Finite Group Databasebeta

← #31907999

Subgroups — #31907999

Order 18176.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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#22590156C211yesmaximal, characteristic
2#22588095C211yesmaximal, characteristic
3#22586334C212yesmaximal
4#22586328C212yesmaximal
5C71×#3177C211yesmaximal, characteristic
6#17985954—21no—
7#17985820—21no—
8#17985781C2212yes—
9#17985777—21no—
10C71⋊#338C2211yescharacteristic
11#17985756C2211yescharacteristic
12#17985755—21no—
13#541×C71—21no—
14#499×C71C411yescharacteristic, complemented
15#494×C71C411yescharacteristic, complemented
16#491×C71—21no—
17#470×C71C2212yes—
18#467×C71—21no—
19#441×C71C412yescomplemented
20C4×Q16×C71—21no—
21#405×C71C2211yescharacteristic
22C2×#11765849—21no—
23#14556087—21no—
24#14556068—22no—
25#14556038—22no—
26C8×Dic71—22no—
27C2×Q16×C71—21no—
28C2×Q16×C71C2×C412yes—
29C2×M16×C71C2311yescharacteristic
30C22×C8×C71D411yescharacteristic
31C4⋊Q8×C71—21no—
32#126×C71—42no—
33#125×C71—21no—
34#125×C71—42no—
35C22⋊Q8×C71—22no—
36C4×Q8×C71—22no—
37#117×C71D411yescharacteristic
38#107×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—22no—
49C4⋊Dic71—41no—
50C22⋊Dic71—41no—
51C4×Dic71—22no—
52C71⋊C16—42no—
53C2×Q8×C71—22no—
54C22×C4×C71C2×D411yescharacteristic
55Q16×C71—42no—
56Q16×C71—41no—
57Q16×C71—41no—
58M16×C71—41no—
59C2×C8×C71—21no—
60C2×C8×C71—21no—
61C2×C8×C71C22⋊C411yescharacteristic
62C2×C8×C71C22⋊C411yescharacteristic
63C2×C8×C71C22×C411yescharacteristic
64C2×C8×C71C2×D411yescharacteristic
65C4⋊C4×C71—21no—
66C4⋊C4×C71—22no—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—42no—
71C22⋊C4×C71—21no—
72C22⋊C4×C71—42no—
73C42×C71—42no—
74C42×C71C22⋊C411yescharacteristic
75C42×C71C22⋊C411yescharacteristic
76C23×C71AGL(1,ℤ/8)11yescharacteristic
77Q8×C71—42no—
78Q8×C71—42no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71C2×C22⋊C411yescharacteristic
86C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
87C2×Dic71—42no—
88C8×C71—21no—
89C8×C71—21no—
90C8×C71—21no—
91C8×C71#10411yescharacteristic
92C8×C71#10411yescharacteristic
93C71⋊C8—42no—
94C22×C71—21no—
95C22×C71—41no—
96C22×C71#41811yescharacteristic
97C4×C71—21no—
98C4×C71—41no—
99C4×C71—41no—
100C4×C71—82no—
101C4×C71—82no—
102C4×C71#41011yescharacteristic
103C4×C71C2×#10411yescharacteristic
104Dic71—82no—
105#11771—711nomaximal, Sylow
106C2×C71—21no—
107C2×C71—41no—
108C2×C71#150511yescharacteristic
109#3177D7111yescharacteristic
110#2166—711no—
111#1747—711no—
112#1342—712no—
113#1340—712no—
114C71#1177111yescharacteristic, complemented, Sylow
115#541—21no—
116#499Dic7111yescharacteristic, complemented
117#494Dic7111yescharacteristic, complemented
118#491—21no—
119#470C2×D7112yes—
120#467—21no—
121#441Dic7112yescomplemented
122C4×Q16—21no—
123#405—1421no—
124#405C2×D7111yescharacteristic
125#364—1421no—
126#349—712no—
127#347—1421no—
128#338—711no—
129H(3,ℤ/4)—711no—
130H(3,ℤ/4)—1421no—
131C2×Q16—21no—
132C2×Q16C2×Dic7112yes—
133C2×M16—1421no—
134C2×M16C22×D7111yescharacteristic
135C22×C8C22⋊D7111yescharacteristic
136C4⋊Q8—21no—
137#126—42no—
138#125—21no—
139#125—42no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142#117—1421no—
143#117C22⋊D7111yescharacteristic
144M32—1422no—
145#107—21no—
146#106C2×Dic7112yes—
147#103—22no—
148#103—22no—
149#97—1422no—
150#97C2×Dic7111yescharacteristic
151#97C22⋊D7111yescharacteristic
152C4×C8—1422no—
153C4×C8C2×Dic7111yescharacteristic
154C4×C8C22⋊D7111yescharacteristic
155C2×Q8—22no—
156C22×C4C2×C22⋊D7111yescharacteristic
157Q16—42no—
158Q16—41no—
159Q16—41no—
160M16—41no—
161M16—2841no—
162M16—2841no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1422no—
166C2×C8C22⋊Dic7111yescharacteristic
167C2×C8C22⋊Dic7111yescharacteristic
168C2×C8C22×Dic7111yescharacteristic
169C2×C8C2×C22⋊D7111yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—2841no—
177C22⋊C4—21no—
178C22⋊C4—42no—
179C22⋊C4—2841no—
180C42—42no—
181C42—1422no—
182C42C22⋊Dic7111yescharacteristic
183C42C22⋊Dic7111yescharacteristic
184C16—2842no—
185C23#1455615411yescharacteristic
186Q8—42no—
187Q8—42no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—2842no—
195C2×C4C2×C22⋊Dic7111yescharacteristic
196C2×C4#1455615411yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—2842no—
201C8#1455605511yescharacteristic
202C8#1455605511yescharacteristic
203C22—21no—
204C22—41no—
205C22#1798600411yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—82no—
211C4—5682no—
212C4#1798597311yescharacteristic
213C4C2×#1455605511yescharacteristic
214C2—21no—
215C2—41no—
216C2#2258684811yescharacteristic
subgroupG/H
centerC2#22586848
commutatorC2×C8×C71C22×C4
FrattiniC2×M16C22×D71
FittingC71×#3177C2
socleC2×C71#1505
prime radicalC23×C71AGL(1,ℤ/8)
cosocleC2×M16×C71C23
hypercenter#3177D71
hypocenterC71#11771
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
14. #499×C71C4semidirect
15. #494×C71C4semidirect
19. #441×C71C4semidirect
114. C71#11771semidirect
116. #499Dic71semidirect
117. #494Dic71semidirect
121. #441Dic71semidirect

Maximal subgroups

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

subgroupindex
#225901562
#225880952
#225863342
#225863282
C71×#31772
#1177171