Finite Group Databasebeta

← #22599500

Subgroups — #22599500

Order 9088.

326 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 117 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
2#17986554C211yesmaximal, characteristic
3#17986395C211yesmaximal, characteristic, complemented
4#17986351C211yesmaximal, characteristic, complemented
5#17986350C211yesmaximal, characteristic, complemented
6#17986342C214yesmaximal
7#17986305C214yesmaximal, complemented
8#17986050C211yesmaximal, characteristic
9#500×C71C211yesmaximal, characteristic, complemented
10#14556128C2211yescharacteristic
11#14556128C2211yescharacteristic
12#14556119—21no—
13#14556119C2212yes—
14#14556108—21no—
15#14556108C2212yes—
16#14556104—21no—
17#14556104—21no—
18Q8×Dic71—21no—
19Q8×Dic71—21no—
20#14556093—21no—
21#14556093C2212yes—
22#14556086—21no—
23#14556086C2212yes—
24C42×D71C2211yescharacteristic
25#14556059C2214yes—
26Dic71⋊C8C2212yes—
27#14556053C2214yes—
28#14556051C2214yes—
29(C2×D71)⋊C8C2212yes—
30C4×C71⋊C8C2211yescharacteristic
31C4⋊Q8×C71C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#107×C71C2214yes—
34C4×C8×C71C2211yescharacteristic
35C2×Dic142—21no—
36C2×Dic142C2312yes—
37C2×C4×D71D411yescharacteristic
38C2×C4×D71D411yescharacteristic
39C2×C4×D71C2311yescharacteristic
40C2×C71⋊C8C2312yes—
41C4⋊Dic71—21no—
42C4⋊Dic71—21no—
43C4⋊Dic71—21no—
44C4⋊Dic71—22no—
45C4⋊Dic71—22no—
46C4⋊Dic71C2312yes—
47Dic71⋊C4—22no—
48Dic71⋊C4—22no—
49(C2×D71)⋊C4—22no—
50(C2×D71)⋊C4—22no—
51C4×Dic71—22no—
52C4×Dic71—22no—
53C4×Dic71D411yescharacteristic
54C4×Dic71D411yescharacteristic
55C4×Dic71C2311yescharacteristic
56C2×Q8×C71—21no—
57C2×Q8×C71—21no—
58C2×C8×C71C2312yes—
59C4⋊C4×C71—21no—
60C4⋊C4×C71—21no—
61C4⋊C4×C71C2312yes—
62C4⋊C4×C71C2312yes—
63C42×C71C2311yescharacteristic
64C22×D71C2×D411yescharacteristic
65Q8×C71—42no—
66Q8×C71—42no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2411yescharacteristic
72C2×Dic71—22no—
73C2×Dic71—22no—
74C2×Dic71C2×D411yescharacteristic
75C2×Dic71C2×D411yescharacteristic
76C2×Dic71C2×D411yescharacteristic
77C4×D71—22no—
78C4×D71—22no—
79C4×D71—22no—
80Dic142—24no—
81Dic142—42no—
82C8×C71—22no—
83C71⋊C8—22no—
84C22×C71C22×D411yescharacteristic
85C2×D71—21no—
86C2×D71—21no—
87C2×D71—21no—
88C4×C71—42no—
89C4×C71—42no—
90C4×C71#13512yes—
91C4×C71#13712yes—
92C4×C712−1+412yes—
93Dic71—21no—
94Dic71—21no—
95Dic71—21no—
96Dic71—42no—
97Dic71—42no—
98C2×C71#53611yescharacteristic
99C2×C71C2×#13511yescharacteristic
100C2×C71C2×#13711yescharacteristic
101D71—41no—
102#3231—711nomaximal, Sylow
103C71#323111yescharacteristic, complemented, Sylow
104#533—711no—
105#533—711no—
106#500D7111yescharacteristic, complemented
107#498—711no—
108#487—711no—
109#487—711no—
110#485—714no—
111#479—714no—
112#431—711no—
113C4⋊Q8—711no—
114C4⋊Q8—711no—
115C4⋊Q8C2×D7111yescharacteristic
116C4⋊Q8C2×D7111yescharacteristic
117#125—712no—
118#125—1421no—
119C22⋊Q8—712no—
120C22⋊Q8—1421no—
121C22⋊Q8—1421no—
122C22⋊Q8—1421no—
123C4×Q8—712no—
124C4×Q8—1421no—
125C4×Q8—1421no—
126C4×Q8—1421no—
127#117—712no—
128#117—1421no—
129C2×C42—711no—
130#107C2×D7114yes—
131#106—714no—
132C4⋊C8—712no—
133#103—714no—
134#103—714no—
135C22⋊C8—712no—
136C4×C8—711no—
137C4×C8C2×D7111yescharacteristic
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—712no—
141C2×Q8—1421no—
142C22×C4—711no—
143C22×C4—711no—
144C22×C4—711no—
145C2×C8—712no—
146C2×C8C22×D7112yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—712no—
150C4⋊C4—1422no—
151C4⋊C4—1422no—
152C4⋊C4—1422no—
153C4⋊C4—1422no—
154C4⋊C4—1421no—
155C4⋊C4—1421no—
156C4⋊C4—1421no—
157C4⋊C4C22×D7112yes—
158C4⋊C4C22×D7112yes—
159C22⋊C4—1422no—
160C22⋊C4—1422no—
161C42—711no—
162C42—711no—
163C42—711no—
164C42—1422no—
165C42—1422no—
166C42C22×D7111yescharacteristic
167C23—711no—
168Q8—42no—
169Q8—42no—
170Q8—2842no—
171Q8—1424no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—711no—
175C2×C4—711no—
176C2×C4—711no—
177C2×C4—1422no—
178C2×C4—1422no—
179C2×C4—1422no—
180C2×C4—1422no—
181C2×C4—1422no—
182C2×C4D4×D7111yescharacteristic
183C2×C4D4×D7111yescharacteristic
184C2×C4C23×D7111yescharacteristic
185C8—22no—
186C8—1422no—
187C22—1421no—
188C22—1421no—
189C22—1421no—
190C22C2×D4×D7111yescharacteristic
191C4—42no—
192C4—42no—
193C4—2842no—
194C4—2842no—
195C4—1421no—
196C4—1421no—
197C4—1421no—
198C4#1455614712yes—
199C4#1455616012yes—
200C4#1455617612yes—
201C2—2841no—
202C2#1798657411yescharacteristic
203C2C2×#1455614711yescharacteristic
204C2C2×#1455616011yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#500×C71C2
socleC22×C71C22×D4
hypercenter#500D71
hypocenterC71#3231
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #17986395C2semidirect
4. #17986351C2semidirect
5. #17986350C2semidirect
7. #17986305C2semidirect
9. #500×C71C2semidirect
103. C71#3231semidirect
106. #500D71semidirect