Finite Group Databasebeta

← #31918326

Subgroups — #31918326

Order 18176.

184 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 51 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
1C2×#17985837C211yesmaximal, characteristic, complemented
2#22590635C211yesmaximal, characteristic, complemented
3#22590466C211yesmaximal, characteristic, complemented
4#22590457C211yesmaximal, characteristic
5#22586543C211yesmaximal, characteristic, complemented
6#22586534C211yesmaximal, characteristic, complemented
7C71×#1419C211yesmaximal, characteristic, complemented
8C2×Dic568C2211yescharacteristic
9#17986415C2211yescharacteristic
10#17986275—21no—
11#17986266—21no—
12#17985837—21no—
13#17985837—21no—
14#17985832—21no—
15C2×C71⋊C32C2211yescharacteristic
16C32×D71—21no—
17#17985829C2211yescharacteristic
18#17985814—21no—
19#17985812—21no—
20C2×D16×C71C2211yescharacteristic
21M64×C71C2211yescharacteristic
22#368×C71C2211yescharacteristic
23#14556161—41no—
24#14556154—41no—
25C2×Dic284—21no—
26#14556134D411yescharacteristic
27Dic568—21no—
28Dic568—21no—
29#14556068D411yescharacteristic
30#14556067—21no—
31C16×D71—21no—
32#14556064—21no—
33#14556046—41no—
34#14556045—41no—
35C71⋊C32D411yescharacteristic
36C71⋊C32D411yescharacteristic
37C2×D8×C71—21no—
38D16×C71—21no—
39D16×C71—21no—
40C2×C16×C71C2311yescharacteristic
41#108×C71—21no—
42C32×C71—21no—
43#11765865D811yescharacteristic
44C2×Dic142—41no—
45Dic284—41no—
46Dic284—41no—
47Dic284—81no—
48#11765854—81no—
49#11765853—81no—
50#11765850—81no—
51#11765849—41no—
52#11765849D811yescharacteristic
53#11765848—21no—
54C8×D71—21no—
55C71⋊C16—21no—
56C2×D4×C71—41no—
57D8×C71—41no—
58D8×C71—41no—
59M16×C71—41no—
60C2×C8×C71C2×D411yescharacteristic
61C16×C71C2×D411yescharacteristic
62C16×C71#4111yescharacteristic
63C23×C71—81no—
64D4×C71—81no—
65D4×C71—81no—
66C2×C4×C71C2×D811yescharacteristic
67C22⋊D71—21no—
68C2×Dic71—81no—
69D284D1611yescharacteristic
70C4×D71—21no—
71Dic142—81no—
72Dic142—81no—
73Dic142D1611yescharacteristic
74C8×C71—81no—
75C8×C71#12111yescharacteristic
76C8×C71#13511yescharacteristic
77C71⋊C8—21no—
78C71⋊C8—81no—
79C22×C71—161no—
80C22×C71—321no—
81C22×C71#50811yescharacteristic
82C2×D71—21no—
83C4×C71#46611yescharacteristic
84C4×C71C2×D1611yescharacteristic
85Dic71—21no—
86Dic71—161no—
87#13702—711nomaximal, Sylow
88C2×C71—21no—
89C2×C71—321no—
90C2×C71#221511yescharacteristic
91D71—41no—
92C2×SD64—711no—
93#2258—711no—
94#2222—711no—
95#2221—711no—
96#1425—711no—
97#1420—711no—
98#1419D7111yescharacteristic, complemented
99C71#1370211yescharacteristic, complemented, Sylow
100C2×Q32—711no—
101C2×D16C2×D7111yescharacteristic
102#504—711no—
103#473—1421no—
104#472—1421no—
105SD64—1421no—
106SD64—1421no—
107M64—1421no—
108M64C2×D7111yescharacteristic
109C2×C32—711no—
110C2×C32—1421no—
111#368—711no—
112#368C2×D7111yescharacteristic
113#362—1421no—
114#361—1421no—
115#137—2841no—
116AGL(1,ℤ/8)—2841no—
117C2×Q16—1421no—
118C2×D8—21no—
119#131—711no—
120Q32—1421no—
121Q32—1421no—
122D16—21no—
123D16—21no—
124M32—711no—
125M32—1421no—
126C2×C16—1421no—
127C2×C16C22×D7111yescharacteristic
128#108—21no—
129#108—1421no—
130#101—2841no—
131#100—2841no—
132C32—21no—
133C32—711no—
134C32—711no—
135#41—711no—
136C2×Q8—2841no—
137C2×D4—41no—
138Q16—2841no—
139Q16—2841no—
140Q16—5681no—
141SD16—5681no—
142SD16—5681no—
143D8—41no—
144D8—41no—
145D8—5681no—
146M16—41no—
147M16—2841no—
148M16—711no—
149M16—1421no—
150C2×C8—1421no—
151C2×C8D4×D7111yescharacteristic
152C16—1421no—
153C16D4×D7111yescharacteristic
154C16#1176586511yescharacteristic
155C23—81no—
156Q8—5681no—
157Q8—5681no—
158Q8—711no—
159D4—81no—
160D4—81no—
161D4—711no—
162D4—1421no—
163C2×C4—5681no—
164C2×C4—1421no—
165C2×C4D8×D7111yescharacteristic
166C8—81no—
167C8—5681no—
168C8—1421no—
169C8#1455610111yescharacteristic
170C8#1455614911yescharacteristic
171C22—161no—
172C22—321no—
173C22—1421no—
174C22#1798643011yescharacteristic
175C4—11361no—
176C4—1421no—
177C4#1798623411yescharacteristic
178C4D16×D7111yescharacteristic
179C2—21no—
180C2—2841no—
181C2—321no—
182C2#2259042511yescharacteristic
subgroupG/H
centerC2#22590425
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC71×#1419C2
socleC2×C71#2215
prime radical#22590457C2
hypercenter#1419D71
hypocenterC71#13702
LCSG ▷ C2×C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1419

Complements

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

NH ≅ G/Ntype
1. C2×#17985837C2semidirect
2. #22590635C2semidirect
3. #22590466C2semidirect
5. #22586543C2semidirect
6. #22586534C2semidirect
7. C71×#1419C2semidirect
98. #1419D71semidirect
99. C71#13702semidirect

Maximal subgroups

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

subgroupindex
C2×#179858372
#225906352
#225904662
#225904572
#225865432
#225865342
C71×#14192
#1370271