Finite Group Databasebeta

← #22599548

Subgroups — #22599548

Order 9088.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986556C211yesmaximal, characteristic
2C2×#14556120C211yesmaximal, characteristic
3#17986408C212yesmaximal, complemented
4#17986368C212yesmaximal, complemented
5#17986335C212yesmaximal
6#17986330C212yesmaximal
7#17986307C214yesmaximal, complemented
8#432×C71C211yesmaximal, characteristic
9#14556128C2211yescharacteristic
10#14556128C2211yescharacteristic
11#14556120—21no—
12#14556120—21no—
13#14556120—21no—
14#14556120C2212yes—
15#14556120C2212yes—
16#14556107—21no—
17#14556107C2212yes—
18C4×Dic142—21no—
19C4×Dic142C2212yes—
20#14556087—21no—
21C2×C4⋊Dic71C2211yescharacteristic
22C2×C4⋊Dic71C2211yescharacteristic
23C2×Dic71⋊C4—21no—
24C2×Dic71⋊C4—21no—
25#14556062C2214yes—
26#14556060C2214yes—
27#14556053C2214yes—
28#14556053C2214yes—
29C2×C42×C71C2211yescharacteristic
30C4⋊C8×C71C2212yes—
31C22⋊C8×C71C2212yes—
32#97×C71C2212yes—
33C2×Dic142—21no—
34C2×Dic142C2312yes—
35C22×Dic71—21no—
36C22×Dic71—21no—
37C4⋊Dic71—21no—
38C4⋊Dic71—21no—
39C4⋊Dic71—21no—
40C4⋊Dic71—21no—
41C4⋊Dic71—21no—
42C4⋊Dic71C2312yes—
43C4⋊Dic71C2312yes—
44C4⋊Dic71C2312yes—
45Dic71⋊C4—22no—
46Dic71⋊C4—22no—
47Dic71⋊C4—22no—
48Dic71⋊C4—22no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51C22⋊Dic71—22no—
52C4×Dic71—22no—
53C22×C4×C71D411yescharacteristic
54C22×C4×C71D411yescharacteristic
55C22×C4×C71C2311yescharacteristic
56C2×C8×C71C2314yes—
57C42×C71D411yescharacteristic
58C42×C71D411yescharacteristic
59C42×C71C2311yescharacteristic
60C42×C71C2311yescharacteristic
61C23×C71C2×D411yescharacteristic
62C2×C4×C71—22no—
63C2×C4×C71—22no—
64C2×C4×C71—41no—
65C2×C4×C71C2×D411yescharacteristic
66C2×C4×C71C2×D411yescharacteristic
67C2×C4×C71C2×D411yescharacteristic
68C2×C4×C71C2×D411yescharacteristic
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2411yescharacteristic
71C2×Dic71—22no—
72C2×Dic71—22no—
73C2×Dic71—22no—
74C2×Dic71—22no—
75C2×Dic71—42no—
76C2×Dic71—42no—
77Dic142—24no—
78Dic142—42no—
79C8×C71—24no—
80C22×C71—21no—
81C22×C71—21no—
82C22×C71#13712yes—
83C22×C71C22×D411yescharacteristic
84C4×C71—21no—
85C4×C71—21no—
86C4×C71—21no—
87C4×C71—21no—
88C4×C71—21no—
89C4×C712−1+412yes—
90Dic71—42no—
91Dic71—42no—
92Dic71—42no—
93Dic71—42no—
94C2×C71—22no—
95C2×C71#53611yescharacteristic
96C2×C71C2×#13711yescharacteristic
97C2×C71#57511yescharacteristic
98#3236—711nomaximal, Sylow
99C71#323611yescharacteristic, complemented, Sylow
100#533—711no—
101C2×#125—711no—
102#501—712no—
103#491—712no—
104#484—712no—
105#483—712no—
106#479—714no—
107#432D7111yescharacteristic
108C4⋊Q8—711no—
109C4⋊Q8—711no—
110#125—712no—
111#125—712no—
112#125—1421no—
113#125—1421no—
114#125—1421no—
115C22⋊Q8—712no—
116C22⋊Q8—1421no—
117C4×Q8—712no—
118C4×Q8—1421no—
119#117—1421no—
120C2×C4⋊C4—711no—
121C2×C4⋊C4—711no—
122C2×C4⋊C4—1421no—
123C2×C4⋊C4—1421no—
124C2×C42C2×D7111yescharacteristic
125#107—714no—
126#106—714no—
127C4⋊C8C2×D7112yes—
128#103—714no—
129#103—714no—
130C22⋊C8C2×D7112yes—
131#97C2×D7112yes—
132C2×Q8—712no—
133C2×Q8—1421no—
134C22×C4—1421no—
135C22×C4—1421no—
136C22×C4D28411yescharacteristic
137C22×C4D28411yescharacteristic
138C22×C4C22×D7111yescharacteristic
139C2×C8C22×D7114yes—
140C4⋊C4—712no—
141C4⋊C4—712no—
142C4⋊C4—712no—
143C4⋊C4—1422no—
144C4⋊C4—1422no—
145C4⋊C4—1422no—
146C4⋊C4—1422no—
147C4⋊C4—1422no—
148C4⋊C4—1422no—
149C4⋊C4—1421no—
150C4⋊C4—1421no—
151C4⋊C4—1421no—
152C4⋊C4—1421no—
153C4⋊C4—1421no—
154C22⋊C4—1422no—
155C42—1422no—
156C42D28411yescharacteristic
157C42D28411yescharacteristic
158C42C22×D7111yescharacteristic
159C42C22×D7111yescharacteristic
160C23C2×D28411yescharacteristic
161Q8—2842no—
162Q8—1424no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—2842no—
167C2×C4—2842no—
168C2×C4—1422no—
169C2×C4—1422no—
170C2×C4—1422no—
171C2×C4—1422no—
172C2×C4C2×D28411yescharacteristic
173C2×C4C2×D28411yescharacteristic
174C2×C4C2×D28411yescharacteristic
175C2×C4C2×D28411yescharacteristic
176C2×C4C2×D28411yescharacteristic
177C2×C4C23×D7111yescharacteristic
178C8—24no—
179C22—21no—
180C22—21no—
181C22#1455616112yes—
182C22C22×D28411yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—2842no—
189C4—2842no—
190C4—2842no—
191C4—2842no—
192C4#1455617512yes—
193C2—22no—
194C2#1798657511yescharacteristic
195C2C2×#1455616111yescharacteristic
196C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#432×C71C2
socleC22×C71C22×D4
hypercenter#432D71
hypocenterC71#3236
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
3. #17986408C2semidirect
4. #17986368C2semidirect
7. #17986307C2semidirect
99. C71#3236semidirect