Finite Group Databasebeta

← #22587513

Subgroups — #22587513

Order 9088.

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

normal subgroups only

This group has 194 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#17986723C211yesmaximal, characteristic
2#17986351C211yesmaximal, characteristic, complemented
3#17986295C211yesmaximal, characteristic, complemented
4#17986290C211yesmaximal, characteristic
5#17985729C211yesmaximal, characteristic, complemented
6#17985726C211yesmaximal, characteristic
7#332×C71C211yesmaximal, characteristic, complemented
8#14556128C2211yescharacteristic
9#14556123—22no—
10#14556117—21no—
11#14556108—21no—
12Q8×Dic71—21no—
13C4×Dic142C2211yescharacteristic
14C4×D284C2211yescharacteristic
15#14556086—21no—
16#14556062—21no—
17#14556060—21no—
18#14556053—21no—
19#14556051—22no—
20#14556050—21no—
21C4×C71⋊C8C2211yescharacteristic
22C4⋊Q8×C71C2211yescharacteristic
23C4⋊C8×C71C2211yescharacteristic
24C4⋊C8×C71C2211yescharacteristic
25C2×Dic142—21no—
26C2×Dic142D411yescharacteristic
27C2×D284D411yescharacteristic
28C2×C4×D71—21no—
29C2×C71⋊C8—21no—
30C4⋊Dic71—22no—
31C4⋊Dic71—21no—
32C4⋊Dic71—21no—
33C4⋊Dic71D411yescharacteristic
34C4⋊Dic71D411yescharacteristic
35Dic71⋊C4—21no—
36(C2×D71)⋊C4—22no—
37(C2×D71)⋊C4—22no—
38(C2×D71)⋊C4—21no—
39C4×Dic71—22no—
40C4×Dic71—21no—
41C2×Q8×C71—21no—
42C2×C8×C71—21no—
43C2×C8×C71—21no—
44C4⋊C4×C71—21no—
45C4⋊C4×C71D412yes—
46C42×C71C2311yescharacteristic
47C22×D71—21no—
48Q8×C71—41no—
49Q8×C71—41no—
50C2×C4×C71—22no—
51C2×C4×C71C2×D411yescharacteristic
52C2×C4×C71C2×D411yescharacteristic
53C2×C4×C71C2×D411yescharacteristic
54C2×Dic71—21no—
55C2×Dic71—21no—
56C2×Dic71—21no—
57C2×Dic71—41no—
58D284—21no—
59D284—21no—
60C4×D71—41no—
61C4×D71—41no—
62Dic142—21no—
63Dic142—21no—
64Dic142—42no—
65C8×C71—41no—
66C8×C71—41no—
67C71⋊C8—41no—
68C22×C71#12011yescharacteristic
69C2×D71—41no—
70C2×D71—41no—
71C2×D71—41no—
72C4×C71—21no—
73C4×C71—42no—
74C4×C71#13511yescharacteristic
75C4×C71#13511yescharacteristic
76C4×C71#13711yescharacteristic
77C4×C71#13711yescharacteristic
78Dic71—41no—
79Dic71—41no—
80Dic71—41no—
81Dic71—81no—
82C2×C71#45211yescharacteristic
83C2×C71#45211yescharacteristic
84C2×C71#45411yescharacteristic
85D71—81no—
86#1638—711nomaximal, Sylow
87C71#163811yescharacteristic, complemented, Sylow
88#556—711no—
89#487—711no—
90#477—711no—
91#476—711no—
92#332D7111yescharacteristic, complemented
93Q8⋊C8—711no—
94D4⋊C8—711no—
95C4⋊Q8—711no—
96C4⋊Q8C2×D7111yescharacteristic
97#126—1422no—
98#124—1421no—
99C22⋊Q8—1421no—
100C4×Q8—711no—
101C4×Q8—1421no—
102C4×D4—711no—
103#117—1421no—
104#107—1421no—
105#106—1421no—
106C4⋊C8C2×D7111yescharacteristic
107C4⋊C8C2×D7111yescharacteristic
108#103—1421no—
109#103—1422no—
110#102—1421no—
111C4×C8—711no—
112C2×Q8—21no—
113C2×Q8—711no—
114C2×Q8—1421no—
115C2×D4—711no—
116C22×C4—1421no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—1421no—
120C4⋊C4—21no—
121C4⋊C4—711no—
122C4⋊C4—711no—
123C4⋊C4—1422no—
124C4⋊C4—1421no—
125C4⋊C4—1421no—
126C4⋊C4—1421no—
127C4⋊C4D28412yes—
128C22⋊C4—1422no—
129C22⋊C4—1422no—
130C22⋊C4—1421no—
131C42—1422no—
132C42—1421no—
133C42C22×D7111yescharacteristic
134C23—1421no—
135Q8—41no—
136Q8—41no—
137Q8—2842no—
138Q8—1421no—
139Q8—1421no—
140D4—1421no—
141D4—1421no—
142C2×C4—22no—
143C2×C4—2841no—
144C2×C4—2841no—
145C2×C4—2841no—
146C2×C4—1421no—
147C2×C4—1421no—
148C2×C4—1421no—
149C2×C4D4×D7111yescharacteristic
150C2×C4D4×D7111yescharacteristic
151C2×C4C2×D28411yescharacteristic
152C8—41no—
153C8—41no—
154C8—2841no—
155C22—2841no—
156C22—2841no—
157C22—2841no—
158C22#1455609811yescharacteristic
159C4—21no—
160C4—42no—
161C4—2841no—
162C4—2841no—
163C4—2841no—
164C4—5681no—
165C4#1455614711yescharacteristic
166C4#1455614911yescharacteristic
167C4#1455615711yescharacteristic
168C4#1455616011yescharacteristic
169C2—5681no—
170C2#1798616211yescharacteristic
171C2#1798616411yescharacteristic
172C2#1798617411yescharacteristic
subgroupG/H
centerC22#14556098
commutatorC42×C71C23
FrattiniC42C22×D71
Fitting#332×C71C2
socleC22×C71#120
hypercenter#332D71
hypocenterC71#1638
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #17986351C2semidirect
3. #17986295C2semidirect
5. #17985729C2semidirect
7. #332×C71C2semidirect
87. C71#1638semidirect
92. #332D71semidirect