Finite Group Databasebeta

← #31918413

Subgroups — #31918413

Order 18176.

170 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 49 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#22590444C211yesmaximal, characteristic
2#22590133C211yesmaximal, characteristic, complemented
3#22586545C211yesmaximal, characteristic, complemented
4#22586531C211yesmaximal, characteristic, complemented
5#22586529C211yesmaximal, characteristic, complemented
6#22586479C211yesmaximal, characteristic, complemented
7C71×#1426C211yesmaximal, characteristic, complemented
8C2×D1136C2211yescharacteristic
9C2×#14556067C2211yescharacteristic
10#17986236—21no—
11#108×D71—21no—
12#17985833C2211yescharacteristic
13#17985829C2211yescharacteristic
14#17985828—21no—
15#17985799—21no—
16M64×C71C2211yescharacteristic
17#368×C71C2211yescharacteristic
18#367×C71C2211yescharacteristic
19C2×D568—21no—
20M16×D71—41no—
21C2×C8×D71D411yescharacteristic
22#14556103—41no—
23D1136—21no—
24D1136—21no—
25#14556067—21no—
26#14556067—21no—
27C2×C71⋊C16D411yescharacteristic
28#14556064—21no—
29#14556063—41no—
30#14556048—41no—
31C71⋊C32—21no—
32C2×C16×C71C2311yescharacteristic
33#108×C71—21no—
34#107×C71—21no—
35C32×C71—21no—
36C2×D284—41no—
37C2×D284—81no—
38C2×C4×D71D811yescharacteristic
39D568—41no—
40D568—41no—
41#11765849—41no—
42#11765848—81no—
43C2×C71⋊C8D811yescharacteristic
44C8×D71—21no—
45C8×D71—21no—
46C8×D71—81no—
47(C2×D71)⋊C4—81no—
48C71⋊C16—21no—
49M16×C71—41no—
50C2×C8×C71C2×D411yescharacteristic
51C4⋊C4×C71—41no—
52C16×C71#4111yescharacteristic
53C16×C71#4111yescharacteristic
54C22×D71—81no—
55C22×D71SD3211yescharacteristic
56C2×C4×C71—81no—
57C2×C4×C71C2×D811yescharacteristic
58C2×Dic71SD3211yescharacteristic
59D284—81no—
60D284—81no—
61D284—162no—
62C4×D71—21no—
63C4×D71—21no—
64C8×C71—81no—
65C8×C71#12311yescharacteristic
66C8×C71#13711yescharacteristic
67C71⋊C8—21no—
68C71⋊C8—81no—
69C22×C71C2×SD3211yescharacteristic
70C2×D71—21no—
71C2×D71—41no—
72C2×D71—161no—
73C2×D71—161no—
74C2×D71—161no—
75C4×C71—161no—
76C4×C71#48011yescharacteristic
77C4×C71#50911yescharacteristic
78Dic71—21no—
79#13717—711nomaximal, Sylow
80C2×C71—21no—
81C2×C71#223311yescharacteristic
82D71—41no—
83D71—321no—
84#2218—711no—
85#2161—711no—
86#1426—711no—
87#1426D7111yescharacteristic, complemented
88#1419—711no—
89#1419—711no—
90#1400—711no—
91C71#1371711yescharacteristic, complemented, Sylow
92C2×D16—711no—
93C2×M32—711no—
94#466—1421no—
95C2×#108—1421no—
96M64—711no—
97M64C2×D7111yescharacteristic
98#368—711no—
99#368—1421no—
100#368C2×D7111yescharacteristic
101#367C2×D7111yescharacteristic
102#357—1421no—
103C2×D8—1421no—
104C2×M16—2841no—
105C22×C8—711no—
106#121—2841no—
107D16—1421no—
108D16—1421no—
109M32—1421no—
110M32—1421no—
111C2×C16—711no—
112C2×C16C22×D7111yescharacteristic
113#108—21no—
114#108—2841no—
115#108—1421no—
116#107—21no—
117#102—2841no—
118C32—21no—
119C32—1421no—
120C2×D4—2841no—
121C2×D4—5681no—
122C22×C4—711no—
123D8—2841no—
124D8—2841no—
125M16—41no—
126M16—2841no—
127M16—5681no—
128C2×C8—5681no—
129C2×C8—711no—
130C2×C8—1421no—
131C2×C8—1421no—
132C2×C8D4×D7111yescharacteristic
133C4⋊C4—41no—
134C22⋊C4—5681no—
135C16—1421no—
136C16#1176586511yescharacteristic
137C16#1176586611yescharacteristic
138C23—5681no—
139C23—711no—
140D4—5681no—
141D4—5681no—
142D4—11362no—
143C2×C4—81no—
144C2×C4—711no—
145C2×C4—1421no—
146C2×C4—1421no—
147C2×C4D8×D7111yescharacteristic
148C8—81no—
149C8—5681no—
150C8—1421no—
151C8#1455611311yescharacteristic
152C8#1455616011yescharacteristic
153C22—2841no—
154C22—11361no—
155C22—11361no—
156C22—11361no—
157C22—1421no—
158C22SD32×D7111yescharacteristic
159C4—161no—
160C4—1421no—
161C4#1798631411yescharacteristic
162C4#1798643611yescharacteristic
163C2—21no—
164C2—2841no—
165C2—22721no—
166C2#2259052711yescharacteristic
subgroupG/H
centerC2#22590527
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC71×#1426C2
socleC2×C71#2233
prime radical#22590444C2
hypercenter#1426D71
hypocenterC71#13717
LCSG ▷ C2×C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1426

Complements

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

NH ≅ G/Ntype
2. #22590133C2semidirect
3. #22586545C2semidirect
4. #22586531C2semidirect
5. #22586529C2semidirect
6. #22586479C2semidirect
7. C71×#1426C2semidirect
87. #1426D71semidirect
91. C71#13717semidirect

Maximal subgroups

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

subgroupindex
#225904442
#225901332
#225865452
#225865312
#225865292
#225864792
C71×#14262
#1371771