteristic, complemented
15#494×C71C411yescharacteristic, complemented
16#487×C71—21no—
17#470×C71—21no—
18#470×C71C2212yes—
19#441×C71—21no—
20#441×C71C412yescomplemented
21C4×M16×C71C2211yescharacteristic
22C2×#11765849—21no—
23#14556087—21no—
24#14556068—22no—
25C2×Q16×C71—21no—
26C2×Q16×C71C2×C412yes—
27C2×M16×C71D411yescharacteristic
28C2×M16×C71C2311yescharacteristic
29C4⋊Q8×C71—21no—
30C4⋊Q8×C71—21no—
31#125×C71—41no—
32C22⋊Q8×C71—22no—
33C22⋊Q8×C71—41no—
34C4×Q8×C71—22no—
35C4×Q8×C71—41no—
36#117×C71—41no—
37C2×C42×C71D411yescharacteristic
38#106×C71—21no—
39#106×C71C2×C412yes—
40#103×C71—22no—
41#103×C71—22no—
42#97×C71C2×C411yescharacteristic
43#97×C71D411yescharacteristic
44C4×C8×C71C2×C411yescharacteristic
45C4×C8×C71D411yescharacteristic
46#11765849—41no—
47#11765849—41no—
48C2×C71⋊C8—41no—
49C4⋊Dic71—41no—
50C22⋊Dic71—41no—
51C4×Dic71—41no—
52C71⋊C16—42no—
53C2×Q8×C71—22no—
54C2×Q8×C71—41no—
55C22×C4×C71—21no—
56C22×C4×C71C2×D411yescharacteristic
57Q16×C71—42no—
58Q16×C71—42no—
59M16×C71—41no—
60M16×C71—41no—
61C2×C8×C71C22⋊C411yescharacteristic
62C2×C8×C71C22⋊C411yescharacteristic
63C2×C8×C71C22×C411yescharacteristic
64C2×C8×C71C2×D411yescharacteristic
65C4⋊C4×C71—22no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—42no—
71C22⋊C4×C71—42no—
72C42×C71—21no—
73C42×C71—42no—
74C42×C71C22⋊C411yescharacteristic
75C42×C71C22⋊C411yescharacteristic
76C23×C71AGL(1,ℤ/8)11yescharacteristic
77Q8×C71—42no—
78Q8×C71—42no—
79Q8×C71—82no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71C2×C22⋊C411yescharacteristic
88C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
89C2×Dic71—81no—
90C8×C71—21no—
91C8×C71—21no—
92C8×C71—21no—
93C8×C71—21no—
94C71⋊C8—81no—
95C22×C71—21no—
96C22×C71—41no—
97C22×C71#41811yescharacteristic
98C4×C71—22no—
99C4×C71—21no—
100C4×C71—41no—
101C4×C71—82no—
102C4×C71—82no—
103C4×C71#41011yescharacteristic
104C4×C71#42111yescharacteristic
105Dic71—161no—
106#11838—711nomaximal, Sylow
107C2×C71—21no—
108C2×C71—41no—
109C2×C71#151411yescharacteristic
110#3185D7111yescharacteristic
111#2165—711no—
112#1841—711no—
113#1342—712no—
114#1340—712no—
115C71#1183811yescharacteristic, complemented, Sylow
116#533—21no—
117#498Dic7111yescharacteristic, complemented
118#494Dic7111yescharacteristic, complemented
119#487—21no—
120#470—21no—
121#470C2×D7112yes—
122#441—21no—
123#441Dic7112yescomplemented
124C4×M16C2×D7111yescharacteristic
125#364—1421no—
126#349—712no—
127#347—1421no—
128#344—1421no—
129#339—1421no—
130#338—711no—
131H(3,ℤ/4)—711no—
132C2×Q16—21no—
133C2×Q16C2×Dic7112yes—
134C2×M16—1421no—
135C2×M16C22⋊D7111yescharacteristic
136C2×M16C22×D7111yescharacteristic
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139#125—41no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C4×Q8—22no—
143C4×Q8—41no—
144#117—41no—
145#117—1421no—
146C2×C42C22⋊D7111yescharacteristic
147M32—1422no—
148#106—21no—
149#106C2×Dic7112yes—
150#103—22no—
151#103—22no—
152#97C2×Dic7111yescharacteristic
153#97C22⋊D7111yescharacteristic
154C4×C8C2×Dic7111yescharacteristic
155C4×C8C22⋊D7111yescharacteristic
156C2×Q8—22no—
157C2×Q8—41no—
158C22×C4—21no—
159C22×C4C2×C22⋊D7111yescharacteristic
160Q16—42no—
161Q16—42no—
162M16—41no—
163M16—41no—
164M16—2841no—
165M16—2841no—
166C2×C8—2841no—
167C2×C8C22⋊Dic7111yescharacteristic
168C2×C8C22⋊Dic7111yescharacteristic
169C2×C8C22×Dic7111yescharacteristic
170C2×C8C2×C22⋊D7111yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—2841no—
178C22⋊C4—42no—
179C22⋊C4—2841no—
180C42—21no—
181C42—42no—
182C42—2841no—
183C42C22⋊Dic7111yescharacteristic
184C42C22⋊Dic7111yescharacteristic
185C16—2842no—
186C23#1455615411yescharacteristic
187Q8—42no—
188Q8—42no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—5681no—
198C2×C4C2×C22⋊Dic7111yescharacteristic
199C2×C4#1455615411yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—5681no—
205C22—21no—
206C22—41no—
207C22#1798600411yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—82no—
212C4—82no—
213C4—11361no—
214C4#1798597311yescharacteristic
215C4#1798601611yescharacteristic
216C2—21no—
217C2—41no—
218C2#2258688711yescharacteristic
subgroupG/H
centerC2#22586887
commutatorC2×C8×C71C22×C4
FrattiniC2×M16C22×D71
FittingC71×#3185C2
socleC2×C71#1514
prime radicalC23×C71AGL(1,ℤ/8)
cosocleC2×M16×C71C23
hypercenter#3185D71
hypocenterC71#11838
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
14. #498×C71C4semidirect
15. #494×C71C4semidirect
20. #441×C71C4semidirect
115. C71#11838semidirect
117. #498Dic71semidirect
118. #494Dic71semidirect
123. #441Dic71semidirect

Maximal subgroups

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

subgroupindex
#225901522
#225884892
#225863342
#225863282
C71×#31852
#1183871