Finite Group Databasebeta

← #22587314

Subgroups — #22587314

Order 9088.

238 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 57 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#17986154C212yesmaximal
2#17985919C211yesmaximal, characteristic
3#17985736C212yesmaximal
4#17985725C211yesmaximal, characteristic
5#544×C71C211yesmaximal, characteristic
6C22×Dic142C2211yescharacteristic
7C2×#11765855—22no—
8C2×Dic71⋊C4—21no—
9#14556051—22no—
10C22⋊(C71⋊C8)C2212yes—
11C71⋊#95—21no—
12C71⋊#95C2211yescharacteristic
13C4⋊Q8×C71—22no—
14#125×C71—22no—
15C22⋊Q8×C71—21no—
16C22⋊Q8×C71C2212yes—
17#117×C71—21no—
18C2×C4⋊C4×C71C2211yescharacteristic
19C2×Dic142—22no—
20C2×Dic142—42no—
21C22×Dic71—21no—
22C22×Dic71—21no—
23#11765855—44no—
24C2×C71⋊C8—22no—
25Dic71⋊C4—42no—
26C2×Q8×C71—21no—
27C2×Q8×C71D412yes—
28C22×C4×C71D411yescharacteristic
29C22×C4×C71D411yescharacteristic
30C22×C4×C71C2311yescharacteristic
31C4⋊C4×C71—21no—
32C4⋊C4×C71—21no—
33C4⋊C4×C71—21no—
34C4⋊C4×C71—22no—
35C4⋊C4×C71—22no—
36C4⋊C4×C71—22no—
37C4⋊C4×C71D412yes—
38C22⋊C4×C71—22no—
39C42×C71—22no—
40C23×C71C2×D411yescharacteristic
41Q8×C71—42no—
42Q8×C71—81no—
43C2×C4×C71—21no—
44C2×C4×C71—21no—
45C2×C4×C71—22no—
46C2×C4×C71—22no—
47C2×C4×C71—21no—
48C2×C4×C71—41no—
49C2×C4×C71—41no—
50C2×C4×C71Q1612yescomplemented
51C2×C4×C71C2×D412yes—
52C2×Dic71—22no—
53C2×Dic71—22no—
54C2×Dic71—41no—
55C2×Dic71—41no—
56C2×Dic71—41no—
57C2×Dic71—41no—
58Dic142—24no—
59Dic142—42no—
60C71⋊C8—42no—
61C22×C71—21no—
62C22×C71—21no—
63C22×C71#12011yescharacteristic
64C22×C71C2×Q1611yescharacteristic
65C22×C71#13711yescharacteristic
66C4×C71—22no—
67C4×C71—22no—
68C4×C71—42no—
69C4×C71—41no—
70C4×C71—42no—
71Dic71—42no—
72Dic71—42no—
73C2×C71—21no—
74C2×C71—21no—
75C2×C71#45111yescharacteristic
76C2×C71#45611yescharacteristic
77C2×C71B(4,𝔽2)11yescharacteristic
78#1605—711nomaximal, Sylow
79C71#160511yescharacteristic, complemented, Sylow
80#544D7111yescharacteristic
81#451—712no—
82#395—711no—
83#328—712no—
84#324—711no—
85C22×Q8—711no—
86C2×Q16—1422no—
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—21no—
90C22⋊Q8C2×D7112yes—
91#117—21no—
92C2×C4⋊C4—1421no—
93C2×C4⋊C4C2×D7111yescharacteristic
94#103—1422no—
95C22⋊C8—712no—
96#95—711no—
97#95—1421no—
98C2×Q8—21no—
99C2×Q8—2842no—
100C2×Q8—1422no—
101C2×Q8C22⋊D7112yes—
102C22×C4—1421no—
103C22×C4—1421no—
104C22×C4C22⋊D7111yescharacteristic
105C22×C4C22⋊D7111yescharacteristic
106C22×C4C22×D7111yescharacteristic
107Q16—2844no—
108C2×C8—1422no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—2842no—
116C4⋊C4C22⋊D7112yes—
117C22⋊C4—22no—
118C42—22no—
119C23C2×C22⋊D7111yescharacteristic
120Q8—42no—
121Q8—81no—
122Q8—2842no—
123Q8—1424no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—2841no—
132C2×C4—2841no—
133C2×C4—2841no—
134C2×C4—2841no—
135C2×C4—1422no—
136C2×C4—1422no—
137C2×C4#1176585512yescomplemented
138C2×C4C2×C22⋊D7112yes—
139C8—2842no—
140C22—21no—
141C22—21no—
142C22#1455609711yescharacteristic
143C22C2×#1176585511yescharacteristic
144C22#1455615911yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—2842no—
151C4—2842no—
152C2—21no—
153C2—21no—
154C2#1798615511yescharacteristic
155C2#1798618211yescharacteristic
156C2#1798618611yescharacteristic
subgroupG/H
centerC22#14556097
commutatorC22×C4×C71C23
FrattiniC22×C4C22×D71
Fitting#544×C71C2
socleC22×C71#120
hypercenter#544D71
hypocenterC71#1605
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
50. C2×C4×C71Q16semidirect
79. C71#1605semidirect
137. C2×C4#11765855semidirect