Finite Group Databasebeta

← #22599487

Subgroups — #22599487

Order 9088.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986554C211yesmaximal, characteristic
2C4⋊Q8×D71C211yesmaximal, characteristic
3#17986404C211yesmaximal, characteristic, complemented
4#17986402C211yesmaximal, characteristic, complemented
5#17986349C211yesmaximal, characteristic, complemented
6#17986333C214yesmaximal
7#17986292C214yesmaximal, complemented
8#17986050C211yesmaximal, characteristic
9#487×C71C211yesmaximal, characteristic, complemented
10C2×Q8×D71—21no—
11#14556128C2211yescharacteristic
12#14556128C2211yescharacteristic
13#14556127—21no—
14#14556127C2212yes—
15#14556126—21no—
16#14556119—21no—
17#14556108—21no—
18#14556104C2212yes—
19Q8×Dic71C2212yes—
20#14556093—21no—
21#14556086—21no—
22C4⋊C4×D71—21no—
23C4⋊C4×D71C2212yes—
24C42×D71C2211yescharacteristic
25#14556062C2214yes—
26#14556061C2214yes—
27Dic71⋊C8C2212yes—
28#14556052C2214yes—
29(C2×D71)⋊C8C2212yes—
30C4×C71⋊C8C2211yescharacteristic
31C4⋊Q8×C71C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#103×C71C2214yes—
34C4×C8×C71C2211yescharacteristic
35C2×Dic142—21no—
36C2×Dic142—21no—
37C2×Dic142—22no—
38Q8×D71—44no—
39C2×C4×D71—22no—
40C2×C4×D71D411yescharacteristic
41C2×C4×D71D411yescharacteristic
42C2×C4×D71C2311yescharacteristic
43C2×C71⋊C8C2312yes—
44C4⋊Dic71—21no—
45C4⋊Dic71—21no—
46C4⋊Dic71—22no—
47C4⋊Dic71C2312yes—
48C4⋊Dic71C2312yes—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52(C2×D71)⋊C4—22no—
53C4×Dic71—22no—
54C4×Dic71D411yescharacteristic
55C4×Dic71D411yescharacteristic
56C4×Dic71C2311yescharacteristic
57C2×Q8×C71—21no—
58C2×Q8×C71C2312yes—
59C2×C8×C71C2312yes—
60C4⋊C4×C71—21no—
61C4⋊C4×C71—21no—
62C4⋊C4×C71—21no—
63C4⋊C4×C71C2312yes—
64C42×C71C2311yescharacteristic
65C22×D71C2×D411yescharacteristic
66Q8×C71—24no—
67Q8×C71—42no—
68C2×C4×C71—22no—
69C2×C4×C71—22no—
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2411yescharacteristic
73C2×Dic71—22no—
74C2×Dic71—22no—
75C2×Dic71C2×D411yescharacteristic
76C2×Dic71C2×D411yescharacteristic
77C2×Dic71C2×D411yescharacteristic
78C4×D71—22no—
79C4×D71—22no—
80C4×D71—44no—
81C4×D71Q1614yes—
82Dic142—44no—
83Dic142—42no—
84Dic142—42no—
85C8×C71—22no—
86C71⋊C8—22no—
87C22×C71C22×D411yescharacteristic
88C2×D71—21no—
89C2×D71—21no—
90C2×D71C2×Q1612yes—
91C4×C71—42no—
92C4×C71—42no—
93C4×C71C2×Q1612yes—
94C4×C71AGL(1,ℤ/8)12yes—
95C4×C712−1+412yes—
96Dic71—21no—
97Dic71—21no—
98Dic71—42no—
99Dic71—42no—
100Dic71C2×Q1612yes—
101C2×C71#53611yescharacteristic
102C2×C71C22×Q1611yescharacteristic
103C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
104D71—22no—
105#3230—711nomaximal, Sylow
106C71#323011yescharacteristic, complemented, Sylow
107#533—711no—
108C2×C4⋊Q8—711no—
109#500—711no—
110#500—711no—
111#487—711no—
112#487D7111yescharacteristic, complemented
113#484—714no—
114#477—714no—
115#431—711no—
116C22×Q8—1421no—
117C4⋊Q8—712no—
118C4⋊Q8—711no—
119C4⋊Q8—711no—
120C4⋊Q8—1421no—
121C4⋊Q8—1421no—
122C4⋊Q8C2×D7111yescharacteristic
123C4⋊Q8C2×D7111yescharacteristic
124#125—1421no—
125C22⋊Q8—712no—
126C22⋊Q8—1421no—
127C4×Q8—712no—
128C4×Q8—1421no—
129#117—1421no—
130C2×C4⋊C4—712no—
131C2×C4⋊C4—1421no—
132C2×C42—711no—
133#107—714no—
134#107—714no—
135C4⋊C8—712no—
136#103—714no—
137#103C2×D7114yes—
138C22⋊C8—712no—
139C4×C8—711no—
140C4×C8C2×D7111yescharacteristic
141C2×Q8—21no—
142C2×Q8—2844no—
143C2×Q8—1422no—
144C2×Q8—1421no—
145C2×Q8—1421no—
146C2×Q8C22×D7112yes—
147C22×C4—711no—
148C22×C4—711no—
149C22×C4—711no—
150C22×C4—1422no—
151C2×C8—712no—
152C2×C8C22×D7112yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—712no—
157C4⋊C4—712no—
158C4⋊C4—1422no—
159C4⋊C4—1422no—
160C4⋊C4—1422no—
161C4⋊C4—1422no—
162C4⋊C4—1421no—
163C4⋊C4—1421no—
164C4⋊C4C22×D7112yes—
165C22⋊C4—1422no—
166C42—711no—
167C42—711no—
168C42—711no—
169C42—1422no—
170C42C22×D7111yescharacteristic
171C23—711no—
172Q8—24no—
173Q8—42no—
174Q8—2844no—
175Q8—2842no—
176Q8—2842no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—2844no—
180C2×C4—711no—
181C2×C4—714no—
182C2×C4—711no—
183C2×C4—711no—
184C2×C4—1422no—
185C2×C4—1422no—
186C2×C4—1422no—
187C2×C4—1422no—
188C2×C4D4×D7111yescharacteristic
189C2×C4D4×D7111yescharacteristic
190C2×C4C23×D7111yescharacteristic
191C8—22no—
192C8—1422no—
193C22—712no—
194C22—1421no—
195C22—1421no—
196C22C2×D4×D7111yescharacteristic
197C4—42no—
198C4—42no—
199C4—2842no—
200C4—2842no—
201C4—712no—
202C4—1421no—
203C4—1421no—
204C4Q16×D7112yes—
205C4#1455615512yes—
206C4#1455617512yes—
207C2—1422no—
208C2#1798656911yescharacteristic
209C2C2×Q16×D7111yescharacteristic
210C2C2×#1455615511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3230
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #17986404C2semidirect
4. #17986402C2semidirect
5. #17986349C2semidirect
7. #17986292C2semidirect
9. #487×C71C2semidirect
106. C71#3230semidirect
112. #487D71semidirect