Finite Group Databasebeta

← #22587463

Subgroups — #22587463

Order 9088.

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

normal subgroups only

This group has 218 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#17986734C211yesmaximal, characteristic
2#17986372C211yesmaximal, characteristic
3#17986295C211yesmaximal, characteristic, complemented
4#17986294C211yesmaximal, characteristic, complemented
5#17985742C211yesmaximal, characteristic
6#17985731C211yesmaximal, characteristic, complemented
7Q8⋊C8×C71C211yesmaximal, characteristic, complemented
8C2×#11765854—22no—
9#14556128C2211yescharacteristic
10#14556125C2211yescharacteristic
11#14556117—21no—
12#14556116—21no—
13#14556113—22no—
14C4×Dic142C2211yescharacteristic
15C4×D284—21no—
16#14556090—21no—
17#14556062—21no—
18#14556061—21no—
19C4⋊(C71⋊C8)C2211yescharacteristic
20#14556053—21no—
21#14556052—21no—
22C4×Q8×C71C2211yescharacteristic
23C4⋊C8×C71C2211yescharacteristic
24C4×C8×C71C2211yescharacteristic
25C2×Dic142—21no—
26C2×Dic142D411yescharacteristic
27C2×D284—21no—
28C2×D284—21no—
29C2×D284D412yes—
30C2×C4×D71—22no—
31#11765854—44no—
32C2×C71⋊C8—21no—
33C4⋊Dic71—21no—
34C4⋊Dic71—21no—
35C4⋊Dic71D411yescharacteristic
36Dic71⋊C4—21no—
37(C2×D71)⋊C4—22no—
38(C2×D71)⋊C4—22no—
39(C2×D71)⋊C4—22no—
40C4×Dic71—21no—
41C2×Q8×C71D411yescharacteristic
42C2×C8×C71—21no—
43C2×C8×C71—21no—
44C4⋊C4×C71—21no—
45C4⋊C4×C71D411yescharacteristic
46C42×C71—21no—
47C42×C71C2311yescharacteristic
48C22×D71—22no—
49Q8×C71—21no—
50Q8×C71—21no—
51C2×C4×C71—21no—
52C2×C4×C71—21no—
53C2×C4×C71C2×D411yescharacteristic
54C2×C4×C71C2×D411yescharacteristic
55C2×C4×C71C2×D411yescharacteristic
56C2×Dic71—21no—
57C2×Dic71—21no—
58C2×Dic71—41no—
59D284—42no—
60D284—41no—
61D284—41no—
62D284—41no—
63D284—41no—
64C4×D71—82no—
65Dic142—21no—
66Dic142—21no—
67Dic142—42no—
68C8×C71—22no—
69C8×C71—41no—
70C71⋊C8—41no—
71C22×C71#12011yescharacteristic
72C2×D71—42no—
73C2×D71—42no—
74C2×D71—42no—
75C4×C71—21no—
76C4×C71—41no—
77C4×C71—41no—
78C4×C71#13511yescharacteristic
79C4×C71#13511yescharacteristic
80C4×C71#13711yescharacteristic
81C4×C71#13711yescharacteristic
82Dic71—41no—
83Dic71—41no—
84Dic71—81no—
85C2×C71#45211yescharacteristic
86C2×C71#45211yescharacteristic
87C2×C71#45311yescharacteristic
88D71—82no—
89#1630—711nomaximal, Sylow
90C71#163011yescharacteristic, complemented, Sylow
91#559—711no—
92#492—711no—
93#477—711no—
94#477—711no—
95#331—711no—
96Q8⋊C8—711no—
97Q8⋊C8D7111yescharacteristic, complemented
98C2×SD16—1422no—
99C4⋊Q8—711no—
100#127—711no—
101#124—1421no—
102#124—1421no—
103#123—1422no—
104C4×Q8—711no—
105C4×Q8C2×D7111yescharacteristic
106C4×D4—1421no—
107C4×D4—1421no—
108#107—1421no—
109#107—1421no—
110C4⋊C8—711no—
111C4⋊C8C2×D7111yescharacteristic
112#103—1421no—
113#103—1421no—
114C4×C8C2×D7111yescharacteristic
115C2×Q8—711no—
116C2×Q8—1421no—
117C2×Q8D28411yescharacteristic
118C2×D4—712no—
119C2×D4—1421no—
120C2×D4—1421no—
121C22×C4—1422no—
122SD16—2844no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—1421no—
126C4⋊C4—21no—
127C4⋊C4—711no—
128C4⋊C4—1421no—
129C4⋊C4—1421no—
130C4⋊C4—1421no—
131C4⋊C4D28411yescharacteristic
132C22⋊C4—1422no—
133C22⋊C4—1422no—
134C22⋊C4—1422no—
135C42—21no—
136C42—1421no—
137C42C22×D7111yescharacteristic
138C23—1422no—
139Q8—21no—
140Q8—21no—
141Q8—2842no—
142Q8—1421no—
143Q8—1421no—
144D4—2842no—
145D4—2841no—
146D4—2841no—
147D4—2841no—
148D4—2841no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—2841no—
152C2×C4—5682no—
153C2×C4—1421no—
154C2×C4—1421no—
155C2×C4D4×D7111yescharacteristic
156C2×C4D4×D7111yescharacteristic
157C2×C4C2×D28411yescharacteristic
158C8—22no—
159C8—41no—
160C8—2841no—
161C22—2842no—
162C22—2842no—
163C22—2842no—
164C22#1455609811yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—2841no—
169C4—2841no—
170C4—5681no—
171C4#1455614911yescharacteristic
172C4#1455615011yescharacteristic
173C4#1455616011yescharacteristic
174C4#1455616111yescharacteristic
175C2—5682no—
176C2#1798616311yescharacteristic
177C2#1798616411yescharacteristic
178C2#1798616811yescharacteristic
subgroupG/H
centerC22#14556098
commutatorC42×C71C23
FrattiniC42C22×D71
FittingQ8⋊C8×C71C2
socleC22×C71#120
hypercenterQ8⋊C8D71
hypocenterC71#1630
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #17986295C2semidirect
4. #17986294C2semidirect
6. #17985731C2semidirect
7. Q8⋊C8×C71C2semidirect
90. C71#1630semidirect
97. Q8⋊C8D71semidirect