Finite Group Databasebeta

← #22587595

Subgroups — #22587595

Order 9088.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#17986722C211yesmaximal, characteristic
2#17986402C211yesmaximal, characteristic, complemented
3#17986201C211yesmaximal, characteristic
4#17985744C211yesmaximal, characteristic, complemented
5Q8⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
6#17985726C211yesmaximal, characteristic
7#475×C71C211yesmaximal, characteristic, complemented
8C2×#11765852—21no—
9#14556128C2211yescharacteristic
10#14556123—22no—
11#14556116—21no—
12#14556104—21no—
13C4×Dic142—21no—
14C4×D284C2211yescharacteristic
15#14556086—21no—
16#14556061—22no—
17C4⋊(C71⋊C8)C2211yescharacteristic
18#14556052—21no—
19C4×C71⋊C8C2211yescharacteristic
20C4⋊Q8×C71C2211yescharacteristic
21C4×Q8×C71C2211yescharacteristic
22#106×C71—21no—
23C4⋊C8×C71C2211yescharacteristic
24#103×C71—21no—
25C2×Dic142—21no—
26C2×D284D411yescharacteristic
27C2×C4×D71—21no—
28#11765852—42no—
29C2×C71⋊C8—21no—
30C2×C71⋊C8—21no—
31C4⋊Dic71—21no—
32C4⋊Dic71D411yescharacteristic
33C4⋊Dic71D412yes—
34Dic71⋊C4—22no—
35(C2×D71)⋊C4—22no—
36(C2×D71)⋊C4—22no—
37(C2×D71)⋊C4—21no—
38C4×Dic71—22no—
39C2×Q8×C71—21no—
40C2×Q8×C71D411yescharacteristic
41C2×C8×C71—21no—
42C4⋊C4×C71—21no—
43C4⋊C4×C71—21no—
44C4⋊C4×C71—21no—
45C4⋊C4×C71D411yescharacteristic
46C42×C71—21no—
47C42×C71C2311yescharacteristic
48C22×D71—21no—
49Q8×C71—21no—
50Q8×C71—21no—
51Q8×C71—41no—
52Q8×C71—41no—
53C2×C4×C71—21no—
54C2×C4×C71—21no—
55C2×C4×C71—41no—
56C2×C4×C71C2×D411yescharacteristic
57C2×C4×C71C2×D411yescharacteristic
58C2×C4×C71C2×D411yescharacteristic
59C2×Dic71—21no—
60C2×Dic71—22no—
61D284—21no—
62D284—21no—
63C4×D71—41no—
64C4×D71—41no—
65Dic142—41no—
66Dic142—41no—
67C8×C71—41no—
68C71⋊C8—22no—
69C71⋊C8—41no—
70C22×C71#12011yescharacteristic
71C2×D71—41no—
72C2×D71—41no—
73C2×D71—41no—
74C4×C71—21no—
75C4×C71—41no—
76C4×C71—41no—
77C4×C71—81no—
78C4×C71#13511yescharacteristic
79C4×C71#13511yescharacteristic
80C4×C71AGL(1,ℤ/8)11yescharacteristic
81C4×C71#13711yescharacteristic
82Dic71—42no—
83Dic71—41no—
84C2×C71#44911yescharacteristic
85C2×C71#45211yescharacteristic
86C2×C71#45611yescharacteristic
87D71—81no—
88#1650—711nomaximal, Sylow
89C71#165011yescharacteristic, complemented, Sylow
90#556—711no—
91#500—711no—
92#475D7111yescharacteristic, complemented
93#461—711no—
94#332—711no—
95Q8⋊C8—711no—
96D4⋊C8—711no—
97C2×SD16—1421no—
98C4⋊Q8—711no—
99C4⋊Q8C2×D7111yescharacteristic
100#126—1422no—
101#124—1421no—
102C22⋊Q8—1421no—
103C4×Q8—1421no—
104C4×Q8C2×D7111yescharacteristic
105C4×D4—711no—
106#117—1421no—
107#107—1422no—
108#106—21no—
109C4⋊C8—711no—
110C4⋊C8C2×D7111yescharacteristic
111#103—21no—
112#103—1421no—
113C4×C8—711no—
114C2×Q8—21no—
115C2×Q8—1421no—
116C2×Q8C22⋊D7111yescharacteristic
117C2×D4—711no—
118C22×C4—1421no—
119SD16—2842no—
120C2×C8—21no—
121C2×C8—1421no—
122C2×C8—1421no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—711no—
127C4⋊C4—712no—
128C4⋊C4—1422no—
129C4⋊C4—1421no—
130C4⋊C4C22⋊D7111yescharacteristic
131C22⋊C4—1422no—
132C22⋊C4—1422no—
133C22⋊C4—1421no—
134C42—21no—
135C42—1422no—
136C42C22×D7111yescharacteristic
137C23—1421no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—2841no—
143Q8—2841no—
144D4—1421no—
145D4—1421no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—2841no—
150C2×C4—2841no—
151C2×C4—1421no—
152C2×C4—1422no—
153C2×C4D4×D7111yescharacteristic
154C2×C4D4×D7111yescharacteristic
155C2×C4C2×C22⋊D7111yescharacteristic
156C8—41no—
157C8—2841no—
158C8—1422no—
159C22—2841no—
160C22—2841no—
161C22—2841no—
162C22#1455609611yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—2842no—
168C4—2841no—
169C4#1455614811yescharacteristic
170C4#1455614911yescharacteristic
171C4#1455615511yescharacteristic
172C4#1455615911yescharacteristic
173C2—5681no—
174C2#1798614311yescharacteristic
175C2#1798615911yescharacteristic
176C2#1798618011yescharacteristic
subgroupG/H
centerC22#14556096
commutatorC42×C71C23
FrattiniC42C22×D71
Fitting#475×C71C2
socleC22×C71#120
hypercenter#475D71
hypocenterC71#1650
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #17986402C2semidirect
4. #17985744C2semidirect
5. Q8⋊(C71⋊C8)C2semidirect
7. #475×C71C2semidirect
89. C71#1650semidirect
92. #475D71semidirect