Finite Group Databasebeta

← #22589745

Subgroups — #22589745

Order 9088.

274 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14556050C212yesmaximal
2#17985901C211yesmaximal, characteristic
3#17985869C211yesmaximal, characteristic, complemented
4#17985762C212yesmaximal, complemented
5#336×C71C211yesmaximal, characteristic, complemented
6C22×D284C2211yescharacteristic
7C2×C4⋊Dic71—21no—
8C2×C4⋊Dic71—21no—
9C2×C4⋊Dic71C2212yes—
10C2×C4⋊Dic71C2211yescharacteristic
11C2×(C2×D71)⋊C4—21no—
12C2×(C2×D71)⋊C4—21no—
13#14556050—24no—
14C22×C8×C71C2212yes—
15C2×C42×C71C2211yescharacteristic
16C23×D71—21no—
17C2×D284—22no—
18C2×D284—22no—
19C2×D284—21no—
20C2×D284D412yes—
21C22×Dic71—21no—
22C22×Dic71—22no—
23C4⋊Dic71—22no—
24C4⋊Dic71—22no—
25C4⋊Dic71—22no—
26C4⋊Dic71—42no—
27C4⋊Dic71—42no—
28(C2×D71)⋊C4—42no—
29(C2×D71)⋊C4—42no—
30C22×C4×C71D411yescharacteristic
31C22×C4×C71D411yescharacteristic
32C22×C4×C71C2311yescharacteristic
33C2×C8×C71—22no—
34C2×C8×C71—22no—
35C2×C8×C71—22no—
36C42×C71—22no—
37C23×C71C2×D411yescharacteristic
38C22×D71—22no—
39C22×D71—41no—
40C22×D71—41no—
41C22×D71—41no—
42C22×D71—41no—
43C22×D71—41no—
44C22×D71—41no—
45C2×C4×C71—21no—
46C2×C4×C71—21no—
47C2×C4×C71—21no—
48C2×C4×C71—21no—
49C2×C4×C71D812yes—
50C2×C4×C71SD1612yes—
51C2×C4×C71C2×D411yescharacteristic
52C2×C4×C71#4112yes—
53C2×C4×C71#4112yes—
54C2×C4×C71#4111yescharacteristic
55C2×Dic71—42no—
56C2×Dic71—41no—
57C2×Dic71—41no—
58C2×Dic71—42no—
59C2×Dic71—41no—
60C2×Dic71—42no—
61D284—24no—
62D284—42no—
63C8×C71—42no—
64C22×C71#12111yescharacteristic
65C22×C71#12311yescharacteristic
66C22×C71#12411yescharacteristic
67C22×C71C2×D811yescharacteristic
68C22×C71C2×SD1611yescharacteristic
69C22×C71AGL(1,ℤ/8)11yescharacteristic
70C22×C71#13711yescharacteristic
71C2×D71—42no—
72C2×D71—42no—
73C2×D71—42no—
74C2×D71—81no—
75C2×D71—81no—
76C2×D71—81no—
77C2×D71—81no—
78C4×C71—22no—
79C4×C71—22no—
80C4×C71#12312yes—
81C4×C71#12612yes—
82Dic71—82no—
83Dic71—81no—
84C2×C71#39711yescharacteristic
85C2×C71#45911yescharacteristic
86C2×C71#46111yescharacteristic
87C2×C71#48011yescharacteristic
88C2×C71#48111yescharacteristic
89C2×C71#48611yescharacteristic
90C2×C71#48911yescharacteristic
91D71—82no—
92#2071—711nomaximal, Sylow
93C71#207111yescharacteristic, complemented, Sylow
94C2×#102—712no—
95#390—711no—
96#384—711no—
97#340—712no—
98#336D7111yescharacteristic, complemented
99C22×D4—711no—
100C22×C8C2×D7112yes—
101C2×C4⋊C4—711no—
102C2×C4⋊C4—712no—
103C2×C4⋊C4—1421no—
104C2×C4⋊C4—1421no—
105C2×C22⋊C4—1421no—
106C2×C22⋊C4—1421no—
107C2×C42C2×D7111yescharacteristic
108#102—1424no—
109C24—1421no—
110C2×D4—712no—
111C2×D4—1422no—
112C2×D4—1422no—
113C2×D4—1421no—
114C22×C4—1421no—
115C22×C4—1422no—
116C22×C4D28411yescharacteristic
117C22×C4D28411yescharacteristic
118C22×C4C22×D7111yescharacteristic
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C4⋊C4—2842no—
123C4⋊C4—2842no—
124C4⋊C4—1422no—
125C4⋊C4—1422no—
126C4⋊C4—1422no—
127C22⋊C4—2842no—
128C22⋊C4—2842no—
129C42—22no—
130C23—2841no—
131C23—2841no—
132C23—2841no—
133C23—2841no—
134C23—2841no—
135C23—2841no—
136C23—1422no—
137C23C2×D28411yescharacteristic
138D4—2842no—
139D4—1424no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—2842no—
145C2×C4—2841no—
146C2×C4—2841no—
147C2×C4—2842no—
148C2×C4—2841no—
149C2×C4—2842no—
150C2×C4D56812yes—
151C2×C4#1176585412yes—
152C2×C4D4×D7111yescharacteristic
153C2×C4#1176586412yes—
154C2×C4#1176586512yes—
155C2×C4#1176586611yescharacteristic
156C8—42no—
157C22—2842no—
158C22—2842no—
159C22—2842no—
160C22—5681no—
161C22—5681no—
162C22—5681no—
163C22—5681no—
164C22#1455610311yescharacteristic
165C22#1455611211yescharacteristic
166C22#1455611611yescharacteristic
167C22C2×D56811yescharacteristic
168C22C2×#1176585411yescharacteristic
169C22#1455615611yescharacteristic
170C22#1455616111yescharacteristic
171C4—22no—
172C4—22no—
173C4—5681no—
174C4—5682no—
175C4#1455611012yes—
176C4#1455612312yes—
177C2—5682no—
178C2#1798592611yescharacteristic
179C2#1798619511yescharacteristic
180C2#1798620511yescharacteristic
181C2#1798631311yescharacteristic
182C2#1798631811yescharacteristic
183C2#1798634611yescharacteristic
184C2#1798635911yescharacteristic
subgroupG/H
centerC23C2×D284
commutatorC22×C4×C71C23
FrattiniC22×C4C22×D71
Fitting#336×C71C2
socleC23×C71C2×D4
hypercenter#336D71
hypocenterC71#2071
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #336

Complements

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

NH ≅ G/Ntype
3. #17985869C2semidirect
4. #17985762C2semidirect
5. #336×C71C2semidirect
93. C71#2071semidirect
98. #336D71semidirect